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Moisture content-matric potential relationship and water flow properties of wood at high moisture contents Fortin, Yves 1980

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MOISTURE CONTENT-MATRIC POTENTIAL RELATIONSHIP AND WATER FLOW PROPERTIES OF WOOD AT HIGH MOISTURE CONTENTS by YVES FORTIN M.Sc, L a v a l U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Fo r e s t r y ( S o i l Science) We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1979 © Yves F o r t i n , 1979 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r a g ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Forestry The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D A T E IH ^RrPTnhPr 1Q7Q i ABSTRACT The water p o t e n t i a l concept and g e n e r a l i z e d Darcy's law f o r unsaturated flow were proposed as a b a s i s f o r the study of the energy s t a t e and flow of water i n wood at high moisture contents. A s e r i e s of experiments was conducted to i n v e s t i g a t e wood p r o p e r t i e s that are c r u -c i a l to the a p p l i c a t i o n of these t h e o r i e s . Much a t t e n t i o n was devoted to the measurement of the moisture content-matric p o t e n t i a l (M-ili ) m r e l a t i o n s h i p by the porous p l a t e methods. Both the boundary drainage and i m b i b i t i o n curves and the drainage curve s t a r t i n g from the green c o n d i t i o n were determined. In a d d i t i o n , independent \b measurements m were made i n some of the specimens e q u i l i b r a t e d on the porous p l a t e s using Wescor PT51-10 thermocouple psychrometers (TCP). The osmotic p o t e n t i a l was then assumed to be n e g l i g i b l e . Transient flow experiments were a l s o conducted f o r determining the water c o n d u c t i v i t y (K) as a f u n c t i o n of both M and IJJ , and the water d i f f u s i v i t y (D) as a f u n c t i o n m of M using an instantaneous p r o f i l e method and the one-step method. The boundary drainage and i m b i b i t i o n curves f o r the l o n g i t u d i n a l d i r e c t i o n of flow were thus obtained. The saturated c o n d u c t i v i t y had to be meas-ured s e p a r a t e l y . Western hemlock sapwood was used f o r a l l determinations which were made at a temperature of 21°C. The r e s u l t s of the porous p l a t e t e s t s confirmed the presence of a considerable h y s t e r e s i s i n the M-ij1 r e l a t i o n s h i p at high moisture m contents, w i t h M at a given greater i n drainage than i n i m b i b i t i o n . The i n k - b o t t l e e f f e c t appears to be the primary cause of t h i s phenomenon. i i Both types of drainage curves obtained showed a tendency to e x h i b i t a plateau at intermediate M's and were found highly dependent on the i n i t i a l moisture content. A f a i r l y good agreement was found between the values imposed m on the specimens on the porous plates and those measured by the TCP's for the ^ m range from -1 to -7 bars i n drainage. These two types of data d i f f e r e d markedly at lower f 's i n drainage and i n imbibition. However, m the differences observed i n imbibition were r e a d i l y explainable and the r e s u l t s of the TCP tests c l e a r l y demonstrated the strong h y s t e r e t i c behavior of the M-i|> r e l a t i o n s h i p i n the wet state. The r e p r o d u c i b i l i t y m of the TCP method during the ty measurements i n wood was rather poor. m The K-M, K — , and D-M curves obtained by the two transient m methods used were very s i m i l a r i n shape. The K(M) function exhibited a considerable hysteresis with K at a given M greater i n imbibition than i n drainage. A change i n K of several orders of magnitude was recorded near f u l l saturation. The K(ty ) and D(M) functions displayed only a p a r t i a l m hy s t e r e s i s . D i s p r o p o r t i o n a l i t y between f l u x and gradient was observed i n i m b i b i t i o n above 75% M. The non-uniqueness of the M-^m r e l a t i o n s h i p with respect to the state of flow was apparently the main cause of t h i s anomaly. i i i TABLE OF CONTENTS Page ABSTRACT . TABLE OF CONTENTS . . . LIST OF TABLES , LIST OF FIGURES . . . . LIST OF IMPORTANT SYMBOLS ACKNOWLEDGEMENTS . . . . INTRODUCTION 1 Chapter 1 - BACKGROUND TO THE STUDY 4 1.1 Water p o t e n t i a l , ty, i n porous media 5 1.1.1 H i s t o r y and d e f i n i t i o n of the ty concept. . . 5 1.1.2 P h y s i c a l p a r t i t i o n of ty i n t o i t s components 9 1.1.3 D e r i v a t i o n of the component p o t e n t i a l s . . . 11 1.1.ii Methods of o b t a i n i n g separate and combined determinations of ty and ty m o at high moisture contents 25 1.2 Isothermal flow of l i q u i d water i n unsaturated porous media 29 1.2.1 Macroscopic flow equations 29 1.2.2 Methods of measuring the fu n c t i o n s K(M), K (^ ) and D(M) 40 m i i i v i i x xv x v i i i i v Page 1.2.2.1 Steady-state methods 41 1.2.2.2 Transient methods 43 Chapter 2 - LITERATURE REVIEW 48 2.1 M—d> r e l a t i o n s h i p at high moisture contents . . . . 48 m 2.1.1 Previous measurements i n wood 48 2.1.2 P o s s i b l e causes of the h y s t e r e s i s 54 2.1.3 Factors of v a r i a t i o n 56 2.1.3.1 Pore s i z e d i s t r i b u t i o n 56 2.1.3.2 Ambient temperature and pressure. . 59 2.1.3.3 Entrapped a i r 60 2.1.3.4 Process dependence 62 2.2 K(M), K ( ^ m ) and D(M) funct i o n s a t high moisture contents 63 2.2.1 Previous measurements i n wood 63 2.2.2 P o s s i b l e causes of the h y s t e r e s i s i n the K(M) f u n c t i o n . 66 2.2.3 Factors of v a r i a t i o n 67 2.2.3.1 Pore s i z e d i s t r i b u t i o n 67 2.2.3.2 Ambient temperature and pressure. . 69 2.2.3.3 Entrapped a i r 70 2.2.3.4 Process dependence 71 Chapter 3 - EXPERIMENTAL PROCEDURE . . 72 3.1 M a t e r i a l s 73 Page 3.1.1 Dimensions and o r i e n t a t i o n of the specimens 73 3.1.2 S e l e c t i o n and prep a r a t i o n of the specimens 74 3.1.3 P r e c o n d i t i o n i n g of the specimens 78 3.2 Methods 80 3.2.1 Measurement of the M-ip r e l a t i o n s h i p . . . . 80 m 3.2.1.1 Porous p l a t e methods 80 3.2.1.2 Thermocouple psychrometer method. . 92 3.2.2 Measurement of the K(M), K0J1 ) and m D(M) fu n c t i o n s 101 3.2.2.1 Instantaneous p r o f i l e method . . . 101 3.2.2.2 One-step method 109 3.2.3 Measurement of the saturated water c o n d u c t i v i t y 110 3.2.4 Measurement of the s p e c i f i c g r a v i t y . . . . 112 Chapter 4 - RESULTS AND DISCUSSION 114 4.1 M-IJJ r e l a t i o n s h i p at high moisture contents . . . . 114 m 4.1.1 Results of the porous p l a t e experiments . . 114 4.1.2 Results of the TCP experiments 127 4.1.3 H y s t e r e s i s i n wood M-ifi r e l a t i o n s h i p m at high moisture contents vs. i n k - b o t t l e e f f e c t 131 v i Page 4.2 K(M), K(i|> ) and D(M) f u n c t i o n s i n the m l o n g i t u d i n a l d i r e c t i o n at high moisture contents 134 4.2.1 Results of the K and D t r a n s i e n t measurements 134 4.2.2 H y s t e r e s i s i n the K(M), K(ty ) and D(M) m f u n c t i o n s vs. h y s t e r e s i s i n the M-IJJ J m r e l a t i o n s h i p 148 4.2.3 Fl u x - g r a d i e n t r e l a t i o n s h i p 149 SUMMARY AND CONCLUSIONS 152 LITERATURE CITED 159 Appendix 1 - R e l a t i o n of water p o t e n t i a l to r e l a t i v e humidity and r a d i u s of curvature of the a i r - w a t e r i n t e r f a c e at 21°C 172 Appendix 2 - C a l i b r a t i o n data of the thermocouple psychrometers used f o r water p o t e n t i a l determinations i n wood at 21°C 174 Appendix 3 - Data p e r t i n e n t to the M-ili determinations . . . . 176 m Appendix 4 - Data p e r t i n e n t to the K and D determinations . . . 183 VITA v i i LIST OF TABLES Table Page Text 1. P e r t i n e n t p h y s i c a l p r o p e r t i e s of the m a t e r i a l s used i n the present study 77 2. Summary of the porous p l a t e experiments c a r r i e d out f o r the measurement of the M-t/1 r e l a t i o n s h i p 91 m Appendix 1 A l . l . R e l a t i o n of water p o t e n t i a l to r e l a t i v e humidity and radius of curvature of the a i r - w a t e r i n t e r f a c e a t 21°C 173 Appendix 2 A2.1. C a l i b r a t i o n data of the thermocouple psychrometers used f o r water p o t e n t i a l determinations i n wood at 21°C 175 Appendix 3 A3.1. Mean moisture contents (M) of western hemlock sapwood specimens obtained by e q u i l i b r a t i o n on porous p l a t e s under various imposed matric p o t e n t i a l s (ty ) at 21°C 177 m V1XX Table Page A3.2 Standard d e v i a t i o n s (SD) and standard e r r o r s (SE) of the M data obtained during the measurement of the M— m r e l a t i o n s h i p by the porous p l a t e methods at 21°C . . . . 178 A3.3 Mean s a t u r a t i o n percentages ( s p) of western hemlock sapwood specimens obtained by e q u i l i b r a t i o n on porous p l a t e s under v a r i o u s imposed matric p o t e n t i a l s (ty ) at 21°C 179 m A3.4 F-values from the a n a l y s i s of variance f o r the M and 3^ data obtained by e q u i l i b r a t i o n on porous p l a t e s under v a r i o u s imposed matric p o t e n t i a l s (ty ) at 21°C. . . 180 m A3.5 Mean matric p o t e n t i a l s (ip ) measured by thermocouple psychrometers (TCP) i n specimens p r e v i o u s l y e q u i -l i b r a t e d on porous p l a t e s at 21°C 181 A3.6 Standard d e v i a t i o n s (SD) and standard e r r o r s (SE) of the ty data obtained by the thermocouple psychrometer (TCP) method from specimens p r e v i o u s l y e q u i l i b r a t e d on porous p l a t e s at 21°c 182 Appendix 4 A4.1 Water c o n d u c t i v i t y (K) and water d i f f u s i v i t y (D) i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood at various moisture contents (M) and matric p o t e n t i a l s (ty ), as obtained by the instantaneous m p r o f i l e method during i m b i b i t i o n at 21°C 184 I X Table Page A4.2 Water c o n d u c t i v i t y (K) and water d i f f u s i v i t y (D) i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood at va r i o u s moisture contents (M) and matric p o t e n t i a l s )> a s obtained by the instantaneous p r o f i l e method during drainage or dry i n g at 21°C 185 A4.3 Water d i f f u s i v i t y (D) and water c o n d u c t i v i t y (K) i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood at v a r i o u s moisture contents (M) and matric p o t e n t i a l s )> a s obtained by the one-step method during drainage and i m b i b i t i o n at 21°C 186 A4.4 Water c o n d u c t i v i t y (K) i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood at f u l l s a t u r a t i o n as a o f u n c t i o n of specimen length and time of flow at 21 C . . . 187 LIST OF FIGURES Figure Page 1. Moisture content as a f u n c t i o n of matric p o t e n t i a l f o r spruce wood at 20°C (adapted from Penner 1963) 50 2. Moisture content as a f u n c t i o n of the r e l a t i v e humidity of the ambient a i r f o r sugar maple at 21°C (adapted from Goulet 1968) 52 3. Moisture content as a f u n c t i o n of the r e l a t i v e humidity of the ambient a i r f o r beech wood at 23°C (from data of Barkas 1936) 53 4. Cumulative pore s i z e d i s t r i b u t i o n of b i r c h wood obtained by the mercury i n t r u s i o n porosimeter method ( a f t e r Heizmann 1970) . 57 5. Water d i f f u s i v i t y as a f u n c t i o n of moisture content i n the r a d i a l d i r e c t i o n of beech during d r y i n g at 30°C (adapted from Voigt et al. 1940), and i n the l o n g i t u d i n a l d i r e c t i o n of b i r c h during i m b i b i t i o n at 20°C (adapted from Heizmann 1970) 65 6. Cross s e c t i o n of b o l t showing l o c a t i o n of specimen b i l l e t s 75 x i Figure Page 7. Schematic diagram of the pressure p l a t e apparatus used f o r M—iii determinations i n drainage between m -0.1 and -1.0 bar \b 81 m 8. Schematic diagram of the pressure p l a t e apparatus used f o r M—iii determinations i n drainage between m -1 and -15 bar \b 82 m 9. Schematic diagram of the te n s i o n p l a t e apparatus used f o r M-ib determinations i n i m b i b i t i o n between m 0 and -0.05 bar i> 85 Tm 10. Schematic diagram of the pressure p l a t e apparatus used f o r ll-ty determinations i n i m b i b i t i o n between Tm -0.1 and -1.0 bar \b 86 m 11. Cross s e c t i o n of the thermocouple psychrometer used f o r water p o t e n t i a l measurements i n wood (adapted from Campbell 1972) 93 12. Block diagram of the setup used f o r water p o t e n t i a l measurements w i t h thermocouple psychrometers 95 13. T y p i c a l c a l i b r a t i o n curves f o r the thermocouple psychrometers used f o r p o t e n t i a l measurements showing the psychrometer emf output as a f u n c t i o n of the water p o t e n t i a l of the c a l i b r a t i n g s o l u t i o n at 20 and 25°C 99 X l l Figure Page 14. Schematic diagram of the setup used f o r saturated water c o n d u c t i v i t y measurements I l l 15. Average moisture content-matric p o t e n t i a l r e l a t i o n s h i p of western hemlock sapwood along the boundary drainage and i m b i b i t i o n curves at 21°C ( L e f t p o r t i o n of broken l i n e s from r e s u l t s of Spalt 1958) 115 16. Moisture content-matric p o t e n t i a l r e l a t i o n s h i p along the boundary drainage and i m b i b i t i o n curves at 21°C, f o r Tree I at the heights of 0.6 and 9.6 m ( L e f t p o r t i o n of broken l i n e s from r e s u l t s of Spalt 1958) . . . 116 17. Average moisture content-matric p o t e n t i a l r e l a t i o n s h i p of western hemlock sapwood along the drainage curve s t a r t i n g from the green c o n d i t i o n at 21°C, shown i n s i d e the main h y s t e r e s i s loop ( L e f t p o r t i o n of broken l i n e s from r e s u l t s of Sp a l t 1958) 117 18. Moisture content-matric p o t e n t i a l r e l a t i o n s h i p along the drainage curve s t a r t i n g from the green c o n d i t i o n at 21°C, shown i n s i d e the main h y s t e r e s i s loop: a) Tree I , 0.6-m height; b) Tree I , 9.6-m height ( L e f t p o r t i o n of broken l i n e s from r e s u l t s of Spalt 1958) 118 x i i i F i gure Page 19. S a t u r a t i o n percentage-matric p o t e n t i a l r e l a t i o n s h i p along the boundary drainage and i m b i b i t i o n curves at 21°C, f o r Tree I at the heights of 0.6 and 9.6 m ( L e f t p o r t i o n of broken l i n e s from r e s u l t s of Sp a l t 1958) 126 20. E f f e c t i v e cumulative pore s i z e d i s t r i b u t i o n s of western hemlock sapwood as i n f e r r e d from the average M-^^ boundary drainage and i m b i b i t i o n curves 133 21. T y p i c a l M and ty p r o f i l e s obtained during i m b i b i t i o n m i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood specimens (Tree I I ) a) M p r o f i l e s ; b) ty p r o f i l e s 135 m 22. T y p i c a l M and ty p r o f i l e s obtained during drainage or d r y i n g i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood specimens (Tree I I ) a) M p r o f i l e s ; b) ty p r o f i l e s 136 m 23. Moisture content as a f u n c t i o n of time of flow i n specimens subjected to i m b i b i t i o n or drainage on porous p l a t e s 138 24. Water c o n d u c t i v i t y - m o i s t u r e content r e l a t i o n s h i p i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood along the boundary drainage and i m b i b i t i o n curves at 21°C 140 Figure 25. Water c o n d u c t i v i t y - m a t r i c p o t e n t i a l r e l a t i o n s h i p i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood along the boundary drainage and i m b i b i t i o n curves at 21°C 26. Water d i f f u s i v i t y - m o i s t u r e content r e l a t i o n s h i p i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood along the boundary drainage and i m b i b i t i o n curves at 21°C 27. Fl u x - g r a d i e n t r e l a t i o n s h i p at various s e l e c t e d values of M during i m b i b i t i o n i n 7.5-cm long western hemlock sapwood specimens a) f l u x vs. ty g r a d i e n t ; b) f l u x vs. M gradient XV LIST OF IMPORTANT SYMBOLS A c r o s s - s e c t i o n a l area, cm 2 °C degree C e l s i u s D water d i f f u s i v i t y of wood or other porous media, cm2/s E t o t a l energy of a system F t e s t of n u l l hypothesis G Gibbs f r e e energy, ergs G s p e c i f i c Gibbs f r e e energy, ergs/g G Q s p e c i f i c Gibbs free energy of the reference s t a t e , ergs/g G s p e c i f i c g r a v i t y of wood based on the green volume G Q s p e c i f i c g r a v i t y of wood based on the oven-dry volume G s p e c i f i c g r a v i t y of wood substance ws K water c o n d u c t i v i t y of wood or other porous media, cm3 (t^O^m/dyn s; k e l v i n K water c o n d u c t i v i t y tensor L length of sample (or length of f l o w ) , cm M moisture content, % (by weight) M^ f i n a l e q u i l i b r i u m moisture content, % M. i n i t i a l moisture content, % 1 M maximum moisture content, % max. M molecular weight of water, 18.01534 g/mol w N mole f r a c t i o n of s o l u t e s s N mole f r a c t i o n of water or solvent w P t o t a l pressure, dyn/cm 2, bars .xvi P e x t e r n a l h y d r o s t a t i c or gas pressure (gage p r e s s u r e ) , dyn/cm 2, bars P m equivalent matric pressure, dyn/cm 2, bars Q t o t a l heat content; flow r a t e , cm^l ^ C O/s R u n i v e r s a l gas constant, 8.3143 x 10 7 ergs/K mol R 2 c o e f f i c i e n t of determination RH r e l a t i v e humidity of the ambient a i r , % S t o t a l entropy of a system s a t u r a t i o n percentage (or degree of s a t u r a t i o n ) , % SD sample standard d e v i a t i o n SE standard e r r o r (or standard d e v i a t i o n of the mean) $Y/x standard d e v i a t i o n of the estimate T temperature, °C, K V t o t a l volume, cm 3 V s p e c i f i c volume, cm3/g V s p e c i f i c volume of water, 1.00000 cm3/g at 4°C w r W work done by a system V work done by a system, excluding the work of expansion against a constant pressure P c s p e c i f i c water c a p a c i t y of wood or other porous media, %/bar g a c c e l e r a t i o n due to g r a v i t y , 980.665 cm/s2; gram h height w i t h respect to the reference l e v e l , cm I dimension of length m dimension of mass m t o t a l mass of a body or a substance p vapor pressure of water i n wood or i n the system under c o n s i d e r a t i o n , mmHg, mbars x v i i P s vapor pressure of pure free water, mmHg, mbars q f l u x of water, cm3(H^O)/cm2s -> q f l u x vector r radius of curvature (or radius of c a p i l l a r y ) , um, cm t dimension of time t time, h, min, s z v e r t i c a l distance from the inflow or outflow end, cm A f i n i t e change V d i f f e r e n t i a l vector operator water p o t e n t i a l gradient av t j-t o t a l volumetric swelling, % t o t a l r a d i a l shrinkage, % u 3rr. t o t a l tangential shrinkage, % •J-\ t Y t o t a l volumetric shrinkage, % surface tension of water, dyn/cm 6 d i f f e r e n t i a l change e g porosity i n the green condition, f r a c t i o n a l e o porosity i n the oven-dry condition, f r a c t i o n a l n dynamic v i s c o s i t y of water, dyn s/cm2, g/cm s X proportion of the t o t a l pore volume, % Pw density of water, g/cm3 water p o t e n t i a l , ergs/g, dyn/cm2, bars g r a v i t a t i o n a l p o t e n t i a l , ergs/g, dyn/cm2, bars *m matric p o t e n t i a l , ergs/g, dyn/cm2, bars osmotic p o t e n t i a l , ergs/g, dyn/cm2, bars pressure p o t e n t i a l , ergs/g, dyn/cm2, bars x v i i i ACKNOWLEDGMENTS The author wishes to express h i s gratitude to Dr. Norman C. Franz for his encouragement and h e l p f u l advice throughout the research phase of the study and preparation of the manuscript. The author i s also deeply indebted to Dr. Jan de V r i e s , Department of S o i l Science, whose t e c h n i c a l guidance and invaluable assistance made t h i s t h e s i s a r e a l i t y . It i s a pleasure to acknowledge the h e l p f u l c r i t i c i s m of Dr. George Bramhall, formerly Senior Research S c i e n t i s t at the Western Forest Products Laboratory, who also provided the incentive to undertake t h i s project. Acknowledgment i s extended to Drs. Robert W. Wellwood and Jack W. Wilson, both of whom made many valuable commments and suggestions i n t h e i r reviews of the manuscript. Special thanks go to Mr. Greg Bohnenkamp, Technician, f o r h i s continued i n t e r e s t and help i n the experimental phase of the study. Acknowledgment i s c e r t a i n l y due to the Faculty of Forestry, the National Research Council of Canada, and the Education Department of Quebec f or providing f i n a n c i a l support which made t h i s research possible. L a s t l y , the author would l i k e to express h i s sincere appreciation to his wife for her help, patience, and understanding over the years that i t took to accomplish t h i s endeavor. To L u c i l l e and hi s daughter, D a l i l a h , the author wishes to dedicate t h i s t h e s i s as a p a r t i a l response to t h e i r love. INTRODUCTION Over the past few decades, a great deal of research has been done on water equilibrium and water movement phenomena i n wood. Despite that, many gaps i n knowledge s t i l l p e r s i s t . Reasons for an apparent slowness i n learning more about some of the problems involved are: 1) most investigations u n t i l very recently have been confined to the lower range of wood moisture contents; 2) i n s u f f i c i e n t v e r i f i c a t i o n by independent experimental investigations and independent methods; and 3) poor exchange of information and ideas between the f i e l d of wood science and technology and the various research f i e l d s conducting s i m i l a r studies on other porous media. The present study was undertaken with the aim of providing a fresh i n s i g h t into the question of water equilibrium and water movement phenomena i n wood at high moisture contents, i . e . , from-the so - c a l l e d f i b e r saturation point up to f u l l saturation. The problem considered was most fundamental. In f a c t , a need was f e l t f o r more basic accurate information with respect to the nature and magnitude of the forces by which water i s held and moved i n wood at high moisture contents, and with respect to the type(s) of flow equation(s) capable of describing and/or p r e d i c t i n g water transport processes i n wood under wet conditions. 2 A comprehensive l i t e r a t u r e review on the general aspects of the problem revealed that considerable progress had been made i n related areas of research based upon concepts which appear to be applicable i n the case of wood as w e l l . Two of these concepts along with inherent methods and techniques of measurement were adopted for the present i n v e s t i g a t i o n purposes. One concept postulates that a thermodynamically derived quantity c a l l e d water p o t e n t i a l can be used conveniently as a s i n g l e r a t i o n a l i n d i c a t o r of the energy state of water i n porous media, and of the d i r e c t i o n of i t s movement under most isothermal or quasi-isothermal flow conditions. The other postulates that the generalized Darcy's law for unsaturated flow and the equation of continuity provide a rigorous t h e o r e t i c a l basis for the d e s c r i p t i o n and p r e d i c t i o n of isothermal or quasi-isothermal mass flow of l i q u i d water i n unsaturated porous media. This unsaturated flow theory recognizes, however, that d i f f u s i v i t y type equations can as well be used conveniently for p r e d i c t i o n purposes under c e r t a i n flow conditions. Water p o t e n t i a l must then be a single-valued function of moisture content. A s e r i e s of experiments was c a r r i e d out i n an attempt to apply the foregoing concepts i n a study of the energy state and flow of water i n wood at high moisture contents. The f i r s t and most important part of the experiments consisted of determining the r e l a t i o n s h i p between matric p o t e n t i a l and moisture content. Matric p o t e n t i a l i s the main component of water p o t e n t i a l under most unsaturated conditions i n wood. In order to study the hysteresis phenomenon i n the moisture content-matric p o t e n t i a l relationship, both the boundary drainage and imbibition curves were determined. The drainage curve starting from the green condition was also obtained. The application of the unsaturated flow theory referred to above requires the determination of the unsaturated water conductivity as a function of moisture content or matric potential. Such measure-ments were carried out in both drainage and imbibition. The water diffusivity-moisture content relationship was also determined using the same specimens. Two independent methods of measurement were used for a l l types of determinations. These were confined to one wood species and one temperature. Knowledge emanating from this pioneering work should serve as a basis for further studies in this important area of research. Even the treatise of the theories involved which i s given in Chapter 1 can certainly be helpful both in teaching and research work. Although the adopted concepts have inherent restrictions which may limit the extent of their application, their usefulness in studies of water equilibrium and water movement phenomena in porous media in general appears well established. Chapter 1 BACKGROUND TO THE STUDY Water equilibrium and water movement phenomena i n porous media have been investigated i n several f i e l d s of natural science and i n a large number of branches of technology. Poor exchange of information and ideas between these various d i s c i p l i n e s , along with pertinent s p e c i f i c needs, has produced numerous approaches and a confusion of terminology i n the l i t e r a t u r e . This holds even within the general f i e l d of s o i l science from which the basic approaches used in the present study have been adopted. A d e t a i l e d discussion of the concepts r e f e r r e d to i n the present work appears then advisable, p a r t i c u l a r l y because these concepts are not very well known i n the wood science l i t e r a t u r e . This i s the object of the present chapter. As complementary information, a number of r e l a t e d methods and techniques of measurement w i l l also be o u t l i n e d . The symbols used and t h e i r meanings are l i s t e d i n the p r e l i m i n a r i e s to t h i s text. 4 5 1.1 Water p o t e n t i a l , ty, i n porous media 1.1.1 H i s t o r y and d e f i n i t i o n of the ty concept H i s t o r i c a l l y , the ty concept f i n d s i t s o r i g i n i n a c l a s s i c paper on the " c a p i l l a r y p o t e n t i a l " , presented at the s t a r t of the century by a p h y s i c i s t , Buckingham (1907). Recognizing h i m s e l f the r e s t r i c t i o n s inherent to h i s mechanical approach to the study of water e q u i l i b r i u m and water movement phenomena i n s o i l , Buckingham suggested a more general approach based on thermodynamics. Edlefsen and Anderson (1943) worked along these l i n e s and presented a notable and comprehensive treatment of the thermodynamics of s o i l moisture. Several other authors attempted to extend f u r t h e r the thermodynamic treatment given by Edlefsen and Anderson, or proposed somewhat d i f f e r e n t approaches (Low 1951, Low and Deming 1953, Babcock and Overstreet 1955, 1957, B o l t and M i l l e r 1958, Taylor and S l a t y e r 1960, Babcock 1963, Noy-Meir and Ginzburg 1967, Iwata 1972a, 1972b, Spanner 1972). A comprehensive and c r i t i c a l review of the thermodynamic approach was published by B o l t and F r i s s e l (1960). Very few references to the ty concept or equivalent notions have been made i n s t u d i e s of wood-water r e l a t i o n s thus f a r . However, the recent p u b l i c a t i o n i n the wood science l i t e r a t u r e of two research papers on the s o - c a l l e d chemical p o t e n t i a l concept ( V i k t o r i n and Cermak 1977, Kawai et al. 1978) i n d i c a t e s along w i t h the present work that the p o t e n t i a l concept i s on the verge of becoming a f a m i l i a r t o o l i n s t u d i e s of moisture behavior i n wood. I t i s s t i l l common p r a c t i c e to c l a s s i f y water i n porous media i n t o forms of " g r a v i t a t i o n a l water", " c a p i l l a r y water" and "hygroscopic water". S t r i c t l y speaking, t h i s i s an inaccurate and somewhat a r b i t r a r y way of d e s c r i b i n g the energy s t a t e of water from place to pl a c e and from time to time w i t h i n porous media. No d e f i n i t e boundaries e x i s t between these v a r i o u s forms of water. For example, the e a r t h g r a v i t a t i o n a l f o r c e f i e l d a f f e c t s a l l the water present i n a porous medium, not only a part of i t ; o r , the laws of c a p i l l a r i t y do not begin or cease at s p e c i f i c values of moisture content, M, or pore s i z e ( H i l l e l 1971). A more r a t i o n a l and convenient c r i t e r i o n that can be used to d e s c r i b e the energy s t a t e of water i n porous media i s the concept of water p o t e n t i a l . The p o s s i b l e values of ty are continuous and do not change a b r u p t l y from one s t a t e to another. The water p o t e n t i a l expresses the f i n i t e d i f f e r e n c e between the s p e c i f i c Gibbs f r e e energy of water i n the s t a t e under c o n s i d e r a t i o n (G) and the s p e c i f i c Gibbs f r e e energy of water i n the standard r e f e r e n c e s t a t e ( G Q ) . 1 The standard r e f e r e n c e s t a t e g e n e r a l l y used i s an h y p o t h e t i c a l pool of pure f r e e water, at atmospheric pressure, at a given and constant e l e v a t i o n , and u s u a l l y at the same temperature as that of water i n the porous m a t e r i a l ( H i l l e l 1971) Under these c o n d i t i o n s , ty i s equal to zero. The basic dimensions of ty are those of energy per u n i t mass: 1 . The Gibbs f r e e energy f u n c t i o n , G, which i s a well-known f u n c t i o n of c l a s s i c a l thermodynamics, i s defined by the expressions Throughout t h i s t e x t the s p e c i f i c v a l u e of an extensive property i s taken to- represent the valu e of the property per u n i t mass of the system. 7 (Gokcen 1975) G = E + PV - TS = H - TS [1] where E i s the t o t a l energy, P the pressure, V the volume, T the absolute temperature, H the enthalpy, and S the entropy of the system under c o n s i d e r a t i o n . I t i s c l e a r that G, l i k e H, i s a purely b r i e f and convenient d e f i n i t i o n a l f u n c t i o n . The j u s t i f i c a t i o n f o r i n t r o d u c i n g t h i s q u a n t i t y w i l l become more evident l a t e r i n the t e x t . Although very u s e f u l i n stu d i e s of water e q u i l i b r i u m and water movement phenomena i n porous media, the ty concept has i t s l i m i t a t i o n s . For example, the p a r t i t i o n of ty i n t o component p o t e n t i a l s that are r e a d i l y measurable and mutually independent may become sometimes a very d i f f i c u l t task. A l s o , a fundamental r e s t r i c t i o n i s found w i t h respect to the use of the ty concept as a c r i t e r i o n of spontaneity, i . e . , as an i n d i c a t o r of the d i r e c t i o n i n which water w i l l tend to move spon-taneously i n a non-equilibrium s t a t e . In f a c t , when d i f f e r e n t i a t e d w i t h respect to displacement ty provides the force tending to r e t u r n the system to e q u i l i b r i u m . But according to theory, t h i s i s true only f o r i sothermal c o n d i t i o n s . The ty gradients i n non-isothermal c o n d i t i o n s cease to be the sol e determinants of the d i r e c t i o n of water movement i n porous media (Spanner 1972). The l a t t e r r e s t r i c t i o n may appear ra t h e r serious at f i r s t s i g h t , knowing that changes i n ty or M i n p r a c t i c e do not o f t e n take place at constant temperature. However, as long as the e f f e c t of the temperature gradients on the o v e r a l l movement of water i s n e g l i g i b l e , the use of the ty concept i n flow problems can be considered legitimate as such. In other words, as long as the ty gradients are the main determinants of the d i r e c t i o n (not the r a t e ) 2 of water movement, the ty concept does not need to be r e s t r i c t e d to isothermal conditions. The water p o t e n t i a l i s expressible p h y s i c a l l y i n at l e a s t three ways ( H i l l e l 1971): 1) energy per unit mass with the dimensions Z-2t 2 ; 2) energy per unit volume with the dimensions ml 2 ; and 3) energy per unit weight with the dimension I. The fundamental expression of ty i s on a mass basis, using units of ergs/g or J/kg. Since water i s almost incompressible, i t s density i s p r a c t i c a l l y independent of ty (or M), except perhaps i n very dry porous media. Hence, there i s a d i r e c t proportion between ij; expressed on a mass basis and ty expressed on a volume basis. The l a t t e r expression y i e l d s the dimensions of pressure. Thus ty can be designated by i t s pressure equivalent which i s usually measured i n terms of dyn/cm2 or bars. From the foregoing i t follows that ty can also be expressed i n terms of an equivalent hydraulic head (weight b a s i s ) , which i s the height of a l i q u i d column corresponding to the given pressure. P h y s i c a l l y speaking, the two l a s t methods of expression may not be very meaningful but they remain more convenient to use than the f i r s t method i n view of the current practices followed i n the l i t e r a t u r e and i n the c a l i b r a t i o n of the measuring equipment. 2 The rate of water movement can be predicted only when both the water p o t e n t i a l gradient and the water conductivity are known. 1.1.2 P h y s i c a l p a r t i t i o n of ty i n t o i t s components Water i n porous media i s subject to a number of fo r c e f i e l d s that r e s u l t from the presence of the s o l i d matrix i t s e l f , s o l u t e s , e x t e r n a l h y d r o s t a t i c and gas pressures, and other v a r i a b l e s . Therefore, the water p o t e n t i a l , which expression i n t h i s t e x t i s merely a b r i e f f o r t o t a l water p o t e n t i a l , can be thought of as the sum of the separate c o n t r i b u t i o n s of these v a r i o u s for c e f i e l d s (Van Haveren and Brown 1972) ty - ty + ty + ty + ty + ty j. + ••• [2] m ro p r g e.f. where ty i s the m a t r i c 3 p o t e n t i a l , ty the osmotic p o t e n t i a l , ty m To P the pressure p o t e n t i a l , ty the g r a v i t a t i o n a l p o t e n t i a l , and ty g e. r. an e x t e r n a l f o r c e component p o t e n t i a l . The dots on the r i g h t s i d e i n d i c a t e that a d d i t i o n a l component p o t e n t i a l s are t h e o r e t i c a l l y p o s s i b l e . The matric p o t e n t i a l represents the combined e f f e c t of c a p i l l a r y and s o r p t i v e forces a r i s i n g from the presence of the s o l i d m a trix, i n c l u d i n g the i n s o l u b l e c o n s t i t u e n t s suspended i n the water. The osmotic p o t e n t i a l accounts f o r the presence of s o l u t e s i n the water. The pressure p o t e n t i a l describes the e f f e c t of a system bulk pressure e i t h e r greater or l e s s than the reference bulk pressure. This e f f e c t may be r e l a t e d to a d i f f e r e n c e i n bulk pressure between the porous medium a i r and the free atmosphere. I t may a l s o be r e l a t e d to a h y d r a u l i c pressure found i n the saturated c o n d i t i o n or to a system 3 A d j e c t i v e form of m a t r i x (Taylor and S l a t y e r 1960). induced bulk pressure r e s u l t i n g from the s w e l l i n g of the matrix against e x t e r n a l loads or r e s t r a i n t s . The g r a v i t a t i o n a l p o t e n t i a l accounts f o r g r a v i t y . The component p o t e n t i a l ty represents the i n t e g r a t e d sum of the e f f e c t s of a l l e x t e r n a l f o r c e f i e l d s , e x c l u s i v e of g r a v i t y . The presence of a d d i t i o n a l terms i n Eq. [2] may be required under s p e c i a l e q u i l i b r i u m or flow c o n d i t i o n s . An i n t e r a c t i o n term, f o r example, may have to be used i n the case of a s o l u t e - m a t r i x i n t e r a c t i o n . Under dynamic flow c o n d i t i o n s , a k i n e t i c energy term must a l s o be present according to theory, as i t appears i n B e r n o u i l l i ' s equation of h y d r o s t a t i c s f o r non-viscous incompressible f l u i d s (Olson 1973). Water flow i n porous media, however, i s q u i t e slow. Thus, the k i n e t i c energy term, which i s p r o p o r t i o n a l to the v e l o c i t y squared, i s g e n e r a l l y considered to be n e g l i g i b l e . This omission would remain q u i t e j u s t i f i e d as long as flow v e l o c i t i e s are l e s s than 1 cm/s (Hubbert 1940). In wood, published data on water p e r m e a b i l i t y (Choong and Kimbler 1971, L i n et al. 1973 , Tesoro et al. 1974) suggest that flow v e l o c i t i e s of t h i s magnitude would r a r e l y occur, except perhaps during saturated flow under high pressure p o t e n t i a l g r a d i e n t s . The primary purpose of p h y s i c a l l y p a r t i t i o n i n g ty i n t o i t s components i s to b e t t e r v i s u a l i z e the q u a n t i t i e s to be measured. Of course, Eq. [2] i s not very p r a c t i c a l unless the v a r i o u s component p o t e n t i a l s i d e n t i f i e d are expressed i n terms of measurable parameters. This i s the aim of the next s e c t i o n . 1.1.3 D e r i v a t i o n of the component p o t e n t i a l s The ty concept depends u l t i m a t e l y upon the f i r s t and second laws of thermodynamics. The f i r s t law i s merely a g e n e r a l i z e d form of the energy conservation law which f o r a closed system1* undergoing a change of s t a t e can be w r i t t e n as dE = dQ - dW [3] where dE i s the change i n the t o t a l energy of the system, dQ the heat input i n t o the system, and dW the work done by the system on the surroundings. I t i s important here to note the above s i g n convention which w i l l be followed throughout the r e s t of t h i s t e x t . I t says that dQ i s p o s i t i v e when heat i s t r a n s f e r r e d from the surroundings to the system, whereas dW i s p o s i t i v e when work i s done by the system on the surroundings. This s i g n convention o r i g i n a t e s from t r a d i t i o n a l thermodynamic c o n s i d e r a t i o n s i n v o l v i n g steam engines and s i m i l a r power devices which perform work when s u p p l i e d w i t h heat. The work term dW i n Eq. [3] i s g e n e r a l l y d i v i d e d i n t o two primary components dW = P dV + dW' [4] where P dV i s the work of expansion ( p o s i t i v e ) against a constant e x t e r n a l pressure which f o r a r e v e r s i b l e process i s equal to the pressure of the system P, and dW' the a l g e b r i c sum of a l l other types of work. Thus, Eq. [3] may be r e w r i t t e n as A system i s s a i d to be cl o s e d i f there i s no interchange of matter w i t h the surroundings. 12 dE = dQ - P dV - dW [5] With regard to the second law of thermodynamics, i t can be summarized by the equation (Gokcen 1975) d S > f [6] where dS i s the increase i n the entropy of the system, dQ the heat input into the system, and T the absolute temperature. The equality sign refers to r e v e r s i b l e processes and the greater than sign holds for i r r e v e r s i b l e processes. For an i n d i v i d u a l system undergoing a r e v e r s i b l e change of state, the increase i n entropy i s therefore equal to the heat which i t absorbs from the surroundings divided by i t s absolute temperature. Based on the f i r s t and second laws of thermodynamics, we can now derive r e l a t i o n s h i p s that w i l l describe the change i n the Gibbs free energy of a system during r e v e r s i b l e processes. D i f f e r e n t i a t i n g the f i r s t expression of Eq. [1], we have dG = dE + P dV + V dP - T dS - S dT [7] Substituting Eq. [5] and the form of Eq. [6] for a r e v e r s i b l e process into Eq. [7] gives dG = V dP - S dT - dW' [8] If the change of state i s taking place at constant temperature, Eq. [8] reduces to dG = V dP - dW' [9] For a process occuring at constant T and P, Eq. [ 8 ] further reduces to dG = -dW or -dG = dW* [10] I f , on the other hand, no work i s performed during the isothermal change but the pressure v a r i e s , we have dG = V dP [11] For a s i t u a t i o n i n which the r e v e r s i b l e change occurs at constant T and P with no work done except work of expansion against P, Eq. [ 8 ] becomes dG = 0 [12] which r e l a t i o n describes the conditions at equilibrium. Equations [10] through [12] constitute the basic energetic r e l a t i o n s from which the component p o t e n t i a l s w i l l be derived below. From t h i s viewpoint, Eq. [10] has a very important meaning. According to t h i s r e l a t i o n , the decrease i n the Gibbs free energy, -dG, of a system during a r e v e r s i b l e isothermal and i s o b a r i c change i s equal to the work done by the system excluding the work of expan-sion against P. In other words, a f i n i t e decrease i n the Gibbs free energy, -AG, i s a measure of the maximum " u s e f u l " work that can be done by the system undergoing a re v e r s i b l e process at constant T and P. A l l natural processes, however, are spontaneous to some extent. Hence, the useful work i n an actual process at constant T and P i s 14 always somewhat l e s s than the decrease In Gibbs f r e e energy (Edlefsen and Anderson 1943). Nevertheless, the foregoing i m p l i c a t i o n s w e l l i l l u s t r a t e the use of the ty concept as a c r i t e r i o n of spontaneity, r e c a l l i n g that ty represents the f i n i t e d i f f e r e n c e between the s p e c i f i c Gibbs f r e e energy of water i n the s t a t e under c o n s i d e r a t i o n and the s p e c i f i c Gibbs f r e e energy of water i n the standard r e f e r e n c e s t a t e . For d e r i v a t i o n purposes l e t us regard the "porous medium-w a t e r - a i r " system as a three phase system, of which only the phase "water" i s considered. The water i s thus assumed to be a system per se, being an homogeneous body of l i q u i d w i t h a d e f i n i t e s urface, which separates the phase from the other phases. The s o l u t e s are a l s o included i n the l i q u i d phase ( B o l t and F r i s s e l 1960). Since the component p o t e n t i a l s i d e n t i f i e d i n Eq. [2] are supposed to be mutually independent, they can be derived s e p a r a t e l y . The component p o t e n t i a l ty^ w i l l be derived f i r s t . For t h i s purpose, we w i l l consider the s i t u a t i o n i n which the change i n the energy s t a t e of water i s due to the so l e change of the e x t e r n a l h y d r o s t a t i c or gas pressure below or above the pressure of water i n the ref e r e n c e s t a t e . I t i s t h e r e f o r e assumed that T i s constant and that no work i s per-formed by the system during the change of s t a t e except P dV ( i . e . , dT = dW' = 0 ) . As seen e a r l i e r , Eq. [11] a p p l i e s i n such c o n d i t i o n s . Since the extensive p r o p e r t i e s G and V do not a l l o w us to deal r e a d i l y w i t h energy changes, Eq. [11] must be r e w r i t t e n i n terms of the s p e c i f i c values of G and V, Then, we have 15 dG = V dP [13] w where dG i s the s p e c i f i c d i f f e r e n t i a l Gibbs free energy of water i n ergs/g ( i n the cgs system of u n i t s ) , V w the s p e c i f i c volume of water i n cm 3/g, and dP the h y d r o s t a t i c pressure d i f f e r e n c e i n dyn/cm 2. The f i n i t e change i n G between water i n the standard reference s t a t e and water i n the s t a t e under c o n s i d e r a t i o n i s found by i n t e g r a t i n g Eq. [13] between the appropriate l i m i t s . As defined e a r l i e r , the standard reference s t a t e i s pure free water i n e q u i l i b r i u m w i t h atmospheric pressure which, by convention, i s taken as zero. Thus, we may w r i t e -G r P ex _ dG = / V dP [14] I w G Jo o where G q i s the s p e c i f i c Gibbs free energy of pure free water, G the s p e c i f i c Gibbs f r e e energy of water i n the s t a t e under c o n s i d e r a t i o n , and P the e x t e r n a l h y d r o s t a t i c or gas pressure (gage p r e s s u r e ) . 6X Assuming th a t the s p e c i f i c volume of water i s constant, Eq. [14] becomes G - G = i|i = V P [15] o p w ex where if> i s the pressure p o t e n t i a l i n ergs/g. I f we d i v i d e Eq. [15] by V , we then ob t a i n a pressure p o t e n t i a l i n terms of energy per w u n i t volume, which i s the equivalent of pressure. As w i l l be r e c a l l e d , a d i r e c t p r o p o r t i o n e x i s t s between t h i s l a t t e r method of expression (volume b a s i s ) and the former method (mass b a s i s ) since V (or the den-w s i t y ) at a given temperature i s p r a c t i c a l l y independent of the p o t e n t i a l . 16 Thus, the pressure p o t e n t i a l may a l s o be expressed as ty = P [16] p ex where ty i s now i n dyn/cm 2 or bars, depending on the u n i t s of P p ex From an a l g e b r a i c p o i n t of view, i t i s c l e a r that ty i n Eq. [16] i s d i f f e r e n t from i n Eq. [ 1 5 ] , although both terms are approximately equal when expressed i n the cgs system of u n i t s . I t must be pointed out that, f o r s i m p l i c i t y , the same symbol w i l l be used throughout t h i s t e x t to express any given component p o t e n t i a l i n these various u n i t s . We w i l l now derive the component p o t e n t i a l ty . An i n d i r e c t m approach has to be used here i n order of express ty i n terms of meas-m urable macroscopic p r o p e r t i e s . I t c o n s i s t s of expressing the matric p o t e n t i a l i n terms of an equivalent matric pressure. An equivalent matric pressure, P , can be measured by pressure p l a t e or membrane methods, which w i l l be described i n the next s e c t i o n . The e f f e c t of the s o l i d m a t r i x on the energy s t a t e of pure f r e e water i s then balanced against the pressure a p p l i e d across a p l a t e br membrane permeable to water and s o l u t e s ( S l a t y e r 1967). I f a gas pressure, here designated as P , i s a p p l i e d on the sample s i d e of the p l a t e or membrane, e q u i l i b r i u m i s reached when P^ has r a i s e d the energy s t a t e of the water i n the sample, G, to that of the pure f r e e water s i d e , G q . Assuming fo r s i m p l i c i t y that no so l u t e s are present i n the water, we can r e w r i t e Eq. [14] as [17] I t should be noted that the l i m i t s of i n t e g r a t i o n f o r the l e f t - h a n d side i n t e g r a l have been reversed. In f a c t , i n t h i s case P increases m the energy s t a t e of water from a given l e v e l G (P = 0) to that of pure free water, G q (P = P ). S o l v i n g Eq. [17] and rearranging f o r signs y i e l d s G - G - ty = -V P [18] o Tm w m where ty^ i s the matric p o t e n t i a l i n ergs/g, and P^ the equivalent matric pressure i n dyn/cm 2. As done above w i t h ty , we may a l s o express ty i n p m pressure u n i t s as *«, = - p™ £19] m m where ty i s now i n dyn/cm 2 or bars, depending on the u n i t s of P . The matric p o t e n t i a l can t h e r e f o r e be set n u m e r i c a l l y equal to the negative of the gas pressure a p p l i e d i n a pressure p l a t e or membrane apparatus, i n order to counterbalance the e f f e c t of the c a p i l l a r y and s o r p t i v e forces on the energy s t a t e of water. The next d e r i v a t i o n i s aimed to y i e l d a combined expression f o r the component p o t e n t i a l s ty^ and ty^. Once again, an i n d i r e c t approach w i l l be used. This approach i s based on the f a c t that the s p e c i f i c Gibbs f r e e energy of two or more phases of a substance i n e q u i l i b r i u m w i t h each other i s the same, i . e . , dG = 0 (Edlefsen and Anderson 1943). We know that both the matrix and the s o l u t e s a f f e c t the vapor pressure of water. I t f o l l o w s , t h e r e f o r e , that a combined measure of ty^ and ty can be obtained by merely determining the e f f e c t of the matrix and the s o l u t e s on the energy s t a t e of the vapor i n e q u i l i b r i u m w i t h the water. Taking the atmospheric pressure as constant and zero, we may-r e w r i t e Eq. [11] as where dG i s the change i n the s p e c i f i c Gibbs f r e e energy of water vapor (or water), V the s p e c i f i c volume of water vapor, and dp the vapor pressure d i f f e r e n c e . The f i n i t e change i n G i s found by i n t e g r a t i n g Eq. [20] between the l i m i t s where G q and p g are r e s p e c t i v e l y the s p e c i f i c Gibbs f r e e energy and the p a r t i a l pressure of water vapor i n e q u i l i b r i u m w i t h pure free water, and G and p the s p e c i f i c Gibbs free energy and the p a r t i a l pressure of water vapor i n e q u i l i b r i u m w i t h water i n the porous medium, r e s p e c t i v e l y . Assuming that water vapor behaves l i k e a p e r f e c t gas, we may now s u b s t i t u t e the i d e a l gas law, pV^ = RT, i n t o Eq. [21]. Since the molar s p e c i f i c volume V' i s i n cm 3/mol, and V i n cm 3/g, i t f o l l o w s that v v V = V'/M where M i s the molecular weight of water i n g/mol. A f t e r v v w w 6 6 s u b s t i t u t i o n of the gas law and rearrangement, Eq. [21]becomes dG = V dp v [20] [21] [22] and hence G - G = ty = l n Z ^23] o Tm+o M p w s • where ty represents the sum of the matric and osmotic p o t e n t i a l s Tm+o r (at constant P) i n ergs/g, R the gas constant (8.3143 x 10 7 ergs/K mol), T the absolute temperature i n K, M the molecular weight of water w (18.01534 g/mol), and p/p g the r e l a t i v e vapor pressure of water i n the porous medium. From the foregoing, i t f o l l o w s that a combined measure of the component p o t e n t i a l s ty^ and ty^ can be obtained by merely determining the r e l a t i v e vapor pressure of the water i n the porous m a t e r i a l . In unsaturated porous media, t h i s u s u a l l y y i e l d s a c l o s e estimate of the t o t a l water p o t e n t i a l . Methods f o r in situ determination of the r e l a t i v e vapor pressure at high moisture contents w i l l be described l a t e r i n the t e x t . The n a t u r a l l o g a r i t h m of a f r a c t i o n being negative, ty i s t h e r e f o r e negative. I t i s a maximum and zero when p/p i s m+o ° v ts equal to u n i t y . Consequently, from a terminology standpoint i t should be kept i n mind that a low water p o t e n t i a l u s u a l l y r e f e r s to ty tending toward a high negative value, whereas a high water p o t e n t i a l u s u a l l y r e f e r s to ty tending toward a low negative value or zero. We may a l s o express ty Q i n pressure u n i t s by using the equation p RT ty x = ~ i r - In *• [24] Tm+o M p w s where ty i s now i n dyn/cm 2, and p (=1/V ) i s the d e n s i t y of water rm+o J w w J 20 at the temperature T, i n g/cm3. A common p r a c t i c e i s to express the p o t e n t i a l i n bars (1 bar = 10 6 dyn/cm 2). Then, we have ty Tm+o p RT w , £ l n M p w s 10 - 6 [25] Obviously, i t i s assumed i n the use of Eqs. [24] and [25] that i s independent of ty. In wood, f o r example, p^ would p r a c t i c a l l y remain constant from the f u l l s a t u r a t i o n down to about 15% M ( S e i f e r t 1972). But one must always bear i n mind that the main reason f o r using t h i s method of expression i s one of convenience. This p r a c t i c e must not lea d to the f a l s e impression that p o t e n t i a l and pressure have common meanings. R e l a t i v e vapor pressure measurements y i e l d a combined measure of ty and ty when s o l u t e s are present i n the water. When the concen-m o t r a t i o n of s o l u t e s i s n e g l i g i b l e , these measurements then provide a measure of ty alone. On the other hand, i f the water s o l u t i o n were m i s o l a t e d from the porous medium, the same method could be used to get a separate estimate of ty . In t h i s l a t t e r case, Eq. [23] may be r e w r i t t e n as ty=~-lnR [26] o M p w s where 4JD i s the osmotic p o t e n t i a l i n ergs/g, and p/p g the r e l a t i v e vapor pressure over the pore f l u i d e x t r a c t or a s o l u t i o n i d e n t i c a l to the s o l u t i o n present i n the porous m a t e r i a l . I f p/p g cannot be measured d i r e c t l y by the pore f l u i d e x t r a c t technique, i t may be evaluated through the use of Raoult's law. This law s t a t e s that f o r an i d e a l s o l u t i o n (Gokcen 1975) £ -w [27] where N i s the mole f r a c t i o n of so l v e n t , namely water i n t h i s case, w Since N = 1 - N , where N i s the mole f r a c t i o n of s o l u t e s , we can w s s ' w r i t e = 1 - N P s [28] The term N i s defined as s n N = s n + n s w [29] where n i s the number of moles of s o l u t e s , and n the number of moles s w of water,. For d i l u t e s o l u t i o n s , l n (1 - N ) i s approximately equal to -N . Hence Eq. [26] can be r e w r i t t e n as s R T XT ty = - — N o M s w [30] where ty i s the osmotic p o t e n t i a l i n ergs/g. Once again, we may express ty^ i n pressure u n i t s as p RT w — - — N M s w [31] where tyQ i s now i n dyn/cm 2. I f tyQ i s to be expressed i n bars, then we must use the r e l a t i o n p RT w — N M s w 10 - 6 [32] F i n a l l y , we w i l l now proceed to the d e r i v a t i o n of the component p o t e n t i a l ij; . For t h i s purpose, i t w i l l be assumed that water undergoes an isoth e r m a l and i s o b a r i c change of s t a t e due to a change of p o s i t i o n i n the g r a v i t a t i o n a l f i e l d . When a body of mass m i s l i f t e d v e r t i c a l l y i n a g r a v i t a t i o n a l f i e l d of i n t e n s i t y g, the work of the downward force -mg exerted on i t i s W' = -mgh [33] where h i s the height of the point i n question r e l a t i v e to some a r b i t r a r y reference l e v e l (h = 0 ) , measured v e r t i c a l l y upward. Thus, the corresponding change i n Gibbs f r e e energy according to Eq. [10] i s dG = mg dh or — = dG = g dh [34] m g I n t e g r a t i n g Eq. [34] between the appropriate l i m i t s [35] and s o l v i n g the i n t e g r a l equation gives G - G = ty - gh [36] ° g g where ty i s the g r a v i t a t i o n a l p o t e n t i a l i n cmz/sz, which i s equivalent to ergs/g, g the a c c e l e r a t i o n due to g r a v i t y (980.665 cm/s 2), and h g the height i n cm between the p o s i t i o n of the water under c o n s i d e r a t i o n and the reference l e v e l . And i f ty i s to be expressed i n u n i t s of pressure, we may use the r e l a t i o n 23 where ty ^ i s i n dyn/cm 2, or T 1 r ^ [38] P gh w g 10 where ty i s i n bars. g I t i s evident from Eqs. [36] through [38] that ty can be p o s i t i v e or negative, depending on the a r b i t r a r y p o s i t i o n of the reference l e v e l . Nevertheless, i t i s customary to set the reference l e v e l at a p o s i t i o n which ensures that ty i s always p o s i t i v e or zero ( H i l l e l 1971). Under saturated c o n d i t i o n , ty = ty f o r most p r a c t i c a l purposes, except i n the presence of s o l u t e s . As soon as the porous medium s t a r t s to l o s e moisture, the r e l a t i v e importance of ty d e c l i n e s r a p i d l y . This completes the d e r i v a t i o n of the p r i n c i p a l component p o t e n t i a l s from c l a s s i c a l thermodynamics. Thus f a r no reference was made to the microscopic s t r u c t u r e of the porous medium. However, such c o n s i d e r a t i o n s may be very u s e f u l . This i s the case, f o r example, of the expression f o r the matric p o t e n t i a l that w i l l be derived next. As w i l l be r e c a l l e d from Eq. [ 1 5 ] , the component p o t e n t i a l due to an e x t e r n a l h y d r o s t a t i c pressure P equals the product V P ex w ex Let us assume that t h i s pressure i s caused by the s o l i d matrix i t s e l f due to c a p i l l a r y phenomena. In t h i s case, P becomes the pressure of water immediately under the surface of the concave a i r - w a t e r i n t e r f a c e s present i n the c a p i l l a r i e s . This pressure may be defined as ex 2 1 r [39] where y i s the surface tension of water i n dyn/cm, and r the radius of curvature of the ai r - w a t e r i n t e r f a c e (or c a p i l l a r y r a d i u s ) i n cm. Equation [39] i s obviously subject to s e v e r a l assumptions ( H i l l e l 1971) 1) the ai r - w a t e r i n t e r f a c e has a uniform radius of curvature or the pores are c y l i n d r i c a l ; 2) there i s a zero contact angle between the meniscus and the c a p i l l a r y w a l l s ; and 3) P r e f l e c t s the so l e e f f e c t ex of the surface tension f o r c e s . Now, s u b s t i t u t i n g Eq. [39] i n t o Eq. [ 1 5 ] , i n which ty becomes P i n the present case ty , we have m 2yV. m [40] where ty i s the matric p o t e n t i a l i n ergs/g. For convenience, we may express ty i n dyn/cm 2 or i n bars m 2y r 2Y r 10 -6 [41] [42] The r e a l usefulness of Eqs. [40] through [ 4 2 ] l i e s p r i m a r i l y i n the determination of the pore s i z e d i s t r i b u t i o n of the porous media based on experimentally obtained ty values. However, t h i s method of m determining the pore s i z e d i s t r i b u t i o n may become u n r e l i a b l e at low M's, where ty r e s u l t s from the combined e f f e c t of the surface tension m forces and the s o r p t i v e f o r c e s . The other assumptions inherent to the 25 d e r i v a t i o n of these r e l a t i o n s a l s o r e s t r i c t t h e i r use f o r t h i s purpose. 1.1.4 Methods of o b t a i n i n g separate and combined determinations of ty and ty at high moisture contents m o Several methods are a v a i l a b l e f o r the separate measurement of ty and ty at high moisture contents. The combined measurement m o of these two component p o t e n t i a l s i s a l s o r e a d i l y f e a s i b l e as a r e s u l t of recent advances i n the area of s o i l - p l a n t - w a t e r r e l a t i o n s research. In general, the methods of measuring ty can be c l a s s i f i e d m i n t o those that are performed under s t a t i c e q u i l i b r i u m c o n d i t i o n s and those performed under t r a n s i e n t c o n d i t i o n s . In the s t a t i c e q u i l i b r i u m methods, samples of the porous m a t e r i a l are allowed to e q u i l i b r a t e w i t h a known imposed pressure, and the corresponding M at e q u i l i b r i u m i s measured g r a v i m e t r i c a l l y or i n f e r r e d from the measured outflow or i n f l o w volume. Two commonly used methods of imposing ty or the pressure are (Chow 1973): 1) Haines' method (Haines 1930) i n which samples r e s t i n g on a water saturated porous p l a t e are exposed to the saturated atmosphere of the chamber above the porous p l a t e , w h i l e the pressure i n the water beneath the porous p l a t e i s kept below the atmospheric pressure by using a hanging water column; and 2) Richards' method (Richards 1947, 1948, 1949) i n which water beneath the pressure p l a t e or membrane i s kept at atmospheric pressure, w h i l e the a i r pressure a c t i n g on the sample i s increased by using a regulated a i r pressure source. 26 The imposed ty w i t h Haines' method i s given by the height of the water column which i s measured v e r t i c a l l y upward between the f r e e surface i n the manometer and the samples on the porous p l a t e ( i . e . , ty - -ty - -gh ). With Richards' method, the imposed ty m g g m i s e q u i v a l e n t to the negative of the a p p l i e d gage pressure, i n accord w i t h Eq. [19]. In general, Haines' method i s used i n the ty range m from 0 to -0.1 bar, and Richards' method i n the ty range from -0.1 m to -15 bars. Pressures as high as 1500 bars can be imposed w i t h the l a t t e r method i f s p e c i a l porous membranes and pressure chambers are used (Coleman and Marsh 1961, W i l l i a m s 1964). These two methods can be used i n both drainage and i m b i b i t i o n modes. 5 In the t r a n s i e n t methods, a probe i s i n s t a l l e d i n the porous sample and ty i s measured on a continuous b a s i s . An instrument m c a l l e d a tensiometer has been developed by Richards (1931) f o r t h i s purpose. A tensiometer c o n s i s t s of a porous cup, g e n e r a l l y of ceramic m a t e r i a l , connected by a continuous water column to a manometer, a vacuum gage, or an e l e c t r i c a l transducer ( H i l l e l 1971) . When the cup i s placed i n contact w i t h a porous m a t e r i a l , the bulk water i n s i d e the cup comes i n t o h y d r a u l i c contact w i t h the water i n the porous m a t e r i a l and an e q u i l i b r i u m s t a t e i s reached. The s u c t i o n exerted by the water i n the porous m a t e r i a l i s i n d i c a t e d on the manometer. This method i s p a r t i c u l a r l y s u i t a b l e f o r porous media of l a r g e volume and high p e r m e a b i l i t y . I t i s a l s o r e s t r i c t e d to the ty range from m 5 The terms "drainage" and " i m b i b i t i o n " are i n t h i s study the r e s p e c t i v e s u b s t i t u t e s f o r the terms "desorption" and "adsorption" commonly used i n the wood science l i t e r a t u r e ; t h i s i s i n accord w i t h the p r i n c i p l e s u s u a l l y i n v o l v e d i n the measurement of the M-<JJ r e l a t i o n s h i p at high moisture contents. 0 to -0.8 bar ( H i l l e l 1971). With regard to ty measurements, they are u n f o r t u n a t e l y d i f f i c u l t to c a r r y out and not always r e l i a b l e . An i n d i r e c t e v a l u a t i o n of ty^ can be obtained from the measurement of pore f l u i d e x t r a c t p r o p e r t i e s such as f r e e z i n g p o i n t depression, r e l a t i v e vapor pressure (Eq. [ 2 6 ] ) , and e l e c t r i c a l c o n d u c t i v i t y ( S l a t y e r 1967). A measurement of can a l s o be c a r r i e d out d i r e c t l y i n the porous medium by means of s a l i n i t y sensors (Oster and Ingvalson 1967, Ingvalson et al. 1970). This l a t t e r method, which c o n s i s t s of measuring the e l e c t r i c a l conduc-t i v i t y of the s o l u t i o n i n the porous medium, i s apparently by f a r the most r e l i a b l e . 6 In many s i t u a t i o n s , however, a separate measurement of the component p o t e n t i a l s ty^ and ty may not be necessary. Instruments c a l l e d thermocouple psychrometers can then be used advantageously to get a combined measure of ty and ty . These instruments have marked m o a notable t u r n i n g p o i n t i n the measurement of water p o t e n t i a l i n porous media. The development of the thermocouple psychrometer (TCP) r e a l l y began w i t h Spanner (1951), who described the use of P e l t i e r c o o l i n g i n thermocouple psychrometry. Spanner's work l e d to a f l u r r y of research a c t i v i t y on development, procedures and a p p l i c a -t i o n s of the TCP (Montieth and Owen 1958, Richards and Ogata 1958, Campbell et al. 1966, Rawlins and Dalton 1967, Brown 1970, Wiebe et al. 1970, Boyer 1972b, Neumann and T h u r t e l l 1972, Campbell et al. 1973, 6 S a l i n i t y Sensors together w i t h a S a l i n i t y Bridge were r e c e n t l y made commercially a v a i l a b l e by S o i l Moisture Equipment Corp., Santa Barbara, C a l i f o r n i a . Chow and de V r i e s 1973, Wiebe et al. 1977). The t h e o r e t i c a l b a s i s on which the TCP operates has a l s o been described by s e v e r a l i n v e s t i g a t o r s (Rawlins 1966, Peck 1968, 1969a, Boyer 1972a). E s s e n t i a l l y , a TCP i s a miniature temperature-sensing instrument which measures the wet-bulb temperature depression. This temperature d i f f e r e n t i a l depends i n turn on the r e l a t i v e vapor pressure of water i n the porous m a t e r i a l , hence on the water p o t e n t i a l (Eq. [ 2 3 ] ) . This instrument c o n s i s t s of a p a i r of thermocouple j u n c t i o n s i n s e r t e d i n t o a small chamber intended to r e c e i v e a sample of the porous m a t e r i a l , or enclosed i n t o a porous cup or screen housing used f o r in situ p o t e n t i a l measurements. Two b a s i c types of TCP's have evolved: 1) the wet-loop type, devised by Richards and Ogata (1958); and 2) the P e l t i e r type, f i r s t d escribed by Spanner (1951). The wet-loop psychrometer i s wetted manually by p l a c i n g a drop of water i n a small s i l v e r r i n g at the sen-si n g j u n c t i o n of the thermocouple. The P e l t i e r psychrometer i s wetted by passing f o r a few seconds a P e l t i e r c o o l i n g current through i t s sensing j u n c t i o n ; t h i s causes the sensing j u n c t i o n to c o o l below the dewpoint, r e s u l t i n g i n condensation of water vapor on i t . Immediately a f t e r the drop of water has been placed i n the r i n g , or a f t e r termina-t i o n of the c u r r e n t , evaporation of the water s t a r t s to occur i n the surrounding atmosphere. This produces a c e r t a i n temperature d i f f e r e n -t i a l between the sensing and the dry reference j u n c t i o n s ( l o c a t e d few mm a p a r t ) , which i n turn causes a minute voltage output from the thermocouple. An emf plateau s i g n a l of v a r i a b l e d u r a t i o n g e n e r a l l y r e s u l t s and can be measured w i t h a s e n s i t i v e galvanometer or micro-voltmeter. The emf output i s then converted i n terms of water p o t e n t i a l , using a c a l i b r a t i o n curve obtained w i t h standard s a l t s o l u t i o n s of known water p o t e n t i a l s . The P e l t i e r type TCP covers the ty range from -1 to about -100 bars at room temperature, which corresponds to the r e l a t i v e vapor pressure range from 0.99926 to 0.9288 (Brown 1970, Chow and de V r i e s 1973). The wet-loop type TCP i s s e n s i t i v e over a wider range of p/p g values ( S l a t y e r 1967), but i t i s not s u i t a b l e f o r in situ p o t e n t i a l measurements. The P e l t i e r type thermocouple psychrometer may a l s o provide a measure of ty based on dewpoint temperature depression measurements. A s p e c i a l microvoltmeter was r e c e n t l y devised f o r t h i s purpose (Wescor, Inc., Logan, Utah). This technique would have the main advantage of y i e l d i n g more r e l i a b l e r e s u l t s under c o n d i t i o n s not a l l o w i n g accurate temperature c o n t r o l (Neumann and T h u r t e l l 1972, Campbell et al. 1973). 1.2 Isothermal flow of l i q u i d water i n unsaturated porous media 1.2.1 Macroscopic flow equations As w i l l be r e c a l l e d from the f i r s t part of t h i s chapter, the change of the water p o t e n t i a l w i t h distance i s the t h e o r e t i c a l d r i v i n g f o r c e r e s p o n s i b l e f o r isothermal flow of water i n porous media. This force i s always d i r e c t e d from a zone of higher to a zone of lower p o t e n t i a l . A l l of the separate p o t e n t i a l gradients may not, however, 30 always be e q u a l l y e f f e c t i v e i n causing flow. In p a r t i c u l a r t h i s would be the case of the ty gradients which would r e q u i r e a semipermeable membrane to induce l i q u i d flow ( H i l l e l 1971). But l i t t l e i s known on the a c t u a l e f f e c t of the ty^ gradients on the mass movement of l i q u i d water i n porous media. I t i s the r e f o r e common p r a c t i c e to assume that osmotic p o t e n t i a l as w e l l as temperature gradients are s m a l l , and that t h e i r e f f e c t s on water movement are n e g l i g i b l e . Water flow i'n porous media i s customarily analysed on the b a s i s of the continuous medium macroscopic approach. The p h y s i c a l v a r i a b l e s i n v o l v e d such as d e n s i t y , c o n c e n t r a t i o n , v e l o c i t y and p o t e n t i a l , are thus considered to be continuous f u n c t i o n s of p o s i t i o n and time. I t i s f u r t h e r supposed that the macroscopic v a r i a b l e s are volume averages of the d i s t r i b u t i o n s of the corresponding microscopic v a r i a b l e s . The average must be taken over a s u f f i c i e n t number of pores, so that the value of the macroscopic v a r i a b l e can be considered to apply to a d i f f e r e n t i a l volume element l o c a t e d at a " p o i n t " i n the medium i n the macroscopic sense (Klute 1973). Darcy (1856) described i n an appendix to h i s t r e a t i s e "Les fon t a i n e s publiques de l a v i l l e de D i j o n " a s e r i e s of experiments on the v e r t i c a l downward flow of water through f i l t e r sand columns. From these experiments o r i g i n a t e d an e m p i r i c a l law d e s c r i b i n g the macro-scopic flow of water i n saturated porous media. This law has become a p p r o p r i a t e l y known as Darcy's law (Hubbert 1940, 1956). Translated i n t o a n o t a t i o n more appropriate to t h i s t e x t , the o r i g i n a l statement of Darcy's law was (Darcy 1856) Q = ^  (h + L - h ) [43] P l P 2 where Q denoted the flow r a t e through the sand column, K "un c o e f f i c i e n t dependant de l a p e r m e a b i l i t e de l a couche", A the c r o s s - s e c t i o n a l area of the column, h and h the equivalent pressure h y d r a u l i c heads at P l p 2 the upper surface and at the base of the column r e s p e c t i v e l y , as measured by open mercury manometers, and L the thic k n e s s of the sand l a y e r . Many expressions given as a statement of Darcy's law have appeared i n published l i t e r a t u r e without the term L found i n s i d e parentheses i n Eq. [43]. S t r i c t l y speaking, t h i s i s only c o r r e c t i f the flow i s h o r i z o n t a l , i n which case the g r a v i t y term L vanishes. Because of these repeated omissions, s i g h t has o f t e n been l o s t of Darcy's own work and of i t s s i g n i f i c a n c e (Hubbert 1940, 1956). Darcy's law as given by Eq. [43] may be ge n e r a l i z e d to any flow d i r e c t i o n as (Ch i l d s 1969) [44] By rearranging symbols and i n t r o d u c i n g a more convenient n o t a t i o n , we may transform Eq. [44] as [45] where q i s the flow r a t e per u n i t c r o s s - s e c t i o n a l area (Q/A), or simply the f l u x , i n cm 3(H^O)/cm 2s, K the saturated c o n d u c t i v i t y i n cm 3(H20)cm/dyn s, Aip (= tyj - ty^) the water p o t e n t i a l d i f f e r e n c e between the outflow and i n f l o w boundaries i n dyn/cm 2, and L the d i f f e r e n c e between the two boundaries i n cm. Equation [45] merely s t a t e s that the macroscopic flow v e l o c i t y i s d i r e c t l y p r o p o r t i o n a l to the macroscopic p o t e n t i a l g r a d i e n t . In the saturated c o n d i t i o n Aty g e n e r a l l y c o n s i s t s of a AiJ; term and a AIJJ term. The minus s i g n i n d i c a t e s that the flow i s P 8 i n the d i r e c t i o n of the decreasing p o t e n t i a l . The saturated c o n d u c t i v i t y K i n Eq. [45] may be replaced by the r a t i o k/n, where k i s the s p e c i f i c p e r m e a b i l i t y of the porous medium, and n the v i s c o s i t y of water. However, the present study i s p r i m a r i l y concerned w i t h flow of water under unsaturated c o n d i t i o n s where the p e r m e a b i l i t y concept becomes l e s s p r a c t i c a l . The concept of unsaturated water c o n d u c t i v i t y discussed below i s a more appropriate choice i n such a case. Darcy's law i s an e m p i r i c a l law by i t s o r i g i n , but u l t i m a t e l y o i t can be shown to be a consequence of the more fundamental Navier-Stokes law d e s c r i b i n g the isothermal flow of incompressible Newtonian f l u i d s (Hubbert 1956, C h i l d s 1969, P h i l i p 1970). E s s e n t i a l l y , i t i s a consequence of the Navier-Stokes equation that the f l u x i s p r o p o r t i o n a l to the p o t e n t i a l gradient when the i n e r t i a l f orces are n e g l i g i b l e i n comparison with the viscous forces ( i . e . , when the pores are s u f f i -c i e n t l y s m all and the microscopic flow v e l o c i t y i s s u f f i c i e n t l y slow). The remaining features of Darcy's law f o l l o w from the s t a t i s t i c a l requirement that the porous medium i s l a r g e enough to be regarded as a uniform body ( C h i l d s 1969). The d e r i v a t i o n of Darcy's law from the Navier-Stokes equation al s o c l e a r l y i n d i c a t e s the c o n d i t i o n s that must be s a t i s f i e d before the law may be a p p l i e d s a f e l y . The c o n d i t i o n s of isothermal laminar flow, water c o n t i n u i t y throughout the flow r e g i o n , water i n c o m p r e s s i b i l i t y , and constant water v i s c o s i t y a l l o r i g i n a t e from the development of the Navier-Stokes equation i t s e l f . The assumptions of macroscopic homogeneity and n e g l i g i b l e i n e r t i a l f orces r e s u l t from the a p p l i c a t i o n of t h i s equation to the s p e c i a l case of macroscopic flow of water i n porous media. There e x i s t s a dimensionless combination of flow c h a r a c t e r i s t i c s , known as the Reynolds number (Re), from which one may judge of the s a f e t y of a p p l y i n g Darcy's law. In s t r a i g h t c i r c u l a r tubes, the c r i t i c a l Re value beyond which turbulence se t s i n has v a r i o u s l y been reported to be of the order of 1000-2200 (Ch i l d s 1969, Scheidegger 1974). In porous media, however, i t would be safe to apply Darcy's law only as long as Re i s smaller than u n i t y (Muskat 1946, H i l l e l 1971). The c o n d i t i o n that the i n e r t i a l f o r c e s i n the Navier-Stokes equation be n e g l i g i b l e would then put a more s t r i n g e n t r e s t r i c t i o n on Re than the c o n d i t i o n of laminar flow i t s e l f (Hubbert 1940, C h i l d s 1969). Where the p o t e n t i a l gradient v a r i e s i n space and time, we must consider l o c a l i z e d f l u x and gradient values r a t h e r than o v e r a l l Values f o r the medium as a whole. A more exact and ge n e r a l i z e d expression of Darcy 1s law i s w r i t t e n , t h e r e f o r e , i n d i f f e r e n t i a l form. For a saturated i s o t r o p i c porous medium, S l i c h t e r (1899) g e n e r a l i z e d Darcy's law i n t o a three-dimensional d i f f e r e n t i a l equation which i n the present n o t a t i o n may be w r i t t e n as q = -K 7-J) [46] ->• -> where q i s the f l u x v e c t o r , K the saturated c o n d u c t i v i t y , and the p o t e n t i a l g r a d i e n t . 7 For an i s o t r o p i c porous medium, K i s a s c a l a r and the d i r e c t i o n of the vec t o r q c o i n c i d e s w i t h that of the v e c t o r F ( = -V\}J) , which i s the d r i v i n g f o r c e a c t i n g on the l i q u i d (Hubbert 1956) . I f the porous medium i s a n i s o t r o p i c , K i s not a s c a l a r but a symmetric tensor of second order. In gene r a l , the two vec t o r s q and F w i l l not c o i n c i d e , except when p a r a l l e l to the p r i n c i p a l axes of the tensor (Hubbert 1956). For an a n i s o t r o p i c porous medium, Eq. [46] then becomes q = -K • Vifi [47] where K i s the c o n d u c t i v i t y tensor. The dot between the terms K and VIJJ i n d i c a t e s that any product of vectors which the m u l t i p l i c a t i o n i n v o l v e s i s c a r r i e d out according to the r u l e s of s c a l a r products. The d i f f e r e n t i a l form of Darcy's law can be extended to unsaturated flow, but w i t h the p r o v i s i o n t h a t K i s now a f u n c t i o n of the moisture content ( i . e . , K = K(M)). In f a c t , the presence of a i r i n the c a p i l l a r y flow system causes a r e d u c t i o n of the e f f e c t i v e 7 Sty -t Sty ? Sty. , , . ,-r <S -f S t S *. Vty = ( l - r ^ + j - r ^ + k -T^) = grad ty, xn which V = ( l — + j - j — + k -r—) <5x J 6y Sz o r , v 6x ^y _^6z i s the d i f f e r e n t i a l v e c t o r operator ( c a l l e d " d e l " ) , and i , j and k the orthogonal u n i t vectors along the d i r e c t i o n s x, y and z, r e s p e c t i v e l y (Malvern 1969). 35 c r o s s - s e c t i o n a l area a v a i l a b l e f o r l i q u i d flow, and increases the t o r t u o s i t y of the remaining flowpaths. U l t i m a t e l y , t h i s causes an i n e v i t a b l e decrease i n K (Swartzendruber 1969, H i l l e l 1971). We may then r e w r i t e Eq. [47] f o r unsaturated flow as q = -K(M) • Vty [48] where K(M) i s the unsaturated c o n d u c t i v i t y tensor, and $ty the p o t e n t i a l gradient (Aty i s now taken as the sum of AI/J and Aty , ty being g e n e r a l l y m g P assumed to be constant or zero under unsaturated flow c o n d i t i o n s ) . 8 Richards (1931) introduced the unsaturated c o n d u c t i v i t y f u n c t i o n K(ty ). He pointed out that, i f ty i s a s i n g l e - v a l u e d f u n c t i o n of M, m m b the choice between the fu n c t i o n s K(M) and K(4> ) i s simply a matter of m mathematical expediency. The three f u n c t i o n s H(ty ), K(M) and K(ty ) may m m a l l be h y s t e r e t i c , but i n p r a c t i c e the h y s t e r e s i s problem can be evaded by l i m i t i n g the use of Eq. [48] to cases i n which ty (or M) i s e i t h e r m i n c r e a s i n g or decreasing continuously ( H i l l e l 1971). For a porous medium having orthogonal p r i n c i p a l axes the tensor K(M) i s symmetric f o r any o r i e n t a t i o n of the coordinate system, and i s diagonal f o r a coordinate system c o i n c i d i n g w i t h the p r i n c i p a l axes ( C o l l i n s 1961) . Therefore, i f the d i r e c t i o n s x, y and z of the coordinate system are chosen to c o i n c i d e w i t h the orthogonal p r i n c i p a l axes of the porous medium, a l l the c o e f f i c i e n t s of the tensor vanish 8 When t o t a l a i r - p r e s s u r e d i f f e r e n c e s cannot be neglected, flow of water i n porous media must be regarded as a two-phase s a t u r a t e d flow problem, which subject i s commonly t r e a t e d , f o r example, i n petroleum engineering l i t e r a t u r e ( C o l l i n s 1961, Morel-Seytoux 1969, Scheidegger 1974). 36 except K (M), K (M), and K (M). Thus, we may write (Childs 1969) xx yy zz J K(M) = 11K (M) '+ J J K (M) + £jk (M) [49] ^ xx y y z z The right-hand side of Eq. [49] i s the dyadic representation of the diagonal tensor K(M) . For s i m p l i c i t y , one may rewrite Eq. [49] as K(M) = txK (M) + J J K (M) + £ & (M) [50] x y z Substituting now Eq. [50] into Eq. [48] and carrying out the dot product y i e l d s (Childs 1969) q M v M > H ) ^ ( v M ) f ) + t ( v » ) ^ ) [51] Since q i n Eq. [51] can also be written as q = iq + + Icq , the x J M y M z scalar components of q, therefore, are: q = - K ( M ) M x x 6x q y = " K y ( M ) f C 5 2 ] q = - K (M) ^ z z v 6z Equations [51] and [52] are two a l t e r n a t i v e formulations of generalized Darcy's law, for three-dimensional unsaturated flow i n anisotropic porous media. To obtain the general flow equation and account f or transient as well as steady flow processes, Darcy's law must be combined with the mass-conservation law, expressed i n the equation of continuity. The mass-conservation law s t a t e s that i f the r a t e of i n f l o w i n t o a d i f f e r e n t i a l volume element i s greater than the r a t e of outflow, then the volume element must be s t o r i n g the excess and i n c r e a s i n g i t s moisture content, and vice versa. For three-dimensional unsaturated flow, the equation of c o n t i n u i t y may be w r i t t e n as ( H i l l e l 1 9 7 1 ) <5M 3 -y . ^ = - V • q [ 5 3 ] where 6M/6t i s the r a t e of change of the moisture content of the d i f f e r e n t i a l volume element w i t h respect to time, and V • q the sum of the r a t e s of change of the three f l u x components q^, q^ and q^ w i t h respect to d i s t a n c e , along the p r i n c i p a l axes x, y and z, r e s p e c t i v e l y . 9 For one-dimensional flow i n the x d i r e c t i o n , Eq. [ 5 3 ] becomes _ ^ x 6t fix [ 5 4 ] In most analyses of flow i n porous media, the f l u x has the dimensions . This i m p l i e s that M i n Eqs. [ 5 3 ] and [ 5 4 ] must be c a l c u l a t e d on a volume b a s i s . I f M i s c a l c u l a t e d on a mass b a s i s , the l e f t - h a n d s i d e of Eqs. [ 5 3 ] and [ 5 4 ] must then be m u l t i p l i e d by the s p e c i f i c g r a v i t y of the porous medium based on i t s volume a t the moisture content i n question. I f now we s u b s t i t u t e Eq. [ 5 1 ] i n t o Eq.. [ 5 3 ] , we o b t a i n 9 V • q &%. 5q„ <Sqr ± -> I 'x - *y - -z I _ ,. -»• 6x + ~&y~ + 6z~ I = d i v q ' T h e e x P r e s s i o n i n s i d e parentheses i£e^in carrying^out t' (Malvern 1 9 6 9 ) . i s ^ a s c a l a r q u a n t i t y since he s c a l a r product V • q, i - i = J-J = k«k = 1 and i - j = j«k = x-k = 0 This i s the general macroscopic equation f o r three-dimensional i s o -thermal unsaturated flow of water, i n non-swelling a n i s o t r o p i c porous media. For one-dimensional flow i n the x d i r e c t i o n , Eq. [55] reduces to 6M _ 6 6t " 6x The r e s t r i c t i o n that Eqs. [55] and [56] apply only to non-s w e l l i n g porous media i s r a r e l y e x p l i c i t l y s t a t e d i n the l i t e r a t u r e , as pointed out by P h i l i p (1970). Reasons f o r t h i s r e s t r i c t i o n are both of h y d r o s t a t i c and hydrodynamic nature. D e t a i l e d d i s c u s s i o n of t h i s t o p i c would, however, i n v o l v e complications beyond the scope of the present study. Reviews of the subject were presented by P h i l i p (1969b, 1970). To s i m p l i f y the mathematical and experimental treatment of unsaturated flow processes, i t may sometimes be advantageous to change the unsaturated flow equations derived above i n t o a form analogous to the equations of d i f f u s i o n and heat conduction, f o r which ready s o l u t i o n s are a v a i l a b l e (Carslaw and Jaeger 1959, Lykov 1968, Crank 1975). In f a c t , i t i s p o s s i b l e under s p e c i a l c o n d i t i o n s to transform the flow equations so that the f l u x becomes r e l a t e d to a moisture content gradient r a t h e r than to a p o t e n t i a l gradient. The ty gradient m must then be the only e f f e c t i v e force causing flow, and ty a s i n g l e -m valued f u n c t i o n of M. K (M) |i x o x [56] The matric p o t e n t i a l gradient Sty^/Sx can be expanded by the chain r u l e of c a l c u l u s as f o l l o w s Sty dty Sx dM 6M 6x [57] where SM/Sx i s the moisture content gradient i n the x d i r e c t i o n . We can now r e w r i t e the f i r s t of the expressions [52] i n t o the form V /T,K\ ^ m SM , ,. 6M q = -K (M) — — — = -D (M) — x x dM Ox x ox [58] where D (M) i s the water d i f f u s i v i t y f u n c t i o n i n the x d i r e c t i o n , x Of course, s i m i l a r expressions can be derived f o r the f l u x components q and q as w e l l . Water d i f f u s i v i t y has the dimensions Z-2t 1, and y z g e n e r a l l y bears the u n i t s of cm 2/s. The f u n c t i o n D(M) i s thus defined as ( H i l l e l 1971) dty D(M) = K(M) m dM [59] where dijWdM i s the r e c i p r o c a l of the s p e c i f i c water c a p a c i t y c(M) ( H i l l e l 1971) c(M) = dM dty [60] F i n a l l y , i n t r o d u c i n g the d i f f u s i v i t y f u n c t i o n i n t o Eq. [ 5 6 ] , we o b t a i n SM St S_ 6x D ( M ) f x Sx [61] A word of c a u t i o n i s now i n order w i t h respect to the use of the d i f f u s i v i t y concept, and a l l r e l a t i o n s h i p s derived from i t , to p r e d i c t l i q u i d water movement i n porous media. Although D has the p h y s i c a l dimensions of the c l a s s i c a l d i f f u s i o n c o e f f i c i e n t , i t does not describe a process of molecular d i f f u s i o n . The process of l i q u i d water movement i n porous media i s not one of d i f f u s i o n but one of mass flow (Swartzendruber 1966, H i l l e l 1971). The d i f f u s i v i t y concept must be looked upon only as a mathematical device, which sometimes can be used to f a c i l i t a t e both the mathematical and the experimental t r e a t -ments of water flow problems i n porous media. This gain i n s i m p l i c i t y i s , however, at the expense of g e n e r a l i t y l o s t by the n e c e s s i t y of imposing the c o n d i t i o n s c i t e d above. The magnitude of the e r r o r s r e s u l t i n g from these r e s t r i c t i o n s must be determined f i r s t i f the d i f f u s i v i t y equations are to be used r e l i a b l y . 1.2.2 Methods of measuring the f u n c t i o n s K(M), K(tp ) and D(M) m Knowledge of the nature of the flow f u n c t i o n s K(M), K(il> ) m or D(M) i s g e n e r a l l y r e q u i r e d before any of the mathematical t h e o r i e s of water movement i n porous media can be a p p l i e d i n p r a c t i c e . Several t h e o r e t i c a l models have been developed f o r p r e d i c t i n g the unsaturated c o n d u c t i v i t y from more fundamental p r o p e r t i e s of the porous medium. P a r t i c u l a r l y popular are the s t a t i s t i c a l models based on the pore s i z e d i s t r i b u t i o n of the porous medium=(Childs and C o l l i s - G e o r g e 1950, Burdine 1953, M a r s h a l l 1958, W y l l i e and Gardner 1958, M i l l i n g t o n and Quirk 1961, F a r r e l and Larson 1972, Campbell 1974, Mualem 1976, Ghosh 1977). A u n i f i e d approach to these v a r i o u s s t a t i s t i c a l models was proposed by Mualem and Dagan (1978). Although such p r e d i c t i v e models 41 may be u s e f u l , t h e i r present degree of development i s f a r from reaching the p o i n t r e q u i r i n g no experimental determination of K. In p r i n c i p l e , any s o l u t i o n to a water-flow boundary-value problem could serve as the b a s i s f o r a method of measuring K or D i f these q u a n t i t i e s appear i n the s o l u t i o n (Swartzendruber 1969). Reviews of the methods a v a i l a b l e f o r measuring K and D i n unsaturated porous media (one-phase flow) were published by Swartzendruber (1969) and Klute (1972). Only the methods which may present some p o s s i b i l i t i e s of a p p l i c a t i o n i n s t u d i e s of wood-water r e l a t i o n s w i l l be o u t l i n e d below. The methods of measuring K and D are g e n e r a l l y d i v i d e d i n t o two types, namely the steady-state and the t r a n s i e n t methods. 1.2.2.1 Steady-state methods The steady-state methods i n v o l v e the establishment of a one-dimensional flow system i n which q, ty and M are constant w i t h time. Darcy's law and the i n f e r r e d d i f f u s i v i t y equation f o r one-dimensional flow are u s u a l l y used i n t h e i r f i n i t e d i f f e r e n c e form to determine K and D, r e s p e c t i v e l y . The p o t e n t i a l d i f f e r e n c e Aty and the moisture content d i f f e r e n c e AM are measured between two p o i n t s separated by a distance Ax = ~ xi/ The computed K and D are then assumed to correspond to the mean p o t e n t i a l ijj (= (ty^ + ty^)/2) or the mean moisture content M (= (M^ + M^)/2) at (x^ + x^)/2. The distance Ax must be kept small s i n c e the i m p o s i t i o n of a p o t e n t i a l gradient normally r e s u l t s i n c u r v i l i n e a r f u n c t i o n s ty(.x) and M(x) (Swartzendruber 1969, K l u t e 1972). 42 One type of steady-state method c o n s i s t s of e s t a b l i s h i n g e i t h e r a steady-state upward or downward flow of water through a long specimen. A constant upward flow can be obtained by immersing the lower end of the specimen i n a c o n s t a n t - l e v e l water supply while s u b j e c t i n g i t s upper end to steady-state evaporation (Moore 1939). A f t e r a t t a i n -ment of a ste a d y - s t a t e , the f l u x i s given by the evaporation r a t e . The bottom of the specimen can a l s o be put i n contact w i t h a porous d i s c or p l a t e connected to a constant-head water r e s e r v o i r ( N i e l s e n et al. 1960). A constant downward flow can be obtained by connecting the upper end of the specimen to a constant-head water supply and then keeping the bottom end below a water surface ( C h i l d s and Co l l i s - G e o r g e 1950), or i n contact w i t h a porous p l a t e to which a water s u c t i o n i s ap p l i e d . Another type of steady-state method c o n s i s t s of h o l d i n g a specimen of f i n i t e length between two saturated porous d i s c s or p l a t e s (Richards 1931, Richards and Moore 1952, Corey 1957, N i e l s e n and Biggar 1961, E l r i c k and Bowman 1964, K l u t e 1965, Talsma 1970, Topp 1971b). The flow can be made to occur i n e i t h e r the v e r t i c a l or h o r i z o n t a l d i r e c t i o n . The mean values of ty and M can be v a r i e d by applying appropriate water su c t i o n s to the porous p l a t e s or by imposing a i r pressures to the specimen from i t s l a t e r a l faces. The i n f l o w and outflow are measured v o l u m e t r i c a l l y by means of constant-head devices. The p o t e n t i a l d i f f e r e n c e Aty can be i n f e r r e d from the p o t e n t i a l d i f f e r e n c e imposed across the ends of the specimen provided t h i s i s known with s u f f i c i e n t accuracy. Otherwise, i t becomes e s s e n t i a l to i n s t a l l tensiometers or thermocouple psychrometers at convenient i n t e r v a l s along the d i r e c t i o n of flow. The moisture content can be i n f e r r e d from a M—ip r e l a t i o n s h i p measured on separate samples, or can be measured d i r e c t l y by non-destructive methods or by s e c t i o n i n g . In general, the steady-state methods are f a i r l y accurate but they have the disadvantage of r e q u i r i n g r e l a t i v e l y long times to e s t a b l i s h steady flow. These methods are t h e r e f o r e best s u i t e d f o r the determination of D and K at high moisture contents near f u l l s a t u r a t i o n , i n which range the establishment of a steady flow i s e a s i e r due to high c o n d u c t i v i t y . 1.2.2.2 Transient methods The t r a n s i e n t methods i n v o l v e the establishment of one-dimensional flow system i n which q, ty and M vary w i t h both p o s i t i o n and time. They may be grouped i n t o (Klute 1972): 1) o u t f l o w - i n f l o w methods; and 2) instantaneous p r o f i l e methods. A p a r t i c u l a r l y i n t e r e s t i n g t r a n s i e n t outflow method i s the s o - c a l l e d "one-step method" which was described by Doering (1965) and based on an a n a l y s i s by Gardner (1962). In t h i s method one l a r g e pressure d i f f e r e n c e i s imposed to a specimen of f i n i t e t h i c k n e s s i n a pressure p l a t e or pressure membrane apparatus, and the rat e of outflow i s measured. I f the porous p l a t e or membrane impedance ( r e s i s t a n c e to flow) i s n e g l i g i b l e and M i s approximately uniform w i t h respect to thickness during the outflow from the specimen, the f o l l o w i n g r e l a t i o n can then be used to c a l c u l a t e D (Gardner 1962) 44 D(M) = - 4L2 dM dt [62] where D i s the water d i f f u s i v i t y at the moisture content M, L the t o t a l l ength of the sample, M the moisture content a f t e r a given time of flow t , M^ the f i n a l e q u i l i b r i u m moisture content, and dM/dt the outflow r a t e measured at the moisture content M. The i n i t i a l and boundary c o n d i t i o n s u n d e r l y i n g the d e r i v a t i o n of t h i s equation are that M i s uniform i n i t i a l l y , that no flow occurs at one boundary, and that M i s constant and equal to M f at the other boundary f o r t > 0. methods of a n a l y s i n g outflow data which have been claimed to be more accurate than Doering's method ( i f D changes monotonically w i t h M), but which are more complicated. Although the one-step method has been used p r i m a r i l y as an outflow method, t e c h n i c a l l y i t can a l s o be used as an i n f l o w method. However, the measurement e r r o r may then i n c r e a s e sub-s t a n t i a l l y s i n c e the assumption of n e a r l y uniform M across the specimen i s apparently not v a l i d i n i m b i b i t i o n (Klute 1972). F i n a l l y , Eq. [62] can a l s o be a p p l i e d to d r y i n g by evaporation as long as the i n i t i a l and boundary c o n d i t i o n s are met. I t should prove to be reasonably accurate f o r the second-half of the t o t a l M change as pointed out by McNamara and Hart (1971), who used a v a r i a t i o n of t h i s equation f o r s o r p t i o n s t u d i e s i n wood. Gupta et al. (1974) and Passioura (1976) provided a l t e r n a t i v e Other o u t f l o w - i n f l o w methods of considerable i n t e r e s t are the Boltzmann's transform methods. These methods are based on the use of an e f f e c t i v e l y s e m i - i n f i n i t e , uniform, one-dimensional flow system i n which the e f f e c t of g r a v i t y i s neglected ( h o r i z o n t a l f l o w ) . The i n i t i a l and boundary moisture contents are assumed constant. The Boltzmann's transform t}> (M) = x t where if i s a c o e f f i c i e n t which v a r i e s w i t h M, i s used to solve Eq. [61] f o r D(M). For the case where M i s measured as a f u n c t i o n of p o s i t i o n at a f i x e d time, we have the r e l a t i o n (Klute 1972) D ( M ) = - £ ( i ) / x d M C63] M , t J M . 1 where dx/dM i s the r e c i p r o c a l of the moisture content gradient measured at the moisture content M and time of flow t , M. the i n i t i a l moisture content, and / x dM the area bounded by the moisture content p r o f i l e (M(x) t) at time t and the moisture contents and M. The method based on Eq. [63] has a l s o been r e f e r r e d to as the Matano's method (Martin and Moschler 1970). The d i r e c t r e s u l t of the o u t f l o w - i n f l o w methods described above i s the d i f f u s i v i t y f u n c t i o n D(M). The c o n d u c t i v i t y f u n c t i o n s K(M) and K(il> ) may be i n f e r r e d from D(M) through Eq. [59] where the m water c a p a c i t y dM/diJj must be obtained from a M-ili r e l a t i o n s h i p m m measured on separate samples. D i r e c t measurement of the c o n d u c t i v i t y f u n c t i o n s can be achieved by the instantaneous p r o f i l e methods. These methods have two other important advantages. F i r s t , the f l u x - g r a d i e n t propor-t i o n a l i t y can be v e r i f i e d d i r e c t l y . P r o p o r t i o n a l i t y between f l u x and gradient i s a b a s i c c o n d i t i o n of the flow theory based on Darcy's law. Second, no s p e c i f i c set of boundary c o n d i t i o n s need to be 46 imposed, except that the f l u x has to be known at one p o s i t i o n . This l a t t e r requirement may be f u l f i l l e d by simply c l o s i n g one end of the flow system (Klute 1972). The instantaneous p r o f i l e methods c o n s i s t e s s e n t i a l l y of determining the water p o t e n t i a l and the moisture content p r o f i l e s along the d i r e c t i o n of flow, and using Darcy's law to c a l c u l a t e the conduc-t i v i t y . There are three options w i t h respect to measurements on the flow system (Klute 1972): 1) M(x)^ and I(J ( x ) ^ are both measured; 2) M ( x ) t i s measured and ^ ( x ) ^ i s i n f e r r e d from an appropriate M~^m r e l a t i o n s h i p ; and 3) ^ ( x ) ^ i s measured and M(x)^ i s i n f e r r e d from a M-^^ r e l a t i o n s h i p measured on separate samples. The f i r s t o p t i o n i s obviously the best choice s i n c e no question a r i s e s as to the a p p l i c a -b i l i t y of M~4> r e l a t i o n s h i p s measured on separate samples, m Several techniques of data a n a l y s i s may be used. One technique of data a n a l y s i s i s based on the equation obtained by d i f f e r e n t i a t i o n and i n t e g r a t i o n of Eq. [56] by parts (Chow 1973) K(M) = • [64] x , t f°jM 6x 'x,t where K i s the unsaturated c o n d u c t i v i t y at M f o r the p o s i t i o n x and s rx the time of flow t , j M dx the f l u x through the imaginary plane x at t , and Sib\ , Xq=0 g the water p o t e n t i a l gradient at x and t . x,t The f l u x through the imaginary plane x at time t can be f X evaluated by p l o t t i n g the i n t e g r a l / M dx as a f u n c t i o n of time, 47 and then measuring the slope of the smoothed curve at t . This i n t e g r a l i s evaluated by determining g r a p h i c a l l y or n u m e r i c a l l y the area bordered by the planes x and x and the M p r o f i l e s at times t and t = 0 (or q=0 a l t e r n a t i v e l y the 0% M a b s c i s s a ) . The p o t e n t i a l gradient can be obtained by g r a p h i c a l or numerical computation procedures. Another equation which i s p a r t i c u l a r l y convenient f o r a l a r g e number of M and i|> p r o f i l e s i s of the form (Chow 1973) K(M) = — ^ [65] x, h+t2 d9 , , < v v — T " x, H+t2 2 where the average f l u x / / d x d t / ( t 2 ~ t ^ ) i s determined by meas-1 =0 u r i n g the area between *"* two clo s e l y - s p a c e d M p r o f i l e s at t ^ and t„ from the planes x n to x, and then d i v i d i n g i t by the time i n t e r v a l I q=U ( t 2 ~ t ^ ) . The corresponding p o t e n t i a l gradient i s the mean value of the two gradients measured at t ^ and t^- The time i n t e r v a l s should not be too small or the surface i n t e g r a l term may not be known w i t h s u f f i c i e n t accuracy; n e i t h e r should i t be too l a r g e , or the use of the mean p o t e n t i a l gradient may lead to serious e r r o r s . Of course, Eqs. [64] and [65] a l s o y i e l d d i r e c t l y the conduc-t i v i t y f u n c t i o n K(ty ). On the other hand, the f u n c t i o n D(M) can be m d i r e c t l y determined from r e l a t i o n s of the form of Eqs. [64] and [65 ] , i n which a moisture content gradient replaces the p o t e n t i a l gradient i n the denominator. Another a l t e r n a t i v e i s to i n f e r D(M) from K(M) through the use of Eq. [ 5 9 ] . Chapter 2 LITERATURE REVIEW 2.1 M-ii r e l a t i o n s h i p a t high moisture contents m 2.1.1 Previous measurements i n wood The water p o t e n t i a l i s the sum of a number of component p o t e n t i a l s among which \b and ty change w i t h M. Both ty and ty decrease rm To rm T o (become more negative) as M decreases. Of course, tyQ vanishes when no water s o l u b l e substances are present i n the porous medium. In wood, sol u t e s are present i n very v a r i a b l e amounts, which i m p l i e s that the r e l a t i v e importance of ty may change markedly among sp e c i e s . The meas-urement of t h i s component p o t e n t i a l i s not, however, w i t h i n the scope of the present study. A few authors have attempted p r e v i o u s l y to conduct K-ty m determinations i n wood at high moisture contents (Penner 1963, Stone and S c a l l a n l 9 6 7 , V i k t o r i n and Cermak 1977). Of p a r t i c u l a r i n t e r e s t are the r e s u l t s of Penner who covered the o v e r a l l M range i n both drainage and i m b i b i t i o n modes. This author used a c y l i n d r i c a l specimen which was about 7.5 cm i n diameter and 0.6 cm i n thickness (along the f i b e r 48 d i r e c t i o n ) . 1 The two M-if; curves published by Penner f o r spruce wood (Pieea spp.) at 20°C are shown i n F i g . 1. The same specimen was used to o b t a i n both curves. The measurement methods used were the te n s i o n p l a t e i n the ty^ range from 0 to -1 bar, the pressure membrane i n the ty range from -1 to -10 bars, and the vacuum d e s i c c a t o r f o r ty 's lower m than -10 b a r s . 2 As can be seen i n F i g . 1, the data obtained from these various measurement methods were co n s o l i d a t e d i n t o one smoothed M-ili m curve. Such p r a c t i c e i s q u i t e j u s t i f i e d s i n c e ty^ i s g e n e r a l l y n e g l i g i b l e compared to ty i n the range covered by the s o r p t i o n methods. A s t r i k i n g f a c t revealed by F i g . 1 i s the very strong h y s t e r e t i c behavior of the M—if; r e l a t i o n s h i p of wood i n the ty range from about m r m b 0 to -10 bars. The shape of the drainage curve c l e a r l y demonstrates that the common c h a r a c t e r i z a t i o n of water i n wood above 30% M (g e n e r a l l y a s s o c i a t e d w i t h the f i b e r s a t u r a t i o n point) as " f r e e water" i s q u i t e erroneous when M i s on a decreasing trend. When M i s on an i n c r e a s i n g trend, water above 30% M then becomes much c l o s e r to the energy s t a t e of pure f r e e water. Nevertheless, water would become completely " f r e e " o n l y when f u l l s a t u r a t i o n i s reached (and i f s o l u t e s are absent). The apparent c l o s i n g of the h y s t e r e s i s loop at f u l l s a t u r a t i o n or zero ty m supports the l a t t e r argument. A stronger i n d i c a t i o n of a c l o s i n g loop can be found by reproducing the i m b i b i t i o n curve on a l a r g e s c a l e l i n e a r 1 Personal communication 2 For the sake of c l a r i t y and ease of f a m i l i a r i z a t i o n w i t h the u n i t s of water p o t e n t i a l , the r e l a t i o n of water p o t e n t i a l to r e l a t i v e humidity as i n f e r r e d from Eq. [25] i s given i n Table A l . l of Appendix 1; Table A l . l a l s o shows the r e l a t i o n of water p o t e n t i a l to the radius of curvature of the ai r - w a t e r i n t e r f a c e i n accord w i t h Eq. [42]. - I 0 4 - I 0 3 - I 0 2 - 1 0 ' - 1 0 ° - I 0 ' 1 H O " 2 - I 0 " 3 MATR IC P O T E N T I A L , ^ (bars) F i g . 1. — Moisture content as a f u n c t i o n of matr i c p o t e n t i a l f o r spruce wood at 20°C (adapted from Penner 1963). p l o t and e x t r a p o l a t i n g to zero ty . 3 m Another f e a t u r e of the M-^m r e l a t i o n s h i p of spruce wood according to F i g . 1 i s the absence of a c l e a r a i r - e n t r y value, which The p l o t t i n g of ty on a l o g a r i t h m i c s c a l e permits the d i s p l a y of the M-ty r e l a t i o n s h i p throughout the o v e r a l l M range w h i l e a v o i d i n g a congestion of the data p o i n t s at high matric p o t e n t i a l s . i s the c r i t i c a l ty value i n drainage at which the l a r g e s t pore of entry begins to empty ( H i l l e l 1971). The f a i l u r e of the drainage curve to e x h i b i t a d i s t i n c t and sharp a i r - e n t r y value i m p l i e s that the pores of entry (or d r a i n i n g r a d i i ) are very v a r i a b l e i n s i z e . On the other hand, the value of dM/di); changes r e l a t i v e l y s harply at about -3 bar ty as m m revealed by slope computations. Table A l . l i n d i c a t e s that at t h i s p o t e n t i a l , the water remaining i n wood would be l o c a l i z e d i n pores of maximum d r a i n i n g r a d i i of about 0.484 um. This feature would therefore correspond w i t h the t r a n s i t i o n between the drainage of the c e l l c a v i t i e s and the drainage of the c e l l w a l l s . The p o s s i b l e values of Penner's r e s u l t s become r e a d i l y apparent when comparing the M-iJ^ curves of F i g . 1 with the s o r p t i o n curves presented i n F i g . 2 f o r sugar maple {Acer saccharum Marsh.) at 21°C (Goulet 1968). The clo s e correspondence between the two diagrams at low RH or ty^ values i s s e l f - e x p l a n a t o r y since both types of curves were derived from s o r p t i o n data which i n the lower humidity range g e n e r a l l y vary very l i t t l e among species. In the high humidity r e g i o n , the wide gap observed i n F i g . 2 between the M-RH adsorption curve and the desorption curves from wet c o n d i t i o n s c l e a r l y supports the l a r g e h y s t e r e -s i s shown i n F i g . 1. However, above 70% RH (- -500 bar ty ) the m h y s t e r e s i s r a t i o which i s the r a t i o of M on the boundary adsorption curve (or i m b i b i t i o n curve) to M on the boundary desorption curve (or drainage curve) at a given RH (or ij1 ) does not f o l l o w q u i t e the same p a t t e r n i n the two f i g u r e s . There i s no evident explanation f o r t h i s discrepancy although the d i f f e r e n c e s i n the c a p i l l a r y systems of the two wood species i n question may have c o n t r i b u t e d to i t . 52 40 ^ 30 UJ 20 o o UJ CL (-V) 2 io 1 — i — ' — i 1 — i 1 — r *Z ~ * DES0RPTI0N FROM WET CONDITIONS DES0RPTI0N AFTER ADSORPTION ABOVE WATER 7 A f ADSORPTION FROM DRYNESS / / 20 40 60 80 RELATIVE HUMIDITY, RH (%) 100 F i g . 2. — Moisture content as a f u n c t i o n of the r e l a t i v e humidity of the ambient a i r f o r sugar maple at 21 C (adapted from Goulet 1968). The suggestion of a wide open h y s t e r e s i s loop i n F i g . 2 at 100% RH i s i n c o n t r a d i c t i o n w i t h Penner's r e s u l t s . This i s a l s o i n c o n t r a d i c t i o n w i t h theory. According to K e l v i n ' s equation, which can be obtained by combining Eqs. [23] and [40] of Chapter 1, a l l the pores should be f i l l e d w i t h water at 100% RH s i n c e the radius of curvature of the water menisci becomes i n f i n i t e (see Table A l . l ) . Therefore, the apparent wide open h y s t e r e s i s loop at 100% RH can only r e s u l t from the la c k of accuracy of the s o r p t i o n methods approaching a i r s a t u r a t i o n . 53 Apparently the f i r s t worker to demonstrate the strong h y s t e r e t i c behavior of the M-RH r e l a t i o n s h i p i n the high humidity region was Barkas (1936). Some t y p i c a l s o r p t i o n data obtained by t h i s author f o r beech wood (Fagus sylvatica L.) at 23°C were p l o t t e d and reported i n F i g . 3. For the same reason as above, Barkas a l s o observed an apparent wide open h y s t e r e s i s loop at 100% RH. His s o r p t i o n method was accurate enough, however, to c l e a r l y demonstrate the presence of a considerable h y s t e r e s i s i n the high humidity r e g i o n . I 4 0 i - A / r 120 ^ 100 K w 80 O o UJ 60 a: z> h-O 40 20 h • D E S 0 R P T I O N FROM F U L L SATURATION O ADSORPTION FROM DRYNESS _L 95 96 97 98 RELATIVE HUMIDITY, RH (%) 99 100 F i g . 3. — Moisture content as a f u n c t i o n of the r e l a t i v e humiditv of the ambient a i r f o r beech wood at 23 C (from data of Barkas 1936). 54 2.1.2 P o s s i b l e causes of the h y s t e r e s i s Past s t u d i e s on the h y s t e r e s i s phenomenon of the M-ij; m r e l a t i o n s h i p f o r porous media i n general have followed mainly two approaches: 1) explanation of the b a s i c causes of the h y s t e r e s i s ; and 2) development of p r e d i c t i v e models. In the l a t t e r approach, simple s t a t i s t i c a l models may be used i n combination w i t h a l i m i t e d amount of experimental data to p r e d i c t unmeasured p o r t i o n s of the h y s t e r e s i s loop or scanning curves w i t h i n the loop. The "independent domain models" of E v e r e t t (1955), Mualem (1973, 1974) and Topp (1971a), and the "dependent domain models" of P o u l o v a s s i l i s and C h i l d s (1971), Mualem and Dagan (1975), and P o u l o v a s s i l i s and El-Ghamry (1978) are t y p i c a l examples of such p r e d i c t i v e models. Although t h e o r e t i c a l models of h y s t e r e s i s may be u s e f u l i n s o l v i n g unsaturated flow problems i n v o l v i n g a l t e r n a t i n g processes of drainage (or drying) and i m b i b i t i o n , t h e i r c o n s i d e r a t i o n i s beyond the scope of the present study. Most p o s t u l a t e d explanations of the s o - c a l l e d s o r p t i o n h y s t e r e s i s i n the lower M range of wood are u l t i m a t e l y r e l a t e d to wood s o r p t i o n and s w e l l i n g phenomena (see reviews by Spalt 1958, Stamm 1964, and Goulet 1967). Since wood at high M's i s an i n e r t and s t a b l e m a t e r i a l , other causes must then be r e s p o n s i b l e f o r the strong h y s t e r e t i c behavior of the M-IJJ^ r e l a t i o n s h i p i n the high humidity r e g i o n . For porous media i n general, two causes are f r e -quently advanced (C h i l d s 1969, H i l l e l 1971, Baver et al. 1972): 1) the i r r e g u l a r shape of the pore space; and 2) the v a r i a t i o n of the contact angle between the s o l i d and water s u r f a c e s . The pore space of many porous media c o n s i s t s of la r g e c a v i t i e s interconnected by narrow channels. During drainage and i m b i b i t i o n , t h i s i r r e g u l a r i t y of shape r e s u l t s i n the s o - c a l l e d " i n k - b o t t l e e f f e c t " , which was the subject of an extensive t h e o r e t i c a l treatment by Haines (1930). Owing to t h i s e f f e c t , the case of decreasing moisture content or drainage tends to be governed by a lower ty value as determined by the narrower s e c t i o n s of the pores, whereas c o n d i t i o n s of i n c r e a s i n g moisture content or i m b i b i t i o n tend to be governed by a higher ty value depending upon the wider s e c t i o n s of the pores m (Haines 1930). The greater the d i s p a r i t y between the s i z e of the c a v i t i e s and the s i z e of the channels, the more marked i s the d i f f e r e n c e between the p o t e n t i a l s of emptying and r e f i l l i n g ( C h i l d s 1969). When a contact angle e x i s t s between the ai r - w a t e r i n t e r f a c e and the s o l i d s u r f a c e s , i t tends to be smaller when the i n t e r f a c e i s r e t r e a t i n g than when i t i s advancing. Other things being equal, t h i s i m p l i e s that the radius of curvature i n a given pore i s greater when the meniscus i s advancing ( f i l l i n g of the pore) than when i t i s receeding (emptying of the pore). Thus, must be greater i n the f i l l i n g than i n the corresponding emptying stages ( C h i l d s 1969) . Both t h e o r i e s c i t e d above present shortcomings which make the assessment of t h e i r r e l a t i v e merit d i f f i c u l t . For example, the i n k - b o t t l e e f f e c t theory assumes a h i g h l y s i m p l i f i e d pore model, and the contact-angle e f f e c t theory i m p l i e s at the s t a r t the exi s t e n c e of 56 contact angles of f a i r l y great magnitudes. The contact-angle e f f e c t theory would be p a r t i c u l a r l y apt to e x p l a i n the time-dependent h y s t e r e s i s occuring i n no n - e q u i l i b r i u m systems (Ngoddy and Bakker-Arkema 1975). 2.1.3 Factors of v a r i a t i o n 2.1.3.1 Pore s i z e d i s t r i b u t i o n A property of the porous medium that most fundamentally a f f e c t s the nature of the r e l a t i o n s h i p i s i t s pore s i z e d i s t r i b u t i o n . An example of pore s i z e d i s t r i b u t i o n f o r wood i s shown i n F i g . 4. This A-r curve was measured by Heizmann (1970) on dry blocks (140 to 200 mm3 i n volume) of b i r c h wood (Betula verrucosa Ehrh.) using the mercury i n t r u s i o n porosimeter method. The term A i s the p r o p o r t i o n (percentage) of pore volume which i s occupied by pores w i t h a radius equal or smaller than a given r a d i u s r . Heizmann c l a s s i f i e d the pores (voids) i n wood i n three ca t e g o r i e s according to t h e i r r a d i u s , namely the submicroscopic pores whose r < 0.1 um, the t r a n s i t i o n a l pores w i t h r between 0.1 and 1.0 um, and f i n a l l y the microscopic pores w i t h r > 1.0 um. Based on F i g . 4, i t would then appear that about 6% of the v o i d space i n b i r c h wood i s formed by sub-microscopic pores, whereas 54% i s occupied by t r a n s i t i o n a l pores and 40% by microscopic pores. I t must be emphasized, however, that the A values i n F i g . 4 were based on the bulk volume i n the dry c o n d i t i o n . Obviously, the above f i g u r e s would change markedly i n the green c o n d i t i o n due to wood s w e l l i n g . For the specimens used to o b t a i n the A-r curve of F i g . 4, the t r a n s i e n t pore volume p r o p o r t i o n (A ) due to wood s w e l l i n g would be 57 100 90 ^° 0 s 80 UJ 3 70 o > uj60 LU 50 40 O F 30 rr o a. O20 o_ 10 "1 i i j 11111 i i i 11 in] i i I 111111 I I i 11 II =ja-rrr -A—A j M J I l l l l l l l I I l l l l l l l I I l l l l l l l I I l l l l l l l V3 I0"2 I0_l 10° I01 i I 11 in PORE RADIUS, r (Mm) F i g . 4. — Cumulative pore s i z e d i s t r i b u t i o n of b i r c h wood obtained by the mercury i n t r u s i o n porosimeter method ( a f t e r Heizmann 1970), approximately 24% (based on the pore volume i n the green c o n d i t i o n ) . ^ This i s l a r g e l y accounted f o r by the t r a n s i e n t c a p i l l a r i e s formed i n the c e l l w a l l s themselves (Stamm 1964). I t i s of i n t e r e s t to note that the above pore volume propor-t i o n a l l o c a t e d to the microscopic pores (40%) i s much smaller than the 1 - ( e o / £ g ( l + a V f c ) ) where At i s the t r a n s i e n t pore volume X = 100 pr o p o r t i o n i n %, a v the t o t a l volumetric s w e l l i n g , and e 0 and e the p o r o s i t i e s based on tthe volumes i n the dry and green c o n d i t i o n s , ^ r e s p e c t i v e l y . 58 pore volume proportion generally assigned to the c e l l c a v i t i e s i n dry wood (Yao 1964, Stayton and Hart 1965). The reason f o r t h i s discrepancy can only be the ink-bottle e f f e c t discussed e a r l i e r . The wood blocks used by Heizmann were probably too thick f o r the lumens to have d i r e c t connection with the mercury source except through the much smaller p i t openings or end perforations (vessel elements). The e f f e c t of these " i n k - b o t t l e " pores i s to assign too small a portion of the pore space to the large pores and too large a part to the small pores, i f the mercury i n j e c t i o n data i s taken at i t s face value (Meyer 1953). It therefore appears that the X-r curve of F i g . 4 does not represent the true pore si z e d i s t r i b u t i o n of b i r c h wood i n the dry state. Nevertheless, for the purpose of p r e d i c t i n g water behavior i n the wood i n question during drainage or drying t h i s r e l a t i o n s h i p may ac t u a l l y be more useful than the true pore si z e d i s t r i b u t i o n i t s e l f . Indeed, since the ink - b o t t l e e f f e c t acts i n the same way on a receding water meniscus as on an advancing mercury meniscus (non-wetting l i q u i d ) the X-r curve of F i g . 4 should be c l o s e l y r e l a t e d to the M-I|J boundary m drainage curve of b i r c h wood. In t h i s connection, the former r e l a t i o n s h i p would be ref e r r e d to as e f f e c t i v e pore si z e d i s t r i b u t i o n . Since the M — r e l a t i o n s h i p i s h y s t e r e t i c , i t i s obvious that the e f f e c t i v e pore si z e d i s t r i b u t i o n i n drainage i s d i f f e r e n t from that i n imbibition. A c t u a l l y , the l a t t e r should be closer to the true pore siz e d i s t r i b u t i o n on the grounds that i t i s governed by the size of the wider sections of the pores, whereas the former depends on the s i z e of the narrower sections of the pores. 2.1.3.2 Ambient temperature and pressure Frequently i n the a n a l y s i s of unsaturated flow systems or i n the measurement of the c o n d u c t i v i t y f u n c t i o n s , ijj data are i n f e r r e d m from a s t a t i c M -^ m r e l a t i o n s h i p measured at a given ambient temperature T and a i r pressure P. I t i s then important to know how f l u c t u a t i o n s v of T and P during the flow process a f f e c t the M—TJJ r e l a t i o n s h i p of the m system considered. Such in f o r m a t i o n i s a l s o valuable f o r the s t a t i c measurement of the M-ip r e l a t i o n s h i p i t s e l f . m One way to p r e d i c t the dependence of ty on T i s to assume that m ty^ a r i s e s predominantly from the surface t e n s i o n forces of the a i r - w a t e r i n t e r f a c e s (see Eqs. [40] through [ 4 2 ] ) . At high moisture contents, t h i s assumption should be q u i t e reasonable. The r a t e of change of ty w i t h T would then f o l l o w the r e l a t i o n (Wilkinson and K l u t e 1962) m 1 6y -vrL'-Ky-sr C 6 6 ] /M where y i s the surface t e n s i o n of water i n dyn/cm. The expression 1 SY — -g7j7 , which denotes the temperature c o e f f i c i e n t of the surface t e n s i o n of water, i s about -2.1 x 10~3/°C (Weast 1975-1976). This represents a r e l a t i v e increase of ty w i t h T (Sty /6T i s p o s i t i v e ) of 0.21%/°C, which m m r i s f a i r l y s m a l l . An increase of ty w i t h T i m p l i e s , i n t u r n , that M at a given ty decreases as T i n c r e a s e s , m Experiments conducted on s o i l s have shown a d e f i n i t e increase of ty w i t h T. However, the measured values of (Sty /<5T)W were found to be m m M up to s e v e r a l times l a r g e r i n magnitude than those p r e d i c t e d by the surface tension-temperature r e l a t i o n s h i p alone.(Gardner 1955, Wi l k i n s o n and K l u t e 1962, Chahal 1965, Haridasan and Jensen 1972). One explanation f o r the apparent discrepancy between experimental r e s u l t s and t h e o r e t i c a l p r e d i c t i o n s was based on the e f f e c t of temperature on the volume of entrapped a i r (Chahal 1964, 1965). S u b s t a n t i a l changes of ty w i t h T i n the presence of entrapped a i r had been o r i g i n a l l y p r e d i c t e d by Peck (1960). . Regarding the dependence of ty on P, the simple surface t e n s i o n m forc e s theory does not p r e d i c t any ty changes w i t h P. However, as pr e d i c t e d by Peck (1960) appreciable changes of ty w i t h P may occur i n m the presence of entrapped a i r . Adam and Corey (1968) reported that changes i n barometric pressure during i m b i b i t i o n experiments had been found to cause n o t i c e a b l e changes i n the volume of gas entrapped at any time throughout the process. The H-ty r e l a t i o n s h i p inherent to a flow m process of t h i s type i s obv i o u s l y not unique. The phenomenon of a i r entrapment w i l l be discussed i n t o more d e t a i l below. 2.1.3.3 Entrapped a i r When a porous medium co n t a i n i n g a i r i s allowed to imbibe water, some of the a i r may become entrapped as mentioned above. This would be caused by the advancement of water through p r e f e r r e d pores or pore sequences, which sometimes r e s u l t s i n the s e a l i n g of a l l paths through which a i r would need to escape to permit the f i l l i n g of another pore or group of pores ( P h i l i p 1957, Peck 1969b). Factors that may determine the i n i t i a l amount of entrapped a i r are the pore s i z e d i s t r i b u t i o n , i n i t i a l moisture content and i m b i b i t i o n r a t e . Subsequent changes i n the i n i t i a l volume of entrapped a i r may be caused by d i s s o l u t i o n of a i r i n the water, by bulk movement or d i f f u s i o n through water, or by v a r i a t i o n s of T and P (Peck 1969b). Q u a n t i t a t i v e measurements conducted on glass beads by de Backer (1967) showed that a i r entrapment occurred mainly during i m b i b i t i o n , reaching a maximum entrapped'air content of about 6% of the t o t a l sample volume. On the other hand, Adam et al. (1969) reported that the amount of a i r i n i t i a l l y entrapped f o l l o w i n g f r e e i m b i b i t i o n (imposed p o t e n t i a l equal to zero) i n t o v a r i o u s types of co n s o l i d a t e d m a t e r i a l s v a r i e d from about 3 to 14% of the sample bulk volume. A l l the entrapped a i r would e v e n t u a l l y d i f f u s e toward the fr e e atmosphere but t h i s may be a matter of s e v e r a l months (Adam and Corey 1968, Adam et al. 1969, McWhorter et al. 1973). At constant T and P, one d i r e c t e f f e c t of entrapped a i r could be a s h i f t i n g of the M - i ^ curves at high M's toward the dry s i d e . In other words, the i m b i b i t i o n curve would not r e t u r n to the s t a r t i n g p o i n t of the i n i t i a l drainage curve ( i . e . , f u l l s a t u r a t i o n ) at zero ijj and, i n tu r n , the second drainage curve would l i e below the i n i t i a l m drainage curve. P o u l o v a s s i l i s (1962, 1970b) observed such behavior of the M-4> r e l a t i o n s h i p on s i n t e r e d glass beads and mixed sand m f r a c t i o n s , and a t t r i b u t e d i t to a i r entrapment during i m b i b i t i o n . Reported dependence of the measured M-ifJ i m b i b i t i o n curve on m the magnitude of the p o t e n t i a l increment (Davidson et al. 1966) may be another consequence of a i r entrapment as suggested by Peck (1969b). On the other hand, expansion of entrapped a i r bubbles a f t e r r e l e a s e of the a i r pressure from a pressure p l a t e or membrane apparatus used f o r K-ty determinations may cause systematic e r r o r s i n the M-iii m m r e l a t i o n s h i p thus obtained. This would r e s u l t i n an " a c t u a l " ty smaller m i n magnitude than the corresponding imposed pressure (Peck 1960). 2.1.3.4 Process dependence As the M-i|> r e l a t i o n s h i p i s assumed to be unique when only i m b i b i t i o n or drainage i s considered, i t i s a l s o assumed to be unique w i t h respect to the s t a t e of flow i n terms of steady-state or t r a n s i e n t flow. However, experiments conducted on sand i n d i c a t e d that the l a t t e r assumption i s not always met (Topp et al. 1961, Rogers and K l u t e 1971, Smiles et al. 1971, Vachaud et al. 1972, Elzeftawy and Mansell 1975). Higher moisture contents were then found at a given ij; i n drainage m during t r a n s i e n t flow than during steady-state flow or s t a t i c e q u i l i b r i u m (zero f l o w ) . The three l a t t e r groups of authors r e l a t e d the process dependence of the M-ili r e l a t i o n s h i p i n drainage to the r a t e of change of ty w i t h m m time ( i . e . , 6 i ^ / 6 t ) during t r a n s i e n t flow. They found that the t r a n s i e n t M-^m r e l a t i o n s h i p was changing w i t h length and so was the r a t e 64>m/6t. The smaller the magnitude of &ty /6t, the smaller was the divergence m between the t r a n s i e n t and s t a t i c M—1(;^ curves. These f i n d i n g s suggest that the use of a s t a t i c M-'J' curve f o r a n a l y s i n g a t r a n s i e n t flow process may be v a l i d only i f the flow process i s extremely slow. Otherwise, d i r e c t ty determinations would have to be c a r r i e d out. m 63 2.2 K(M), K(iti ) and D(M) f u n c t i o n s at high moisture contents m 2.2.1 Previous measurements i n wood The l i t e r a t u r e apparently r e v e a l s no i n f o r m a t i o n on unsatu-rated water c o n d u c t i v i t y measurements i n wood at high moisture contents. Nevertheless, s p e c u l a t i o n s on the nature of the K(M) f u n c t i o n may be made based upon r e s u l t s of r e l a t i v e p e r m e a b i l i t y measurements c a r r i e d out by Tesoro et al. (1974) during saturated two-phase (nitrogen + water) flow i n wood. 5 These authors found that the r e l a t i v e p e r m e a b i l i t y of wood to water i n the l o n g i t u d i n a l d i r e c t i o n during drainage decreased to v i r t u a l l y zero as the s a t u r a t i o n percentage (Sp) decreased below 80%. 6 Although only a p o r t i o n of the i m b i b i t i o n curve was measured, there was a c l e a r tendency f o r the r e l a t i v e p e r m e a b i l i t y at a given to be s u b s t a n t i a l l y greater i n i m b i b i t i o n than i n drainage. I t may be expected from the foregoing r e s u l t s that during unsaturated flow of water ( a i r phase immobile) i n wood K w i l l decrease sharply w i t h decreasing moisture content near f u l l s a t u r a t i o n , and that the f u n c t i o n K(M) w i l l e x h i b i t h y s t e r e s i s . A r a p i d decrease of K at the s t a r t of d e s a t u r a t i o n should normally a r i s e f o r the f o l l o w i n g reasons ( P h i l i p 1969a): 1) the t o t a l e f f e c t i v e cross s e c t i o n f o r flow decreases along w i t h M; 2) the l a r g e s t pores are emptied f i r s t and the c o n t r i b u t i o n to K v a r i e s roughly as the square of pore r a d i u s ; and 5 The expression " r e l a t i v e p e r m e a b i l i t y " denotes the r a t i o of the e f f e c t i v e p e r m e a b i l i t y of a porous medium to a given f l u i d at a given M, to i t s s p e c i f i c p e r m e a b i l i t y (Tesoro et al. 1974). 6 The s a t u r a t i o n percentage, S^, or the degree of s a t u r a t i o n expresses the volume of water present i n a porous medium r e l a t i v e to the t o t a l pore volume ( H i l l e l 1971) . 64 3) as M decreases, the p r o b a b i l i t y increases that water w i l l become trapped i n pores and wedges i s o l a t e d from the general three-dimensional network of water f i l m s and channels. With regard to the water d i f f u s i v i t y f u n c t i o n D(M), the l i t e r a t u r e reports at l e a s t two cases of D determinations i n wood at high moisture contents. T y p i c a l examples of the r e s u l t s then obtained are shown i n F i g . 5. The s o l i d l i n e D-M curve has been measured by Voigt et al.. (1940) during r a d i a l d r y i n g of beech (Fagus s y l v a t i c a L.) heartwood at 30°C. The broken l i n e curve was obtained by Heizmann (1970) during l o n g i t u d i n a l i m b i b i t i o n of water i n b i r c h {Betula verrucosa Ehrh.) sapwood at 20°C. A l l the r e s u l t s were converted to a common ba s i s i n u n i t s of cm 2/s. Both curves i n F i g . 5 e x h i b i t a very sharp decrease of D at the s t a r t of d e s a t u r a t i o n . As M decreases f u r t h e r , D changes r e l a t i v e l y l i t t l e or increases s l o w l y . The above authors a s s o c i a t e d the behavior of these curves to the pore s i z e d i s t r i b u t i o n s of the wood species i n question. The d e f i n i t i o n of D(M) based on Eq. [59] suggests that such c o r r e l a t i o n e x i s t s but i t must be very complex. Indeed, D(M) depends on the two f u n c t i o n s K(M) and c(M), both of which i n turn depend on A(r). For example, i f A(r) causes dK/dty to decrease at a f a s t e r r a t e than m K as M decreases, the u l t i m a t e r e s u l t w i l l be an increase i n D. This shows that the behavior of the D(M) f u n c t i o n may be q u i t e d i f f e r e n t from that of the K(M) f u n c t i o n . Therefore, considerable c a u t i o n must be e x e r c i s e d i n g i v i n g any p h y s i c a l i n t e r p r e t a t i o n to the shape of the D-M curves at high moisture contents. Fig. 5. — Water diffusivity as a function of moisture content in the radial direction of beech during drying at 30 C (adapted from Voigt et al. 1940), and in the longitudinal direction of birch during imbibition at 20 C (adapted from Heizmann 1970). One of the basic premises upon which unsaturated flow theory depends is that of proportionality between flux and potential gradient or between flux and moisture content gradient. Of course, any flow system that i s used to measure the functions K(M), K(i|> ) and D(M) must m also exhibit flux-gradient proportionality. No calculations were reported by either Voigt et al. or Heizmann to show that this condition had been satisfied by the transient flow systems used to obtain the curves reported i n F i g . 5. N o n - l i n e a r i t y between f l u x and gradient at a given ty or M during flow of water i n unsaturated porous media appears to be r e l a -t i v e l y frequent as demonstrated by experiments (Swartzendruber 1963, Hadas 1964, Thames and Evans 1968). P o s s i b l e causes of dispropor-t i o n a l i t y during unsaturated flow are non-Darcian behavior, non-uniqueness of the M-I|J r e l a t i o n s h i p , and n o n - n e g l i g i b l e vapor t r a n s f e r (Swartzendruber 1969). Non-Darcian behavior r e f e r s to the f a c t that the f l u x i s e i t h e r " l e s s than" or "more than" p r o p o r t i o n a l to the grad i e n t . The former case may a r i s e from i n e r t i a l e f f e c t s whereas the l a t t e r case may be due to non-Newtonian behavior of the moving water ( P h i l i p 1969a). The non-uniqueness of the M~^m r e l a t i o n s h i p has been discussed e a r l i e r w i t h respect to entrapped a i r and process dependent e f f e c t s . For the vapor t r a n s f e r , i t may become s i g n i f i c a n t w i t h respec to l i q u i d flow wherever important temperature gradients are present. 2.2.2 P o s s i b l e causes of the h y s t e r e s i s i n the K(M) f u n c t i o n I t was deduced above that the K(M) f u n c t i o n i n wood should d i s p l a y a f a i r l y l a r g e h y s t e r e s i s w i t h K at a given M greater i n i m b i b i t i o n than i n drainage. A s i m i l a r behavior of K(M) was observed i n loamy s o i l s by Staple (1965) and Dane and Wierenga (1975) . A considerable h y s t e r e s i s was a l s o found i n sand by P o u l o v a s s i l i s (1969, 1970a), Dane and Wierenga (1975), and P o u l o v a s s i l i s and Tzimas (1975), but K was then greater i n drainage than i n i m b i b i t i o n . On the other hand, Topp and M i l l e r (1966), Talsma (1970) and Topp (1971b) reported a s m a l l or i n c o n c l u s i v e h y s t e r e s i s i n K(M) f o r glass beads, sand and loam, r e s p e c t i v e l y . P o u l o v a s s i l i s (1969) advanced two hypotheses to e x p l a i n the apparently very v a r i a b l e h y s t e r e t i c behavior of the K(M) f u n c t i o n . On account of the h y s t e r e s i s i n the M-<1> r e l a t i o n s h i p , t h i s author m assumed that water at a given M i s contained i n pores not common to both the i m b i b i t i o n and drainage s t a t e s and that the d r a i n i n g r a d i i of these pores are greater i n i m b i b i t i o n than i n drainage. On t h i s b a s i s , the h y s t e r e s i s i n the M-i|» r e l a t i o n s h i p may a f f e c t K i n two ways. F i r s t l y , the d i f f e r e n c e s i n the d r a i n i n g r a d i i of the pores c o n t a i n i n g the water would favor a greater K i n i m b i b i t i o n than i n drainage f o r the same moisture l e v e l . Secondly, the d i f f e r e n c e s of the geometric coordinates of these pores, which may i n f l u e n c e the t o r t u o s i t y p a t t e r n of the flowpaths and consequently K, may have e i t h e r a s i m i l a r or an opposite e f f e c t . Therefore, K i n the i m b i b i t i o n s t a t e can be greater, equal, or even l e s s than i n the drainage s t a t e , depending on the p a r t i c u l a r porous body. 2.2.3 Factors of v a r i a t i o n 2.2.3.1 Pore s i z e d i s t r i b u t i o n The pore s i z e d i s t r i b u t i o n of a porous medium fundamentally a f f e c t s i t s water c o n d u c t i v i t y f u n c t i o n s . In s o i l physics i n p a r t i c u l a r t h i s has been the subject of numerous t h e o r e t i c a l and experimental i n v e s t i g a t i o n s . Since C h i l d s and C o l l i s - G e o r g e (1950) have launched the idea of p r e d i c t i n g K from the pore s i z e d i s t r i b u t i o n and w r i t t e n the b a s i c philosophy behind i t , many authors have developed equations f o r t h i s purpose or have t e s t e d the proposed p r e d i c t i v e models against experimental data (see Secti o n 1.2.2). The pore s i z e d i s t r i b u t i o n s used i n these p r e d i c t i v e models are g e n e r a l l y i n f e r r e d from e x p e r i -mentally determined M -4 ; m curves using the c a p i l l a r y r i s e equation (Eqs. [40] through [ 4 2 ] ) . I t i s not w i t h i n the scope of the current study to review and analyse the f u n c t i o n a l r e l a t i o n s h i p s proposed i n the l i t e r a t u r e f o r c a l c u l a t i n g the unsaturated water c o n d u c t i v i t y from pore s i z e d i s t r i -b u t i o n data. I t may be s u f f i c i e n t to say t h a t , despite t h e i r l i m i t a t i o n s , these p r e d i c t i v e models have given so f a r very promising r e s u l t s (Gardner 1974). This approach of p r e d i c t i n g K i s , f o r example, more r a t i o n a l and much more e f f i c i e n t than that based on the t o t a l p o r o s i t y of the porous medium. The f a c t that t o t a l p o r o s i t y bears l i t t l e or no c o r r e l a t i o n to the unsaturated water c o n d u c t i v i t y deserves some comments. Indeed, i f at s a t u r a t i o n the most conductive porous media are those w i t h a high p r o p o r t i o n of l a r g e and continuous pores, the very opposite may be true when the porous media are unsaturated. In a porous medium w i t h a high p r o p o r t i o n of v o i d volume occupied by l a r g e pores, these pores q u i c k l y empty and become nonconductive as M decreases, thus s t e e p l y decreasing the i n i t i a l l y high c o n d u c t i v i t y . In a porous medium w i t h small pores, many of the pores remain f u l l and conductive even at low 69 moisture contents. Thus, the c o n d u c t i v i t y does not decrease as s t e e p l y and at the same moisture l e v e l may a c t u a l l y be greater than that of a porous medium w i t h l a r g e pores ( H i l l e l 1971). Support f o r the above argument i n the case of wood i s given by the r e s u l t s of Tesoro e t al. (1974). The shape of the r e l a t i v e p e r m e a b i l i t y - s a t u r a t i o n percentage r e l a t i o n s h i p was indeed found to be more a f u n c t i o n of wood s t r u c t u r e than of i t s s p e c i f i c p e r m e a b i l i t y . 2.2.3.2 Ambient temperature and pressure By d e f i n i t i o n the saturated water c o n d u c t i v i t y of a porous medium i s i n v e r s e l y p r o p o r t i o n a l to the v i s c o s i t y of water (n). Since n decreases as the temperature i n c r e a s e s , the c o n d u c t i v i t y must then increase w i t h the temperature. On t h i s b a s i s alone, K at s a t u r a t i o n would increase almost three times between 5 and 50°C. The unsaturated water c o n d u c t i v i t y should behave s i m i l a r l y . Haridasan and Jensen (1972) found f o r two s i l t loam s o i l s that the increase i n K w i t h T at a given M was almost e n t i r e l y accounted f o r by the decrease i n n. However, K at a given \b d i d not r e f l e c t m any s i g n i f i c a n t temperature dependence. Haridasan and Jensen explained t h i s behavior of the K(iii ) f u n c t i o n as f o l l o w s . As T in c r e a s e s , M at m a given \> decreases causing K to decrease , and n decreases causing K to i n c r e a s e . These two opposite e f f e c t s seem to counterbalance each other to keep K ( ^ m ) independent of temperature. The dependence of D(M) on temperature can a l s o be i n f e r r e d from theory. The d i f f u s i v i t y should be d i r e c t l y p r o p o r t i o n a l to the surface t e n s i o n of water (y) and i n v e r s e l y p r o p o r t i o n a l to the v i s c o s i t y of water (Jackson 1963). This can e a s i l y be seen from Eqs. [42] and [59]. In experiments conducted on wood, Voigt et al. (1940) observed that the dependence of D on T between 30 and 50°C was very w e l l explained on the b a s i s of T dependence of the r a t i o y/r). Results reported by Jackson f o r three d i f f e r e n t s o i l s subjected to temperatures between 5 and 42.5°C appear to bear out the same c o n c l u s i o n . The i n f l u e n c e of the pressure on K(M), K(ili ) and D(M) i s , m however, more d i f f i c u l t to p r e d i c t . Wood s p e c i f i c p e r m e a b i l i t y data reported by Kelso et al. (1963) and B a i l e y and Preston (1970) i n d i c a t e that t h i s i n f l u e n c e i s s m a l l under the saturat e d c o n d i t i o n . I t may be expected that the pressure has a l s o a n e g l i g i b l e e f f e c t below f u l l f s a t u r a t i o n , except i n presence of entrapped a i r under which c o n d i t i o n t h i s e f f e c t could be g r e a t l y a m p l i f i e d . 2.2.3.3 Entrapped a i r As explained by P o u l o v a s s i l i s (1970) the phenomenon of a i r entrapment may a f f e c t K i n at l e a s t two ways. F i r s t l y , entrapped a i r decreases the number of pores that should normally take p a r t i n the water t r a n s f e r . Secondly, i t increases the t o r t u o s i t y of the remaining flowpaths. The phenomenon of a i r entrapment can a l s o occur under the saturated c o n d i t i o n due to the n u c l e a t i o n of a i r bubbles from the a i r d i s s o l v e d i n the water. Kelso et al. (1963) conducted a d e t a i l e d study 71 of the phenomenon of a i r blockage i n wood during the measurement of water p e r m e a b i l i t y at s a t u r a t i o n . I t was demonstrated by these authors that a i r entrapment at s a t u r a t i o n can be avoided by using proper deaeration and u l t r a f i l t r a t i o n techniques w i t h f r e s h l y d i s t i l l e d water. 2.2.3.4 Process dependence Very l i t t l e appears to be known regarding the d i r e c t i n f l u e n c e of the s t a t e of flow on the K(M) , K(ip ) and D(M) f u n c t i o n s . Haridasan m and Jensen (1972) s y s t e m a t i c a l l y obtained f o r two s o i l s greater K values at a given M w i t h a t r a n s i e n t outflow method than w i t h a steady-state method. L i n and Lancaster (1973) made s i m i l a r observations on saturated wood. However, i n n e i t h e r one of these cases was the above phenomenon a t t r i b u t e d e n t i r e l y to the process dependence of the c o n d u c t i v i t y . Apparently, d i f f e r e n c e s i n the techniques of measurement may have been re s p o n s i b l e f o r some of the v a r i a t i o n observed. Chapter 3 EXPERIMENTAL PROCEDURE The objectives of the experiments conducted i n the current study were to measure the M-ty r e l a t i o n s h i p and the K(M), K(ty ) and D(M) m m functions of wood at high moisture contents. Both the M-ip boundary m drainage and imbibition curves and the drainage curve s t a r t i n g from the green state were determined. The porous plate methods were used f or t h i s purpose. Psychrometric p o t e n t i a l determinations were also c a r r i e d out i n some of the specimens e q u i l i b r a t e d on the porous pla t e s . The K and D determinations were r e s t r i c t e d to the boundary drainage and imbibition curves i n the l o n g i t u d i n a l d i r e c t i o n of flow. Two independent methods of measurement were used, namely an instantaneous p r o f i l e method and the one-step method. In addition, K measurements were conducted separately at f u l l saturation. Specimens of western hemlock (Tsuga heterophylla (Raf.) Sarg.) sapwood were used for a l l determinations. The choice of sapwood was based c h i e f l y on i t s high green moisture content and i t s low percentage of ex t r a c t i v e s . The temperature of the experiments was 21°C. 7 2 73 3.1 M a t e r i a l s 3.1.1 Dimensions and o r i e n t a t i o n of the specimens The specimens used f o r M-iii determinations and f o r the measure-in ment of K and D by the one-step method were 1.5 cm long i n the f i b e r d i r e c t i o n and 4.5 cm square across the g r a i n . The specimen o r i e n t a t i o n was s e l e c t e d i n such a way as to fo r c e flow to occur i n the l o n g i t u d i n a l d i r e c t i o n (specimen placed f l a t w i s e on the porous p l a t e ) . This allowed both types of determinations above to take place simultaneously, and ensured sh o r t e r e q u i l i b r a t i o n times. The 1.5-cm length along the g r a i n was aimed at minimizing e q u i l i b r a t i o n times w h i l e ensuring the macro-scopic homogeneity of the wood i n t h i s d i r e c t i o n and p r o v i d i n g f o r other experimental c o n s t r a i n t s . The dimensions across the g r a i n corresponded approximately to the minimum thickness of the sapwood zone. The d i s -t i n c t i o n between sapwood and heartwood was made on the green cross s e c t i o n s based on the degree of surface b r i g h t n e s s . The specimens used f o r the determination of K and D by the instantaneous p r o f i l e method were about 7.5 cm long i n the l o n g i t u d i n a l d i r e c t i o n and 4.5 cm square across the g r a i n . The 7.5-cm length i n the l o n g i t u d i n a l d i r e c t i o n appeared as a reasonable compromise between a number of experimental c o n s t r a i n t s . In p a r t i c u l a r t h i s dimension was s e l e c t e d to provide s u f f i c i e n t data p o i n t s f o r accurate f l u x and gradient determinations. The saturated c o n d u c t i v i t y measurements were conducted on specimens 2.0 and 5.0 cm long i n the l o n g i t u d i n a l d i r e c t i o n . The r a d i a l 74 and t a n g e n t i a l dimensions were both 4.5 cm. The reason f o r using two specimen lengths was to check whether the saturated c o n d u c t i v i t y was length dependent. 3.1.2 S e l e c t i o n and p r e p a r a t i o n of the specimens Four stem s e c t i o n s of 0.6 m i n l e n g t h were obtained from two western hemlock t r e e s , two s e c t i o n s being cut at stump height (0.6 m) and the two others at about 9.6 m above the ground l e v e l . The trees were f e l l e d i n e a r l y autumn i n the U.B.C. Research Forest, Haney, B.C., a h i g h - r a i n f a l l area of the c o a s t a l r e g i o n . Immediately a f t e r f e l l i n g , the b o l t s were cut, put i n polyethylene bags and transported to the l a b o r a t o r y the same day. On t h e i r a r r i v a l , the b o l t s were c a r e f u l l y wrapped i n Mylar p l a s t i c f i l m , sealed i n polyethylene bags and stored at 3°C u n t i l processing. The specimen tr e e s were 74 and 91 years o l d at stump height and t h e i r diameters at the same l e v e l were 45.7 and 38.9 cm, r e s p e c t i v e l y . The processing of the b o l t s occurred f i v e months a f t e r t h e i r e x t r a c t i o n from the f o r e s t . Sporadic spots of d i s c o l o r a t i o n s were observed on the end cross s e c t i o n s but they were only s u p e r f i c i a l . Although some aging of the wood could have taken place, the consequences could not be s e r i o u s s i n c e the character of the experiments was p r i m a r i l y comparative. Nine specimen b i l l e t s 4.5 by 4.5 cm i n the r a d i a l and t a n g e n t i a l d i r e c t i o n s were f i r s t sawn from each b o l t . Location of the b i l l e t s around the sapwood zone was marked beforehand on the small end cross s e c t i o n of the b o l t as i l l u s t r a t e d i n F i g . 6. The cross s e c t i o n was d i v i d e d i n t o twelve 30° s e c t o r s , and three chords were drawn between the 0, 120 and 240° p o s i t i o n s (0° p o s i t i o n f i x e d a r b i t r a r i l y ) on the periphery. These chords marked the l o c a t i o n of the chain-saw r i p - c u t s which produced three t h i c k slabs c o n t a i n i n g the sapwood zone. Each slab was recut on a band-saw i n t o three rough b i l l e t s which were then taken to a bench-saw f o r f u r t h e r shaping and o r i e n t a t i o n . The f i n a l dimensions of 4.5 by 4.5 cm i n cross s e c t i o n were obtained by s u r f a c i n g i n a planer on a l l four s i d e s . F i g . 6. — Cross s e c t i o n of b o l t showing l o c a t i o n of specimen b i l l e t s . 76 The b i l l e t s obtained from the 60, 180 and 300° p o s i t i o n s were cut i n t o specimens 1.5 cm i n len g t h . The cross-cuts were made by using a hollow-ground saw. This gave a smooth surface which was re q u i r e d f o r good h y d r a u l i c contact between specimen and porous p l a t e . The place occupied by each specimen i n a given b i l l e t was i d e n t i f i e d . Whether a s e r i e s of specimens was progressing upwards or downwards w i t h respect to the d i r e c t i o n of the b i l l e t i n the tree was not marked. This allowed a b e t t e r randomization of the twenty-six groups of twelve specimens formed from the twelve s e r i e s i n question. The groups were formed by t a k i n g i n succession one specimen from each s e r i e s composed of about t h i r t y specimens. Thus each group was composed of four subgroups representing two heights and two t r e e s . The specimens w i t h defects were discarded. Twenty of the above groups were used to obt a i n seven data points on both M-ifi drainage curves and s i x data p o i n t s on the i m b i b i t i o n curve. The others were reserve groups. The K and D determinations by the one-step method took place on a few of the groups r e t a i n e d f o r the M—iii determinations. m The above sampling procedure was devised to provide represen-t a t i v e data w i t h a good o v e r a l l p r e c i s i o n w h i l e o b t a i n i n g at the same time some i n f o r m a t i o n about two important f a c t o r s of v a r i a b i l i t y . As i n d i c a t e d by p r e l i m i n a r y experiments the p r e c i s i o n of porous p l a t e measurements seemed to j u s t i f y t h i s sampling design. Indeed, M-iJj measurements conducted on end-matched specimens i n drainage showed a c o e f f i c i e n t of v a r i a t i o n l e s s than 5% among M data obtained at any given imposed p o t e n t i a l between 0 and _13 bars. The b i l l e t s remaining from the two b o l t s at stump l e v e l were cross- c u t i n t o specimens of about 7.5 cm i n length using a micro-saw. Twelve s e r i e s of seven end-matched specimens were thus obtained. Four of these s e r i e s were f i n a l l y s e l e c t e d f o r K and D determinations based on the absence of defects and t h e i r higher and uniform green moisture content. The use of good r e p l i c a t e s was c r u c i a l to the success of these experiments. One specimen from two of the s e l e c t e d s e r i e s served f o r the saturat e d c o n d u c t i v i t y measurements. P e r t i n e n t p h y s i c a l p r o p e r t i e s of the m a t e r i a l s used i n t h i s study are presented i n Table 1. Each value i n the three columns on the r i g h t represents a mean based on a l l the specimens used f o r the TABLE 1. — P e r t i n e n t p h y s i c a l p r o p e r t i e s of the m a t e r i a l s used i n the present study. S p e c i f i c Width of Green Tree Height g r a v i t y 1 growth zones M (m) (mm) <%) 0.6 0.429 3.1 175.4 I 9.6 0.390 3.2 156.5 0.6 0.419 1.9 172.6 I I 9.6 0.398 2.0 146.8 1 Based on mass and volume i n the oven-dry c o n d i t i o n . measurement of the r e l a t i o n s h i p . Both s p e c i f i c g r a v i t y based on oven-dry volume and green moisture content e x h i b i t a decrease w i t h i n c r e a s i n g height i n the t r e e . The former behavior appears to f o l l o w a general trend (Kollmann and Cote 1968) but no support was found i n the l i t e r a t u r e f o r decreasing sapwood green moisture content w i t h height. A higher p r o p o r t i o n of inner sapwood, which would tend to be d r i e r than the outer sapwood (Linzon 1969), i n the specimens at the 9.6-m height or seasonal v a r i a t i o n of the green moisture content i n the standing t r e e s are p o s s i b l e explanations. 3.1.3 P r e c o n d i t i o n i n g of the specimens The groups a l l o c a t e d f o r the measurement of the M-ip drainage m curve from the green c o n d i t i o n r e q u i r e d no p a r t i c u l a r p r e c o n d i t i o n i n g . A l l other s i m i l a r groups were f i r s t d r i e d slowly i n a c o n d i t i o n i n g cabinet placed i n s i d e a constant-temperature room (21 ± 0.5°C) using saturated s a l t s o l u t i o n s or phosphorus pentoxide (P^O^) to c o n t r o l the humidity of the d r y i n g a i r . About 1 month was required to g r a d u a l l y dry the specimens below 3% M. The specimens were then subjected to r e - a d s o r p t i o n by g r a d u a l l y i n c r e a s i n g RH of the humid a i r from 50 to near 100%. The groups a l l o c a t e d f o r M-^ determinations on the boundary m drainage curve were removed from the c o n d i t i o n i n g cabinet a f t e r 1 month of a d s o r p t i o n i n a n e a r l y saturated atmosphere. They were then subjected to a s a t u r a t i o n c y c l e of 12-h vacuum and 12-h atmospheric pressure w h i l e being kept submerged i n f r e s h l y d i s t i l l e d and M i l l i p o r e - f i l t e r e d water. 79 A second i d e n t i c a l s a t u r a t i o n c y c l e was a p p l i e d j u s t p r i o r to each e x t r a c t i o n run. The groups conditioned f o r M-ii determinations on the m boundary i m b i b i t i o n curve reached an apparent e q u i l i b r i u m w i t h the saturated vapor a f t e r 2 months of adsorption. Their average M was then 27.5%. I m b i b i t i o n was s t a r t e d i n the swollen c o n d i t i o n to avoid l o s s of h y d r a u l i c contact between the specimen and the porous p l a t e . This a l s o precluded the development of temperature gradients r e s u l t i n g from the s u b s t a n t i a l r e l e a s e of heat of wetting at low moisture contents. The reason f o r using once-dried specimens i n the measurement of the boundary drainage curve was to have specimen groups with a c o n d i t i o n i n g h i s t o r y comparable to that of the groups used to o b t a i n the i m b i b i t i o n curve. The s e r i e s of end-matched specimens a l l o c a t e d f o r the determi-n a t i o n of K and D by the instantaneous p r o f i l e method were saturat e d d i r e c t l y from the green c o n d i t i o n , or conditioned to the swollen c o n d i t i o n (M - 30%) without p r i o r d e s i c c a t i o n . Due to the l e n g t h of the specimens, c o n d i t i o n i n g times w i t h the same m i l d desorption and adsorption treatments as above would have been p r o h i b i t i v e . Five consecutive s a t u r a t i o n c y c l e s of 12-h vacuum and 12-h atmospheric pressure were re q u i r e d to reach a degree of s a t u r a t i o n between 98 and 100%. F r e s h l y d i s t i l l e d and f i l t e r e d water was used f o r each c y c l e . The c o n d i t i o n i n g to about 30% M took place i n an Aminco t e s t chamber set at 21°C and 90% RH. An apparent e q u i l i b r i u m was reached a f t e r 3 weeks of d r y i n g . 80 During the period of time from the end of the specimen precon-d i t i o n i n g and the s t a r t of a given experiment, the specimens were kept t i g h t l y packed i n sealed polyethylene bags and stored i n a n e a r l y saturated atmosphere at 21°C. 3.2 Methods 3.2.1 Measurement of the M-iti r e l a t i o n s h i p m 3.2.1.1 Porous p l a t e methods Schematic diagrams of the setups that were devised to determine the M-ij;^ r e l a t i o n s h i p i n drainage are shown i n F i g s . 7 and 8. These setups represent two pressure p l a t e apparatuses of d i f f e r e n t gage a i r pressure ranges, namely from 0.1 to 1.0 bar ( F i g . 7) and from 1 to 15 bars ( F i g . 8 ) . 1 The pressure p l a t e apparatus of F i g . 7 c o n s i s t s of two pressure p l a t e c e l l s enclosed i n a pressure chamber, two vented outflow b u r e t t e s , a constant bleed pressure r e g u l a t o r , a vapor s a t u r a t o r , a mercury manometer, and v a r i o u s a i r f i l t e r i n g and s a f e t y devices. The pressure p l a t e c e l l s were commercially a v a i l a b l e u n i t s ( S o i l Moisture Equipment Co., Santa Barbara, C a l i f o r n i a ) , both of which c o n s i s t i n g of 1-bar ceramic pressure p l a t e of about 28 cm i n diameter sealed on one side by a t h i n neoprene gasket. An i n t e r n a l f i n e mesh screen keeps the diaphragm 1 The equipment shown i n these diagrams i s s i m i l a r to that commercially a v a i l a b l e from S o i l Moisture Equipment Co., Santa Barbara, C a l i f o r n i a . P O R O U S N E O P R E N E C E R A M I C P L A T E D I A P H R A G M F i g . 7. — Schematic diagram of the pressure plate apparatus used for M-u> determinations i n drainage between -0.1 and-1.0 bar il> . m m oo PRESSURE REGULATOR BY-PASS VALVE SAFETY RELIEF PRECISION PRESSURE GAGE (0-60 psi) PRECISION PRESSURE GAGE (0-300 psi) STANDARD 1/4 - in. GALVANIZED >IPE AND FITTING CONNECTING HOSE PRESSURE VAPOR CHAMBER SATURATOR COMPRESSED AIR CYLINDER PRESSURE PLATE CELL CLAMPING BOLT FLEXIBLE TUBING OUTFLOW BURETTE SCREEN -TUBING CLAMP POROUS CERAMIC PLATE NEOPRENE DIAPHRAGM F i g . 8. — Schematic diagram of the pressure plate apparatus used for M-i|i determinations i n drainage between -1 and -15 bar \b m m oo from c l o s e contact w i t h the p l a t e and provides a passage f o r flow of water to the o u t l e t stem running through the p l a t e - The pressure chamber was a pressure cooker (Presto Model 7B, 15.14 l i t e r s ) m odified s l i g h t l y f o r t h i s work. A Norgren pressure r e g u l a t o r (0-60 p s i range) maintained the a i r pressure i n s i d e the chamber to w i t h i n ±5 mbars of the imposed value. The l a b o r a t o r y a i r supply was used as a i r source. The b u r e t t e s f o r measuring the volume of outflow were accurate to ±0.1 cm 3. The vapor s a t u r a t o r c o n s i s t e d of a 500-ml heavy w a l l f i l t e r i n g f l a s k c o n t a i n i n g a mixture of small g l a s s beads and d i s t i l l e d water. The purpose of the vapor s a t u r a t o r (Tanner and E l r i c k 1958) was to minimize evaporative moisture l o s s e s from the specimens caused by back d i f f u s i o n i n the pressure l i n e or by a i r l e a k s from the mounting. The pressure p l a t e apparatus of F i g . 8 i s composed of two pressure p l a t e c e l l s enclosed i n a high-pressure chamber, two vented outflow b u r e t t e s , a compressed a i r c y l i n d e r , a pressure r e g u l a t o r , a vapor s a t u r a t o r , two pressure gages w i t h d i f f e r e n t ranges, and valves of v a r i o u s f u n c t i o n s . The pressure p l a t e c e l l s were u n i t s s i m i l a r to those described above except f o r the bubbling pressure of the ceramic p l a t e s which was e i t h e r 3 or 15 bars depending on the imposed pressure. The high-pressure chamber c o n s i s t e d of two 1.25-in. t h i c k s t e e l p l a t e s and a seamless s t e e l c y l i n d e r w i t h 0.500-in. w a l l t h i c k n e s s , 12.25-in. outside diameter, and 4.0-in. height. The c y l i n d e r was f i t t e d w i t h a groove at both ends accomodating s y n t h e t i c rubber "0" r i n g s that provide the a i r s e a l . The assembly was fastened by eight 7/8-in. b o l t s . A F i s h e r m u l t i - s t a g e c y l i n d e r r e g u l a t o r (0-250 p s i range) permitted a pressure c o n t r o l accuracy of about ±1% i n the pressure range covered. This was approximately the accuracy of the pressure gages employed (0-60 p s i range; 0-300 p s i range). The vapor s a t u r a t o r was f a b r i c a t e d from a small compressed gas c y l i n d e r . Regarding the determination of the M-ip r e l a t i o n s h i p i n m i m b i b i t i o n , schematic diagrams of the setups used f o r t h i s purpose are shown i n F i g s . 9 and 10. The diagram of F i g . 9 represents a tension p l a t e apparatus w i t h a tension range of about 0 to 50 cmR^O (0 to 0.05 ba r ) . The diagram of F i g . 10 represents a pressure p l a t e apparatus w i t h an a i r pressure range of 0.1 to 1.0 bar. The tension p l a t e apparatus of F i g . 9 i s composed of a tension p l a t e c e l l , a hanging water column, and a large c a p a c i t y i n f l o w b urette a c t i n g as a constant-head water r e s e r v o i r . The te n s i o n p l a t e c e l l c o n s i s t s of a base c o n t a i n i n g the porous p l a t e , a frame, and an en c l o s i n g box. The porous p l a t e measured about 22 cm square and was made of a b l o t t i n g paper and a 4-mm t h i c k l a y e r of f i n e sand on top of i t . The sand l a y e r allowed good h y d r a u l i c contact between specimen and porous p l a t e and a l s o p r o t e c t e d the f r a g i l e paper membrane. A f i b e r g l a s screen kept the membrane from c l o s e contact w i t h the base. The pressure s e a l of the membrane at i t s outer edge was made w i t h the p l e x i g l a s base on one s i d e , and w i t h a neoprene gasket and the frame on the other s i d e . This assembly was clamped together w i t h s i x t e e n s m a l l brass b o l t s . The p l e x i g l a s e n c l o s i n g box was clamped to the bottom part w i t h wing nuts. The i n f l o w b urette was accurate to about ±0.25 cm 3. ENCLOSING BOX INFLOW BURETTE BUBBLER TUBING CLAMP F i g . 9. — Schematic diagram of the tension p l a t e apparatus used f o r M-ip determinations i n i m b i b i t i o n between 0 and -0.05 bar 4) . m m oo Ln P R E S S U R E C H A M B E R F i g . 10. — Schematic diagram of the pressure plate apparatus used for M-ifi determinations i n imbibition between -0.1 and -1.0 bar ty . m m oo 87 The pressure p l a t e apparatus of F i g . 10 i s s i m i l a r to that shown i n F i g . 7 except f o r the pressure p l a t e c e l l , the l a r g e c a p a c i t y i n f l o w b u r e t t e and the a i r purging system. The pressure p l a t e c e l l was f a b r i -cated from a 1-bar ceramic pressure p l a t e f i t t e d f l u s h and sealed i n t o a p l e x i g l a s base. A clearance was allowed at the p l a t e bottom by g l u i n g small copper w i r e s i n symmetrical curved grooves s c r i b e d i n the bottom of the base c a v i t y between the two o u t l e t tubes. The a i r purging system (Tanner and E l r i c k 1958) c o n s i s t e d of an a i r trap of l a b o r a t o r y glassware and a long connecting tube made of amber l a t e x tubing. I t s purpose was to remove the a i r that could accumulate under the porous p l a t e during an i m b i b i t i o n run. Now to c a r r y out the determinations the f o l l o w i n g pro-cedures were adopted. In drainage, the procedures employed were very s i m i l a r f o r both setups of F i g s . 7 and 8. I t was f i r s t e s s e n t i a l to f u l l y deaerate and saturate the porous p l a t e . This was accomplished by a s a t u r a t i o n c y c l e of 12-h vacuum and 12-h atmospheric pressure. Next, a saturated c l a y l a y e r about 2 mm t h i c k was formed on the p l a t e to ensure good h y d r a u l i c contact between specimen and porous p l a t e . A rubber diaphragm f i x e d w i t h 5-cm square holes was used to f a c i l i t a t e t h i s o p e r a t i o n . A f t e r two pressure p l a t e c e l l s had been prepared i n t h i s manner, 2 one of them was i n s t a l l e d i n p o s i t i o n i n the pressure chamber and the i n s i d e outflow tube connected to the c y l i n d e r o u t l e t . I f any a i r had 2 Group p a i r s of s a t u r a t e d and green specimens were t e s t e d simultaneously i n the same pressure p l a t e apparatus. 88 a c c i d e n t l y flowed back i n t o the space under the porous p l a t e during the above manipulations, i t was a s p i r a t e d using mouth s u c t i o n or a syringe at the outside outflow tube w h i l e t i l t i n g the c e l l s l i g h t l y i n the chamber. Next, the specimens were placed m e t i c u l o u s l y on the porous p l a t e , being pressed f i r m l y against the c l a y bed f o r good h y d r a u l i c contact. Once the two pressure p l a t e c e l l s w i t h the specimens adhering to the c l a y l a y e r were ready i n p o s i t i o n , the chamber was h e r m e t i c a l l y close d and the pressure a p p l i e d g r a d u a l l y . For the f i n a l adjustment of the pressure, the a i r was forced to bubble through the vapor s a t u r a t o r i n order to humidify the atmosphere of the chamber. Burette readings were then s t a r t e d as soon as the outflow r a t e had dropped markedly and continued u n t i l attainment of e q u i l i b r i u m . These outflow data permitted the approach and attainment of e q u i l i b r i u m to be r e a d i l y a s c e r t a i n e d , and were a l s o r e q u i r e d f o r K and D determinations (one-step method). A f t e r the outflow had ceased or become n e g l i g i b l e , 3 clamps were put on the outflow tubes to prevent backflow of water to the specimens and the pressure was released s l o w l y . The chamber was then opened, the specimens removed from the porous p l a t e and taken f o r weighing. Any adhering c l a y on the specimens was scraped o f f w i t h a k n i f e blade. The procedures adopted f o r M-ty determinations i n i m b i b i t i o n m 3 In general, e q u i l i b r i u m f o r the porous p l a t e t e s t s was considered to have been reached when the outflow or i n f l o w r a t e was s m a l l e r than 0.03 cm 3/day/specimen; the maximum e r r o r thus induced was about ±0.25% A M f o r each a d d i t i o n a l day that the flow would otherwise have p e r s i s t e d at t h i s given r a t e . 89 were s i m i l a r to those i n drainage except f o r a few important changes. Regarding the t e n s i o n p l a t e apparatus of F i g . 9, the porous p l a t e was conditioned f o r a few minutes at the d e s i r e d water t e n s i o n before p l a c i n g the specimens on top of i t . This precaution i s necessary i n i m b i b i t i o n t e s t s to prevent the specimen of becoming too wet a c c i d e n t l y before the water t e n s i o n or the a i r pressure i s a p p l i e d (Coleman and Marsh 1961). Good h y d r a u l i c contact between specimen and sand l a y e r was maintained by p l a c i n g 250-g weights on top of each specimen. The height of the water column was measured w i t h a cathetometer. With the pressure p l a t e apparatus of F i g . 10, the c l a y l a y e r on top of the porous p l a t e was a l s o c onditioned to the d e s i r e d a i r pressure before p l a c i n g the specimens i n contact w i t h i t . Another important operation w i t h t h i s setup was the p e r i o d i c " m i l k i n g " of the connecting i n f l o w tube f o r the removal of the a i r accumulated under the porous p l a t e . By running a r o l l e r over the connecting tube r e s t i n g on the l a b o r a t o r y bench, the water was forced to c i r c u l a t e under the p l a t e thereby purging any accumulated a i r which was then c o l l e c t e d i n the a i r t r a p . To avoid e r r o r s i n the b u r e t t e readings, the l i q u i d l e v e l i n the a i r trap was adjusted to a reference l e v e l mark a f t e r each " m i l k i n g " by using an hypodermic syringe connected to the upper end of the b u r e t t e . F r e s h l y d i s t i l l e d , f i l t e r e d and deaerated water was used f o r a l l porous p l a t e experiments. These t e s t s were conducted i n a constant-temperature room at 21 ± 0.5°C. Temperature d e v i a t i o n s of +1°C occurred s p o r a d i c a l l y but always f o r very short d u r a t i o n s . For the moisture content determinat i o n s , the specimens were oven-dried slowly i n a forced a i r c i r c u l a t i o n oven f o r one day at 50°C, one day at 75°C, and f i n a l l y two days at 103°C. This m i l d schedule permitted the specimens to be desiccated v i r t u a l l y f r e e of d e f e c t s . This improved subsequently the accuracy of the s p e c i f i c g r a v i t y measurements which w i l l be described l a t e r . A l l mass determinations were made on an a n a l y t i c a l balance of ±0.0001 g accuracy. Specimens were h e l d i n d e s i c c a t o r s except during a c t u a l weighing i n order to reduce moisture changes. Moisture gains (on oven-dry specimens) or moisture l o s s e s (on wet specimens) caused by handling and a i r exposure were estimated to be l e s s than 0.015 g. Based on the l a t t e r f i g u r e and on a specimen oven-dry mass of 11 g, the absolute e r r o r AM of an i n d i v i d u a l moisture content determination was estimated from a propagation of e r r o r formula f o r systematic e r r o r s . 4 The values thus obtained v a r i e d from 0.3% AM at 30% M to 0.6% AM at f u l l s a t u r a t i o n . Although the signs of the elemental systematic e r r o r s a s s o c i a t e d w i t h the mass measurements were ignored, i t i s evident that the above systematic e r r o r s f o r M were almost e n t i r e l y on the negative s i d e . For a f u n c t i o n of two v a r i a b l e s w = f ( x , y ) , the absolute e r r o r f o r the systematic e r r o r i n w i s given by (Ku 1966): Aw Aw _6f 6x Ax A i 1 6y Ay assuming that Ax and Ay are small such that second and higher order terms i n Ax and Ay are c o l l e c t i v e l y n e g l i g i b l e i n the Taylor s e r i e s expansion. A summary of the porous p l a t e experiments c a r r i e d out f o r the measurement of the K-ty r e l a t i o n s h i p i s given i n Table 2. As can be m seen, two drainage runs were conducted at an imposed ty of about -14 m bars. Due to unexpected r e s u l t s , a v e r i f i c a t i o n run was f e l t necessary. On the other hand, the imposed ty of -0.0015 bar i n i m b i b i t i o n was the m r e s u l t of an experimental e r r o r . This t e s t was f i r s t intended to be conducted at an imposed ty of zero. However, i t was found at the end of the run that the porous p l a t e was a c t u a l l y subjected to a tension of about 1.5 cmH^O due to inaccurate p o s i t i o n i n g of the bubbler (see F i g . 9). TABLE 2. — Summary of the porous p l a t e experiments c a r r i e d out f o r the measurement of the M-IJJ r e l a t i o n s h i p . m E q u i l i b r a t i o n mode I n i t i a l M (%) Imposed (bars) E q u i l i b r a t i o n time (days) Drainage 215.1 162.9 1 -0.110 4 " 216.1 162.3 -0.470 15 " 216.6 162.8 -0.984 14 217.6 165.5 -1.95 15 " 217.9 163.8 -2.87 23 217.4 161.1 -6.95 22 " 215.2 160.0 -13.8 21 " 215.7 160.3 -13.9 31 I m b i b i t i o n 26.9 -0.0015 38 " 28.4 -0.0098 82 28.0 -0.0294 29 " 27.5 -0.091 48 27.7 -0.499 22 " 26.3 -0.994 12 1 Saturated and green M, r e s p e c t i v e l y . Regarding the e q u i l i b r a t i o n times obtained, Table 2 shows that they tended to be s h o r t e r and more uniform i n drainage than i n i m b i b i t i o n . A s i m i l a r p a t t e r n has been observed by Coleman and Marsh (1961). The very long e q u i l i b r a t i o n time obtained at -0.0098 bar ip i n i m b i b i t i o n was apparently caused by poor h y d r a u l i c contact between two specimens and the porous p l a t e . Only one minor a i r leak was experienced during a l l the porous p l a t e measurements. Of course, some a i r was a l s o l o s t through the pressure p l a t e s s i n c e a i r t r a n s f e r always occurs through wet pressure p l a t e s or membranes. This i s a consequence of Henry's law which says that the s o l u b i l i t y of a i r i n water i s p r o p o r t i o n a l to the pressure (Richards 1965). Therefore, the c o n c e n t r a t i o n of d i s s o l v e d a i r i n the p l a t e water on the sample side i s always higher than on the outflow s i d e . This a i r moves through the p l a t e during l i q u i d outflow and even a f t e r the outflow has ceased through molecular d i f f u s i o n . The maximum a i r t r a n s f e r measured at the outflow tube at the end of a run was 0.01 cm 3/min/bar. 3.2.1.2 Thermocouple psychrometer method The thermocouple psychrometer method was used i n the current study as a means of checking some of the M-i|> data obtained by the porous p l a t e methods. For t h i s purpose d i r e c t p o t e n t i a l measurements were c a r r i e d out w i t h TCP's on 15 of the 22 groups of specimens that had been p r e v i o u s l y e q u i l i b r a t e d at known i / j m * s o n the porous p l a t e s . Obviously, the comparison of these two types of data i m p l i e d that ip^ 93 was n e g l i g i b l e w i t h respect to ty . This assumption w i l l be discussed m l a t e r i n the t e x t . Eight commercially a v a i l a b l e s i n g l e - l o o p P e l t i e r type TCP's (Model PT51-10, Wescor, Inc., Logan, Utah) were used f o r the water p o t e n t i a l measurements. A cross s e c t i o n of t h i s type of TCP i s shown i n F i g . 11. This miniature psychrometer c o n s i s t s of 0.0025-cm diameter chromel-constantan wires welded together at one end (sensing or wet j u n c t i o n ) , and attached at the other end to two gold p l a t e d brass pins t i g h t l y wedged w i t h the thermocouple wires In holes made through a t e f l o n plug. Each of these p i n s , which serve as heat sin k s (Campbell 1972), i s i n turn soldered at i t s outer end to a copper lead wire about COPPER - CONSTANTAN THERMOCOUPLE GOLO PLATED BRASS PINS F i g . 11. — Cross s e c t i o n of the thermocouple psychrometer used f o r water p o t e n t i a l measurements i n wood (adapted from Campbell 1972). 3 m long t e r m i n a t i n g w i t h a s o l i d copper end. The three TCP j u n c t i o n s are sealed i n a hollow porous cup of 0.64 cm i n diameter extending about 1.3 cm beyond the TCP rubber j a c k e t which f i r m l y holds together a l l the components. Also encased i n the TCP j a c k e t i s a copper-constantan thermocouple which i s used i n conjunction w i t h an i c e bath or other i c e - p o i n t reference system f o r ambient temperature measurements. A block diagram of the setup used f o r p o t e n t i a l measurements wi t h TCP's i s shown i n F i g . 12. The setup i s composed of a micro-voltmeter, a switchbox, a recorder, a constant-temperature water bath, a specimen chamber, the psychrometer and the temperature thermocouple w i t h the reference j u n c t i o n s immersed i n an i c e bath. In order to c l e a r l y show the thermocouple c i r c u i t s , only one specimen chamber was drawn on the diagram whereas i n r e a l i t y e i g h t of these u n i t s were immersed together i n the water bath. The microvoltmeter was a K e i t h l e y model 155 ( K e i t h l e y Instrument, Inc., Cleveland, Ohio), a battery-operated high p r e c i s i o n voltmeter which enabled readings to be made ot the nearest 0.1 uV. The switchbox (Chow and de V r i e s 1973) c o n s i s t e d of a 1.35-V mercury b a t t e r y , a 250-fi v a r i a b l e r e s i s t o r and a 0-10 mA meter making up the c o o l i n g c i r c u i t , and a 4 pole 3 throw make before break r o t a r y switch used to connect the TCP c i r c u i t to e i t h e r the c o o l i n g c i r c u i t or the output measuring c i r c u i t . The recorder was a Hewlett-Packard Model 7100B two-pen s t r i p chart recorder. One channel was used f o r r e c o r d i n g the TCP output s i g n a l and the other channel served to make ambient temperature checks 95 COPPER LEAD WIRES CONSTANTAN SWITCHBOX CONSTANTAN F i g . 12. — Block diagram of the setup used for water p o t e n t i a l measurements with thermocouple psychrometers. with the copper-constantan thermocouple. Ambient temperature readings (±0.2°C accuracy) were pr i m a r i l y useful to assess the approach of temperature equilibrium i n s i d e the specimen chamber at the s t a r t of a measurement run. The water bath was a Haake Model FK2 which was found capable of c o n t r o l l i n g the temperature to within ±0.001°C i n a submersed polyethylene b o t t l e f i l l e d with water at an operating temperature of 21°C. A platinum resistance thermometer connected to a Wheatstone bridge c i r c u i t was used for t h i s t e s t . The reason for using a water bath was that changes i n ambient temperature during a ip measurement could cause temperature gradients between the psychrometer j u n c t i o n s other than that due to evaporative c o o l i n g of the sensing j u n c t i o n (Chow and de V r i e s 1973). A c t u a l l y , very small temperature gradients may always p e r s i s t between the v a r i o u s j u n c t i o n s of the TCP c i r c u i t but those do not a f f e c t the ip measurements as long as they remain constant f o r the d u r a t i o n of the TCP output readings. The specimen chamber c o n s i s t e d of a bottom p a r t w i t h i n s i d e dimensions j u s t s l i g h t l y l a r g e r than those of the specimen, and a f l a t top p a r t ( l i d ) f i t t e d w i t h a 9-cm long tube through which the TCP lea d wires were introduced i n the chamber. The l i d was clamped to the bottom p a r t w i t h wing nuts. A l l p a r t s were made of p l e x i g l a s . To prepare the TCP-specimen-chamber assembly f o r a lp measurement, the psychrometer was f i r s t passed through the l i d of the specimen chamber and then the ceramic cup was i n s e r t e d by a t w i s t motion i n t o a hole bored f o r t h i s purpose at the center of a r a d i a l face of the specimen. Te f l o n tape was used to improve the a i r s e a l at the outer edge of the hole. Next, the specimen was i n s e r t e d i n the bottom part of the chamber and the l i d fastened i n p l a c e . The upper end of the l i d tube was then sealed w i t h a rubber stopper through which the copper lead wires extended outside the chamber. A temporary vent was created by i n s e r t i n g a needle between the rubber stopper and the tube i n s i d e w a l l to ensure pressure u n i f o r m i t y w i t h the atmosphere. F i n a l l y , stopcock grease was put on a l l the mounting j o i n t s f o r good water s e a l . Once e i g h t of these assemblies were completed, they were attached to a rack f o r complete immersion i n 97 the water bath. The d r i l l i n g of the hole i n the specimen took place before the porous p l a t e t e s t s . The hole was 0.64 cm i n diameter and about 1.5 cm deep. A d r i l l press and a short shank 1/4-in. diameter wood d r i l l i n g b i t was used f o r t h i s purpose. In order to prevent moisture l o s s e s , hence p o t e n t i a l changes, the specimens were kept wrapped i n s e v e r a l l a y e r s of Saran f i l m at a l l times between the end of the porous p l a t e t e s t s and the completion of the psychrometric p o t e n t i a l measurements. Despite t h a t , the specimens s u f f e r e d an average M l o s s of about 0.6% during t h i s p e r i o d of time. To c a r r y out a reading of the TCP output, the TCP c i r c u i t was i n i t i a l l y switched v i a the switchbox t o the readout c i r c u i t (micro-voltmeter on "read" p o s i t i o n ) , the microvoltmeter was zeroed and then the TCP c i r c u i t immediately switched to the c o o l i n g c i r c u i t . In t h i s p o s i t i o n a c o o l i n g current of 5 mA was passed through the sensing j u n c t i o n i n the d i r e c t i o n from constantan to chromel f o r a d u r a t i o n of 30 s. F o l l o w i n g t h i s , the TCP c i r c u i t was immediately switched back to the microvoltmeter and the emf output generated by evaporative c o o l i n g read on the panel meter. The emf output was read w i t h i n 30 s a f t e r the t e r m i n a t i o n of the c o o l i n g current and c o i n c i d e d w i t h the observation of a p l a t e a u on the output t r a c e . The magnitude and d u r a t i o n of the c o o l i n g current had been determined experimentally. The s e l e c t e d 5 mA-30 s combination provided a readable emf p l a t e a u at the lowest p o t e n t i a l s measured while minimizing the d u r a t i o n of the c o o l i n g c u r r e n t . In order to a s c e r t a i n the approach and attainment of e q u i l i b r i u m , the above reading procedure was repeated at various time i n t e r v a l s w i t h each psychrometer a f t e r the i n s t a l l a t i o n of the specimen chamber i n the water bath. The response times v a r i e d from 6 to 48 h f o r the specimens e q u i l i b r a t e d i n drainage and from 48 to 120 h f o r those e q u i l i b r a t e d i n i m b i b i t i o n . The water bath temperature was set at 21 ± 0.1°C f o r a l l the measurement runs w i t h the a i d of a p r e c i s i o n thermometer. Furthermore, the e n t i r e setup shown i n F i g . 12 was i n s t a l l e d i n a constant-temperature room at 21 ± 0.5°C. Since the a c t u a l psychrometer output reading i s a v o l t a g e , a c a l i b r a t i o n of t h i s instrument i s necessary to i n t e r p r e t i t s output readings i n terms of water p o t e n t i a l s . In the current study t h i s was achieved w i t h standard molal (m) sodium c h l o r i d e (NaCl) s o l u t i o n s of known water p o t e n t i a l s (Lang 1967). A c a l i b r a t i o n p o i n t at a given m o l a l i t y and temperature was obtained by immersing the TCP's i n the s o l u t i o n contained i n a polyethylene b o t t l e placed i n the water bath. The TCP outputs were read p e r i o d i c a l l y to assess attainment of tem-perature and vapor e q u i l i b r i u m . E q u i l i b r a t i o n times were about 3 h. Two m u l t i - p o i n t c a l i b r a t i o n s were conducted, one p r i o r to the p o t e n t i a l measurements i n wood and one a f t e r t h e i r completion. The c a l i b r a t i o n temperatures were 20 and 25°C i n the former case, and 21°C i n the l a t t e r case. The above c a l i b r a t i o n procedure has the dual advantage of requiring a simple experimental setup and r e l a t i v e l y short e q u i l i b r a t i o n times. A c a l i b r a t i o n check conducted with the ceramic cup mounted over the s o l u t i o n i n s i d e sealed test tubes yielded the same r e s u l t s within ±10%. However, the response times were up to twenty times longer. T y p i c a l examples of the c a l i b r a t i o n curves (TCP emf output vs. water p o t e n t i a l of the c a l i b r a t i n g solution) obtained at 20 and 25°C are presented i n F i g . 13. These curves were hand f i t t e d and forced through the o r i g i n . They are l i n e a r , as curves of t h i s type -30 -25 -20 -15 -10 WATER POTENTIAL, (bars) F i g . 13. — Typ i c a l c a l i b r a t i o n curves for the thermocouple psychrometers used for p o t e n t i a l measurements showing the psychrometer emf output as a function of the water p o t e n t i a l of the c a l i b r a t i n g s o l u t i o n at 20 and 25 C. usually are between 0 and about -40 bars (Brown 1970, Meyn and White 1972, Wheeler et al. 1972). The absolute value of t h e i r slope represents the TCP s e n s i t i v i t y i n pV/bar for the temperature under consideration. The s e n s i t i v i t y was found to increase of 0.0087 uV/bar°C between 20 and 25°C. For the same type of psychrometer, Wheeler et al. observed a s i m i l a r change of s e n s i t i v i t y with temperature between 10 and 30°C. The TCP s e n s i t i v i t i e s computed from the two multi-point c a l i b r a t i o n s are given i n Table A2.1 (Appendix 2). A l l computations were made for a common temperature of 21°C. For the f i r s t c a l i b r a t i o n , the c a l c u l a t i o n s were based on the data obtained at 20°C a f t e r making the proper temperature corrections. Two TCP units were c a l i b r a t e d only once as a consequence of t h e i r f a i l u r e during the course of the p o t e n t i a l measurements. I t can be seen from Table A2.1 that the TCP s e n s i t i v i t i e s changed s u b s t a n t i a l l y with use. Such observations were also reported by Brown (1970). Accordingly, the best estimate of ty was considered m to be the mean of the two values calculated from both the s e n s i t i v i t i e s before and a f t e r use. As a second method of p r e d i c t i n g ty from the measured TCP outputs, m l i n e a r regression equations were computed from the c a l i b r a t i o n data using the l e a s t squares method. For t h i s purpose, the water p o t e n t i a l of the s a l t s o l u t i o n was chosen as the dependent v a r i a b l e as t h i s choice proved to be i r r e l e v a n t . The regression equations computed from the two m u l t i -point c a l i b r a t i o n s , and the values of the s t a t i s t i c s S , , F and R 2 Y/X obtained from the analysis of variance for the regression are reported i n Table A2.1. The magnitude of the s t a t i s t i c s F and R 2 c l e a r l y show 101 the high degree of s i g n i f i c a n c e of the r e g r e s s i o n equations. Of a p a r t i c u l a r i n t e r e s t are the values of $Y/X w k i c n t^ i e standard d e v i a t i o n of the estimate. This s t a t i s t i c can be taken as a measure of the e r r o r w i t h which ty was p r e d i c t e d from an i n d i v i d u a l TCP emf m measurement. 5 The average value of §Y/X "*"n Table A2.1 i s 0.46 bar. This c a l i b r a t i o n e r r o r i s s i m i l a r to that reported by Lang (1968) and Meyn and White (1972). I t i s about twice the estimated i n s t r u m e n t a l e r r o r , which was based on the accuracy of the microvoltmeter (±0.1 uV) and the TCP s e n s i t i v i t y . 3.2.2 Measurement of the K(M), K(ty ) and D(M) f u n c t i o n s 3.2.2.1 Instantaneous p r o f i l e method As was discussed i n the f i r s t chapter, the use of the instantaneous p r o f i l e methods f o r K and D determinations i n v o l v e s the measurement of M p r o f i l e s and/or ty p r o f i l e s on t r a n s i e n t flow systems. In the present study, only the M p r o f i l e s . were d i r e c t l y measured, whereas the ty p r o f i l e s were i n f e r r e d from s t a t i c K-ty m r e l a t i o n s h i p s obtained on separate specimens. I t was t h e r e f o r e assumed that ty was a unique f u n c t i o n of M. This assumption was not m of the type to be r e a d i l y a s c e r t a i n e d , but d i r e c t ty measurements m 5 S t r i c t l y speaking, the e r r o r w i t h which a value of Y can be p r e d i c t e d from an i n d i v i d u a l observation of X i s always somewhat greater than ^Y/X' ^ t '""s § i v e n t n e s Q u a r e root of the sum of the variance about the r e g r e s s i o n , S 2,^, and the variance of the p r e d i c t e d "mean" value of Y f o r the given value of X (Draper and Smith 1966). 102 w i t h TCP's would have brought about an even greater degree of uncer-t a i n t y . The very long response times observed during s t a t i c p o t e n t i a l measurements suggested that the use of TCP's under dynamic flow c o n d i t i o n s could l e a d to s e r i o u s e r r o r s . Furthermore, the TCP's cannot provide accurate p o t e n t i a l measurements i n the range from 0 to -1 bar. To r e a l i z e flow i n the l o n g i t u d i n a l d i r e c t i o n and through only one end of the specimen (see Section 3.1.1), a moisture b a r r i e r was formed. F i r s t , f r e s h surfaces were exposed on both ends of the specimen w i t h a sharp razor blade, and then these ends were covered w i t h a double l a y e r of aluminum f o i l . A 2-mm t h i c k neoprene gasket kept the f o i l from c l o s e contact w i t h wood at one end. This precaution was to ensure that pressure e q u a l i z a t i o n through a p i n h o l e made at the center of the end f o i l would take place f r e e l y during the flow run. The e n t i r e assembly was f i r m l y pressed between two wood p l a t e s by a C-clamp. Next the l a t e r a l faces of the specimen were f l a s h - d r i e d w i t h a h a i r dryer and coated twice w i t h Saran Resin 310 (Dow Chemical of Canada, L t d , S a r n i a , Ontario) at 15-min i n t e r v a l s . The outer edge of the aluminum f o i l was then folded against the s i d e faces and a t h i r d coat a p p l i e d . A s t r i p of aluminum f o i l was f i n a l l y put on top of the r e s i n c o a t i n g , and a l l j o i n t s sealed w i t h e l e c t r i c a l tape. P r e l i m i n a r y t e s t s revealed that t h i s moisture b a r r i e r would prevent any s i g n i f i c a n t moisture l o s s e s at a l l times during any of the flow experiments that were to be conducted. 103 Two s e r i e s of end-matched specimens (see Section 3.1.2) were teste d from the s a t u r a t e d c o n d i t i o n . One s e r i e s was subjected to drainage i n a pressure p l a t e apparatus, whereas the other was subjected to evaporative d r y i n g i n a controlled-environment chamber. Two important c o n d i t i o n s had to be met by both flow systems: 1) M had to remain high at the outflow end to avoid impeding d i f f u s i o n e f f e c t s ; and 2) the zero f l u x boundary had to be maintained at the opposite end. To c a r r y out a flow run w i t h the pressure p l a t e technique, the aluminum f o i l was f i r s t cut o f f from both ends of the specimen, which was then weighed and placed on a 15-bar pressure p l a t e (one specimen at a time). Saturated c l a y served as c a p i l l a r y contact medium. Using the apparatus shown i n F i g . 8, a pressure of 75 p s i (5.2 bars) was a p p l i e d g r a d u a l l y so that i t was f u l l y e s t a b l i s h e d a f t e r 5 min. A f t e r what appeared to be a s u i t a b l e i n t e r v a l of time (based on the measured outflow volume), the pressure was r e l e a s e d , the specimen removed from the porous p l a t e , weighed, and then cut i n t o about 5-mm t h i c k wafers f o r determining the M p r o f i l e . F o l l o w i n g the c u t t i n g o p e r a t i o n , which took about 5 min, the wafers were weighed and t h e i r t h i c k n e s s measured w i t h a c a l i p e r . F i n a l l y , the wafers were oven-dried at 103°C. The above procedure was repeated f o r each of the r e p l i c a t e s of the s e r i e s t e s t e d . Since one specimen of t h i s s e r i e s served f o r saturated c o n d u c t i v i t y determinations, there were not enough remaining specimens to use one of them f o r d i r e c t measurement of the i n i t i a l M p r o f i l e . The i n i t i a l M d i s t r i b u t i o n was then assumed to be uniform 104 and to correspond to the average estimated i n i t i a l M of the r e p l i c a t e s . The wafers were not cut w i t h a micro-saw as planned because of unforseen l a s t minute t e c h n i c a l d i f f i c u l t i e s . The use of a back saw appeared the most s u i t a b l e a l t e r n a t i v e i n the c o n d i t i o n s then p r e v a i l i n g . With the a i d of a m i t r e box type of specimen holder, wafers of f a i r l y uniform thickness were obtained. The a c t u a l thickness was measured w i t h an estimated e r r o r l e s s than ±0.1 mm. Despite some shortcomings, t h i s c u t t i n g technique had the b i g advantage of causing very l i t t l e h e a t i n g to the wood during the c u t t i n g process, hence very l i t t l e change i n the a c t u a l M d i s t r i b u t i o n . For the flow measurements by the evaporative d r y i n g technique, the f o l l o w i n g procedure was adopted. While one r e p l i c a t e was kept a s i d e to determine the i n i t i a l M p r o f i l e , a l l the others were put at one time i n an Aminco c o n d i t i o n i n g chamber f o r slow d r y i n g at 21°C (dry-bulk temperature) and 68% RH. The dry-bulb and wet-bulb temperatures were c o n t r o l l e d at ±0.1°C. The specimens were placed i n the u p r i g h t p o s i t i o n w i t h the upper end f u l l y exposed to the d r y i n g a i r . The lower end was sealed at i t s outer edge on top of a small vented p l a s t i c box c o n t a i n i n g water i n the bottom. This p r e c a u t i o n was to ensure that no evaporative moisture l o s s e s would occur through the p i n h o l e vent made i n the end aluminum f o i l . A f t e r various time i n t e r v a l s w i t h i n the constant d r y i n g r a t e p e r i o d , a specimen was removed from the t e s t chamber and the M p r o f i l e determined by the procedure described above. Now regarding the flow measurements i n i m b i b i t i o n , they were conducted on the two other s e l e c t e d s e r i e s of end-matched specimens w i t h the tension p l a t e apparatus shown i n F i g . 9. I n i t i a l l y a l l the specimens of a given s e r i e s , except f o r the r e p l i c a t e reserved f o r the measurement of the i n i t i a l M p r o f i l e , were placed simultaneously on the porous p l a t e subjected to a tension of about 3 cmH^O. The top part of the tension p l a t e c e l l was then secured i n place. A f t e r v a r i o u s time i n t e r v a l s , t h i s cover was t e m p o r a r i l y removed and a specimen withdrawn from the porous p l a t e f o r determining the M p r o f i l e . Weights of 250 g were put on top of the specimens to maintain b e t t e r h y d r a u l i c contact between the bottom face and the porous p l a t e . F r e s h l y d i s t i l l e d , f i l t e r e d and deaerated water was used f o r each flow run. The c a l c u l a t e d M of each wafer was taken to represent the M at the midpoint of i t s t h i c k n e s s . The absolute e r r o r | A M | a s s o c i a t e d w i t h t h i s M determination was estimated from the propagation of e r r o r formula given e a r l i e r . The e r r o r c a l c u l a t i o n s were based on the gains and l o s s e s of moisture caused by handling and a i r exposure during the weighing process (-0.015 g), and on the average oven-dry mass of the wafers (-3.3 g) . The r e s u l t s thus obtained i n d i c a t e d that the c a l c u -l a t e d wafer M was not i n e r r o r by more than 1% A M at 30% M and 2% A M at f u l l s a t u r a t i o n . Once again, these systematic e r r o r s tended mostly toward the negative s i d e and were probably much greater than any of the random e r r o r s . To c a l c u l a t e K and D from the experimental data, an appropriate number of moisture content p r o f i l e s ( M ( z ) ^ ) and matric p o t e n t i a l p r o f i l e s ( i p m ( z ) ^ ) were f i r s t obtained. An i n t e r p o l a t i o n procedure was used f o r 106 t h i s purpose. F i r s t f o r each time of flow r e a l i z e d , the average M of each wafer was p l o t t e d on a l a r g e s c a l e graph as a f u n c t i o n of the wafer p o s i t i o n i n the specimen. The data points were then j o i n e d by s t r a i g h t l i n e segments. From t h i s rough graph, smooth M(t) curves were constructed by p l o t t i n g M as a f u n c t i o n of t f o r each of the t h i r t e e n wafer average p o s i t i o n s z. By s e l e c t i n g next appropriate time i n t e r v a l s on t h i s l a t t e r l a r g e s c a l e M ( t ) z graph, as many M ( z ) t curves as needed f o r the c a l c u l a t i o n of K and D were der i v e d . From these smoothed M(z) curves, an equivalent number of ty (z) curves were t m t obtained by s u b s t i t u t i n g the M values by t h e i r corresponding ty values i n f e r r e d from a s t a t i c M-ty^ r e l a t i o n s h i p measured on specimens of the same source. Since the flow runs i n i m b i b i t i o n were s t a r t e d from a drainage s t a t e (==30% M), the use of a boundary i m b i b i t i o n M-ty curve to derive m the ty p r o f i l e s was not q u i t e c o r r e c t . However, the e r r o r s thus m produced were c e r t a i n l y very small due to the small h y s t e r e s i s i n the M-ii) r e l a t i o n s h i p at low moisture contents. On the other hand, the m component p o t e n t i a l ty was not taken i n t o account i n the p l o t t i n g of g the p o t e n t i a l p r o f i l e s because the ty gradients were found n e g l i g i b l e compared to the ty gradients i n the M range covered by the c a l c u l a t i o n s . The unsaturated water c o n d u c t i v i t y K was determined by g r a p h i c a l a n a l y s i s of the M and ij; p r o f i l e s by applying the general flow equation as expressed by Eq. [64]. This r e l a t i o n i s r e w r i t t e n here i n the form used f o r the c a l c u l a t i o n s 107 K(M) = — [67] z,t I Sty \ m 6z / z,t where K(M) i s the unsaturated c o n d u c t i v i t y at M f o r the p o s i t i o n z, t time of flow t i n cm 3(H^O)cm/dyn s, M the moisture content i n the matric p o t e n t i a l gradient at z and t i n dyn/cm2cm, and l 6 Z lz,t G the green s p e c i f i c g r a v i t y (based on the green volume). The reason f o r using the f a c t o r G /100 i n Eq. [67] was to obta i n K i n proper u n i t s while M was expressed i n % (by weight). The value assigned to G was the average value c a l c u l a t e d f o r the wood of the source i n question. The reference l e v e l z = 0 was set at the specimen end through which water was e n t e r i n g or l e a v i n g the specimen. The plane of s f z "zero f l u x " z _ was set at the opposite end. The f l u x -7— | M dz q=0 ot 4 J z q = 0 through the imaginary plane z at t was determined by p l o t t i n g the f Z i n t e g r a l I M dz as a f u n c t i o n of time and measuring the slope of the smoothed curve at t . The i n t e g r a l was evaluated by g r a p h i c a l i n t e g r a t i o n ( g r i d method) of the area defined by the M p r o f i l e s at t = 0 and t , and the planes z _ and z. For the matric p o t e n t i a l 6*m) ' - 7 — , i t was estimated from the smoothed curve of Sty /Sz , oz m I / z,t vs. t at z. This procedure improved the accuracy of the c a l c u l a t e d gradient 108 p o t e n t i a l gradient values, and permitted the c a l c u l a t i o n of K at any predetermined value of M. A p l o t of M vs. t at z was used to deduce the times of flow at which K was to be c a l c u l a t e d . For the conduc-t i v i t i e s K(ty ) _ , they were simply i n f e r r e d from the c o n d u c t i v i t i e s m z, t J r J K(M) by using a p l o t of iii vs. t at z. z, t m The water d i f f u s i v i t y D was obtained from the f o l l o w i n g approximation M dz D(M) z,t [68] where D(M) i s the water d i f f u s i v i t y at M f o r the p o s i t i o n z and Z, £ time of flow t i n cm 2/s, M the moisture content i n %, and 6z the moisture content gradient at z and t i n %/cm. z,t The f l u x -jj--; I M dz was the same as that used f o r K c a l c u l a -t i o n s . The moisture ^ ^content gradient the smoothed curve of 6M/6z VS. t at Z . i vSz . I / z,t was assessed from Since the g r a v i t y e f f e c t was not taken i n t o c o n s i d e r a t i o n i n e i t h e r the c a l c u l a t i o n of K or D, and sin c e the ty p r o f i l e s were i n f e r r e d m from the M p r o f i l e s , i t i s evident that K(M) could simply have been i n f e r r e d from D(M) by usin g Eq. [ 5 9 ] . The reasons f o r p l o t t i n g the ty m p r o f i l e s and c a l c u l a t i n g K d i r e c t l y from Eq. [67] were to increase 109 the accuracy of K determinations and to check the f l u x - p o t e n t i a l gradient r e l a t i o n s h i p . 3.2.2.2 One-step method The one-step method was used f o r K and D determinations f o r two p r i n c i p a l reasons: 1) to o b t a i n a second independent assessment of the nature of the K ( M ) , K(ip ) and D(M) f u n c t i o n s ; and 2) because m these K and D determinations could be made simultaneously w i t h the the measurement of the M-ip r e l a t i o n s h i p without i n v o l v i n g a d d i t i o n a l m experimental work, except c o l l e c t i n g i n f l o w - o u t f l o w data. One porous p l a t e c h a r a c t e r i s t i c not discussed p r e v i o u s l y i s the a c t u a l p l a t e impedance (or r e s i s t a n c e to f l o w ) . Whereas t h i s c h a r a c t e r i s t i c should a f f e c t M-ip determinations very l i t t l e , i t becomes m very c r u c i a l to the determination of K and D by the one-step method. Porous p l a t e saturated c o n d u c t i v i t y measurements were t h e r e f o r e conducted i n order to ensure that the K and D data obtained by t h i s method would not be a f f e c t e d by the c o n d u c t i v i t y of the porous p l a t e i t s e l f . The saturated c o n d u c t i v i t y of the ceramic p l a t e s ranged from 0.2 x 1 0 - 1 0 ( 1 5 - b a r pressure p l a t e ) to 8 x 1 0 ~ 1 0 (1-bar pressure p l a t e ) cm 3 (R^O^m/dyn s. These values are i n very good agreement w i t h those reported by Richards (1965). The saturated c o n d u c t i v i t y of the porous p l a t e used w i t h the t e n s i o n p l a t e apparatus of F i g . 9 was about 8 x 10-• cm 3(H20)cm/dyn s. The procedure f o r computing K and D from the porous p l a t e i n f l o w - o u t f l o w data was simple. I n i t i a l l y , the t o t a l volume of water 110 absorbed or removed during a given t e s t run was determined by sub-t r a c t i n g the specimen masses measured before and a f t e r the run. Using next the i n f l o w or outflow data, the average cumulative i n f l o w or outflow volume was c a l c u l a t e d backwards i n time beginning at the end of the run. This was done f o r only the second h a l f of the t o t a l M change i n which range Eq. [62] would best apply (see Section 1.2.2.2). Based on t h i s cumulative i n f l o w or outflow volume, the average M (%) of the specimens a f t e r v a r i o u s times of flow t was c a l c u l a t e d . These values were then p l o t t e d against t and a smooth curve was constructed. The slopes dM/dt obtained from t h i s l a r g e s c a l e graph were i n turn p l o t t e d against M. The values of dM/dt defined by the l a t t e r curve were then used i n Eq. [62] to c a l c u l a t e D(M). The c a l c u l a t i o n of K(M) and K(ty ) r e a d i l y followed using Eq. [59] and the appropriate K-ty m m r e l a t i o n s h i p . 3.2.3 Measurement of the saturated water c o n d u c t i v i t y Because n e i t h e r the instantaneous p r o f i l e method nor the one-step method were apt to provide K data that could be r e a d i l y e x t r a p o l a t e d to f u l l s a t u r a t i o n (or to zero ty ), saturated c o n d u c t i v i t y m measurements were c a r r i e d out s e p a r a t e l y . The setup devised f o r saturated c o n d u c t i v i t y measurements i s shown i n F i g . 14. I t i s composed of a constant-head water r e s e r v o i r , a c o n d u c t i v i t y c e l l , a hanging water column te r m i n a t i n g w i t h an overflow device, a beaker and a top l o a d i n g balance. The c o n d u c t i v i t y c e l l made of p l e x i g l a s c o n s i s t e d of two i d e n t i c a l h a l f - c e l l s 2.5 cm deep, 3.7 by 3.7 cm i n s i d e cross s e c t i o n , provided w i t h a 1/4-in. brass f i t t i n g I l l CONSTANT- HEAD WATER RESERVOIR WOOD SPECIMEN CLAMPING BOLT NEOPRENE GASKET WATER COLUMN TUBING CLAMP OVERFLOW DEVICE BALANCE F i g . 14. — Schematic diagram of the setup used f o r saturated water c o n d u c t i v i t y measurements. o u t l e t . The balance had a maximum ca p a c i t y of 1200 g and was accurate w i t h i n ±0.1 g. The setup was i n s t a l l e d i n a constant-temperature room at 21 ± 0.5°C. To prepare the cell-specimen assembly f o r a c o n d u c t i v i t y measurement, both ends of the t e s t specimen (see Secti o n 3.1.1) were 112 trimmed w i t h a sharp razor blade, then the specimen was placed between the two h a l f - c e l l s and the assembly f i r m l y clamped with the a i d of four brass b o l t s and wing nuts. A neoprene gasket was used to e s t a b l i s h the water s e a l between the c e l l and the outer edge of the specimen ends. Once both halves of the c e l l had been f i l l e d w i t h water, the l a t e r a l faces of the specimen were f l a s h - d r i e d w i t h a h a i r dryer and s e v e r a l coats of Saran Resin 310 were a p p l i e d . F i n a l l y , a t h i c k coat of white s i l i c o n e rubber sealant was put on top of the r e s i n and a l l j o i n t s of the mounting to ensure a t i g h t s e a l to a i r and water. To conduct a measurement, a few drops of water were f i r s t allowed to d r i p from the overflow device, then the balance was q u i c k l y readjusted to a convenient reference number, and time zero was noted. The flow readings were taken at 30-s i n t e r v a l s f o r the f i r s t 10 min of flow and at 5-min i n t e r v a l s subsequently. The d u r a t i o n of a measurement run was 1 h. The above procedure was repeated on four specimens, two of which were 2.0 cm long and the two others 5.0 cm long. A constant p o t e n t i a l d i f f e r e n c e ( F i g . 14) of 118 cmH^O (0.115 bar) was maintained between the ends of the specimens throughout the d u r a t i o n of the experiments. Fr e s h l y d i s t i l l e d , f i l t e r e d and deaerated water was used f o r the water c o n d u c t i v i t y measurements. The c a l c u l a t i o n of K was based on Eq. [45]. 3.2.4 Measurement of the s p e c i f i c g r a v i t y S p e c i f i c g r a v i t y determinations were made on a l l the specimens used f o r the measurement of the M-ip r e l a t i o n s h i p . For convenience, 113 only the oven-dry s p e c i f i c g r a v i t y G q was d i r e c t l y measured. The green s p e c i f i c g r a v i t y G was simply derived from the measured values of G . g o The water immersion method was used to o b t a i n G . For immersion o i n the water, the oven-dry specimen was dipped i n hot p a r a f f i n wax and placed i n s i d e a wire cage attached to a balance accurate w i t h i n ±0.1 g. The r e l a t i v e e r r o r of a s i n g l e G q determination was estimated to be l e s s than 1.0%. The green s p e c i f i c g r a v i t y was estimated from the r e l a t i o n Gg = G q ( 1 - $ v ) where gy i s the t o t a l volumetric shrinkage (Kollmann and Cote 1968) . The t o t a l volumetric shrinkage was c a l c u l a t e d from the approximation 3 = g + 3 - g g where g_, and g_ are the V t R t T t R t T t R t T t t o t a l l i n e a r shrinkages i n the r a d i a l and t a n g e n t i a l d i r e c t i o n s , r e s p e c t i v e l y . The values of g and g were based on the r a d i a l and R t T t t a n g e n t i a l dimensions of the specimens at the green s t a t e and a f t e r oven-drying. The r e l a t i v e e r r o r of an i n d i v i d u a l G determination was estimated to be l e s s than 2.0%. Chapter 4 RESULTS AND DISCUSSION 4.1 M-iii r e l a t i o n s h i p at high moisture contents rm 4.1.1 Results of the porous p l a t e experiments Results of the M-ip^ determinations by the porous p l a t e methods are summarized i n Table A3.1 (Appendix 3). The M values i n the l a s t column on the r i g h t represent group means, and those i n the preceding next four columns represent subgroup means. For a n a l y s i s purposes, part of the M-ip data presented i n m Table A3.1 are reproduced g r a p h i c a l l y i n F i g s . 15 through 18. Figure 15 shows the boundary drainage and i m b i b i t i o n curves f o r a l l specimens. Figure 16 e x h i b i t s the boundary drainage and i m b i b i t i o n curves f o r Tree I at the heights of 0.6 m ( t r i a n g l e s ) and 9.6 m (squares). Figure 17 d i s p l a y s the drainage curve s t a r t i n g from the green c o n d i t i o n f o r a l l specimens, and F i g . 18 shows the corresponding curves f o r Tree I at the heights of 0.6 m ( F i g . 18a) and 9.6 m ( F i g . 18b). A l s o drawn as p a i r s of s o l i d l i n e s i n F i g s . 17 and 18 i s the main h y s t e r e s i s loop f o r 114 115 F i g . 15. — Average moisture content-matric p o t e n t i a l r e l a t i o n s h i p of western hemlock sapwood along the boundary drainage and i m b i b i t i o n curves at 21°C ( L e f t p o r t i o n of broken l i n e s from r e s u l t s of Spalt 1958) . 116 F i g . 16. — Moisture content-matric p o t e n t i a l r e l a t i o n s h i p along the boundary drainage and i m b i b i t i o n curves at 21°C, f o r Tree I at the heights of 0.6 and 9.6 m ( L e f t p o r t i o n of broken l i n e s from r e s u l t s of Spalt 1958) . 117 F i g . 17. — Average moisture content-matric p o t e n t i a l r e l a t i o n s h i p of western hemlock sapwood along the drainage curve s t a r t i n g from the green c o n d i t i o n at 21°C, shown i n s i d e the main h y s t e r e s i s loop ( L e f t p o r t i o n of broken l i n e s from r e s u l t s of Spalt 1958) . 118 I I I I I I 11 I N | I I I |lll I | I I I [III I | l l I [III I l l l l flTTTT ILU*II "7~ I 0 4 - 1 0 * - 1 0 ' - 1 0 ° - 1 0 " M A T R I C P O T E N T I A L , <|/ (bar s ) 1 0 " j l l l l I | I I I [Ml I | I I I fTTTTf lu-H-rrT"! i - L L J I l l l l l l l I I l l l l l l l M i l l I I I l u i I I -\0* - 1 0 * - 1 0 ' - 1 0 " - 1 0 " - 1 0 " - 1 0 - I 0 " 3 M A T R I C P O T E N T I A L , * (bar s ) 18. — Moisture content-matric p o t e n t i a l r e l a t i o n s h i p along the drainage curve s t a r t i n g from the green c o n d i t i o n at 21°C, shown i n s i d e the main h y s t e r e s i s loop: a) Tree I , 0.6-m height; b) Tree I , 9.6-m height ( L e f t p o r t i o n of broken l i n e s from r e s u l t s of Spalt 1958). the wood of the source i n question. Results f o r Tree I I were not p l o t t e d s e p a r a t e l y because very l i t t l e d i f f e r e n c e was found i n the nature of the corresponding M-ijj curves between the two t r e e s . m The broken l i n e s i n Figs. 15 through 18 are e x t r a p o l a t i o n s of the M-ip curves beyond the experimentally determined data p o i n t s . The m p o r t i o n of these broken l i n e s i n the ty range from -5 x 10 2 to -1 x 101* m bars was i n f e r r e d from s o r p t i o n data published by Spalt (1958) f o r western hemlock at 32°C. The RH values were converted i n terms of ty m values at 21°C w i t h the a i d of Eq. [25]. A temperature c o r r e c t i o n f a c t o r v a r y i n g from +0.03 to +0.1% AM per degree C e l s i u s d i f f e r e n c e from 21°C was a p p l i e d to the o r i g i n a l M data. This f a c t o r was derived from a comparison of s o r p t i o n isotherms obtained on s i m i l a r wood species at the r e s p e c t i v e temperatures of 21°C (Hedlin 1967) and 32°C (Spalt 1958). A l l curves were hand f i t t e d to the data p o i n t s . Assuming that ty was known without e r r o r , the u n c e r t a i n t y a s s o c i a t e d w i t h the K-ty m m curves presented above can be v i s u a l i z e d through the standard e r r o r s (SE) of the M data given i n Table A3.1. These SE values are l i s t e d i n Table A3.2 along w i t h those of the standard d e v i a t i o n (SD). One must poi n t out here that a t o t a l of f i v e observations were r e j e c t e d from the groups on the drainage curves based on a t e s t f o r r e j e c t i o n of extreme observations (Snedecor and Cochran 1967). The presence of abnormal wood as revealed by extremely high d e n s i t y values caused the r e j e c t i o n of seven other observations from groups on the i m b i b i t i o n curve. In a d d i t i o n , two subgroups were incomplete due to a l a c k of m a t e r i a l . Table A3.2 shows that the SE values f o r the group means are reasonably small as they vary from 1.1 to 2.9% AM on the drainage curve, and from 0.1 to 2.7% AM on the i m b i b i t i o n curve. These f i g u r e s were based on the mean square e r r o r (mean square w i t h i n subgroups) obtained from an a n a l y s i s of v a r i a n c e . This method of computing SE was j u s t i f i e d by the type of sampling procedure used f o r the M-ip determinations m (see S e c t i o n 3.1.2). The f a c t that i n such case the v a r i a b i l i t y between subgroups does not c o n t r i b u t e to the e r r o r of the group mean r e s u l t s i n a s u b s t a n t i a l g ain i n p r e c i s i o n (Snedecor and Cochran 1967). Indeed, the mean square e r r o r between subgroups was from three to ten times greater than that w i t h i n subgroups. Based on the magnitude of the SE values f o r the group means, i t can e a s i l y be shown that the average M - ,J J m curves of F i g . 15 are f a i r l y good f i t s . This i s p a r t i c u l a r l y true because the slope of the M-ip^ curves i s very steep over most of the ty range covered by the data p o i n t s . A higher degree of u n c e r t a i n t y i s o b v i o u s l y a s s o c i a t e d w i t h the M-ty curves of F i g . 16. The reason i s t h a t , unless s t a t e d m otherwise (Table A3.2), the M means i n the l a t t e r case were based on three observations i n s t e a d of twelve. Regarding the drainage curves of F i g s . 17 and 18, the high degree of s c a t t e r of the data points may throw some doubts on t h e i r r e l i a b i l i t y . An explanation w i l l be given l a t e r i n t h i s chapter to j u s t i f y the shape of these curves. As f a r as the systematic e r r o r s a s s o c i a t e d w i t h the g r a v i m e t r i c determination of M (see Section 3.2.1.1) are concerned, they were undoubtedly too small to a f f e c t s i g n i f i c a n t l y e i t h e r the shape of the H-ty curves or the s i z e of m the h y s t e r e s i s loop. 121 The s e l e c t i o n of the specimens at p o s i t i o n s 120° apart around the sapwood zone c e r t a i n l y c o n t r i b u t e d to minimize the b i a s i n the M-iJ; m data. The v a r i a b i l i t y measured w i t h i n subgroups was l i k e l y very c l o s e to the true c i r c u m f e r e n t i a l v a r i a b i l i t y of the t r e e s . In t h i s respect, i t can be shown from the SD values of Table A3.2 that the c o e f f i c i e n t of v a r i a t i o n of the M data w i t h i n subgroups was from two to three times greater on average than that claimed e a r l i e r f o r end-matched specimens (see S e c t i o n 3.1.2). Proceeding now w i t h the p h y s i c a l a n a l y s i s of the r e s u l t s obtained, i t i s of i n t e r e s t to compare f i r s t the average K-ty m r e l a t i o n s h i p shown i n F i g . 15 w i t h the r e s u l t s of Penner (1963) reported i n F i g . 1. A number of s i m i l a r i t i e s are brought out by t h i s comparison. For example, both f i g u r e s show the presence of a l a r g e h y s t e r e s i s i n the M-iJ>m r e l a t i o n s h i p at high M's or high ty^1 s. Both diagrams a l s o support the c l o s i n g of the h y s t e r e s i s loop at zero ty . m The small remaining gap between the drainage and i m b i b i t i o n curves near zero ty^ i n F i g . 15 would be due i n great part to the e r r o r made i n determining the imposed ty value of -0.0015 bar. However, one cannot m r u l e out the p o s s i b i l i t y of a f a i l u r e to a t t a i n true e q u i l i b r i u m caused by the phenomenon of a i r entrapment. The above comparison a l s o r e v e a l s a few important d i f f e r e n c e s . One notable d i f f e r e n c e i s the shape of the i m b i b i t i o n curve, which i s asymptotic i n F i g . 1 and sigmoid i n F i g . 15. Another important d i f f e r -ence e x i s t s i n the drainage curve. This curve i n F i g . 15 tends to e x h i b i t a p l a t e a u at intermediate moisture contents, whereas i n F i g . 1 122 the t r a n s i t i o n between high and low moisture contents i s very gradual. L a s t l y , the drainage curve of F i g . 15 e x h i b i t s a more d i s t i n c t a i r entry value (see Section 2.1.1) than that of F i g . 1. The reason f o r the above discrepancy between the i m b i b i t i o n curves of F i g s . 1 and 15 i s a matter f o r conjecture. I t appears u n l i k e l y that i t can be s o l e l y explained on the b a s i s of anatomical d i f f e r e n c e s between the species concerned. A c t u a l l y , one may express some doubts as to the r e p r e s e n t i v i t y of Penner's r e s u l t s . In general, M-iJ;^ i m b i b i t i o n curves f o r porous media other than wood tend to e x h i b i t a sigmoid shape ( P o u l o v a s s i l i s 1962, 1970a, Morrow and H a r r i s 1965, Topp and M i l l e r 1966, Mualem and Dagan 1975). Of considerable i n t e r e s t i s the tendency of the drainage curves i n F i g s . 15 through 18 to e x h i b i t a plateau at intermediate moisture contents. This i s i n c o n t r a d i c t i o n w i t h the r e s u l t s of both F i g s . 1 and 3. This becomes more apparent i f one compares the values of the h y s t e r e s i s r a t i o (see Section 2.1.1) i n the ty range from -5 to -20 m bars. I n F i g . 15 the h y s t e r e s i s r a t i o over t h i s ty range remains almost constant and below 0.40. In F i g s . 1 and 3 i t increases substan-t i a l l y and reaches 0.70 or more at -20 bar ty (98.5% RH). On the other m hand, i t can be shown on the same b a s i s that the r e s u l t s of F i g . 2 do support the existence of the above phenomenon. Other desorption data obtained by Hart et al. (1974) on i n i t i a l l y green specimens of h i c k o r y (Carya tomentosa Nutt.) seem to bear out the same c o n c l u s i o n . A number of f a c t s r e l a t e d to the present experiments a l s o tend to s u b s t a n t i a t e the above behavior of the drainage curve, which would 123 imply some.kind of a i r entry anomaly at intermediate moisture contents. F i r s t , s i m i l a r r e s u l t s were obtained from both e x t r a c t i o n runs conducted at the imposed ty of about =14 bars. Second, the data p o i n t at -7 bar ty a l s o suggests a break i n the drainage curve. This i s p a r t i c u l a r l y apparent i n F i g . 16. L a s t l y , the i n d i v i d u a l drainage curves f o r subgroups a l l e x h i b i t the behavior i n question and yet at very d i f f e r e n t moisture contents. As w i l l be seen l a t e r from the r e s u l t s of psychrometric ty Tm determinations, no absolute proof was found as to the attainment of true e q u i l i b r i u m during the e x t r a c t i o n runs at about =14 bar ty . I t m i s , however, very u n l i k e l y that d e v i a t i o n s from e q u i l i b r i u m of as much as +50% AM (see F i g s . 1 and 16) could have occurred. Although the s a t u r a t i o n percentage of the clay used as contact medium had dropped to about 50% at the end of the 14-bar e x t r a c t i o n runs, the p o s s i b i l i t y of a complete water c o n t i n u i t y breakdown i n the c l a y appears to be r u l e d out according to the outflow data f o r the runs i n question. whether the above feature of the drainage curve i s r e a l or not, i t has at l e a s t one important i m p l i c a t i o n . Indeed, i t would represent the p r a c t i c a l l i m i t of e f f i c i e n c y of moisture removal from wood by mechanical processes. Any f u r t h e r appreciable r e d u c t i o n i n moisture content would r e q u i r e extremely high pressures and long periods of time. This would apply to the c e n t r i f u g e method as w e l l as to the pressure p l a t e or pressure membrane methods. Concerning the s c a t t e r of the data p o i n t s noted e a r l i e r f o r the drainage curves s t a r t i n g from the green c o n d i t i o n ( F i g s . 17 and 18), r e s u l t s of p r e l i m i n a r y experiments conducted on s i m i l a r specimens i n d i c a t e that M measured at the imposed ill of -2.87 bars i s by f a r too m high. This i s s e l f - e v i d e n t i n F i g . 18. This may be due i n part to the heavy d i s c o l o r a t i o n s observed on the outflow face of the specimens concerned at the end of the run. The same anomaly was observed on the specimens e x t r a c t e d under the pressure of 1.95 bars. Changes i n the p r o p e r t i e s of wood c a p i l l a r y system or other unknown e f f e c t s would then have caused the e q u i l i b r i u m s t a t e to be reached at a higher M than normally. Each drainage curve s t a r t i n g from the green c o n d i t i o n i n F i g s . 17 and 18 would represent only one example among the multitude of p o s s i b l e scanning curves i n s i d e the main h y s t e r e s i s loop. I t may a l s o be expected that numerous scanning drainage curves could be measured from the green c o n d i t i o n alone due to the seasonal v a r i a t i o n of M i n standing trees (Linzon 1969). This may have important p r a c t i c a l i m p l i c a t i o n s . For example, depending on the green M i n the t r e e s at the time of f e l l i n g , and depending whether M before f e l l i n g i s toward an i n c r e a s i n g or a decreasing trend, the e f f e c t i v e n e s s of a given drainage or d r y i n g process may be q u i t e d i f f e r e n t . I t i s a l s o evident that the a p p l i c a b i l i t y of data obtained on i n i t i a l l y green wood i s f a i r l y r e s t r i c t e d . As i l l u s t r a t e d by F i g s . 17 and 18, scanning and boundary drainage curves merge very slowly i n t o one another. A s i m i l a r behavior of the scanning drainage curves has been reported f o r other porous media as w e l l ( P o u l o v a s s i l i s 1962, 1970a, Morrow and H a r r i s 1965, Topp and M i l l e r 1966, Topp 1971b, Mualem and Dagan 1975). 125 An interesting fact revealed by Table A3.1 is the effect of height on the M~ m^ data, particularly in the case of the boundary drainage curve. This i s also well illustrated by Fig. 16. The change of density with height (Table 1) appears to be the most plausible cause of this effect. It should be noted that the choice of the index M as indicator of the degree of wetness in wood contributed to amplify the above effect. In order to demonstrate that, the M-ip data of Table A3.1 were converted m in terms of S -\b data. Above 30% M, S was obtained by merely dividing p rm p J J b the measured value of M by the maximum moisture content M of the max specimen in question. For the i n i t i a l l y saturated specimens, the M max values were obtained directly from the formula M max G G - 1 0 0 ' V g ws/ where G is the specific gravity based on green volume and G the g . ws specific gravity of the wood substance (Kollmann and Cote 1968). A value of 1.53 was used for G . This figure was derived from the ws b i n i t i a l l y saturated specimens using the above formula. Below 30% M, the approximation S P max ~ KV '"X M t M/(I4 - 30 3TR /B_7 ) -100, where 3TT /3TT is V ' "V M t the ratio of the volumetric shrinkage at M to the total volumetric shrinkage, was used. The coefficient 30 i s taken to represent M at which shrinkage starts. The subgroup and group means computed from the obtained individual values of S^  are presented in Table A3.3 (Appendix 3). It can be seen from Table A3.3 that the substitution of M by 5^ decreased considerably the vari a b i l i t y between heights in the case of the boundary drainage curve. On the other hand, this had relatively 126 l i t t l e e f f e c t on the v a r i a b i l i t y between heights i n the case of the boundary i m b i b i t i o n curve. This can be best appreciated by comparing the M-ty curves of F i g . 16 to the corresponding S -ty curves shown i n m p m F i g . 19 below. The above c o n s i d e r a t i o n s c l e a r l y demonstrate that the S -ty p m boundary drainage curve f o r woods of s i m i l a r anatomy i s more s t a b l e than the corresponding M-ip curve. This i s r e a d i l y e x p l a i n a b l e . Indeed, m whenever there i s a change i n de n s i t y there i s a change i n M which, to b max -I04 -I03 -I02 -10' -10° -10"' -ICT2 -I0"3 M A T R I C P O T E N T I A L , * (bars) F i g . 19. — S a t u r a t i o n percentage-matric p o t e n t i a l r e l a t i o n s h i p along the boundary drainage and i m b i b i t i o n curves at 21°C, f o r Tree I at the heights of 0.6 and 9.6 m (L e f t p o r t i o n of broken l i n e s from r e s u l t s of Spalt 1958). 127 i n t u r n , w i l l a l t e r to some extent the shape of the M-ip boundary drainage m curve. The same change i n M w i l l a l s o a f f e c t the shape of the S -ty max p m boundary drainage curve but to a much l e s s e r degree as can be deduced from the d e f i n i t i o n of S above. This i s p a r t i c u l a r l y true i n the S P P range from 50 to 100%. Of course, the foregoing argument does not apply i n the case of a d r y i n g or drainage curve s t a r t i n g from the green c o n d i t i o n since the green M i s not n e c e s s a r i l y r e l a t e d to M . The choice between to max the K-ty and S -ty r e l a t i o n s h i p s becomes then i r r e l e v a n t . This i s evident m p m from Tables A3.1 and A3.3. The v a r i a b i l i t y between he i g h t s and t r e e s f o r the M-ty and m Sp"~^m data obtained by the porous p l a t e methods can a l s o be appreciated by examining the F values presented i n Table A3.4 (Appendix 3). These F values s u b s t a n t i a t e a l l the previous conclusions drawn on the subject. 4.1.2 Results of the TCP experiments R e s u l t s of the ty determinations by the thermocouple psychrometer method are summarized i n Table A3.5 (Appendix 3). The ty values measured by the psychrometers are shown i n comparison w i t h those imposed to the porous p l a t e s during e q u i l i b r a t i o n of the specimens under c o n s i d e r a t i o n . Subgroup means were not computed since a l l the specimens of a given group were supposed to have the same ty value. Since the TCP method i s m poorly accurate between 0 and -1 bar (see Section 3.2.1.2), no ty meas-m urements were c a r r i e d out on the specimens e q u i l i b r a t e d under an imposed ty greater than -1 bar. There were two exceptions i n i m b i b i t i o n i n which 128 cases i t was f e l t important to v e r i f y at l e a s t the order of magnitude of ty . From the two sets of ty values derived from the TCP emf outputs, m Tm one was based on the TCP s e n s i t i v i t i e s (TCP output conversion Method 1) and the other was based on the l i n e a r r e g r e s s i o n equations f i t t e d to the c a l i b r a t i o n data (TCP output conversion Method 2). Taking i n t o account the accuracy of the TCP method, Table A3.5 shows a f a i r l y good agreement between measured and imposed ty values m f o r the groups e q u i l i b r a t e d between -1 and -7 bar ty^ i n drainage. This i s p a r t i c u l a r l y true i n the case of the TCP data based on conversion Method 2. On the other hand, there e x i s t s a considerable discrepancy between imposed and measured ^ values f o r the groups e q u i l i b r a t e d a t about -14 bar ijj i n drainage and two other groups i n m i m b i b i t i o n . This i s regardless of the conversion method used. The f a c t that conversion Method 1 apparently produced a greater b i a s than conversion Method 2 at high p o t e n t i a l s i s not s u r p r i s i n g . This was l i k e l y the r e s u l t of f o r c i n g the psychrometer c a l i b r a t i o n curves through the o r i g i n i n the determination of t h e i r s e n s i t i v i t i e s . The SD and SE values of the TCP data given i n Table A3.5 are reported i n Table A3.6 (Appendix 3). P a r t i c u l a r l y i n t r i g u i n g at f i r s t glance i s the magnitude of the SD values which are up to s e v e r a l times greater than the c a l i b r a t i o n accuracy of the TCP's (see Section 3.2.1.2). This i n d i c a t e s that the r e p r o d u c i b i l i t y of the TCP method was r a t h e r poor. I t i s p o s s i b l e , however, that some v a r i a t i o n e x i s t e d i n the a c t u a l ty values of- the t e s t specimens themselves. Assuming that M measured m at the removal of the specimens from the porous p l a t e s was w i t h i n ±5% of the true e q u i l i b r i u m M (see Section 3.1.2), the amount of v a r i a t i o n i n ty a f t e r the porous p l a t e t e s t s can be estimated by m u l t i p l y i n g the average M measured at any given imposed ty by 0.05, and by the slope dty /dM at ty . In the case of the drainage curve shown i n F i g . 15, f o r m m example, the values thus obtained f o r ty equal to -1, -3 and -7 bars are m ±0.24, ±0.36 and ±3.0 bars, r e s p e c t i v e l y . The corresponding SD values i n Table A3.6 are on average 0.64, 1.2 and 3.6 bars, r e s p e c t i v e l y . I t t h e r e f o r e appears that the poor r e p r o d u c i b i l i t y of the TCP method, as f a r as the drainage s t a t e i s concerned, d i d not r e s u l t from the v a r i a t i o n that may have e x i s t e d i n the a c t u a l ty values of the m specimens at the end of the e x t r a c t i o n runs, except perhaps at ^ m ' s lower than -7 bars. On the other hand, the long e q u i l i b r a t i o n times recorded during the psychrometric ty determinations i n wood (see Section 3.2.1.2) suggest that the c o n d i t i o n s of measurement d i d not match pro p e r l y the c o n d i t i o n s p r e v a i l i n g during the c a l i b r a t i o n of the TCP's. Regarding the l a r g e discrepancy that was recorded between the TCP data and the imposed ty values of -13.8 and -13.9 bars, there are m at l e a s t two p o s s i b l e explanations. One of these i s a f a i l u r e to a t t a i n true e q u i l i b r i u m on the porous p l a t e s during water e x t r a c t i o n . According to F i g s . 15 and 17, a d e v i a t i o n of about +5% AM from true e q u i l i b r i u m would have r e s u l t e d i n ty values comparable to the TCP data. The other m p o s s i b l e e x p l a n a t i o n i s a change of the e q u i l i b r i u m s t a t e during the r e l e a s e of the pressure from the e x t r a c t i o n chamber at the end of the drainage runs i n question (see Section 2.1.3.3). 130 The discrepancy between the TCP data and the imposed ty values m of -0.499 and -0.994 bar i n i m b i b i t i o n can be a t t r i b u t e d to s e v e r a l causes. For example, the f l a t n e s s of the M-ip boundary i m b i b i t i o n curve m i n the ty range from -1 to -50 bars suggests that the s l i g h t e s t d e v i a t i o n from true e q u i l i b r i u m on the porous p l a t e s would have produced a con-s i d e r a b l e d i f f e r e n c e i n the a c t u a l ty obtained. The moisture l o s s e s m s u f f e r e d during the psychrometric ty determinations (-0.6% AM) could m a l s o e x p l a i n i n part the observed discrepancy. F i n a l l y , poor h y d r a u l i c contact between TCP ceramic cup and wood may have prevented the establishement of e q u i l i b r i u m across the cup w a l l at t h i s low moisture content. Despite the l a r g e e r r o r a s s o c i a t e d w i t h the ty value measured m on the group e q u i l i b r a t e d under an imposed ty of -0.0912 bar i n i m b i -m b i t i o n , t h i s r e s u l t remains very important. Indeed, i t c o n s t i t u t e s a sound independent proof of the existence of the l a r g e h y s t e r e s i s i n the M-iLi r e l a t i o n s h i p at high M's or * 's. I t can be seen from Table rm . m A3.5 that approximately the same ty value was measured at M equal to 54% i n i m b i b i t i o n and at more than 150% i n drainage. As pointed out e a r l i e r , the use of the TCP method f o r ty m determinations was based on the assumption that ty was n e g l i g i b l e . In order to check t h i s assumption, the magnitude of ty^ was estimated w i t h the a i d of Eq. [32]. The p o s s i b l e c o n c e n t r a t i o n of s o l u t e s present i n the t e s t specimens was estimated from data published by Barton and Gardner (1966) on the e x t r a c t i v e s of western hemlock sapwood. These e x t r a c t i v e s c o n s i s t e d mainly of l i g n a n s and other phenolic 131 substances. Monomeric leucoanthocyanidins and catachins accounted f o r about 0.05% and 0.04% r e s p e c t i v e l y of the dry mass of e x t r a c t i v e - f r e e wood. Unknown phenolic glucosides v a r i e d between 0.01 and 0.06%. Lignans accounted f o r approximately another 0.01%. I t was thus assumed f o r the c a l c u l a t i o n s that the e x t r a c t i v e content of the t e s t specimens was approximately 0.2% of the dry mass. I t was a l s o assumed that a l l these e x t r a c t i v e s were s o l u b l e i n c o l d water and had an average molecular weight of 250. This y i e l d e d a value of -0.090 bar f o r the osmotic p o t e n t i a l a t f u l l s a t u r a t i o n (-216.5% M). For M equal to 200, 100 and 30%, the c a l c u l a t e d tyQ values were -0.098, -0.195 and -0.650 bar, r e s p e c t i v e l y . The foregoing f i g u r e s i n d i c a t e t h a t , i n general, the magnitude of the component p o t e n t i a l ip Q was too s m a l l to s i g n i f i c a n t l y a f f e c t the TCP data shown i n Table A3.5. Moreover, the magnitude of these ijj^ values may have been s e v e r a l times overestimated s i n c e a hi g h percen-tage of the s o l u t e s present i n the water was probably leached from the wood specimens during e q u i l i b r a t i o n on the porous p l a t e s . Furthermore, the s o l u b i l i t y of the e x t r a c t i v e s i n question i n c o l d water i s probably not as complete as supposed. 4.1.3 H y s t e r e s i s i n wood M-ip r e l a t i o n s h i p a t hig h moisture contents vs. i n k - b o t t l e e f f e c t The experiments conducted i n t h i s study w e l l demonstrated the strong h y s t e r e t i c behavior of wood M-ip r e l a t i o n s h i p a t high moisture contents. Ignoring or o v e r l o o k i n g t h i s phenomenon may have 132 s e r i o u s consequences i n many problems d e a l i n g w i t h e q u i l i b r i u m or flow of water i n wood. Since the c a p i l l a r y system of wood c o n s i s t s of c a v i t i e s interconnected by narrow channels, i t appears reasonable to consider the i n k - b o t t l e e f f e c t (see S e c t i o n 2.1.2) as the primary cause of h y s t e r e s i s at high moisture contents. To s u b s t a n t i a t e t h i s b e l i e f , e f f e c t i v e pore s i z e d i s t r i b u t i o n s of the type shown i n F i g . 4 were der i v e d from the M-i[> r e l a t i o n s h i p of F i g . 15. The A values were m obtained from the S values of Table A3.3. The r values were c a l c u -P l a t e d from the r e l a t i o n r = 1.4518/ii) (Eq. [ 4 2 ] ) , where r i s i n um m and TII i n bars. The two A-r curves thus obtained are presented i n m F i g . 20. As i n d i c a t e d by F i g . 20, the A-r curve derived from the M - T p m boundary i m b i b i t i o n curve i s c e r t a i n l y much c l o s e r to the a c t u a l pore s i z e d i s t r i b u t i o n than that derived from the M-ip boundary drainage m curve. For example, the former d i s t r i b u t i o n suggests that i n the swollen c o n d i t i o n 13.0% of the t o t a l pore volume of western hemlock sapwood i s occupied by pores of radius smaller than 0.1 um, 1.5% by pores of radius between 0.1 and 1 ym, and 85.5% by pores of radius greater than 1 um. The corresponding values f o r the l a t t e r d i s t r i b u t i o n are 36.5%, 26.5% and 3 7 . 0 % , r e s p e c t i v e l y . In accord w i t h the theory of the i n k - b o t t l e e f f e c t these f i g u r e s show that the M-TJ J^ boundary i m b i b i t i o n curve i s governed by the s i z e of the c e l l c a v i t i e s , whereas the M~n) boundary drainage curve i s governed by the s i z e of the channels connecting the lumina. 1 3 3 F i g . 2 0 . — E f f e c t i v e cumulative pore s i z e d i s t r i b u t i o n s of western hemlock sapwood as i n f e r r e d from the average M - IJ J boundary drainage and imb i b i t i o n curves I t i s evident, however, that the lower curve i n F i g . 2 0 should be s h i f t e d somewhat toward lower r values to best represent the actual pore siz e d i s t r i b u t i o n of the wood i n question. In f a c t , the value of X at r equal to the maximum radius of the c e l l c a v i t i e s should be approaching 1 0 0 % . The maximum radius of western hemlock l o n g i t u d i n a l tracheids i s about 2 5 um (Panshin and de Zeeuw 1 9 7 0 ) . At t h i s r value on the lower curve of F i g . 2 0 , X amounts to only 3 5 % . 134 The above discrepancy may be due i n part to the use of an i n c o r r e c t r e l a t i o n s h i p between r and ty . The assumptions of zero m contact angle, c i r c u l a r c a p i l l a r i e s , and a surface t e n s i o n of the water i n wood equal to that of pure free water d i d not perhaps match qu i t e the r e a l i t y . I t i s a l s o p o s s i b l e that a c e r t a i n "delay" i n the f i l l i n g of some of the c e l l c a v i t i e s occurred due to the presence of entrapped a i r or other unknown phenomena. 4.2 K ( M ) , K(IIP ) and D ( M ) f u n c t i o n s i n the l o n g i t u d i n a l d i r e c t i o n m at high moisture contents 4.2.1 Results of the K and D t r a n s i e n t measurements T y p i c a l M and ty p r o f i l e s obtained during the measurement of rm K and D by the instantaneous p r o f i l e method are presented i n F i g s . 21 and 22 f o r i m b i b i t i o n and drainage, r e s p e c t i v e l y . The M p r o f i l e s are shown along w i t h the o r i g i n a l data p o i n t s from which they were derived. One set of data p o i n t s i s missing from both F i g . 21a and F i g . 22a. For unknown reasons, these two sets of data p o i n t s proved to be completely erroneous and were t h e r e f o r e r e j e c t e d . In i m b i b i t i o n , a l l the M d i s t r i b u t i o n s ( F i g . 21a) were ex t r a p o l a t e d to a common M at z = 0. This value of M corresponded to the ty value imposed to the m tension p l a t e (--0.003 b a r ) . L i k e w i s e , the ty d i s t r i b u t i o n s ( F i g . 21b) were e x t r a p o l a t e d to ty - -0.003 bar at z = 0. In drainage, the M p r o f i l e s ( F i g . 22a) and the ty p r o f i l e s ( F i g . 22b) were traced as m s t r a i g h t l i n e segments. The reason f o r using t h i s procedure was to ease f l u x and gradient computations. 135 F i g . 21. — T y p i c a l M and tym p r o f i l e s obtained during i m b i b i t i o n i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood specimens (Tree I I ) a) M p r o f i l e s ; b) ty p r o f i l e s . 136 a) 60 ti ORIGINAL TIME DATA POINTS OF FLOW • 0.5 h O 3 1 A 14 h a 43 h • 89 h 3 4 5 D I S T A N C E F R O M T H E O U T F L O W E N D (cm) b) D I S T A N C E F R O M T H E O U T F L O W E N D (cm) 2 3 4 5 30 t> 1 40 ' 50 h ' 60 h ' 70 h / 80 h s ' 90 h / 100 h s F i g . 22. — Typic a l M and i p m p r o f i l e s obtained during drainage or drying i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood specimens (Tree II) a) M p r o f i l e s ; b) il) p r o f i l e s , m Two planes were considered f o r f l u x and gradient c a l c u l a t i o n s i n i m b i b i t i o n , namely z = 0.6 cm and z = 2.4 cm. This permitted the determination of K and D over a broader M range. For both flow experiments conducted i n i m b i b i t i o n , the slope of the M(z)^_ curves changed s i g n near the zero f l u x end when M reached about 50%. This phenomenon was p o s s i b l y caused by the presence of entrapped a i r . The plane z = 0.25 cm was chosen f o r f l u x and gradient computa-t i o n s i n drainage. This choice was aimed at minimizing the b i a s i n the gradient estimates. The M p r o f i l e s obtained by drainage on the porous p l a t e s ( F i g . 22a) were e s s e n t i a l l y i d e n t i c a l to those obtained by evaporative d r y i n g . This would be an i n d i c a t i o n of the c l o s e resemblance between these two types of t r a n s i e n t flow processes. The shape of the M p r o f i l e s i n F i g . 22a would be charac-t e r i s t i c of a x i a l flow of water i n permeable woods. Choong et at. (1973) observed the same type of M d i s t r i b u t i o n s during l o n g i t u d i n a l d r y i n g of 5-cm long blocks of eucalypt (Eucalyptus vegnans F. Muell) heartwood. Tiemann (1952) a l s o reported a remarkable u n i f o r m i t y i n the M d i s t r i b u t i o n throughout a 30-cm long block of i n i t i a l l y wet red gum (Liqui-dambav styvaciflua L.) sapwood a f t e r eight days of slow a x i a l d r y i n g . Such moisture behavior was i n t e r p r e t e d by these authors as an i n d i c a t i o n of mass movement of l i q u i d water i n the l o n g i t u d i n a l d i r e c t i o n at high moisture contents. The shape of the M p r o f i l e s near the zero f l u x end i n F i g . 22a i s r a t h e r i n t r i g u i n g . The same behavior of the M p r o f i l e s was observed 138 on the specimens subjected to evaporative drying. The reasons for t h i s behavior are not known, but at l e a s t the p o s s i b i l i t y of moisture losses by evaporation can be eliminated. Some of the inflow-outflow data used for the determination of K and D by the one-step method are also shown below i n F i g . 23, where they appear as curves of M vs. time of flow. Each data point on the diagram represents an average for the specimen group used for the drainage or the imbibition run i n question. The l a s t data point on F i g . 23. — Moisture content as a function of time of flow i n specimens subjected to imbibition or drainage on porous plates . 139 each curve corresponds to the end of the run. In a l l , outflow data from four drainage runs and i n f l o w data from two i m b i b i t i o n runs were required to determine K and D over a s u i t a b l e M range. Two of the four outflow r a t e curves that were used f o r the c a l c u l a t i o n s are not shown i n F i g . 23 due to s c a l e problem. As can be seen i n F i g . 23, e q u i l i b r i u m was a t t a i n e d much more g r a d u a l l y i n i m b i b i t i o n than i n drainage. For example, 90% of the t o t a l M change during drainage g e n e r a l l y occurred w i t h i n the f i r s t t h i r d of the e x t r a c t i o n run. To reach the same percentage of the t o t a l M change i n i m b i b i t i o n , i t took from 40 to 60% of the run du r a t i o n . A s i m i l a r behavior of the i n f l o w - o u t f l o w r a t e curves was observed f o r a c l a y s o i l by Coleman and Marsh (1961). The K and D data computed from the types of raw data i l l u s t r a t e d i n F i g s . 21 through 23 are presented i n Tables A4.1 through A4.3 (Appendix 4 ) . Tables A4.1 and A4.2 show the K and D data obtained by the instantaneous p r o f i l e method f o r i m b i b i t i o n and drainage, respec-t i v e l y . Table A4.3 shows the K and D data obtained by the one-step method. For a n a l y s i s purposes, part of the r e s u l t s given i n Tables A4.1 through A4.3 are shown g r a p h i c a l l y i n F i g s . 24, 25 and 26 which d i s p l a y the f u n c t i o n s K(M), K(ip ) a n d D(M*), r e s p e c t i v e l y . Since the r e p l i c a t e d measurements conducted by the instantaneous p r o f i l e method y i e l d e d s i m i l a r r e s u l t s , at l e a s t q u a l i t a t i v e l y speaking, only the K and D data computed from the M and ty p r o f i l e s shown i n F i g s . 21 and 22 140 40 60 80 100 120 140 160 180 200 220 MOISTURE CONTENT, M (%) F i g . 24. — Water c o n d u c t i v i t y - m o i s t u r e content r e l a t i o n s h i p i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood along the boundary drainage and i m b i b i t i o n curves at 21°C . F i g . 25. — Water c o n d u c t i v i t y - m a t r i c p o t e n t i a l r e l a t i o n s h i p i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood along the boundary drainage and i m b i b i t i o n curves at 21°C. 142 40 60 80 100 120 140 160 180 200 220 MOISTURE CONTENT, M (%) F i g . 26. — Water d i f f u s i v i t y - m o i s t u r e content r e l a t i o n s h i p i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood along the boundary drainage and i m b i b i t i o n curves at 21°C. 143 were p l o t t e d . The measured p o r t i o n s of the K-M curves i n F i g . 24 were j o i n e d by broken l i n e s to the mean K value measured at f u l l s a t u r a t i o n . On the other hand, the measured p o r t i o n s of the K—TJ> and D-M curves m i n F i g s . 25 and 26 were e x t r a p o l a t e d (broken l i n e s ) to zero ty or f u l l m s a t u r a t i o n based on the completed K-M curves and an appropriate M-ty m r e l a t i o n s h i p . R e s ults of the separate saturated c o n d u c t i v i t y measurements are presented i n Table A4.4. These r e s u l t s show no s i g n i f i c a n t length e f f e c t on the saturated c o n d u c t i v i t y . However, a systematic decrease of the saturated water c o n d u c t i v i t y w i t h the time of flow was recorded. The o v e r a l l mean K value and the mean K value f o r Tree I I measured a f t e r 60 min of flow were the saturated c o n d u c t i v i t i e s p l o t t e d i n F i g . 24. These K values are of the same order of magnitude as those reported p r e v i o u s l y by L i n et al. (1973) f o r western hemlock sapwood. To help the i n t e r p r e t a t i o n of the K and D data obtained i n the present study, an e r r o r a n a l y s i s was conducted. The propagation of e r r o r formula given e a r l i e r (see Section 3.2.1.1) was used f o r t h i s purpose. The estimated l a r g e s t range ( F r i t t o n 1974) was chosen as the elemental e r r o r i n the various input v a r i a b l e s f e a t u r i n g i n the r e l a t i o n s used to c a l c u l a t e K and D (Eqs. [ 6 2 ] , [67] and [ 6 8 ] ) . I t must be emphasized that a simple s t a t i s t i c a l a n a l y s i s of the K and D data presented i n Tables A4.1 through A4.3 was not p o s s i b l e because each value given i n these t a b l e s represents only one measurement. This i s , of course, a consequence of the t r a n s i e n t character of these experiments. 144 For the K and D data obtained by the instantaneous p r o f i l e method (Eqs. [67] and [ 6 8 ] ) , the random component of the e r r o r i n the input v a r i a b l e s ( f l u x and gradient) was assumed to be much l a r g e r than the systematic component. The range f o r the f l u x q was a s s o c i a t e d w i t h three times the value of the standard e r r o r of the slope from which i t was c a l c u l a t e d . The r e l a t i v e e r r o r s i n q thus estimated v a r i e d from ±3 to ±10%. The elemental e r r o r i n the gradients Sty /Sz and <SM/6z m was a s s o c i a t e d w i t h the l a r g e s t d e v i a t i o n from the smoothed Sty I ST. m vs. t and 5M/<SZ VS. t curves. The r e l a t i v e e r r o r s thus estimated f o r Sty ISz ranged from ±6 to ±12%. Those i n 6M/6z v a r i e d from ±3 to ±11%. m In the former case the r e l a t i v e e r r o r increased w i t h M f o r both i m b i b i t i o n and drainage. In the l a t t e r case t h i s trend was observed f o r drainage only. Based on the foregoing f i g u r e s , the maximum p o s s i b l e e r r o r i n K and D obtained by the instantaneous p r o f i l e method ranged from ±9 to ±22% and from ±6 to ±21%, r e s p e c t i v e l y . These values should represent maximum estimates i f the elemental e r r o r s i n the input v a r i a b l e s were c o r r e c t l y chosen. Obviously, i f systematic e r r o r s were introduced by i n f e r r i n g ty from a s t a t i c VL-ty r e l a t i o n s h i p i n s t e a d of m m measuring i t d i r e c t l y , they are not r e f l e c t e d i n the above e r r o r estimates. As w i l l be discussed l a t e r , t h i s type of systematic e r r o r s may have been the cause of the discrepancy that can be observed i n Table A4.1 between the r e s u l t s obtained at the c a l c u l a t i o n plane z = 0.6 cm and those obtained at z = 2.4 cm. On the other hand, the e r r o r a n a l y s i s conducted on the D data 145 obtained by the one-step method (Eq. [62]) showed a maximum p o s s i b l e e r r o r i n D v a r y i n g from ±5 to ±30% depending on the stage of the i m b i b i t i o n or drainage run at which the i n f l o w - o u t f l o w data were c o l l e c t e d . The e r r o r i n K was not estimated but was c e r t a i n l y greater than that i n D si n c e i t included e r r o r s i n both D and dM/dip (Eq. [ 5 9 ] ) . m The above e r r o r estimates may appear e x c e s s i v e l y high at f i r s t glance, but they compare fav o r a b l y w i t h other e r r o r estimates given i n the l i t e r a t u r e f o r these types of measurements (Chow 1973, F l i i h l e r e t al. 1976). As pointed out by F l u h l e r e t al. who a l s o used an instantaneous p r o f i l e method, any other method would most l i k e l y end up w i t h s i m i l a r u n c e r t a i n t i e s . Bearing i n mind the foregoing f i g u r e s , i t i s evident that s i g n i f i c a n t q u a n t i t a t i v e d i f f e r e n c e s e x i s t between the r e s u l t s obtained by the instantaneous p r o f i l e method and those obtained by the one-step method (Tables, A4.1 through A4.3). S i g n i f i c a n t q u a n t i t a t i v e d i f f e r e n c e s can a l s o be observed between the r e s u l t s of the r e p l i c a t e d measurements c a r r i e d out by the instantaneous p r o f i l e method. In t h i s l a t t e r case, the two s e r i e s of t e s t specimens used i n e i t h e r i m b i b i t i o n or drainage were not true r e p l i c a t e s . Therefore, the noted d i f f e r e n c e s may be l a r g e l y due to d i s s i m i l a r i t i e s i n the specimen p r o p e r t i e s . This may als o e x p l a i n i n great p a r t the dis c r e p a n c i e s observed between the two methods of measurement. F i n a l l y , the p o s s i b i l i t y of a porous p l a t e impedance e f f e c t on the r e s u l t s obtained by the one-step method i s about non-existent as judged from a comparison between the K values measured on wood and those measured on the saturated porous p l a t e s used f o r the 146 flow measurements (see Section 3.2.2.2). On the other hand, the shape of the K-M, K-ib and D-M curves m i n F i g s . 24 through 26 changes very l i t t l e from one method of measure-ment to the other. Such s i m i l a r i t y c e r t a i n l y makes the contents of these f i g u r e s much more c r e d i b l e . One notable feature of the K-M r e l a t i o n s h i p shown i n F i g . 24 i s i t s strong h y s t e r e t i c behavior, w i t h K at a given M up to two orders of magnitude greater i n i m b i b i t i o n than i n drainage. The two-phase (n i t r o g e n + water) flow data of Tesoro et al. (1974) suggested such a trend i n wood. The r e s u l t s of these authors a l s o support the abrupt decrease of K as wood s t a r t s to desaturate. In c o n t r a s t , conventional water p e r m e a b i l i t y measurements (single-phase flow) do not u s u a l l y show t h i s d r a s t i c e f f e c t of d e s a t u r a t i o n on the p e r m e a b i l i t y values (Comstock 1965) . The reason f o r t h i s apparent i n c o n s i s t e n c y i s that the flow mechanism i n v o l v e d during conventional water p e r m e a b i l i t y measurements below f u l l s a t u r a t i o n i s q u i t e d i f f e r e n t from the true unsaturated flow mechanism. The main d i f f e r e n c e between these two flow mechanisms would l i e i n the r e s p e c t i v e s p a t i a l d i s t r i b u t i o n s of a i r i n the wood during the flow measurements. Another i n t e r e s t i n g f e a t u r e of the K-M r e l a t i o n s h i p i s the f l a t n e s s of the i m b i b i t i o n curve from about 60% M up to near f u l l s a t u r a t i o n . This behavior would imply that the transport of water over t h i s broad M range i s c a r r i e d out by n e a r l y a l l the same continuous flow channels. In other words, an increase i n M i n s i d e t h i s moisture 147 range would have l i t t l e e f f e c t on the e f f e c t i v e c r o s s - s e c t i o n a l area of flow. The complexity of the wood c a p i l l a r y system i s f u r t h e r r e f l e c t e d i n the i n t r i g u i n g nature of the K-^m r e l a t i o n s h i p shown i n F i g . 25. The K(IIJ ) f u n c t i o n would a l s o be h y s t e r e t i c , but only i n the m ip range from 0 to about -0.3 bar w i t h K greater i n drainage than i n m i m b i b i t i o n . Outside t h i s ty range, there i s i n d i c a t i o n that the m drainage and i m b i b i t i o n curves c o i n c i d e . This i m p l i e s that K at a given ty would be equal i n both i m b i b i t i o n and drainage d e s p i t e the great d i f f e r e n c e i n M between these two moisture s t a t e s ( F i g . 15). Also very i n t e r e s t i n g i s the nature of the D ( M ) f u n c t i o n shown i n F i g . 26. This f u n c t i o n e x h i b i t s a p a r t i a l h y s t e r e s i s w i t h D greater i n drainage than i n i m b i b i t i o n . This i s the opposite of the h y s t e r e s i s i n the K ( M ) f u n c t i o n ( F i g . 24). Another notable feature of the D-M r e l a t i o n s h i p i s the decrease of D w i t h M i n i m b i b i t i o n from about 40% M up to near f u l l s a t u r a t i o n . The drainage curve would behave s i m i l a r l y but only i n the f i r s t t h i r d of t h i s range. This appears to be an inherent c h a r a c t e r i s t i c of the wood D-M r e l a t i o n s h i p as i n d i c a t e d by the s i m i l a r i t y between these f i n d i n g s and the r e s u l t s of Voigt et at-. (1940) and Heizmann (1970) reported i n F i g . 5. This behavior i s apparently c h a r a c t e r i s t i c of other porous media as w e l l (Swartzendruber 1969). The r e s u l t s of F i g . 5 al s o support the abrupt change of D near f u l l s a t u r a t i o n observed i n F i g . 26. At f u l l s a t u r a t i o n , water d i f f u -s i v i t y i s not defined as can be seen from the d e f i n i t i o n of D given i n Eq. [ 59]. 148 4.2.2 H y s t e r e s i s i n the K ( M ) , K(ip ) and D ( M ) f u n c t i o n s vs. h y s t e r e s i s i n the M-ib r e l a t i o n s h i p rm The f a c t that the M — r e l a t i o n s h i p of wood i s s t r o n g l y m h y s t e r e t i c at high moisture contents would not n e c e s s a r i l y i n d i c a t e that h y s t e r e s i s i s als o present i n the K - M r e l a t i o n s h i p . According to P o u l o v a s s i l i s (1969), the presence of h y s t e r e s i s i n the M-(p r e l a t i o n s h i p m im p l i e s that the pores c o n t a i n i n g water at a given M are not common to both i m b i b i t i o n and drainage, and that the e f f e c t i v e d r a i n i n g r a d i i of these pores are greater i n the former s t a t e than i n the l a t t e r s t a t e . The combined e f f e c t of these two c h a r a c t e r i s t i c s on K ( M ) would be very v a r i a b l e even w i t h i n s i m i l a r porous media (see Section 2.2.2). For the r e s u l t s of t h i s study, i t seems that the l a t t e r c h a r a c t e r i s t i c alone could e a s i l y e x p l a i n the h y s t e r e s i s i n K ( M ) . In f a c t , at a moisture l e v e l of 50% S^, f o r example, i t can be seen from F i g . 20 that the e f f e c t i v e pore ra d i u s i s about two orders of magnitude greater i n imb i -b i t i o n than i n drainage. Thus, knowing that K i s roughly p r o p o r t i o n a l to the square of pore r a d i u s , i t i s evident that such d i f f e r e n c e i n e f f e c t i v e pore radius could l a r g e l y e x p l a i n i n both d i r e c t i o n and s i z e the h y s t e r e s i s shown i n F i g . 24. On the other hand, the connection between the h y s t e r e t i c behavior of the K( IJ J ) and D ( M ) f u n c t i o n s ( F i g s . 25 and 26) and that of m the M-ii» r e l a t i o n s h i p i s not very apparent. The h y s t e r e s i s i n both m K(ip ) and D ( M ) would a c t u a l l y be the complex r e s u l t of the s u p e r p o s i t i o n m of two hystereses, namely that i n the K ( M ) f u n c t i o n and that i n the M-ip r e l a t i o n s h i p , m 149 4.2.3 Flu x - g r a d i e n t r e l a t i o n s h i p As noted e a r l i e r , p r o p o r t i o n a l i t y between f l u x and gradient i s one of the b a s i c c o n d i t i o n s upon which the a p p l i c a t i o n of the unsaturated flow theory depends and t h i s c o n d i t i o n must be met by any flow system used to determine the flow c h a r a c t e r i s t i c s . T y p i c a l q vs. Sty /6z and q vs. 6M/6z curves derived from the M and ty p r o f i l e s obtained m rm i n i m b i b i t i o n are shown i n F i g . 27. The data p o i n t s were obtained from c a l c u l a t i o n planes l o c a t e d between 0.5 and 4.5 cm from the i n f l o w end A I 30 MATRIC POTENTIAL GRADIENT, (I03dyn/cmzcm) 40 MOISTURE CONTENT GRADIENT, |y- (cm3(hU5M:m3cm) (a) (b) F i g . 27. — Flux-gradient r e l a t i o n s h i p at va r i o u s s e l e c t e d values of M during i m b i b i t i o n i n 7.5-cm long western hemlock sapwood specimens a) f l u x vs. ty gradient; b) f l u x vs. M gradient, m 150 ( F i g . 21). The c l o s e r the data p o i n t s to the o r i g i n i n F i g . 27, the more d i s t a n t were the c a l c u l a t i o n planes to the i n f l o w end. Fl u x and gradient computations f o r t h i s purpose were c a r r i e d out at four d i f f e r e n t M's w i t h the a i d of Eq. [65]. Such computations were not f e a s i b l e i n the case of the flow experiments conducted i n drainage due to the f l a t shape of the M and ty p r o f i l e s then obtained ( F i g . 22). m A s i m i l a r trend developed i n the derived q vs. 6iii /<5z curves m ( F i g . 27a) and q vs. <5M/Sz curves ( F i g . 27b). This appears normal owing to the f a c t that ty was i n f e r r e d from a M-ty r e l a t i o n s h i p . F l u x - e r a d i e n t m m p r o p o r t i o n a l i t y can be observed at 50% M only. At 75% M or more, there i s a systematic d e v i a t i o n from l i n e a r i t y i n the z range from 0 to about 2 cm from the i n f l o w end. Non-uniqueness of the M-ty r e l a t i o n s h i p (see Section 2.2.1) m i s the most p l a u s i b l e explanation f o r the f l u x - g r a d i e n t d i s p r o p o r t i o n a l i t y observed i n t h i s study. Concerning d i s p r o p o r t i o n a l i t y between f l u x and ty g r a d i e n t , i t may therefore have o r i g i n a t e d frOm an over s i m p l i -f i e d experimental procedure. In other words, the f a c t of i n f e r r i n g ty from a s t a t i c M-ty r e l a t i o n s h i p i n s t e a d of measuring i t d i r e c t l y may m m have l e d to erroneous ty estimates f o r the t r a n s i e n t flow systems i n m question. Presence of entrapped a i r or process dependence e f f e c t s would have been r e s p o n s i b l e f o r the non-uniqueness of the M-ty r e l a t i o n s h i p . m The l a t t e r p o s s i b i l i t y i s advanced on the b a s i s that the change rat e of iii w i t h time was found to decrease w i t h distance from the i n f l o w end m (see Section 2.1.3.4). 151 The above deviations from f l u x - p o t e n t i a l gradient propor-t i o n a l i t y do not, therefore, n e c e s s a r i l y imply a v i o l a t i o n of Darcy's law. However, they c e r t a i n l y suggest serious l i m i t a t i o n s i n the a p p l i c a t i o n of the d i f f u s i v i t y concept at high moisture contents be cause of the assumption that ty must be a single—valued function of m M. Of course, the a l t e r n a t i v e use of the general flow equations based on Darcy's law would present advantages insofar as M and ty can be m measured independently. The maximum r e l a t i v e error i n K and D caused by flux-gradient nonproportionality was about 300%. This systematic error did not appear to a f f e c t s i g n i f i c a n t l y the nature of the K(M), K(ty ) and D(M) functions, m except perhaps the s i z e of the hysteresis i n D(M). In c l o s i n g t h i s discussion one may quote some i n t e r e s t i n g remarks made by Swartzendruber (1969): "From a s c i e n t i f i c and academic viewpoint, nonproportionality i s not altogether a disadvantage. P o t e n t i a l i t y f or new knowledge i s provided by i n v e s t i g a t i o n of the phenomenon, unless, of course, i t i s ultimately shown to be merely an a r t i f a c t of experimental procedure. If eventually shown to a r i s e from such causes as modified-water properties, porous-medium f a b r i c changes, e l e c t r i c a l streaming p o t e n t i a l , or salt-osmotic e f f e c t s , nonproportionality would l i k e l y become i n i t s own r i g h t an i n t e r e s t i n g i n t e r a c t i o n property of the flow l i q u i d and the porous medium.". SUMMARY AND CONCLUSIONS This study was designed to investigate water equilibrium and water movement phenomena i n wood at high moisture contents with the major emphasis placed on the experimental determination of the M-ty re l a t i o n s h i p and the flow functions K ( M ) , K(IJJ ) and D(M) . A l l experi-m ments were conducted on western hemlock sapwood at a temperature of 21°C. The M-ty boundary drainage and imbibition curves were both m determined as well as the M-ty drainage curve s t a r t i n g from the green m condition. The porous plate methods were used for these determinations. For comparison purposes, psychrometric p o t e n t i a l measurements were also conducted on some of the specimen groups equil i b r a t e d on the porous plates. The K and D determinations under unsaturated conditions were ca r r i e d out by two independent transient methods, namely an instantaneous p r o f i l e method and the one-step method. They were r e s t r i c t e d to the boundary drainage and imbibition curves i n the l o n g i t u d i n a l d i r e c t i o n of flow. Conductivity measurements were also c a r r i e d out separately i n the saturated condition. The determination of the M-ty r e l a t i o n s h i p by the porous plate m methods constituted the main part of the experiments conducted i n t h i s study. The p r e c i s i o n of the r e s u l t s obtained proved to be f a i r l y good. 152 153 The SE values f o r the group M means v a r i e d from 1.1 to 2.9% AM i n drainage and from 0.1 to 2.7% AM i n i m b i b i t i o n . The estimated systematic e r r o r s i n M were g e n e r a l l y n e g l i g i b l e compared to these f i g u r e s . Of major importance i s the c o n f i r m a t i o n by the porous p l a t e t e s t s of the strong h y s t e r e t i c behavior of the wood M-ip r e l a t i o n s h i p m at high moisture contents. This phenomenon may have numerous i m p l i c a -t i o n s . For example, important h y s t e r e s i s e f f e c t s may r e s u l t wherever M changes are not monotonic, i . e . , where M increases and decreases simultaneously or s e q u e n t i a l l y i n v a r i o u s p a r t s of the wood pie c e . Another i m p l i c a t i o n i s that the use of M-ip data obtained on separate m specimens f o r flow p r e d i c t i o n s i n woods of unknown i m b i b i t i o n and drainage (or drying) h i s t o r y may l e a d to serious e r r o r s . The i n k - b o t t l e e f f e c t appears to be the primary cause of the h y s t e r e s i s i n the wood M-ip r e l a t i o n s h i p at high moisture contents. In other words, the wide gap between the boundary i m b i b i t i o n and drainage curves would be l a r g e l y accounted f o r by a "delay" i n the emptying of the c e l l c a v i t i e s as t h i s process would be c o n t r o l l e d by the s i z e of the narrow channels i n t e r c o n n e c t i n g them together. Other important c h a r a c t e r i s t i c s of the M - ^ m r e l a t i o n s h i p were brought out by the porous p l a t e t e s t s . I t i s of i n t e r e s t to note the sigmoid shape of the i m b i b i t i o n curve, and the tendency of the drainage curve to e x h i b i t a p l a t e au at intermediate moisture contents. The l a t t e r c h a r a c t e r i s t i c may have a marked e f f e c t on the drainage or d r y i n g p r o p e r t i e s of wet wood. However, f u l l independent experimental evidence 154 of i t s existence was not provided i n t h i s study. The M-ty^ drainage curve s t a r t i n g from the green condition was found s i m i l a r i n shape to the boundary drainage curve but l a i d w e ll inside the hysteresis loop u n t i l a r e l a t i v e l y low M was reached. D i f f e r e n t imbibition and drainage h i s t o r i e s of the te s t specimens concerned would be l a r g e l y responsible for the gap observed between the two drainage curves. Since the green M i s very v a r i a b l e , i t i s evident that there may e x i s t f o r a given wood species a multitude of M-ty^ drainage curves s t a r t i n g from the green condition. This implies, for example, that the response of green wood to a given drainage or drying process may vary considerably. The M-ty^ boundary drainage curve at high moisture contents can also be expected to change markedly within the same wood species. In fa c t , the r e s u l t s of the porous plate experiments showed a high degree of v a r i a b i l i t y i n t h i s r e l a t i o n s h i p with respect to height i n the tree. This was c l e a r l y r e l a t e d to the changes i n the maximum moisture content with height. Moreover, the s u b s t i t u t i o n of the index M by the index S^ (-M/M ) as indicato r of the degree of wetness i n wood reduced consider-max bly the v a r i a b i l i t y between height i n the case of the boundary drainage curve. The fac t that the S -ty boundary drainage curve i s more stable p m b than the M-ty^ boundary drainage curve suggests that the former r e l a t i o n -ship may be advantageously used to describe the energetic properties of water i n wood at high moisture contents. A f a i r l y good agreement was found between the M-tym data obtained by the porous plate methods and those obtained by the thermocouple 155 psychrometer method In the ty^ range from -1 to -7 bars i n drainage. Conversely, a l a r g e discrepancy between imposed and measured ty^ values was observed f o r the specimens e q u i l i b r a t e d at about -14 bar ty i n m drainage. A f a i l u r e to a t t a i n true e q u i l i b r i u m during water e x t r a c t i o n on the porous p l a t e s , or a change of the e q u i l i b r i u m s t a t e during the r e l e a s e of the pressure from the e x t r a c t i o n chamber at the end of the drainage runs i n question, are two p o s s i b l e explanations f o r the observed d i f f e r e n c e s . In i m b i b i t i o n , a very poor agreement between imposed and measured ty values was a l s o noted i n the case of two specimen groups m but the d e v i a t i o n s observed were e a s i l y e x p l a i n a b l e . Nevertheless, one r e s u l t i n i m b i b i t i o n turned out to be very meaningful as i t provided an independent proof of the existence of a large h y s t e r e s i s i n wood M-ty r e l a t i o n s h i p at high moisture contents, m The r e p r o d u c i b i l i t y of the TCP method proved to be r a t h e r poor. In f a c t , the SD values f o r the psychrometric ty determinations were up m to s e v e r a l times greater than the c a l i b r a t i o n accuracy of the TCP's estimated at ±0.5 bar. The v a r i a t i o n that may have e x i s t e d i n the a c t u a l ty^ values of the t e s t specimens f o l l o w i n g e q u i l i b r a t i o n on the porous p l a t e s apparently was not s o l e l y r e s p o n s i b l e f o r t h i s poor r e p r o d u c i b i l i t y of the TCP method. This had c e r t a i n l y something to do w i t h the method of measurement i t s e l f . For example, the long e q u i l i b r a t i o n times recorded during the psychrometric ty determinations i n wood suggest that the m c o n d i t i o n s of measurement d i d not match p r o p e r l y the c o n d i t i o n s p r e v a i l i n g during the c a l i b r a t i o n of the TCP's. 156 Concerning the flow measurements i n the l o n g i t u d i n a l d i r e c t i o n the K-M, K-il; and D-M curves obtained by the instantaneous p r o f i l e method rm were found very s i m i l a r i n shape to those obtained independently by the one-step method. However, s i g n i f i c a n t q u a n t i t a t i v e d i f f e r e n c e s were observed between the r e s u l t s of the two measurement methods. The f a c t that the specimens used i n these r e s p e c t i v e cases were not q u i t e comparable may l a r g e l y e x p l a i n the observed d i s c r e p a n c i e s . The maximum p o s s i b l e e r r o r i n K and D obtained by the i n s t a n -taneous p r o f i l e method ranged from ±9 to ±22% and from ±6 to ±21%, r e s p e c t i v e l y . The maximum p o s s i b l e e r r o r i n D obtained by the one-step method v a r i e d from ±5 to ±30%. The e r r o r i n K i n t h i s case was not estimated but was c e r t a i n l y somewhat greater than that i n D. These e r r o r estimates may appear f a i r l y high but as revealed by the l i t e r a t u r e any other method of flow measurement would l i k e l y lead to s i m i l a r or even greater u n c e r t a i n t i e s . The K(M) f u n c t i o n was found to e x h i b i t a l a r g e h y s t e r e s i s w i t h K at a given M up to two orders of-magnitude greater i n i m b i b i t i o n than i n drainage. A s u r p r i s i n g f e a t u r e of the K-M r e l a t i o n s h i p i s the f l a t n e s s of the i m b i b i t i o n curve from about 60% M up to near f u l l s a t u r a t i o n . Such behavior would be a s s o c i a t e d w i t h the presence of a i r i n c r i t i c a l l o c a t i o n s of the wood c a p i l l a r y system during i m b i b i t i o n . But as wood approaches a f u l l y saturated s t a t e , a sudden change i n the flowpaths must occur s i n c e the r e s u l t s obtained i n d i c a t e an increase i n K of s e v e r a l orders of magnitude w i t h i n only a few M percents. The h y s t e r e s i s i n K(M) appears to be c l o s e l y r e l a t e d to that i n 157 the M-ty^ r e l a t i o n s h i p . A p o s s i b l e explanation f o r t h i s h y s t e r e s i s i s that the e f f e c t i v e d r a i n i n g r a d i i of the pores c o n t a i n i n g water at a given M would be greater i n i m b i b i t i o n than i n drainage. P a r t i c u l a r l y complex i s the nature of the K(ty ) and D(M) m f u n c t i o n s . E x t r a p o l a t i o n s of the measured K-ty^ curves suggest that the K(ty^) f u n c t i o n i s h y s t e r e t i c but only i n the ty^ range from 0 to -0.3 bar, w i t h K greater i n drainage than i n i m b i b i t i o n . L i k e w i s e , the D(M) f u n c t i o n e x h i b i t s a p a r t i a l h y s t e r e s i s from about 100% M to f u l l s a t u r a -t i o n w i t h D greater i n drainage than i n i m b i b i t i o n . For both K(ty ) and m D(M), the h y s t e r e s i s would a c t u a l l y be the complex r e s u l t of the s u p e r p o s i t i o n of two hystereses, namely that i n the K(M) f u n c t i o n and that i n the M-ty r e l a t i o n s h i p . As was reported f o r other porous media, m D i n i m b i b i t i o n was found to decrease w i t h i n c r e a s i n g M u n t i l the approach of f u l l s a t u r a t i o n where i t increased a b r u p t l y . The f l u x - g r a d i e n t r e l a t i o n s h i p , which i s an important aspect of the flow measurements under unsaturated c o n d i t i o n s , was a l s o i n v e s t i -gated i n t h i s study. F l u x - g r a d i e n t d i s p r o p o r t i o n a l i t y was observed i n i m b i b i t i o n at 75% M or more i n the zone from 0 to about 2 cm from the i n f l o w end. The non-uniqueness of the M-ty^ r e l a t i o n s h i p appears to be the most p l a u s i b l e explanation f o r t h i s anomaly. In other words, the a c t u a l M-ty r e l a t i o n s h i p p r e v a i l i n g during the t r a n s i e n t flow experiments m would have been d i f f e r e n t from the s t a t i c M-ty r e l a t i o n s h i p used to m d e r i v e the ty p r o f i l e s from the M p r o f i l e s . A i r entrapment or process m dependence e f f e c t s may have been re s p o n s i b l e f o r t h i s behavior. The e r r o r s i n K and D caused by f l u x - g r a d i e n t n o n p r o p o r t i o n a l i t y 158 were f a i r l y large but did not appear to a f f e c t s i g n i f i c a n t l y the nature of the K(M), K(ty ) and D(M) functions, except perhaps the si z e of the m hysteresis i n D(M). Since M and ty for the flow measurements were not m determined independently, i t i s evident that the observed deviations from flux-gradient p r o p o r t i o n a l i t y do not neces s a r i l y imply a v i o l a t i o n of Darcy's law. However, they do suggest serious l i m i t a t i o n s i n the a p p l i c a t i o n of the d i f f u s i v i t y concept at high moisture contents. This important assumption of flux-gradient p r o p o r t i o n a l i t y i n unsaturated flow theory should obviously be the object of further i n v e s t i -gation. Much remains also to be learned about the other properties discussed or investigated i n t h i s study. 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Yao, J . 1964. Determination of lumen s i z e d i s t r i b u t i o n i n softwood by the mercury i n j e c t i o n method. Forest Prod. J . 14/4): 167-170. Appendix 1 Relation of water p o t e n t i a l to r e l a t i v e humidity and radius of curvature of the air-water i n t e r f a c e at 21°C TABLE A l . l . — R e l a t i o n of water p o t e n t i a l to r e l a t i v e humidity and radius of curvature of the a i r - w a t e r i n t e r f a c e at 21°C. Water p o t e n t i a l (bars) (ergs/g) R e l a t i v e humidity (%) Radius of curvature (urn) 0.000 0.000 100.0000 OO -0.001 -0.001 X 10 6 99.9999 1451.8 -0.010 -0.010 X 10 6 99.999 145.18 -0.050 -0.050 X 10 6 99.996 29.038 -0.100 -0.100 X 10 6 99.993 14.519 -0.200 -0.200 X 10 6 99.985 7.259 -0.300 -0.301 X 10 6 99.978 4.840 -0.400 -0.401 X 10 6 99.970 3.630 -0.500 -0.501 X 10 6 99.963 2.904 -0.600 -0.601 X 10 6 99.956 2.420 -0.700 -0.701 X 10 6 99.948 2.074 -0.800 -0.802 X 10 6 99.941 1.815 -0.900 -0.902 X 10 6 99.933 1.613 -1.000 -1.002 X 10 6 99.926 1.452 -2.000 -2.004 X 10 6 99.852 0.726 -3.000 -3.006 X 10 6 99.778 0.484 -4.000 -4.008 X 10 6 99.705 0.363 -5.000 -5.010 X 10 6 99.631 0.290 -6.000 -6.012 X 10 6 99.557 0.242 -7.000 -7.014 X 10 6 99.483 0.207 -8.000 -8.016 X 10 6 99.409 0.181 -9.000 -9.018 X 10 5 99.335 0.161 -10.000 -10.020 X 10 5 99.262 0.145 -11.000 -11.022 X 10 6 99.188 0.132 -12.000 -12.024 X 10 6 99.114 0.121 -13.000 -13.026 X 10 6 99.040 0.112 -14.000 -14.028 X 10 6 98.966 0.104 -15.000 -15.030 X 10 6 98.892 0.097 -16.000 -16.032 X 10 6 98.818 0.091 -17.000 -17.034 X 10 6 98.745 0.085 -18.000 -18.036 X 10 6 98.671 0.081 -19.000 -19.038 X 10 6 98.597 0.076 -20.00,0 -20.040 X 10 6 98.523 0.073 -25.000 -25.051 X 10 6 98.154 0.058 -50.000 -50.101 X 10 6 96.375 0.029 -100.000 -100.20 X 10 6 92.882 0.014 -500.000 -501.01 X 10 5 69.120 0.0029 -1000.000 -1002.0 X 10 6 47.774 0.0014 -5000.000 -5010.1 X 10 6 2.459 0.00029 -10000.000 -10.020 X 10 9 0.061 0.00014 174 Appendix 2 C a l i b r a t i o n data of the thermocouple psychrometers used f o r water p o t e n t i a l determinations i n wood at 21°C TABLE A2.1. — C a l i b r a t i o n data of the thermocouple psychrometers used f o r water p o t e n t i a l determinations i n wood at 21°C. TCP No. Conditions of c a l i b r a t i o n Psychrometer s e n s i t i v i t y (uV/bar) Regression equation P r e d i c t e d ty (bar) TCP emf output (urn) A n a l y s i s of variance f o r the r e g r e s s i o n °Y/X (bar) S i g n i f i c a n t at the 0.01 l e v e l . Rz 1 0.38 Y -0.24 - 2.6 X 0.05 113,454 ** 0.999 2 Before use; 0.1, 0.40 Y = 0.38 - 2.6 X 0.57 909. g** 0.998 3 0.2, 0.4 & 0.6 molal 0.40 Y — 0.66 - 2.6 X 0.43 1,634 ** 0.999 4 NaCl s o l u t i o n s at 20°C 0.40 Y — 0.55 - 2.6 X 0.34 2,555 ** 0.999 5 ( o r i g i n a l data c o r r e c t e d 0.43 Y = 2.73 - 2.8 X 0.83 431. 0.995 6 f o r 21°C) 0.41 Y 0.63 - 2.6 X 0.12 19,912 ** 0.999 7 0.43 Y = 0.90 - 2.5 X 0.56 963. 8** 0.998 8 0.41 Y 0.94 - 2.7 X 0.32 2,942 ** 0.999 1 0.44 Y 0.18 - 2.3 X 0.45 2,310 ** 0.999 2 A f t e r use; 0.0, 0.1, 0.43 Y = 0.41 - 2.4 X 0.27 6,369 ** 0.999 4 0.2, 0.4 & 0.6 molal 0.46 Y 0.59 - 2.3 X 0.39 3,128 ** 0.999 5 NaCl s o l u t i o n s at 21°C 0.35 Y -0.80 - 2.7 X 0.91 566. 2** 0.995 6 0.53 Y 0.65 - 2.0 X 0.68 1,020 ** 0.997 8 0.49 Y - 0.30 - 2.1 X 0.50 1,855 ** 0.998 176 Appendix 3 Data pertinent to the M-ii) determinations m 177 TABLE A3.1. — Mean moisture contents (M) of western hemlock sapwood specimens obtained by e q u i l i b r a t i o n on porous p l a t e s under various imposed matric p o t e n t i a l s (up ) a t 21°C. . — j Subgroup M E q u i l i b r a t i o n Imposed ~ ~ ~ ~ Group , , Tree I Tree I I mode ^ o c o c n c o c M m 0.6 m 9.6 m 0.6 m 9.6 m m (bars) (%) (%) (%) (%) (%) -0. ,110 193. ,8 -0. ,470 162. ,2 Boundary -0. ,984 136. ,6 -1. ,95 112. ,1 drainage -2. ,87 84. ,1 curve -6. ,95 68. ,4 -13. ,8 62. ,1 -13. ,9 67. ,6 -0. ,110 174. ,5 Drainage -0. .470 138. ,8 curve -0. ,984 112. ,6 s t a r t i n g -1. ,95 103. ,6 from -2. ,87 99. .3 green -6. .95 59. .4 c o n d i t i o n -13. .8 63. .2 -13. .9 63. .2 -0. .0015 198. .5 Boundary -0. -0. .0098 .0294 187, 124. .9 .4 i m b i b i t i o n -0. .091 50. . 3 curve -0. .499 31. .8 -0. .994 31, .3 210.2 197.4 215. 4 204.2 186.7 172.9 184. 5 176.6 161.7 144.9 162. 1 151.3 144.7 125.1 141. 8 130.9 117.6 93.2 124. 2 104.8 97.6 67.4 100. 5 83.5 96.6 78.9 103. 0 85.1 90.8 73.5 102. 2 83.5 157.0 173.0 133. 6 159.5 124.9 141.8 122. 0 131.9 97.2 114.7 98. 3 105.7 91.7 108.9 95. 2 99.9 85.3 95.0 86. 1 91.4 64.1 55.2 64. 6 60.8 66.7 59.2 61. 4 62.6 70.4 55.6 62. 7 63.0 216.9 201.3 213. 6 207.6 196.4 184.5 174. 6 185.8 103.8 128.1 101. 6 114.5 61.2 50.6 53. 8 54.0 33.9 32.5 33. 1 32.8 32.0 32.1 32. 4 31.9 TABLE A3.2. — Standard d e v i a t i o n s (SD) and standard e r r o r s (SE) of the M data obtained during the measurement of the M-ty r e l a t i o n s h i p by the porous p l a t e methods at 21°C. Subgroups u i l i b r a t i o n mode Imposed 0.6 m Tree I 9.6 m 0.6 m T r e e l l 9.6 m Groups m SD SE SD SE SD SE SD SE SD 1 SE 2 (bars) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) -0. 110 3. 5 2. 0 6. 6 3. 8 1. 3 0. 8 5. 7 3 4.0 3 10. 5 1. 4 -0. 470 5. 2 3. 0 4. 8 2. 8 1. 7 1. 0 9. 9 5.7 11. 5 1. 8 Boundary drainage curve -0. 984 9. 9 5. 7 6. 5 3. 8 4. 4 2. 5 10. 5 6.1 13. 5 2. 4 -1. 95 3. 8 2. 2 1. 6 0. 9 2. 0 1. 2 10. 0 5.8 14. 6 1. 6 -2. 87 9. 9 5. 7 14. 1 8. 1 4. 7 3 3. 3 3 4. 6 2.7 19. 2 2. 2 -6. 95 12. 6 7. 3 10. 3 5. 9 3. 2 1. 8 12. 2 7.0 18. 5 2. 9 -13. 8 2. 3 1. 3 9. 6 5. 5 6. 3 3. 6 7. 3 4.2 17. 7 2. 0 -13. 9 8. 7 5. 0 6. 4 3. 7 2. 4 1. 4 4. 3 2.5 15. 2 1. 7 -0. 110 9. 2 5. 3 2. 6 1. 5 5. 7 3. 3 _4 4 19. 7 2. 0 Drainage -0. 470 9. 5 5. 5 6. 6 3. 8 12. 8 7. 4 6. 0 3.5 11. 9 2. 6 curve -0. 984 13. 6 3 9. 6 3 7. 9 4. 6 10. 2 5. 9 6. 8 3.9 11. 7 2. 8 s t a r t i n g -1. 95 7. 1 4. 1 11. 4 6. 6 7. 7 4. 4 7. 2 4.2 10. 4 2. 6 from -2. 87 5. 0 2. 9 9. 2 5. 3 14. 1 8. 1 4. 3 2.5 9. 9 2. 6 green -6. 95 4. 0 2. 3 4. 9 2. 8 5. 5 3. 2 3. 4 2.0 5. 2 1. 1 c o n d i t i o n -13. 8 1. 2 3 0. 8 3 8. 7 5. 0 12. 9 7. 4 13. 1 7.6 6. 5 2. 1 -13. 9 7. O 3 4. 9 3 11. O3 7. 8 3 3. 7 2. 1 2. 6 1.5 7. 8 1. 9 -0. 0015 8. 0 4. 6 10. 8 6. 2 2. 9 1. 7 4. 4 2.5 10. 3 2. 1 Boundary i m b i b i t i o n -0. 0098 7. 7 4. 4 2. 6 3 1. 8 3 11. 9 6. 9 4. 4 3 3.1 3 11. 3 2. 7 -0. 0294 5. 6 3. 2 12. 6 7. 3 1. 1 0. 6 6. 8 3.9 14. 0 2. 2 -0. 091 3. 9 2. 3 3. 3 3 2. 3 3 1. 7 1. 0 0. 9 3 0.6 3 5. 5 0. 9 curve -0. 499 0. 2 0. 1 0. 5 3 0. 4 3 0. 2 0. 1 0. 3 3 0.2 3 0. 9 0. 1 -0. 994 0. 6 0. 3 0. 2 0. 1 0. 1 0. 06 0. 3 3 0.2 3 0. 6 0. 1 Square root of t o t a l mean square. Based on the mean square e r r o r (mean square w i t h i n subgroups). Two t e s t specimens only. One t e s t specimen only. 179 TABLE A3.3 — Mean s a t u r a t i o n percentages (Sp) of western hemlock sapwood specimens obtained by e q u i l i b r a t i o n on porous p l a t e s under various imposed matric p o t e n t i a l s (ty ) at 21°C. m Subgroup Sp E q u i l i b r a t i o n Imposed z — — Group , Tree I Tree I I m ° m^ 0.6 m 9.6 m 0.6 m 9.6 m P (bars) (%) (%) (%) (%) (%) -0 .110 95 .3 94 .2 96. .2 94. .3 95. .0 -0 .470 79 .8 81 .6 83. .2 82. ,1 81. .7 Boundary -0 .984 68 .7 69 .3 70. .4 71. .0 69. ,8 -1 .95 55 .5 61 .1 60. .9 62. ,9 60. .1 drainage -2 .87 41 .6 49 .5 45. .4 54. .8 47. .8 curve -6 .95 34 .0 41 .2 32. ,8 44. ,7 38. .2 -13 .8 31 .2 41 .4 38. .3 46. ,2 39. .3 -13 .9 33 .8 40 .2 35. .3 44. ,9 38. .5 -0 .110 85 .1 67 .7 84. ,5 57. ,3 73. .6 Drainage -0 .470 68 .1 53 .6 67. .7 53. .9 60. ,8 curve -0 .984 56 .2 41 .4 55. .7 43. ,6 49. ,2 s t a r t i n g -1 .95 51 .0 38 .6 53, .1 41. .9 46, .1 from -2 .87 49 .2 35 .8 46, .2 38. ,4 42. ,4 green - 6 .95 29 .1 27 .2 26, .6 28. ,6 27. .9 c o n d i t i o n -13 .8 31 .8 29 .0 28, .7 27. .5 29. ,2 -13 .9 30 .9 31 .2 27, .0 27. .5 29. .2 - o .0015 97 .6 95 .9 96, .8 95. ,7 96, .5 Boundary Imbibition -0 .0098 92 .4 83 .3 88. .4 80. ,4 86. .1 - o .0294 60 .5 42 .5 61. .8 44. .6 52. ,3 -0 .091 25 .0 26 .2 24, .4 24. ,7 25, .1 curve -0 .499 15 .6 14 .5 15. .5 15. .4 15, .2 - 0 .994 15 .5 13 .8 15. .5 14. .6 14, .9 TABLE A3.4. — F-values from the a n a l y s i s of variance f o r the M and Sp data obtained by e q u i l i b r a t i o n on porous p l a t e s under various imposed matric p o t e n t i a l s (ty ) at 21°C. m E q u i l i b r a t i o n mode Imposed ty m (bars) F-values Sources of v a r i a t i o n Height M S Tree M Height x Tree M S P -0 .110 41, .79** 20. 7 0 * * 2, .76 O 4- • 30 0. 08 1 .23 -0 .470 25, .92** 0. 25 1. .43 6. 14* 3. 31 3 .36 Boundary drainage -0 .984 19. .90** 0. 26 0. .83 1. 97 0. 68 0 .00 -1 .95 59, .89** 10. 20* 2. ,56 9. 15* 6. 20* 2 .29 -2 .87 59, .08** 17. 49** 3. .50 5. 00 0. 09 0 .13 curve -6 .95 28, . 39** 12. 0, .03 0. 18 0. 11 0 .75 -13 .8 53, 29. 28** 8. ,52* 12. 52** 1. 73 0 .47 -13 .9 56, .69** 23. 03** 6. ,34* 3. 51 0. 64 0 .91 -0 .110 58, .61** 181. 96** 11. ,39** 11. 02* 8. 66* 8 .74* Drainage -0 .470 10, .18* 32. 70** 0. .00 0. 00 0. 31 0 .02 curve -0 .984 8. .76* 48. 44** 0. ,09 0. 17 0. 01 0 .48 s t a r t i n g -1 .95 6, .27* 25. 08** 0. .74 1. 33 0. 03 0 .07 from -2 .87 4, .81 25. 33** 0. ,11 0. 01 0. 25 1 .73 green -6 .95 9, .74* 0. 00 0. .67 0. 27 1. 11 3 .37 c o n d i t i o n -13 .8 0, .50 1. 47 1. ,41 1. 88 0. 02 0 .21 -13 .9 4, .40 0. 04 4. ,99 3. 69 0. 00 0 .00 -0 .0015 13, .07** 5. 48 0. ,00 0. 83 0. 52 0 .31 Boundary i m b i b i t i o n -0 .0098 0, .02 10. 12* 6. .71* 1. 66 3. 60 0 .04 -0 .0294 28, .15** 238. 56** 0. .03 2. 23 0. 44 0 .12 -0 .091 18, .79** 1. 04 4. ,81 2. 26 5. 56 0 .40 curve -0 .499 65, .33** 6. 61* 0. ,14 3. 09 19. 60** 4 .12 -0 .994 8. .02* 46. 73** 11. .11* 3. 96 0. 45 3 .97 * S i g n i f i c a n t at the 0.05 l e v e l . ** S i g n i f i c a n t at the 0.01 l e v e l . 181 TABLE A3.5. — Mean matric p o t e n t i a l s (ty m) measured by thermocouple psychrometers (TCP) i n specimens p r e v i o u s l y e q u i l i b r a t e d on porous p l a t e s at 21°C. Measured ty. E q u i l i b r a t i o n mode Imposed m Group M TCP o u t p u t 1 conversion TCP o u t p u t 2 conversion Method 1 Method 2 (bars) (%) (bars) (bars) -0.984 151.3 -1.2 -0.8 Boundary drainage curve -1.95 -2.87 -6.95 -13.8 130.9 104.8 83.5 85.1 -2.7 -3.3 -6.8 -7.0 -2.3 -2.9 -6.5 -6.8 -13.9 83.5 -5.1 -4.8 Drainage -0.984 105.7 -1.4 -1.0 curve -1.95 99.9 -2.9 -2.4 s t a r t i n g -2.87 91.4 -3.4 -3.1 from -6.95 60.8 -7.1 -6.8 green -13.8 62.6 -5.6 -5.3 c o n d i t i o n -13.9 63.0 -7.4 -7.3 Boundary -0.091 54.0 -1.7 -1.2 i m b i b i t i o n -0.499 32.8 -14.4 -14.5 curve -0.994 31.9 -30.9 -31.5 Based on the measured TCP s e n s i t i v i t i e s . Based on the re g r e s s i o n equations f o r the c a l i b r a t i o n data. TABLE A3.6. — Standard d e v i a t i o n s (SD) and standard e r r o r s (SE) of the i p m data obtained by the thermocouple psychrometer (TCP) method from specimens p r e v i o u s l y e q u i l i b r a t e d on porous p l a t e s at 21°C. E q u i l i b r a t i o n Imposed Group mode tli M m TCP output 1 TCP o u t p u t 2 conversion conversion Method 1 Method 2 SD SE SD SE (bars) (%) (bars) (bars) (bars) (bars) -0. .984 151, .3 0, .59 0, .17 0. .68 0, .20 Boundary -1. .95 130, .9 1, .40 0, .40 1. .41 0. .41 -2. .87 104, .8 1, .10 0, .33 1. .24 0. ,37 drainage -6. .95 83, .5 3. .49 1, .01 3. .61 1, .04 curve -13. .8 85, .1 3. .39 1, .38 3. .39 1. .38 -13. .9 83, .5 2, .97 0, .86 3. .07 0. .89 Drainage -0. .984 105. .7 0. .57 0. .17 0. .56 0. .17 curve -1. .95 99, .9 1. .15 0. .33 1. .29 0. ,37 s t a r t i n g -2. .87 91. .4 1. .65 0. .48 1. .74 0. .50 from -6. .95 60, .8 2. .57 0. .74 2. .51 0. ,72 green -13. .8 62, .6 1. .74 0. .78 1. .65 0. .74 c o n d i t i o n -13. .9 63, .0 3. .98 1. .26 4. .65 1. .47 Boundary -0.091 54.0 0.97 0.31 0.89 0.28 i m b i b i t i o n -0.499 32.8 5.39 1.70 5.94 1.88 curve -0.994 31.9 6.48 1.95 6.57 1.98 Based on the measured TCP s e n s i t i v i t i e s . Based on the r e g r e s s i o n equations f o r the c a l i b r a t i o n data. 183 0 Appendix 4 Data pertinent to the K and D determinations 184 TABLE A4.1. — Water conductivity (K) and water d i f f u s i v i t y (D) i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood at various moisture contents (M) and matric po t e n t i a l s (tym), as obtained by the instantaneous p r o f i l e method during imbibition at 21°C. Or i g i n of the z1 M m K x i o1 0 D x i o 5 specimens (cm) (%) (bar) (cm 3(H 00)cm/dyn s) (cm 2/s) 30.5 2 50 -0.0918 0.34 2.92 60 -0.0729 0.59 2.45 2.4 70 -0.0631 0.81 2.25 80 -0.0531 1.13 2.04 90 -0.0460 1.37 1.92 Tree I 100 -0.0398 1.47 1.74 0.6-m height 90 -0.0460 0.46 0.63 100 -0.0398 0.51 0.63 110 -0.0355 0.58 0.66 0.6 120 -0.0307 0.67 0.68 130 -0.0266 0.75 0.70 140 -0.0231 0.78 0.69 150 -0.0200 0.85 0.72 160 -0.0172 0.92 0.74 29.5 2 50 -0.0918 0.39 3.34 60 -0.0688 0.59 2.50 2.4 70 -0.0562 0.78 2.14 80 -0.0501 0.96 1.92 90 -0.0447 1.03 1.55 Tree I I 100 -0.0398 0.96 1.17 0.6-m height 90 -0.0447 0.37 0.60 100 -0.0398 0.41 0.58 110 -0.0355 0.44 0.56 0.6 120 -0.0316 0.48 0.55 130 -0.0282 0.49 0.54 140 -0.0251 0.50 0.51 150 -0.0224 0.46 0.43 160 -0.0194 0.47 0.41 Distance of the c a l c u l a t i o n plane from the inflow end. Average i n i t i a l moisture content. 185 TABLE A4.2. — Water c o n d u c t i v i t y (K) and water d i f f u s i v i t y (D) i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood at various moisture contents (M) and matric p o t e n t i a l s (ty m), as obtained by the instantaneous p r o f i l e method during drainage or d r y i n g at 21°C. Tree II 0.6-m height ° ^ i g ; n Z l M ty K x 1 0 1 2 D x 10 5 of the m specimens (cm) (%) (bars) (cm 3(li^COcm/dyn s) (cm 2/s) 203.O4 -0.026 190 -0.212 3.26 1.76 185 -0.282 1.81 1.26 180 -0.365 1.45 1.09 170 -0.531 0.84 0.81 160 -0.750 0.62 0.64 2 0.25 150 -0.972 0.50 0.55 140 -1.22 0.43 0.53 130 -1.54 0.41 0.51 120 -1.88 0.39 0.50 110 -2.24 0.40 0.51 100 -2.58 0.41 0.54 90 -3.07 0.43 0.62 2 0 2 . 5 4 - 0 . 0 3 6 175 - 0 . 4 4 7 4 . 8 3 3 .54 170 - 0 . 5 3 1 3 .01 2 .38 160 - 0 . 7 5 0 1 .77 1 .52 150 - 0 . 9 7 2 1 .40 1 .27 Tree I I i i e e i i Q > 2 5 1 4 Q -1.22 1 .10 1 .07 0.6-m hexght 1 3 Q ^ Q > 9 3 120 - 1 . 8 8 0 .79 0 .94 110 - 2 . 2 4 0 .79 0 .96 . 100 - 2 . 5 8 0 .80 1 .06 95 - 2 . 8 2 0 .82 1 .13 Distance of the c a l c u l a t i o n plane from the outflow or dryi n g end. Evaporative d r y i n g technique. Pressure p l a t e technique. Average i n i t i a l moisture content. 186 TABLE A4.3. — Water d i f f u s i v i t y (D) and water conductivity (K) i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood at various moisture contents (M) and matric po t e n t i a l s (tym), as obtained by the one-step method during drainage and imbibition at 21°C. Porous plate t e s t s p e c i f i c a t i o n s Imposed m M f M m D x i o5 = D (dM/dip ) m (bars) (%) (%) (%) (bars) (cm 2/s) (cm 3(H 20)cm/dyn s) Drainage -0.110 215.1 204.2 207 206 -0.0794 -0.0891 7.77 2.25 25.9 7.50 -0.470 216.1 176.6 195 190 -0.211 -0.274 1.30 0.81 3.70 2.17 180 -0.422 0.87 1.94 -0.984 216.6 151.3 170 -0.596 0.59 1.03 160 -0.817 0.38 0.51 150 -1.12 0.46 0.48 140 -1.45 0.40 0.38 130 -1.83 0.34 0.31 -6.95 217.4 83.5 120 -2.24 0.30 0.27 110 -2.66 0.25 0.18 100 -3.25 0.20 0.073 95 -3.76 0.18 0.040 Imbibition 75 -0.0531 0.23 8.05 -0.0294 28.0 114.5 80 90 -0'.0487 -0.0410 0.22 0.20 9.12 10.7 100 -0.0354 0.16 10.7 120 -0.0266 0.11 11.0 130 -0.0231 0.10 11.9 140 -0.0200 0.095 12.8 -0.0015 26.9 207.6 150 160 -0.0173 -0.0154 0.090 0.085 13.4 13.6 170 -0.0130 0.085 13.6 180 -0.0109 0.098 14.9 190 -0.0084. 0.13 16.7 1 8 7 TABLE A4.4. — Water c o n d u c t i v i t y (K) i n the l o n g i t u d i n a l d i r e c t i o n of western hemlock sapwood at f u l l s a t u r a t i o n as a f u n c t i o n of specimen le n g t h and time of flow at 21°C. O r i g i n of the specimens Specimen len g t h Time of flow K x 10 8 (cm) (min) (cm 3(H 20)cm/dyn s) 1 14.2 5 13.3 2.0 10 12.4 20 30 11.8 11.5 Tree I I 60 11.0 0.6-m height 1 5 16.8 14.5 5.0 10 13.4 20 30 60 1 5 12.4 11.9 11.3 4.2 3.6 2.0 10 3.5 20 30 3.3 3.3 Tree I 60 3.3 0.6-m height 1 5 6.2 5.3 5.0 10 5.1 20 30 60 4.6 4.2 4.0 

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