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An analysis of transient heat flow through a composite wall McDonald, James W. 1962

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AN ANALYSIS OF TRANSIENT HEAT FLOW THROUGH A COMPOSITE WALL by JAMES W. MCDONALD B . A . S c , University of British Columbia, 1958  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of MECHANICAL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1962  In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study.  I further agree that permission  for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  The University of British Columbia, Vancouver 8, Canada. Date  ABSTRACT The object of t h i s i n v e s t i g a t i o n was heat flow through a composite w a l l .  to examine the transient  This wall was  represent the type used i n house construction.  chosen t o  I t consisted of  a f i r frame, covered on one side with hardboard and on the with cedar, and the space between the hardboard and cedar f i l l e d with f i b r e g l a s s i n s u l a t i o n . A vapour b a r r i e r was as i t would o f f e r l i t t l e resistance to heat flow.  was not  included  This structure,  therefore, offered resistances to heat flow i n series and The t h e o r e t i c a l analysis was  other  parallel.  numerical owing to the anisotropic  properties of the materials and to the composite structure of the wall.  Two  analyses were made of the transient heat flow, an exact  analysis and an approximate analysis which neglected the e f f e c t of the frame.  The heat flow was three dimensional i n the f i r s t analysis  owing to the difference i n the magnitude of the p a r a l l e l resistances and was  one dimensional i n the approximate a n a l y s i s .  solutions both showed exponential  The two t h e o r e t i c a l  cooling rates and agreed within f i v e  percent of each other, which shows that the e f f e c t of the frame i s n e g l i g i b l e when i t s surface area i s small as compared to the t o t a l surface area of the w a l l .  The r a t i o of t o t a l wall surface area to  frame area f o r the wall studied was  9.6 to  1.0.  The wall was mounted i n a guarded hot-box apparatus and experiments were performed i n order to v e r i f y the r e s u l t s of the theoretical analysis.  The experiments consisted of e s t a b l i s h i n g  a steady state temperature gradient across the wall and then eliminating the heat source.  The ensuing transient temperatures  were measured hy thermocouples and were compared with those predicted by theory.  The experimental results varied from the  exact solution by I** percent and from the approximate solution by 18 percent.  The experimental results indicated that the tests  were consistent. The difference between the theoretical and experimental results was attributed to: ( l ) contact resistances, (2) nonhomogeneous wall materials, (3) nonuniform surface coefficients of heat transfer, and (k) the effect of neglecting certain heat capacities which actually were not negligible. The results indicated that the transient temperatures varied according to the equation T = T i e  r  where T represents temperature,  •£ represents time, and 7> is the time constant.  The results also  showed that the method of analysis was acceptable and that the approximate analysis is suitable for walls with small frame areas.  ACKNOWLEDGMENT The writer wishes to express his gratitude to Professors W. A. Wolfe and J . L . Wighton for their supervision of the thesis, to W. Hancock of The British Columbia Forest Products Laboratory for his assistance, to W. 0 . Richmond, and the entire Department of Mechanical Engineering, University of B r i t i s h Columbia. This research was made possible by funds supplied by the National Research Council, under a grant to Professor W. 0 . Richmond.  V  TABLE OF CONTENTS PAGE INTRODUCTION  1  DESCRIPTION OF THE APPARATUS  2  The Wall  2  The Test Section  3  The Guard Section  4  The Exit Section  5  The Temperature Control System  5  The Thermocouples  7  TEST PROCEDURE  9  Preparation for the Test  9  Performance of the Test  10 i  ANALYSIS  12  The Exact Analysis  13  The Approximate Analysis  16  RESULTS  18  OBSERVATIONS AND CONCLUSIONS  19  BIBLIOGRAPHY  23  APPENDICES  24  Appendix A.  List of Symbols  Appendix B.  General Physical Properties of Wood and other Wall Materials  Appendix C.  2k  27  Determination of the Physical Properties of the Wall Materials  3^  vi PAGE Appendix D.  Physical Properties of the Wall Materials  h2  Appendix £ .  Summary of Theoretical Equations  Appendix F .  Data  h$  Appendix G.  Figures  58  hj>  1 INTRODUCTION A study was started ten years ago in the Department of Mechanical Engineering at the University of British Columbia of the heat transfer conditions involved in the transient heating of buildings and houses.  The i n i t i a l phase was the investigation of  the transient response of heated a i r in an enclosure with heat losses through a concrete slab.  Experiments were performed under the  direction of Mr. G. Green, on the air enclosure and slab in a guarded hot-box apparatus using step and cyclic changes i n the heat supply.  A paper covering the analytic investigation was written  by Professor W. A. Wolfe and published in 1959  This paper, which  considered the heat capacity of the f l u i d in the enclosure, predicted the transient temperatures of the air enclosure and the inside surface of the slab.  This was a refinement on a previous paper by E . G. 2  Smith which did not consider the heat capacity of the enclosed f l u i d . The results of the paper by W. A. Wolfe showed that when the f l u i d is a i r , the heat capacity can be neglected.  In the discussion of his  paper, i t was suggested that this type of problem could be solved numerically i f a multilayer wall were available. • Thus, from this suggestion, i t was decided to investigate the heat transfer properties of a frame wall typical of house construction.  T~. W. A. Wolfe, "Transient Response of Heated A i r in an Enclosure With Heat Losses", Journal of Heat Transfer, 8 l : 19-23, February, 1959• 2. E . G. Smith, "A Simple and Rigorous Method for the Determination of Heat Requirements of Simple Intermittently Heated Exterior Walls", Journal of Applied Physics, 12:638-642, 1941.  DESCRIPTION OF THE APPARATUS The apparatus, shown i n Figures 2 and J>, consisted of four principle parts:  (l) the wall, (2) the test section, (3) the guard  section, and (k) the exit section. The Wall The wall consisted of a f i r frame, covered on one side with hardboard, and on the other with a cedar panel.  The studs were  spaced on centres 16 inches apart, and the girts were spaced on centres 28 inches apart, in the section of the frame in the region of the test section (Figures h and 5)« The components of the frame were carefully fitted together in order to reduce contact resistances.  The components of the frame were selected with a  straight grain, i n order that the heat flow would be either normal to or parallel with the grain.  This is important as the grain causes  3 anisotropy i n wood. Hardboard was glued and tightly screwed to the hot side of the frame i h an effort to reduce contact resistances produced by air spaces. of the frame.  A cedar panel was securely fastened to the cold side Access to the interior of the wall was facilitated  by constructing the cedar panel in such a manner that i t could be 3. Appendix B.  3 removed i n t a c t .  With t h i s arrangement, i n s u l a t i o n could be  changed, and thermocouples relocated. A f i b r e g l a s s i n s u l a t i o n was i n s t a l l e d f o r the experiments performed i n t h i s study, and was compressed into place t o ensure a good thermal contact with the f i r , hardboard and cedar.  Due  to the anisotropic properties of wood, edge cut cedar with a straight grain was selected f o r the cedar panel.  The panel was  also constructed to minimize the e f f e c t s of a i r spaces between the boards. The Test Section The temperature measurements were taken i n the t e s t section, which was the centre portion of the w a l l , plus an adjacent a i r enclosure (Figures 2 and 5 ) -  The sides of the t e s t section were  away from the edge of the wall i n order that the heat losses at the edges would not a f f e c t the temperatures i n the t e s t section.  A  control system prevented any t r a n s f e r of heat across the boundaries of the a i r enclosure. Thus, a l l the heat i n the t e s t section passed d i r e c t l y through the w a l l . The casing of the t e s t section consisted of aluminum coated heavy k r a f t paper, insulated by an aluminum l i n e d f i b r e g l a s s i n s u l a t i o n , with the l i n i n g on the outside t o prevent r a d i a t i o n to the t e s t section.  This i n s u l a t i o n was used t o decrease the  response of the t e s t section to the surrounding guard section. Two heaters, one of four ohms resistance, and the other of s i x ohms, were suspended i n the t e s t section.  They could be used  separately, together, i n p a r a l l e l , or s e r i e s , depending on the power required.  The voltage t o the heaters was controlled by a v a r i a c ,  and t h i s voltage determined the temperature difference across the wall.  These heaters had r a d i a t i o n shields i n order t o prevent  r a d i a t i o n t o the w a l l , since only conduction heat t r a n s f e r was desired. A b a l s a wood fan was placed i n the t e s t section t o mix the a i r , and create a uniform surface c o e f f i c i e n t over the surface of the wall.  This fan was driven by an e l e c t r i c motor, which was on the  outside of the apparatus. The Guard Section The function of the guard section was t o i s o l a t e the t e s t section.  I t surrounded the t e s t section, and was enclosed by a  plywood casing, with two p l a s t i c windows f o r viewing the inside of the apparatus (Figures 1, 2, and 3).  This casing was l i n e d  with f i b r e g l a s s i n s u l a t i o n i n order t o reduce heat l o s s e s . Ribbon heaters, supported by wooden rods with porcelain i n s u l a t o r s , e n c i r c l e d the t e s t section t o give a uniform heat generation throughout the guard section.  There were t e n of these  heaters i n the guard section which could be used on a continuous 110 v o l t s , and f i v e that were connected t o the output of the variac i n the control system.  This arrangement gave a better temperature  control than that given by connecting a l l the heaters t o the v a r i a c .  Two fans, mounted in opposite corners of the apparatus, circulated the a i r and produced a uniform temperature throughout the guard section.  Aluminum coated paper was attached to wire,  supports to form a radiation barrier between the ribbon heaters and the test section.  This barrier directed the a i r to encircle  the test section. The Exit Section The exit section was an a i r enclosure on the cold side of the wall, formed by covering a steel frame with aluminum coated paper.  The aluminum covering opposite the wall was coated with  brown paper, i n order to prevent radiation from the heaters being reflected to the wall.  The temperature i n this enclosure was  controlled by a system consisting of a thermostat, relay switch, variac and heaters with a radiation shield (Figures 6 and 10). Two fans were used to circulate the a i r and create a uniform surface coefficient over the cold surface of the wall. The Temperature Control System The function of the temperature control system was to maintain equal temperatures i n the guard and test section.  As shown i n  Figure 7> sensing elements of a bridge system were placed i n the guard and test sections. to temperature changes.  These resistors were extremely sensitive The difference i n the resistances of the  elements caused by a temperature difference between the test and  g u a r d s e c t i o n s , p r o d u c e d an u n b a l a n c e d b r i d g e s y s t e m .  This lack  o f b a l a n c e i n d u c e d a s m a l l v o l t a g e between t h e motor p o t e n t i o m e t e r w i p e r and t h e g r o u n d , t h a t i s , a c r o s s t h e i n p u t t e r m i n a l s o f relay (Figure 8).  the  I n t h e event t h a t t h e g u a r d s e c t i o n t e m p e r a t u r e  was h i g h e r , t h e r e l a y w o u l d i d e n t i f y t h e s i g n a l v o l t a g e as due t o an i n c r e a s e i n t h e r e s i s t a n c e o f T 3 , and c l o s e t h e r e l a y between t e r m i n a l s one a n d t h r e e .  contacts  This closed the c i r c u i t of the  c o u n t e r - c l o c k w i s e w i n d i n g , and s t a r t e d t h e m o t o r .  As t h e motor  t u r n e d c o u n t e r - c l o c k w i s e , t h e motor p o t e n t i o m e t e r w i p e r moved towards t h e " G " end o f t h e w i n d i n g u n t i l t h e b a l a n c e o f t h e b r i d g e c i r c u i t was r e s t o r e d . The s h a f t  The r e l a y t h e n b r o k e t h e c o n t a c t and s t o p p e d t h e m o t o r .  o f t h e motor was c o n n e c t e d t o t h e h a n d l e o f a v a r i a c , and  as t h e motor t u r n e d , i t r e d u c e d t h e o u t p u t v o l t a g e o f t h e v a r i a c , and d e c r e a s e d t h e power s u p p l i e d t o t h e g u a r d s e c t i o n h e a t e r s .  With  the output voltage of the v a r i a c reduced, the temperature i n the guard s e c t i o n f e l l below t h a t i n the t e s t s e c t i o n .  This a c t i v a t e d the c o n t r o l  and t u r n e d t h e motor i n t h e c l o c k w i s e d i r e c t i o n , c a u s i n g an i n c r e a s e i n t h e power s u p p l i e d t o t h e h e a t e r s , balance.  and r e s t o r i n g t h e  temperature  As t h e motor t u r n e d f r o m t h e maximum c o u n t e r - c l o c k w i s e  p o s i t i o n t o t h e maximum c l o c k w i s e p o s i t i o n , t h e o u t p u t o f t h e v a r i a c r a n g e d f r o m 35 t o 110 v o l t s .  This cyclic control  c a u s e d a maximum v a r i a t i o n o f 0.2 d e g . F . i n t h e s t e a d y temperature i n the t e s t  section.  voltage  state  The Thermocouples Thermocouples were u s e d t o measure t h e t e m p e r a t u r e s i n t h e r e g i o n s o f o n e , two and t h r e e d i m e n s i o n a l h e a t f l o w , and t o check the o p e r a t i o n of the c o n t r o l system.  Copper-constantan thermocouples  were s e l e c t e d w i t h t h e l a r g e gauge number o f 30 i n o r d e r t o m i n i m i z e t h e mass o f w i r e i n t h e w a l l , s i n c e a p p r o x i m a t e l y 120 were i n s t a l l e d .  thermocouples  They were c o n n e c t e d t h r o u g h a s w i t c h box t o a 16  p o i n t r e c o r d e r , which a u t o m a t i c a l l y converted the output of the t r a n s d u c e r s t o degrees F a h r e n h e i t ( F i g u r e 9).  Each thermocouple  was s o l d e r e d t o one o f t h e l6 l o c a t i o n s on one o f t h e 10 c o n n e c t o r p l u g s mounted i n t h e s w i t c h b o x .  The t e r m i n a l s o f t h e r e c o r d e r  were s o l d e r e d t o a male p l u g w h i c h c o u l d be a t t a c h e d t o any one o f t h e female c o n n e c t o r s t h e r e b y e n a b l i n g t h e r e c o r d e r t o measure t h e t e m p e r a t u r e s i n any one o f t h e t h r e e r e g i o n s o f h e a t f l o w .  The  m u l t i p l e p o i n t r e c o r d e r measured t h e o u t p u t o f one o f t h e t h e r m o c o u p l e s e v e r y 15 seconds w i t h an a c c u r a c y o f 0.2  deg.F.  The r e l a x a t i o n method was u s e d w i t h e s t i m a t e d p h y s i c a l p r o p e r t i e s o f t h e w a l l m a t e r i a l s t o d e t e r m i n e t h e t e m p e r a t u r e p r o f i l e s on a  k p l a n e n o r m a l t o t h e a x i s o f a component o f t h e f r a m e .  These  t e m p e r a t u r e p r o f i l e s were u s e d as a g u i d e t o l o c a t e t h e t h e r m o c o u p l e s 4 . W . H . G i e d t , P r i n c i p l e s o f E n g i n e e r i n g Heat T r a n s f e r , New Y o r k , D.Van N o s t r a n d C o . I n c . , 1957, pp 65-71.  8 i n the w a l l .  The thermocouple leads were taken along isothermal  l i n e s f o r two inches before branching away from the w a l l .  This  prevented heat conduction from the hot junction, along the wire, causing an error i n temperature measurement. Thermocouples were placed i n the guard and t e s t section a i r enclosures i n order t o check the operation of the temperature system.  control  Five thermocouples i n the guard section were connected i n  p a r a l l e l , and t h e i r signal was read on a potentiometer.  This reading  was compared with the output of a thermocouple located i n the t e s t section and when the average value of each s i g n a l was equal, the controls were functioning properly.  TEST PROCEDURE  Preparation  f o r the Test  The fans and the control system were started and the control point adjustment was set at the maximum p o s i t i o n .  This caused the  variac t o supply the maximum voltage t o the control heaters i n the guard section.  A l l of the heater c i r c u i t s were then closed  to give maximum heating.  One or both of the heater c i r c u i t s i n  the t e s t section was closed, and the variac was set t o give the voltage required f o r a p a r t i c u l a r temperature drop across the wall. was  For the t e s t s performed i n t h i s study, the s i x ohm heater  used, and the variac was set at four v o l t s t o give a temperature  difference across the wall of approximately 56 deg.F. The thermostat i n the exit section was set t o the desired p o s i t i o n , approximately 20 deg.F. above room temperature.  As  the room temperature varied between 70 deg.F. and 90 deg.F., the thermostat was set at 95 deg.F. and the heater c i r c u i t s i n the exit section were closed. When the temperature i n the t e s t section was near the desired value, the control point s e t t i n g was reduced i n order to make the heating and cooling periods of the control heaters equal.  With  equal heating and cooling times, the apparatus was able t o reach steady state, since the heat l o s t and gained by the t e s t section a i r enclosure was equal during each control c y c l e .  I t was  extremely d i f f i c u l t t o both acquire and maintain the steady  s t a t e c o n d i t i o n due t o i n a d e q u a t e s e n s i t i v i t y i n t h e c o n t r o l system.  The c o n t r o l p o i n t s e t t i n g was a f f e c t e d b y t h e t e m p e r a t u r e  d i f f e r e n c e between t h e g u a r d s e c t i o n and t h e room, t h u s as t h e t e m p e r a t u r e i n t h e a p p a r a t u s approached t h e d e s i r e d v a l u e , and as t h e room t e m p e r a t u r e changed, t h e c o n t r o l p o i n t h a d t o be t o m a i n t a i n e q u a l h e a t i n g and c o o l i n g p e r i o d s .  adjusted  When s t e a d y  h a d been o b t a i n e d , t h e a p p a r a t u s was r e a d y f o r t h e  state  test.  Performance o f t h e T e s t The t e m p e r a t u r e s were measured t h r o u g h o u t t h e t e s t w h i l e the steady s t a t e e x i s t e d i n order t o determine the temperatures f o r the t h e o r e t i c a l a n a l y s i s .  section initial  Following t h i s ,  several  t e r m i n a l s i n t h e r e c o r d e r were a t t a c h e d d i r e c t l y t o t h e r m o c o u p l e s i n t h e two and t h r e e d i m e n s i o n a l h e a t f l o w r e g i o n s .  The male c o n n e c t o r  was t h e n a t t a c h e d t o t h e c o n n e c t o r w i t h t h e t h e r m o c o u p l e s I n t h e one dimensional region.  Thus, the t r a n s i e n t temperatures i n the t h r e e  a r e a s o f h e a t f l o w c o u l d be measured s i m u l t a n e o u s l y t h r o u g h o u t t h e test.  The c i r c u i t o f t h e t e s t s e c t i o n h e a t e r was opened i n o r d e r  t o begin the  test.  D u r i n g t h e t e s t , t h e c o n t r o l p o i n t adjustment and t h e power s u p p l y t o t h e g u a r d s e c t i o n h e a t e r s were v a r i e d i n o r d e r t o p r e v e n t a t e m p e r a t u r e d i f f e r e n c e o c c u r i n g between t h e h o t a i r a n d t h e surface of the w a l l .  T h i s c o n d i t i o n was m a i n t a i n e d , as i t  corresponded  t o the assumption i n the a n a l y s i s t h a t the heat c a p a c i t y o f the a i r .  was n e g l i g i b l e .  A t e m p e r a t u r e d i f f e r e n c e was m a i n t a i n e d a c r o s s  the w a l l of the t e s t s e c t i o n a i r enclosure' i n order t o prevent h e a t f r o m f l o w i n g i n t o t h e t e s t s e c t i o n and c a u s i n g a d e c r e a s e the c o o l i n g r a t e .  in  T h i s t e m p e r a t u r e d i f f e r e n c e was measured b y a  p o t e n t i o m e t e r , and was not a l l o w e d t o become g r e a t e r t h a n 2 d e g . F . , as t h e t e m p e r a t u r e symmetries a t t h e b o u n d a r i e s o f t h e t e s t would be d i s t u r b e d .  section  T h i s t e m p e r a t u r e d r o p was m a i n t a i n e d b y a d j u s t i n g  t h e c o n t r o l p o i n t and v a r y i n g t h e h e a t  supply.  I f the temperature d i f f e r e n c e across the w a l l o f the a i r e n c l o s u r e became t o o l a r g e , t h e t e m p e r a t u r e o f t h e a i r i n t h e t e s t s e c t i o n w o u l d become l o w e r t h a n t h e t e m p e r a t u r e i n t h e w a l l ' s surface.  T h i s w o u l d cause h e a t t o f l o w i n t h e wrong d i r e c t i o n  and t h u s i n c r e a s e t h e c o o l i n g r a t e o f t h e w a l l .  I n order t o  c o r r e c t t h i s s i t u a t i o n , t h e c o n t r o l p o i n t s e t t i n g must be r a i s e d T h u s , two i t e m s were c o n t r o l l e d s i m u l t a n e o u s l y : ( l ) t h e t e m p e r a t u r e o f t h e a i r i n t h e t e s t s e c t i o n , and ( 2 ) t h e t e m p e r a t u r e between t h e g u a r d and t e s t s e c t i o n s .  difference  F i n a l l y , when a t e m p e r a t u r e  g r a d i e n t no l o n g e r e x i s t e d a c r o s s t h e t e s t w a l l , t h e t e s t was terminated.  ANALYSIS The numerical method of finite differences was used to 5 determine the thermal response of the wall. This was used "because of the structure of the wall and the anisotropic c properties of i t s materials.  There were several axes of temperature  symmetry occuring at the centre of the f i r members of the frame and at the midpoints between them (Figures 11 and 12). These axes of symmetry simplified the problem by making i t possible to analyse only a small portion of the wall. The heat flow was one, two and three dimensional owing to the presence of the f i r frame.  The two dimension heat flow at  the studs and girts was due to the difference in the thermal conductivity of the f i r and fibreglass insulation.  The effect  of a frame component on i t s surrounding temperature distribution did not extend beyond four inches from the centreline.  Thus, as  shown i n Figure 12, there was a region enclosed by the studs and girts where the heat flow was one dimensional.  Where the studs  and girts intersected at right angles, the heat flow was three dimensional.  At a sufficient distance away from the intersection,  along a stud or g i r t , the temperature distribution on successive 5- G. M. Dusiriberre, Numerical Analysis of Heat Flow, New York, McGraw-Hill Book Co. Inc., 1949. 6.  Appendix B.  13 planes did not change, and the heat flow became two dimensional. An analysis was performed neglecting the frame, i n order to show its effect on the thermal response of the wall, and to obtain an approximate solution to the problem. This was a one dimensional heat flow analysis. The exact analysis of the problem was performed on the region of three dimensional heat flow. was chosen to make the boundaries  The size of this region  the two dimensional heat flow  regions and the corners the one dimensional regions. The Exact Analysis The small portion of the wall analysed i n the exact analysis was divided into a grid for which the numerical equations were derived (Figure 13).  Two important items had to be considered  when this grid was selected: (l) the distance between the nodes had to be such that the heat flow between them was not falsely reduced due to high thermal resistances caused by large internodal distances, and (2) the number of nodes had to be such that the size of the problem was within the capacity of the computer available for the calculations.  To satisfy both conditions, a grid system of 150  nodes was selected. The derivation of the finite difference equations was based on the following assumptions:  (1) The contact resistances were negligible. (2) The surface coefficients of heat transfer were uniform. (3) The heat capacity of the a i r and apparatus components i n the test section was negligible. (k) The materials were homogeneous. (5) The boundaries of the region were adiabatic. The f i r s t assumption was based on the fact that the wall was constructed to minimize contact resistances.  The surfaces  of the 2 by h inch boards were smoothed by planing and the films 7 of glue were made thin enough not to have any effect.  The  second assumption was reasonable since fans were used to circulate the a i r over both sides of the wall.  The heat capacity of the  apparatus components i n the test section was minimized by using light materials with low specific heats.  The effect of the heat  capacity of the a i r was shown to be negligible i n an analysis by 8 W. A. Wolfe.  The fourth assumption was good i n the case of the  hardboard, cedar and f i r , as the variation in their specific weights was small.  However, for the fibreglass insulation, which had a  variation of 12.2 percent i n its specific weight, the assumption was not as valid.  It was reasonable however, since 12.2 percent  variation was tolerable, and necessary since the variation i n the 7. Brown and Marco, Introduction to Heat Transfer, McGraw H i l l Book Co. Inc., 1958. 8. Wolfe, op_. c i t . , p.23.  15 s p e c i f i c weight throughout the wall was not known. The  fifth  assumption was based on the f a c t that the temperature gradients normal to the boundaries were n e g l i g i b l e (Figure  13).  Considering the law of conservation of energy, the following f i n i t e difference equation was written to express the heat flow at node  a 25a:  • ^ ^ ( T 2 0 a - T25a)<5#+  - ^ C ^ b  ^£2££< ( 4a - T25a)*#"+ T2  =  l?  €h Cf  U  -  T25a)<^  (T25a« - T25a)  The l e f t side of the equation represents the heat flowing i n t o and element during a time i n t e r v a l , dt,  and the r i g h t side i s the  change i n heat content during that time i n t e r v a l .  Rearranging  the above equation to solve f o r T 2 5 a ' . T 2 5 a  »  + [1 - ( 2 + k  = ^ ^ ( T 2 0 a + T24a) + f 'S25b Z  tJhdt  b^l)°^^\  Substituting values and l e t t i n g dt - 1 minute.  T25a' = 0.069 T20a +  O.O69  T24a +  Q.Jjh  T25b + 0.488 T 25a  The c o e f f i c i e n t of T25a was termed the s e l f influence c o e f f i c i e n t , since i t a f f e c t s i t s own future temperature.  The s e l f influence  c o e f f i c i e n t must be p o s i t i v e , otherwise an i n s t a b i l i t y w i l l a r i s e i n the equations. ^ This i n s t a b i l i t y i s produced by a thermo1  dynamically  9. 10.  impossible condition, where the future temperature  Dusiriberre, ep. c i t . , p. I b i d . , p.  116.  115.  T25a  a t t h e end o f t h e t i m e i n t e r v a l ,  w i l l become l o w e r as t h e  t e m p e r a t u r e a t t h e b e g i n n i n g o f t h e i n t e r v a l becomes h i g h e r . Also, the eoefficent  should not equal z e r o , as t h i s i s e q u i v a l e n t  t o n e g l e c t i n g t h e h e a t c a p a c t y o f t h e element and t h e node w o u l d t h e n have no e f f e c t on i t s f u t u r e t e m p e r a t u r e s .  The most c o n v e n i e n t  t i m e i n t e r v a l , w h i c h made a l l o f t h e s e l f i n f l u e n c e c o e f f i c i e n t s p o s i t i v e was one m i n u t e .  The e q u a t i o n f o r node l a governed t h e  s e l e c t i o n of t h i s time i n t e r v a l .  The r e m a i n i n g 149 e q u a t i o n s were  d e r i v e d i n t h e same manner."^ The t r a n s i e n t t e m p e r a t u r e s were c a l c u l a t e d b y s u b s t i t u t i n g t h e i n i t i a l t e m p e r a t u r e s i n t o t h e e q u a t i o n s , and c a l c u l a t i n g t h e t e m p e r a t u r e s a t t h e end o f t h e f i r s t t i m e i n t e r v a l . These t e m p e r a t u r e s became t h e i n i t i a l v a l u e s f o r t h e n e x t t i m e i n t e r v a l and t h e c a l c u l a t i o n was r e p e a t e d . was c o n t i n u e d u n t i l t h e t e m p e r a t u r e d i f f e r e n c e was n e g l i g i b l e .  T h i s procedure  across the w a l l  The c a l c u l a t i o n s were p e r f o r m e d on t h e U n i v e r s i t y  o f B r i t i s h Columbia's computer, ALWAC I I I E. The i n i t i a l t e m p e r a t u r e s were o b t a i n e d b y d i r e c t measurement, and b y i n t e r p o l a t i n g v a l u e s f r o m t h e c u r v e s drawn f r o m t h e measured temperatures  The A p p r o x i m a t e A n a l y s i s The e q u a t i o n f o r one d i m e n s i o n a l h e a t f l o w f o r node 25a  11.  A p p e n d i x E.  12.  A p p e n d i x F.  17 was simplified from the three dimensional case to:  -^pL*(T25b - T25a)^ = ^ ^ ( 1 2 ^ ' rearranging terms and letting  M=  - T25a)  —  The maximum value for d£ was found hy letting the self influence  2  coefficient, ( l Therefore,  equal zero. M= 2  By arbitrarily letting oLt' = 2 minutes and substituting values the equation became:  T25a* = 0.748 T25b + 0.252 T25a The equation for node 25b was:  -^^- (T25a - T25b)^ + 2  (T25c - T25b )cii = -|-  (uehch  +vet  -r )(T25b' - t25a) (  and reduced to:  T25b' = 0.572 T25a + O.O5U T25c + 0.374 T25b similarily,  T25c' = 0.116 T25b + 0.116 T25d + O.768 T25c T25d = 0.116 T25c + 0.116 T25e + O.768 T25d f  T25e* = 0.040 T25d + 0.130 T25f + 0.830 T25e T25f' * 0.156 T25e + 0.626 To + 0.218 T25f The i n i t i a l temperatures were measured and the calculations were performed,as in the exact analysis, ,on the ALWAC I I I E computer.  18 RESULTS The theoretical and experimental results are in good agreement (Figures 17 and 20).  The difference between the experimental results  and the exact analysis was 14 percent and between the experimental results and the approximate analysis was 18 percent. The theoretical and experimental results, when plotted on semi-logarithmic graph paper, showed the cooling of the wall was exponential (Figures 18 and 19).  The approximate analysis showed  a faster cooling rate than that predicted by the exact analysis, and the experimental curves exhibited a slower cooling rate for approximately the f i r s t ten hours than that shown by the theoretical results, and a faster cooling rate after the f i r s t ten hours. The determination of the exact transient period for the wall was d i f f i c u l t , since the curves asymtotically approach zero, but the period was approximately 24 hours for an i n i t i a l temperature difference across the wall of 55-6 deg. F.  The agreement between the  two tests indicated that the experimental results were consistent.  19  OBSERVATIONS AND CONCLUSIONS The i n c r e a s e i n t h e c o o l i n g r a t e o f t h e e x p e r i m e n t a l t e s t s a f t e r a p p r o x i m a t e l y t e n h o u r s o f c o o l i n g was due t o h e a t b e i n g l o s t from the i n s i d e surface o f the t e s t w a l l t o the guard s e c t i o n . T h i s h e a t l o s s was caused b y t h e l a c k o f s e n s i t i v i t y i n t h e c o n t r o l s y s t e m , w h i c h a l l o w e d t h e t e s t s e c t i o n a i r t e m p e r a t u r e t o become l o w e r t h a n t h e t e m p e r a t u r e on t h e w a l l ' s s u r f a c e .  T h i s p o o r c o n t r o l was  caused b y a c o m b i n a t i o n o f two c h a r a c t e r i s t i c s o f t h e (l)  apparatus:  a good t h e r m a l r e s p o n s e between t h e q u a r d and t e s t  and (2)  sections  t h e l a r g e t e m p e r a t u r e d i f f e r e n c e r e q u i r e d between t h e two  r e s i s t o r s t o a c t i v a t e the r e l a y s .  As t h e t e m p e r a t u r e i n t h e g u a r d  s e c t i o n gradually f e l l during the c o o l i n g p o r t i o n of a c o n t r o l c y c l e , the temperature i n the t e s t s e c t i o n r e a d i l y f o l l o w e d i t .  Therefore,  the temperature d i f f e r e n c e r e q u i r e d t o a c t i v a t e the c o n t r o l s occured a f t e r a long period of time.  The t i m e p e r i o d r e q u i r e d d u r i n g t h e  h e a t i n g p o r t i o n was s h o r t e r s i n c e t h e h e a t i n g power was l a r g e  enough  t o cause t h e g u a r d s e c t i o n t e m p e r a t u r e t o r i s e r a p i d l y above t h e t e s t s e c t i o n temperature. control cycle.  T h u s , more h e a t was l o s t t h a n added d u r i n g a  I n s u l a t i o n was added t o t h e w a l l o f t h e t e s t  section  i n and e f f o r t t o reduce t h e t h e r m a l r e s p o n s e , b u t t h e r e d u c t i o n p r o d u c e d was n o t s u f f i c i e n t .  Too much i n s u l a t i o n c o u l d n o t be  added as i t would increase the heat capacty of the t e s t section, thereby slowing down the cooling of the w a l l . The cooling was exponential f o r the f i r s t nine hours of t e s t one, but afterwards the slope of the curve gradually increased because of the poor control (Figure 19). portion of the the curve was extrapolated and new were obtained.  The  straight  temperatures  The same was done t o the second experimental curve  which showed a sudden change a f t e r ten hours of cooling.  The values  obtained from the extrapolated portions were p l o t t e d and the r e s u l t i n g curves exhibited very good agreement (Figure 20).  These new  curves  were more r e a l i s t i c as they allowed f o r the heat l o s t from the inner surface of the w a l l .  As these curves are exponential, they may  expressed by the equation:  -P T - To = ( T i - To) e  *  t  where *?• , the time constant, depends only on the c h a r a c t e r i s t i c s of the w a l l .  The average value of t obtained from the slope of  the logarithmic curves was 6.334 hours.  Thus, the equation can  be used t o determine the cooling of the w a l l f o r any temperature  initial  difference across the w a l l .  The difference between the experimental and t h e o r e c t i c a l r e s u l t s were a t t r i b u t e d l a r g e l y t o the non-homogeneity of the  be  materials, p a r t i c u l a r i l y i n the case of the i n s u l a t i o n where the v a r i a t i o n i n the s p e c i f i c weight was 12.2  percent.  The e f f e c t  of any contact resistances and of the heat capacity of the t e s t section would be to slow down the cooling rate i n the t e s t s . Although t h i s e f f e c t was  indeterminate,  i t was  probably small  owing to the c a r e f u l construction of the apparatus.  I t was  l i k e l y that the surface c o e f f i c i e n t of heat t r a n s f e r on the cool side of the wall was not uniform since the two fans were not capable of producing uniform a i r c i r c u l a t i o n over the e n t i r e surface of the w a l l .  The surface c o e f f i c i e n t would be lower i n  the regions where the a i r movement was  l e s s over the surface.  Thus, the actual mean value of the c o e f f i c i e n t was l e s s than the value used, since the value of the c o e f f i c i e n t could only be measured i n the region of one dimensional heat flow, where the a i r c i r c u l a t i o n was  good.  A l l of these factors would cause the  experimental cooling rate to be l e s s than the t h e o r e t i c a l rate. Inaccuracies  i n the thermocouples would account f o r some of  the difference between theory and experiment.  However, t h i s error  i s not l i k e l y to be more than two percent since c a l i b r a t i o n of the thermocouples showed errors of 1.5 readings.  deg. F. and l e s s i n 100 deg. F  I t i s possible that thermocouples may become loose a f t e r  i n s t a l l a t i o n and cause large errors. was  However, no i n d i c a t i o n of t h i s  observed on temperature records, but detection would be  i f thermocouples detached between t e s t s .  difficult  The agreement "between the exerimental and theoretical curves was good and therefore indicated that the method of analysis was satisfactory.  The results of the exact analysis  showed a slower cooling rate than the one predicted by the approximate analysis.  This was due to the f i r having a lower  thermal dlffusivity than the insulation.  The approximate analysis  would be acceptable for walls with a smaller ratio of frame area to total surface area.  The ratio for the wall studied was 1.0  to 9-6. However, the approximate analysis could be justified for this wall and other walls with similar frames since:  (l)  the  maximum temperature difference between the one dimensional analysis and the three dimensional analyis was only four percent of the i n i t i a l temperature difference across the wall, and (2) the one dimensional analysis was less laborious than the exact analysis. The exact analysis was very lengthy owing to the large number of equations derived and the large amount of computer time required. Thus, the results indicated the following: (1) The method of analysis and experimental work was satisfactory. (2) The approximate analysis, which neglects the effect of the frame, gives sufficiently accurate results for walls with small ratios of frame area to surface area. (3) The cooling of this type of wall can be represented by the equation T = T i e , where t = 6.334 hours for the wall investigated in this thesis. t / I f  (4) The control system used was not adequate and should be improved i f more tests are to be conducted.  BIBLIOGRAPHY  Brown and Marco,.Intoduction t o Heat Transfer, New York, McGrawH i l l Book Co. Inc., 19587 Batchelor, G. K i , "Heat Transfer by Free Convection Across a Closed Cavity Between V e r t i c a l Boundaries at D i f f e r e n t Temperatures , Quarterly of Applied Mathematics, 12: No. 3, October, 1954. ,,  Campbell, W. G., Form and S t y l e i n Thesis Writing, Boston, Houghton M i f f l i n Co., 1954 Dussiriberre, G. M., Numerical Analysis of Heat Flow, New York, McGraw-Hill Book Cq. Inc., 1949Giedt, W. H., P r i n c i p l e s of Engineering Heat Transfer, New York, D. Van Nostrand Co. Inc., 1957. Kollmann, F., Technologie Des Holzes und der B e r l i n , Springer, Vol.1, 1951.  Holzwerkstoffe,  MacLean, J . D., "Thermal Conductivity of Wood", Heating, Piping, and A i r Conditioning, V o l . I l l , No. 6, June, 1941. MacLean, J . D., "Rate of D i s i n t e g r a t i o n of Wood Under D i f f e r e n t Heating Conditions", Proceedings of American Wood-Preservers Assoc., 47:155-68, 1951. Smith, E. G., "A Simple and Rigourous Method f o r the Determination of Heat Requirements of Simple. Intermittently Heated E x t e r i o r Walls", Journal of Applied Physics, 12:638-642,  194l.  Tiemann, H. D., Wood Technology, New York, Pitman Publishing Corp.,  1951.  United States Department of A g r i c u l t u r e , Forest Products Forest Service, Wood Handbook, No. 72, 1955.  Laboratory,  United States Department of A g r i c u l t u r e , Forest Products Laboratory, Forest Service, The Rate of Temperature Changes i n Wood Panels Heated Between Hot Plates, No. 1299, June, 1955. Wolfe, W. A. "Transient Response of Heated A i r i n an Enclosure with Heat Losses", Journal of Heat Transfer, 8 l r 19-23, February, 1959•  APPENDIX  2k  APPENDIX A.  LIST OF SYMBOLS  2 A C  area,ft. specific heat of wet wood,B.T.U./lb. deg. F.  Q  specific heat of cedar, B . T . U . / l b . deg. F.  C  specific heat of f i r , B . T . U . / l b . deg. F.  f  specific heat of hardboard, B . T . U . / l b . deg. F . Q  specific heat of fibreglass insulation,B.T.U./lb. deg. F .  Cm  mean specific heat of dry wood, B . T . U . / l b . deg. F.  Cp  specific heat of dry wood, B . T . U . / l b . deg. F.  D  weight after oven-drying.  dT  increment of temperature.  dtt  increment of time.  dTx-y  temperature difference between planes x and y.  dx  increment of length.  e  base of natural logarithms.  h  surface coefficient of heat transfer for the hot surface, B.T.U./hr. f t . deg. F .  h  surface coefficient of heat transfer for the cold surface, B.T.U./hr. f t . deg. F.  k  thermal conductivity, B . T . U . / i n . / h r . f t . deg. F.  0  k  c  thermal conductivity of cedar tangential and normal to the grain, B.T.U./hr. f t . deg. F .  kf  thermal conductivity of f i r normal and tangential to the grain, B.T.U./hr. f t . deg. F.  k^  thermal conductivity of hardboard normal to the plane of the hardboard, B.T.U./hr. f t . deg. F .  ki  thermal conductivity of fibreglass insulation, B . T . U . / h r . f t . deg. F .  25 kj  thermal conductivity of cedar parallel to the grain, B.T.U./hr. f t . deg. F .  c  kjf  thermal conductivity of f i r parallel to the grain, B.T.U./hr. f t . deg. F.  kj^  thermal conductivity of the hardboard in the plane of the hardboard, B.T.U./hr. f t . deg. F .  ln  natural logarithm.  M  moisture content, percent.  M.V.x  thermocouple reading at plane x, millivolts, m.v.  Q  heat flux, B.T.U./hr  S  specific gravity of wood based on volume at current moisture content and weight when oven dried.  t  time, hr.  T  temperature, deg. F .  Ti  i n i t i a l hot surface temperature, deg. F .  To  cold a i r temperature, deg. F.  T  temperature of surrounding f l u i d , deg. F .  W  original wet weight.  *  cartesian coordinate  o<  thermal dlffuslvity, k/ c, f t . / h r .  ft.  2  p  <=< c  thermal diffusivity of cedar normal and tangential to the grain, f t . / h r . thermal diffusivity of f i r normal and tangential to the grain, f t . / h r .  c*^  °<i CXJ  C  o<^  thermal diffusivity of hardboard normal to the plane of the hardboard, f t . / h r . 2 thermal diffusivity of fibreglass insulation, f t . / h r . 2 thermal diffusivity of cedar parallel to the grain,, f t . / h r . 2 thermal diffusivity of f i r parallel to the grain, f t . / h r .  thermal d l f f u s i v i t y of hardboard i n the plane of hardboard, f t . / h r . m i l l i v o l t equivalent, specific  weight,  the  m.v./deg. F.  lb./ft. •z  specific  weight of cedar,  lb./ft.  specific  weight of f i r ,  specific  weight of hardboard,  lb./ft.^ lb./ft.^ •7.  specific  weight of f i b r e g l a s s  time constant  of w a l l ,  hr.  insulation,  lb./ft.  27 APPENDIX B .  GENERAL PHYSICAL PROPERTIES OF WOOD AND OTHER WALL MATERIALS  G e n e r a l P r o p e r t i e s o f Wood A.  Thermal C o n d u c t i v i t y The t h e r m a l c o n d u c t i v i t y o f wood i s a f f e c t e d (a) (b) (c) (d) (e) (f)  (a)  by:  the d i r e c t i o n of the g r a i n the s p e c i f i c g r a v i t y t h e m o i s t u r e c o n t e n t and i t s d i s t r i b u t i o n structural characteristics heat temperature  The D i r e c t i o n o f t h e G r a i n The t h e r m a l c o n d u c t i v i t y i n t h e 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 i s a p p r o x i m a t e l y t h e same, b u t i t i s  generally  2.25 t o 2.75 t i m e s g r e a t e r a l o n g t h e g r a i n t h a n i n t h e  transverse  13 directions (Figure l 4 ) . (b)  The S p e c i f i c  J  T h u s , wood i s a n a n i s o t r o p i c m a t e r i a l .  Gravity  The t h e r m a l c o n d u c t i v i t y i n c r e a s e s w i t h t h e gravity.  specific  Temperature v a r i e s more s l o w l y i n woods w i t h a  h i g h s p e c i f i c g r a v i t y t h a n i t does i n woods w i t h a l o w specific gravity.  T h i s i s due t o t h e i n c r e a s e i n t h e  specific  h e a t p e r u n i t volume b e i n g g r e a t e r t h a n t h e i n c r e a s e i n t h e thermal c o n d u c t i v i t y .  Thus, the thermal d i f f u s i v i t y  d e c r e a s e s as t h e s p e c i f i c g r a v i t y  generally  increases.  13. U n i t e d S t a t e s Department o f A g r i c u l t u r e , F o r e s t L a b o r a t o r y , F o r e s t S e r v i c e , Wood Handbook, No. 7 2 , 1955*  Products  (c)  The Moisture Content and Its Distribution The thermal conductivity of wood can be calculated from  the following formula when the moisture content is less than 40 percent." " * 1  1  k = S (1.39 + 0.028M) + 0.165 The moisture content is defined by the equation  M= (  W  ^ ^)100.  The above equation for thermal conductivity applies to the wood i n the wall, since the moisture content was about seven percent.  It can be observed from the equation that the  conductivity increases with an increase i n the water content. When the heating medium is below the boiling point of water, there is no significant difference i n the rate of heating wood at different moisture contents ranging up to 15  approximately 20 percent.  The moisture content affects the  rate of temperature rise at heating temperatures well above 212 deg. F . , since part of the heat entering the wood evaporates the water.  Thus, the heating medium in the apparatus was  never permitted to reach a temperature of 212 deg. F . Studies of moisture distrubution have shown that when wood with a uniform moisture content was subjected to a 14. United States Department of Agriculture, Forest Products Laboratory, Forest Service, Wood Handbook, No. 72, 195515. United States Department of Agriculture, Forest Products Laboratory, Forest Service, The Rate of Temperature Changes in Wood Panels Heated Between Hot Plates, No. 1299, June, 1955-  temperature gradient, there were often marked increases i n l6 the moisture content near the cold side of the specimen. These variations in moisture distribution were due to differences in vapour pressure produced by the difference in temperatures. The variation was mainly influenced by the original amount of water i n the wood and by the magnitude of the temperature gradient between the faces of the specimen.  These studies  have also shown that the changes i n moisture distribution were comparatively small when the average i n i t i a l moisture content was approximately ten percent or less. There was only a small temperature gradient across the cedar and the hardboard in the wall of the transient heat transfer apparatus, and the moisture content i n both was under ten percent.  Thus, the change in the thermal conductivity  of the hardboard and cedar due to the variation i n the moisture distribution was negligible.  The f i r boards had  large temperature gradients across then, but since they had a moisture content of approximately six percent, the effect of a non-uniform moisture content was again negligible, (d)  Structural Characteristics Knots, checks, and cross grain structure have no  appreciable effect on the conductivity of wood when they T&~. J . D . MacLean, "Thermal Conductivity of Wood", Heating, Piping, and A i r Conditioning, V o l . I l l , No. 6, June, 1941.  are not numerous.  Large knots have a tendency to increase  the conductivity, and small checks'have l i t t l e or not effect. Wood with pronounced cross grain has an increased conductivity in the direction of the cross grain, (e)  Heat It was found by J . D. MacLean that the effect of heat  on the physical properties of wood depends upon several factors which include the temperature to which the wood Is 18  exposed, and the time the temperature is maintained.  The  oven dry weight decreases i f wood is subjected to a high temperature for a long period of time because of charring. The rate and amount of this decrease depends upon the temperature.  The average reduction i n the oven dry weight of  wood was found to be 2 . 7 percent for a heating period of one year, and a temperature of 200 deg. F . These results indicated that the temperature should not be appreciably higher than 150 deg. F . i f a good service l i f e i s desire. ( f)  T emperat ure There is a slight increase i n the thermal conductivity  with an increase in the average wood temperature.  It was  found in conductivity tests by J . D. MacLean that the conductivity varied from nearly zero to a maximum value of less than four 17.  MacLean, op_. c i t .  1 8 . J . D. Maclean, "Rate of Disintegration of Wood Under Different Heating Conditions", Proceedings of American WoodPreservers Assoc., 4 7 : 1 5 5 - 6 8 , 1 9 5 1 .  percent with temperature differences across a specimen ranging from 22 deg. F . to 96 deg. F.  Thus, the effect of  the temperature on the thermal conductivity may be neglected since the largest temperature difference across any board was kO deg. F. B.  Specific Weight The specific weight of wood of any species varies  considerably from tree to tree and even within the same 19 tree.  There is usually considerable variation i n the  specific weight of the veneer cut from a single log.  Thus,  wood is generally a non-homogenous material. C.  Specific Heat The specific heat varies with temperature according to  20 the following formula:  Cp = 0.266 + 0.00064MT - 32) The average specific heat over a particular temperature interval is given by:  Ti  The specific heat is nearly independent of the specific gravity.  The moisture content has a marked effect on the  specific heat, since the specific heat of wood is close to that  19. United States Department of Agriculture, Forest Products Laboratory, Forest Service, The Rate of Temperature Changes in Wood Panels Heated Between Hot Plates, No. 1299, June, 1955. 20. F . Kollmann, Technologie Des Holzes und der Holzwerkstoffe, Berlin, Springer, Vol.1, 1951-  of a i r at 32 deg. F.  The average s p e c i f i c heat which now  includes the moisture content i s given by:  C = W'/THS T ? M/100 + 1  M  S p e c i f i c Properties of Individual Wall Materials A.  F i r and Cedar Boards The species of the wood used i n the wall were f i r and  cedar.  A l l of the boards of both species were chosen with  a straight grain running lengthwise along the board and p a r a l l e l t o the edge.  They were also chosen free of knots  and checks and were oven d r i e d .  Since the thermal  conductivities i n the r a d i a l and tangential d i r e c t i o n s are nearly equal, i t was assumed that the thermal conductivity across the board and through the board were equal (Figure l h ) . Both species of wood were assumed t o be homogeneous i n the analysis as the varation i n the s p e c i f i c g r a v i t y of the cedar was h.J percent and of the f i r was 6.9 percent as 21 determined by t e s t s . B.  Hardboard The hardboard i s a fibrous material with wooden f i b r e s  randomly oriented i n the plane of the hardboard. the hardboard i s also an anisotropic material.  Thus, The conductivity  i n any d i r e c t i o n i n the plane of the hardboard was assumed t o be equal but the conductivity through the hardboard was l e s s since the heat flowing through i t crossed the g r a i n . 21.  Appendix C.  The h a r d b o a r d was assumed t o be homogeneous s i n c e t h e v a r i a t i o n i n t h e s p e c i f i c w e i g h t o f t h e samples was o n l y 2.5 p e r c e n t .  The s p e c i f i c h e a t cannot be c a l c u l a t e d b y t h e  f o r m u l a u s e d f o r wood, s i n c e t h e h a r d b o a r d f i b r e s  are  bonded t o g e t h e r by g l u e . C.  Fibreglass Insulation The f i b r e g l a s s i n s u l a t i o n was assumed t o be an i s o t r o p i c  m a t e r i a l s i n c e i t d i d n o t appear t o have any d i r e c t i o n a l properties.  The i n s u l a t i o n was t r e a t e d as a homogeneous  m a t e r i a l although the v a r i a t i o n i n the s p e c i f i c weight of the samples f r o m t h e average v a l u e was 12.5 p e r c e n t .  APPENDIX C .  DETERMINATION OF THE PHYSICAL PROPERTIES OF THE WALL MATERIALS  Steady S t a t e D e t e r m i n a t i o n o f k^, kj , k ,  kf,  c  The p r o p e r t i e s k , A- , k , and k , h  h  0  c  t  f  ho and h  and t h e s u r f a c e  and h were d e t e r m i n e d b y a s t e a d y s t a t e t e m p e r a t u r e  of the w a l l .  coefficients analysis  The h e a t f l u x was measured a t t h e c e n t r e o f t h e one  d i m e n s i o n a l h e a t f l o w r e g i o n w i t h a heat t r a n s d u c e r .  The D . C .  m i l l i v l o l t s i g n a l f r o m t h e t r a n s d u c e r was a m p l i f i e d and t h e n r e c o r d e d on an o s c i l l o g r a p h .  The average s i g n a l was d e t e r m i n e d by  e v a l u a t i n g t h e a r e a under t h e c u r v e w i t h a p l a n i m e t e r , and d i v i d i n g t h e a r e a b y t h e base l e n g t h .  The average m i l l i v o l t  s i g n a l was t h e n c o n v e r t e d t o u n i t s o f h e a t f l u x b y u s i n g a c a l i b r a t i o n c u r v e and c o n v e r s i o n f a c t o r s u p p l i e d b y t h e m a n u f a c t u r e r . The t e m p e r a t u r e d i f f e r e n c e s fibreglass  were measured a c r o s s t h e h a r d b o a r d ,  i n s u l a t i o n , and cedar by t h e r m o c o u p l e s . ki> and k were t h e n c a l c u l a t e d f r o m  The c o n d u c t i v i t i e s k ,  c  h  F o u r i e r ' s equation.  U s i n g t h e d a t a f r o m t e s t one and t h e f o l l o w i n g  22 equations:  , _  n  dx  dT dra-b « f o r c e d a r , dT =  ,  6  - - r -v  ^ " • CO 1  ,  7  ^  f o r h a r d b o a r d , dT =  ,  7  6  5  1  Appendix F .  a  M v b  = 4.56  2  f o r i n s u l a t i o n , dT =  22.  M  Q  0  ^  <  " ^  6  ^  ,  T  2  ^  deg. F .  = U5.i3.deg. F. =1.14  deg. F .  for cedar, dx = 0.75 inches for insulation, dx = 3-25 inches for hardboard, dx = 0.244 inches Thus, k = 0.0633 B.T.U./hr. f t . deg. F . c  kl = 0.0277 B.T.U./hr. f t . deg. F . kf, = 0.0844 B.T.U./hr. f t . deg. F . The average value determined for ki from tests one to seven was 0.0282 B.T.U./hr. f t . deg. F . with a deviation of 2.8 percent. Since the variation was very small, the average temperature gradient across the insulation was used to determine the heat flux for tests eight,nine, and ten. The temperature difference across the convection film on the inside surface was measured and the surface coefficient was calculated from Newton's equation for surface convection. h = <L dT From test one, dT =  1  ,  8  l  9 Q  ^ '  7  6  ?  = 2.29 deg. F.  h = | ^ | | = 2.02 B.T.U./hr. f t .  2  The average temperature difference across the convection film on the outside surface was determined by evaluating the average height of the curve given by a trace of the temperature difference on an oscillograph.  This average temperature difference and heat  flux given by tests 11 to 20 determined the outside surface / 23 coefficient, "o • The heat flow was one dimensional along the centreline of a frame component since the temperature distribution was symmetric about the centre axis (Figure 11).  Thus, the temperature  differences were measured across the hardboard, f i r and cedar at the centreline of a f i r board.  The heat flux was determined by  using the previously calculated conductivites of the hardboard and cedar.  The conductivity, A/ , was then evaluated.  This method of determining the conductivities and surface coefficients appeared to be reliable, since the calculated value of kh agreed within one percent of the manufacturer's value. The average physical properties were tabulated i n Appendix D. Steady State Oven Tests to Determine kih , k  jc  In order to obtain  , and kjt  , k , and kpf, i t was necessary to Jc  test specimens with the heat flowing parallel to the grain. Specimens were mounted in a plywood frame and placed in the doorway of an oven with the grain oriented perpendicular to the plane of the door (Figure 15).  The oven a i r was well circulated  by a fan to produce a uniform surface coefficent.  The dimensions,  heat flow, and temperature difference for each specimen were  23-  Appendix F .  measured and the longitudinal conductivity was calculated from Fourier's equation.  The heat flow was determined by using a heat  flux transducer as described on page jk of this appendix. Determination of the Specific Weight and Moisture Content The volume and the wet weight, which is defined as the weight under current moisture content, was measured for each specimen removed from the cedar and f i r boards.  These samples were oven  dried for 2k hours at a temperature of 105 deg. C. and weighed. From these values the moisture contents were calculated. Since the moisture content of the wall materials was expected to be reduced by the higher temperatures i n the apparatus, the moisture content of some cedar samples was determined after they were subjected to the higher temperatures.  Tests one to eight for  moisture content were made on the cedar samples not subjected to the conditions in the apparatus and tests nine to twelve were performed on specimens subjected to the temperatures.  It was  impossible to measure the moisture content of the f i r under test conditions since i t was an interior wall material.  However, the  moisture content of the f i r was considered negligible, as the i n i t i a l moisture contents of the f i r and cedar were approximately equal and since the f i r was subjected to test temperatures of 117 deg. F. to 160 deg. F . while the cedar was subjected to lower temperatures of 107 deg. F . to 117 deg. F . and had a moisture  content of only 4.22 percent.  The moisture content of the hardboard  was also considered insignificant since i t was subjected to an average temperature of l 6 l deg. F . The specific weight of the hardboard, cedar, and f i r was based on the oven dried weight.  The specific weight of the hardboard  was 55-92 l b . / f t . ^ with a variation of 2.7 percent, and the cedar •z  was 20.22 l b . / f t .  with a variation of 5-5 percent.  weight of the fibreglass insulation was 4 l b . / f t .  The specific but i t was  compressed to a thickness of 5'25 inches in the wall and thereby increased i t s specific weight to 4.92 l b . / f t .  The variation i n  the weight of the insulation was 12.2 percent. Determination of Specific Heat The specific heat of the f i r was calculated from the following 24 equation, since the moisture content was considered negligible.  90  = 0.320 B . T . U . / l b . deg. F . For the cedar with a moisture content of 4.22 percent and at an average temperature of 105 deg. F . the following method was used: M/100 + Cp M/100 + 1  .0422 + .266 + .000644(105 - 52) 1.0422  = 0.341 B . T . U . / l b . deg. F .  W.  Appendix B.  The specific heat of the fibreglass insulation was stated by the manufacturer to be 0 . 2 0 B . T . U . / l b . deg. F .  The specific  heat of the hardboard was found to be 0 . 2 5 5 B.T.U./lb deg. F . in the following section of this appendix. Determination of the Thermal Diffusivity of Hardboard The thermal diffusivity of the hardboard was found directly by experiment.  The experiment was based upon the following analysis  which showed that the slope of the transient cooling curve, plotted on semi-logarithmic graph paper, for a point at the 25 midpoint of a plate was related to the thermal diffusivity. The following assumtions were made for a plate of thickness, 2 I ,which was assumed to be homogenous. (a)  —(T - T^) = 0 at X - 0 , the centre of the plate  (b)  h is very large, such that k//j tends to zero  (c)  -kg|(T - T^) = h (T - T«,) at DC =J  (d)  T - TL* = T i - T^ at t = 0  (e) the solution is of the form T(x,f) = X(X) T(t) Applying (e) to the heat diffusion equation: the following solution was obtained: T  from (a)  25.  ~ oo T  =  e  £ " C  COS/TJX  + C sinmX]  C =0 2  Giedt, o £ . c i t . , pp. 293 - 2 9 7 .  2  from (c)  k e~ °^C s i n m i = h e*"*^, cos m£ m  t  thus  cotmi' = mk ^ /  from (b)  cot ml = 0  and m = ( " 2  +  1  2  )  -j-  Upon considering (d), the solution becomes: oo  Ti -  Too"  f2.ru-1  ^Tr^o? f  „=o "  ^  2  x  Now,as £ becomes large, terms with n >s I are negligible as compared to the n=o term.  Thus, by taking the natural logarithm  of both sides when t is large l n (T -  Too)  t  - l n (Ti - T«J = l n C 0  For a thermocouple, the voltage, M.V., is proportional to the temperature difference for a finite range. l n (M.V. - M.V. ) =  Thus  Tr o< z  When the natural logarithm of (M.V. - M.V^) time, the slope of the line is equal to  is plotted against  -  Two sheets of hardboard, six inches square, were glued together with a thermocouple at the centre of the interface. The effect of the heat loss at the edges was negligible since the ratio of the length of the sides to the thickness was large. The ratio was approximately twelve to one.  The plate was  waterproofed with a coating of varnish, which was made as thin as possible i n order to minimize the effect of the varnish on the diffusivity.  The specimen was heated to a uniform temperature and  immersed i n a cold stream of water.  The flow was made as large  as possible i n order to achieve assumption (b).  The cooling  temperatures were measured on a potentiometer and plotted. From the f i r s t experiment (Figure l 6 ) : /  =0.25 inches  M.V. - M.V.^ = 0.309 m.v. at 7J = 5 minutes M.V. - M.V^ = 0.016 m.v. at t = 11.75 minutes and solving the above equation for log.  0.509 - log. 0.016 11-75 - 5-0  7t- °< a  =  4(2.505)(.25)  2  = 17-142 o<  T h u s , = 0.00465 f t . / h r . 2  Also,0, =  = 0.255 B . T . U . / l b . deg. F.  From the second experiment, J! =0.25 inches M.V. - M.V.^ = 0.478 m.v.  at t  = 4.0 minutes  M.v. - M.V.  at t  ='8.0 minutes  = 0.082 m.v.  Thus,«*X = 0.00465 f t . / h r . 2  and  Cf, - 0.254 B . T . U . / l b . deg.'F.  The average values used were: cx^ = 0.00464 f t . / h r . 2  C  h  = 0.255 B . T . U . / l b . deg. F.  42 APPENDIX D.  PHYSICAL PROPERTIES OF THE WALL MATERIALS  = 0.541 B . T . U . / l b . deg. , F .  <v  = 0.520 B . T . U . / l b . deg. . F. = 0.255 B . T . U . / l b . deg. . F .  Q  = 0.20 B . T . U . / l b . deg. F.  *c  = 0.065 B.T.U./hr. f t . deg. F.  k<  = 0.06l B.T.U./hr. f t . deg. F.  ki,  = 0.066 B.T.U./hr. f t . deg. F.  ki  = 0.028 B.T.U./hr. f t . deg. F.  kf  = 0.147 B.T.U./hr. f t . deg. F.  (  k&  = 0.501 B.T.U./hr. f t . deg. F.  kjtu = 0.526 B.T.U./hr. f t . deg. F. h  = 2.02 B.T.U./hr. :ft. deg. ;F. 2  = 4.04 B.T.U./hr. ft. deg. :F. 2  - 0.00914 f t . / h r . 2  = 0.00545 f t . / h r . 2  = 0.00464 f t . / h r . 2  <*< = 0.0284 f t . / h r . 2  °<*  = 0.0215 f t . / h r . 2  = 0.0268 f t . / h r . 2  = O.0229 f t . / h r . 2  = 20.22 l b . / f t .  5  = 55.07 l b . / f t . ; 5  = 55.92,lb./ft. = 4.92 l b . / f t .  5  5  43  APPENDIX E. SUMMARY OF THEORETICAL EQUATIONS Tla  .273T2a + •273T6a + •373Tlb + ..08lTla  1  T2a'  = .091Tla + .046T3a + .274T7a + .,374T2b + .215T2a  T3a«  = .034T2a + .034T4a + .275T8a + ,•375T3b + .282T3a  Tlj-a  = .034T3a + .034T5a + .275T9a + .•375T4b + .282T4a  1  T5a«  .069T4a + • 274T10a -t• .374T5b + .283T5a = .091Tla + .046Tlla + .274T7a + •374T6b + ,.215T6a  T6a« TTa'  .091T6a + .091T2a + .046T8a + ,,046T12a + ,374T7b + .352T7a  T8a'  = .034T7a + •034T9a + .046T13a + .091T3a + .37^T8b + .421T8a  T9a'  = .034T8a + .034T10a + .046T14a -(• .091T4a + .37UT9b + .421T9a  TlOa' = .069T9a + .046T15a + .091T5a + •374T10b + .420T10a Tlla  1  = .03U-T6a + . 0 3 4 T l 6 a + .275T12a + .374Tllb + .283Tlla = .03l+T7a + .034T17a + .046T13a •>.I- .091Tlla + .374T12b + .421T12a  T12a'  T15a*  .034T8a + .034T12a + .034T14a -I- . 0 3 4 T l 8 a + .374T13b + .490T13a  T14a'  .03+T9a + .034T13a + .034T15a + .034T19a + .374T14b + .49CT14a i  T15a* = .069T14a + .034T10a + .034T20a + .37^15b + .489T15a Tl6a' = .034Tlla + .034T21a + .274T17a + .374Tl6b + . 2 8 4 T l 6 a T17a = .034T12a + .034T22a + . 0 4 6 T l 8 a + .091Tl6a + .37^T17b + .421T17a f  Tl8a» = .034T17a + .034T13a + .034T19a + .034T23a + -374Tl8b + . 4 9 0 T l 8 a T19a*  .03 +Tl8a + •034T14a + .034T20a + ,034T24a + .37 +T19b + . 4 9 0 T l 9 a )  1  T 2 0 a ' = .069T19a + .034T15a + T21a'  .034T25a + .37^T20b + ,489T20a  = •069Tl6a + .274T22a + .37^T21b + .283T21a  T22a* = .069T17a + .046T23a + .091T21a + .374T22b + .420T22a  T23a' = .069Tl8a + .034T22a + .034T24a +  .374T23b + .489T23a  T24a'  •069T19a + .034T23a + .034T25a + .374T24b + .489T24a  T25a'  .069T20a +  .069T24a + .374T25b + .488T25a  kk  Tib'  = .083Tla + .OlTTlc + •lllT2b + .310T6b + ,479Tlb  T2b'  = .083T2a + .,017T2c + .03TTlb + .052T3b + .178T7b + ,633T2b  T3b'  = .083T3a + .,017T3c + .039T2b + .039T4b + .lllT8b + .711T3b  T4b'  = .083T4a + ..01TT4c + .039T3b + .039T5b + .lllT9b + .711T4b  T5b'  = .083T5a + .,017T5c + .078T4b + .lllTlOb + .711T5b  T6b'  = .083T6a + .,01TT6c + .lllT7b + .103Tlb + . 0 5 2 T l l b + .634T6b  TTb'  = .121TTa + .,019TTc + .054T6b + .039T8b + .087T2b + .039T12b  T8b'  = .158T8a + .021T8c + •038T9b + .038T7b + .071T3b + .027Tl3b + .6kTS&b = .156T9a + .021T9c + .038T10b + .038T8b + ,071T4b + .027T14b  +  T9b'  +  .64lT7b  .647T9b  TlOb* = .15&T10a + .021T10c + .071T5b •+• .076T9b + .027T15b + .6U7T10b T l l b ' = .083Tlla + .OlTTllc + •039T6b ••i- .039Tl6b + .lllT12b + .711Tllb = .15&T12a + .021T12c + .038T17b + .03&T7b + .071Tllb + .027T13b  T12b'  + .647T12b  T l j b ' = .286T13a + .027T13c + .036T8b •4- .036T12b + . 0 3 6 T l 8 b + .036T14b + .543T13b  T14b'  = .286T14a + •027T14C + •036T9b + .036Tl3b + .036T19b + .036Tl5b + .543T14b T15b* •= .286T15a + .027T15c + .072T14b + .036T10b + .036T20b + .543T15b T l 6 b ' = . 0 8 3 T l 6 a + .017Tl6c + .039Tllb + .039T21b + . l l l T 1 7 b + .711Tl6b  TlTb* = .158T17a + + .647T17b Tl8b* = .286Tl8a + + .543Tl8b T19b' = .286T19a + + .543T19b T20b = .286T20a + f  •021T17C  + .038T22b + .038T12b + .071Tl6b + .027Tl8b  •027Tl8c + .036T13b + .036T17b + .036T23b +  .036T19b  .02TT19C + .036Tl4b + .036Tl8b + .036T24b + .036T2Gb .027T20c + .072T19b + .036T15b + .036T25b +  .543T20b  T21b = .083T21a + .017T21c + .076Tl6b + .lllT22b + .7HT21b f  T22b = .158T22a + .021T22c + •071T21b + .072T17b + .027T23b + .647T22b !  T 2 3 b ' = .286T23a + .027T23C + .072Tl8b + .036T22b + .036T24b +  .543T23b  T2kb *  = .286T24a + .027T24C + .072T19b + .036T23b + .036T25b + .543T24b  T25b  = .286T25a + .027T25c + .072T24b + .072T20b + .543T25b  f  45 Tic  = • O l l T l b + • O l l T l d + .065T2c + •321T6c + . 5 9 2 T 1 C  1  T2c'  = .011T2b + .011T2d + .022Tlc + .053T3c + .151T7c + .752T2c  T3c'  = .011T3b + .011T3d + .040T2c + .04OT4C + .065T8c + .833T3c  T4c'  = .011T4b + .011T4d + .040T3c + .040T5c + . 0 6 5 T 9 C + .833T4c  T5c'  .OUT 5b + .011T5d + .080T4c + .065TIOC + .833T5C  T6c'  = .011T6b + •011T6d + •107T1C + .053T11C + .065T7C + .753T6c  T7c'  .Ol4T7b + . O l 4 T 7 d + .l80T6c + .084T2c + .036T8c + .036T12c +  .636T7C  .0l8T8b + .Ol8T8d + .055T3C + •04lT7c + . 0 4 l T 9 c + . 0 1 3 T 1 3 C  T8c«  + .8I4T8C .Ol8T9b + .0l8T9d + .055T4c + •04IT8C + ,04lT10c + .013Tl4c + .8l4T9c T10c» = .Ol8T10b + •Ol8T10d + •055T5C + . 0 1 3 T 1 5 C + .082T9C + .8l4T10c T9c'  Tile  = • O l l T l l b + . O l l T l l d + .040T6c + .040Ti6c + .065Tl2c + .833Tllc  1  .Ol8T12b + .0l8T12d + .055TIIC + .04lT7c + . 0 4 l T i 7 c + .0l3T13c + .8l4T12c T13c* = .058T13b + .058T13d + .043T8c + . 0 4 3 T l 4 c + .043Tl8c + .043T12c + .712T13c Tl4c' .058Tl4b + . 0 5 8 T l 4 d + .043T9c •+ .043T15c + .043Tl9c + .043Tl3c T12c'  —  + .712Tl4c T15c' = .056T15b + .056Tl5d + .086Tl4c + .043T10<c + .043T20c + .712T15C T l 6 c ' = .011Tl6b + .011Tl6d + .c40Tllc + .040T21c + .065T17c + .833Tl6c T17c'  —  .Ol8Tl7b + .Ol8T17d + .055Tl6c + .04lT12c + .04lT22c + .013Tl8c  + .8l4T17c  T l 8 c ' = .058Tl8b + • 05&Tl8d + .043T13C + .043T19^c + .043T23c + .043Tl7c T19c'  + .712Tl8c .05&Tl9b + . 0 5 8 T l 9 d + .043Tl4c + .0U3T2Oc + .o43T24c + .043Tl8c +  .712T19C  T20c' = .05&T20b + .058T20d + .086T19C + .043T15-c + .043T25c + .712T20c T21c' = •011T21b + .011T21d + .080Tl6c + .065T22c + . 8 3 3 T 2 1 C T22c'  .0l5T22b + . 0 l 8 T 2 2 d + .055T21c + .013T23c + .082T17c + .8l4T22c  T23C = •.058T23b + ,058T23d + .086Tl8c + .043T22c + .043T24c + .712T23c 1  T24c' = .058T24b + .058T24d + .086T19C + .043T23c + .C43T25C + .712T24c T25c'  .05&T25b + .05&T25d + . 0 8 6 T 2 0 C + .086T24c + . 7 1 2 T 2 5 C  Tld  = . O l l T l c + .OllTle + .065T2d + .321T6d + .592Tld  1  T2d'  = .011T2c + .011T2e + .022Tld + . 0 5 3 T 3 d + .151T7d + .752T2d  T3d'  = ,011T3c + .011T3e + .040T2d + .OkOlkd + .065T8d + .833T3d  T4d'  = .OllT^c + .OllT^e + .C4ai?3d + .C40T5d + .065T9d + .833T4d  T5d'  = .OllTSc + .011T5e + .0803?4d + .065T10d + .833T5d  T6d'  = .011T6c + .011T6e + .lCTTld + .053Tlld + .065T7d + .753T6d  T7d'  = .Ol4T7c + .Ol4T7e + .l8OT6d + .084T2d + .036T8d + .036T12d  + .636T7d - .Ol8T8c + .Ol6T8e + .055T3d + .c4lT7d + .c4lT9d + .013T13d  T8d'  + .8l4T8d  T9d*  = .Ol6T9c + .01&T9e + .055T4d + .04lT8d + .O^lTlOd + .013Tl4d + .8l4T9d  TlOd' = .OloTlOc + .Ol8T10e + .055T5d + .013T15d + .082T9d + .8l4T10d Tlld  1  T12d  f  = . O l l T l l c + . O l l T l l e + .c40T6d +  .0U0Tl6d  + .065T12d + .833Tlld  = .0l8T12c + .Ol6T12e + .055Tlld + .c4lT7d + .c4lT17d + .013T13d + .8l4T12d  T13d' = .05&Ti3c + .056Tl3e + .C43l8d + .c43TlUd + .c43Tl8d + .c43T12d + .712T13d Tl4d« = .056Tl4c + .058Tl4e + .c43T9d + .c43Tl5d + .c43Tl9d + .c43T13d + .712Tl4d T15d' = .05&T15c + .058T15e + .086Tl4d + .c43T10d + .c43T20d + .712T15d T l 6 d ' = .011Tl6c + .011Tl6e +  .c4OTlld + .c4OT21d + .065T17d + .833Tl6d  T17d' = .Ol8T17c + .Ol8T17e + .055Tl6d + .c4lT12d + .c4lT22d + .013Tl8d + .8l4T17d Tl8d' = .058Tl8c + .05&Tl8e + .C43T13d + .c43T19d + .043T23d + .043T17d + .712Tl8d T19d' = .055T19C + .058T19e + .C43Tl4d + .043T20d + .C43T24d + .c43Tl8d + .712T19d  T20d'  = . 0 5 6 T 2 0 C + .05&T20e + .086T19d + .c43T15d + .c43T25d + .712T20d  T21d'  =  T22d'  = .01&T22c + .0lST22e + .055T21d + .013T23d + .082T17d + .8l4T22d  .011T21C  + .011T21e  + .080Tl6d + .065T22d + ,833T21d  T23d' = . 0 5 8 T 2 3 C + .058T23e + .086Tl8d + .c43T22d + .C43T24d + .712T23d T24d' = .058T24c + .056T24e + .086T19d + .c43T23d + .c45T25d + .712T2l+d T25d' = . 0 5 6 T 2 5 C  + .05&T25e + .086T20d + .086T2Ud + .712T25d  kl Tie'  —  T2e'  = .015T2d + .023T2f + ,040Tle + .05OT3e + .138T7e + .73^T2e  .015Tld + .023Tlf  + .122T2e + .258T6e + .582Tle  T3e'  ,015T3d + .023T3f + •037T2e + .OJTShe + .078T8e + .8lOT3e  T4e'  ,015T4d + .023T4f + .037T3e + .037T5e + .078T9e + .8lOT4e  T5e'  .015T5d + .02JT5f +. .OlhUhe +,078T10e + .8l0T5e  T6e'  = .015T6d + ,023T6f + .086Tle + • C 4 3 T l l e + .122T7e + . 7 H T 6 e  TTe'  zz .01?T7d + .025T7f + .064T2e + .057T6e + .038T8e + .028T12e  T8e'  —  T9e'  -  + .771T7e .017T8d + .040T8f + .036T7e + •036T9e + .045T3e + . 0 l 6 T 1 3 e + .8lOT8e .017T9d + .04OT9f + .036T8e + .036T10e + .OkjSke + .0l6T14e + .8lOT9e  TlOe' zz .017T10d + •040T10f + .072T9e •+• .045T5e + •Ol6T15e + .8lOT10e •• .032Tl6e + .122T12e + .776Tlle T i l e ' zz .015Tlld + .023Tllf + •032T6e + T12e zz .017T12d + .C40T12f + .OTlTlle + .027T7e + .027T17e + .028T13e + .790T12e T15e* — .020T13d + •065T13f + .034T12e + .034T14e + .019T8e + .019Tl8e + .809T13e T14e' zz •020T14d + .065T14f + .034T13e + .034T15e + .019T9e + .019T19e f  + .809T14e  - .020T15d + .065Tl5f + .068T14e + .019T10e + .019T20e + .809T15e  T15e'  Tl6e* = .015Tl6d + . 0 2 3 T l 6 f + .032Tlle + .032T21e + .122T17e + .776Tl6e .017T17d + .040T17f + + .790T17e Tl8e* = .02OTl8d + .065Tl8f + + .809Tl8e .020T19d + .065T19f + T19e' + .809T19e .020T20d + .065T20f + T20e' T17e'  T21e  f  . 0 7 1 T l 6 e + .027T12e + .027T22e + .02&Tl8e .034T17e + .034T19e + ,019T13e + .019T23e .034Tl8e + ,034T20e + .019T14e + .019T24e  .068T19e + .019T15e + .019T25e + .809T20e  .015T21d + .023T21f + .065Tl6e + .122T22e + .775T21e  T22e = .017T22d + .040T22f + .071T21e + .054T17e + .028T23e + .790T22e !  T23e' = . 0 2 0 T 2 3 d + .065T23f + .034T22e + .034T24e + .038Tl8e + .809T23e T2i+e' = .020T24d + .065T24f + .034T23e + .034T25e + .03&T19e + .809T24e T25e'  ,02OT25d + .065T25f + .03&T20e + .068T24e + .809T25e  kd Tlf•  = .076Tle + .315TO + . 2 5 5 T 2 f + . 1 0 9 T 6 f + .245Tlf  T2f'  = .07&T2e + .313To + .085Tlf + .C42T3f + .109T7f + .373T2f  T3f*  = .078T3e + .313To + .032T2f + .032T4f + .109T8f + .436T3f  Thf'  = ,076Ti+e + .313TO + .032T3f + .032T5f + .109T9f + .436T4f  T5f'  = .07&T5e + .313TO + .06kTkf  + .109T10f + .436T5f  T6f *• = .07&T6e + .313T0 + .037Tlf + .Ol&Tllf + .255T7f + .299T6f = .078T7e + .313To + .037T2f + .0l8T12f + .085T6f + .c43T8f + .426T7f T8f = .078T8e + .313To + .032T7f + .032T9f + .037T3f + .Ol8Tl3f + .490T8f T9f = .078T9e + .313To + .032T8f + .032T10f + .OJfEkf + .0l8T14f + .4-9ca?9f TlOf' = .078T10e + .313To + .037T5f + .Ol8T15f + .064T9f + .490T10f T7f  1  1  1  T l l f ' = . 0 7 8 T l l e + .313To + .013T6f + .015Tl6f + .255T12f + . 3 2 6 T l l f T12f' = .078T12e + + .453T12f T15f* = .078T13e + + .519T13f T14f = .07&T14e + + ,519T14f T 1 5 f * = .078T15e + 1  .313To + .085Tllf + .043T13f + .014T7f + .014T17f .313To + .013T8f + .013Tl8f + .032T14f + .032T12f .313To + .015T9f + . 0 1 5 T 1 9 f + .032T15f + .032T13f  .313To + .013T20f + .064T14f + .OlJTlOf + .519T15f  T l 6 f * = .078Tl6e + .313TO + .013T21f + ,255T17f + .013Tllf + .32&Tl6f T17f' = .07&T17e + .313To + .085Tl6f + .043Tl8f + .C04T12f + .014T22f + .453T17f T l S f ' = .Q78Tl8e + .313T0 + .013T13f + .013T23f + .032T19f + .032T17f + .5igri8f T19f = .078T19e + .313TO + .013T14f + .013T24f + .032T20f + .032Tl8f + ,5l9Tl9f 1  T20f  1  = .078T20e + .313To + .064T19f + .Q13T15f + .013T25f + .519T20f  T 2 l f = .078T21e + .313To + .027Tl6f' + .255T22f + ,327T2lf T22f' = .078T22e + -313To + .027T17f + .085T21f + .043T23f + .454T22f  T23f  1  = .078T23e + .313To + .032T22f + .032T24f + .027Tl8f + .5l8T23f  T24f' = .07&T24e + .313To + .032T23f + .032T25f + .027T19f + .5l8T24f T25f' = .078T25e + .313To + .027T20f + .064T24f + »5l8T25f  49 APPENDIX F . DATA Data to Determine h  0}  Test 1 2 3 4 5 6 7 8  10  l)>  k and k c  c  M.V.i  M.V.a  M.V.b  M.V.e  M.V.f  . 1.819 1.828 1.823 1.820 1.809 1-771 1-790  1.763 1.758 1.750 1.760 I.766 1.706 1-738 1.675 1.667 1.650  1-735 1.723 1.718 1.723 1.727 1.675 1.698 1.643 1.627 1.617 1.617 1.616 1.616 1.615 1.618 1.617 1.619 1.600 1.605 1.605 1.605 1.603 1.604  0.652 0.653 0.642 o.64i 0.637 0.623 0.619 0.578 0.554 0.540 0.538 0.537 0.537 0.538 0.543 0.543 0.546 o.54o 0.538 0.538 0.541 0.538 0.538  0.547 0.551 0.541 0.542 0.538 0.530 0.520 0.487 O.457 0.4n 0.427 0.412 0.434 0.424 0.445 0.434 0.453  1.65^  9  h ,k  1.688 1.678 1.688 1.695 1.697 1.688  1-655 1.653 1.655 I.656 1.654 I.656 I.638 1.632 1.637 I.636 1.635 I.636  0.330 0.329 0.328 0.326 0.326 0.326 0.323 0.321 0.315 0.309  M.V.  0.301 0.303 0.302 0.300 0.299 0.296 0.296 0.293 0.289 0.280  Q 4.62 4.77 4.74 4.69 4.78 4.52 4.63 4.54 4.54 4.54 4.54 4.54 4.54 4.54 4.54 4.54 4.54 4.49 4.49 4.49 4.49 4.49 4.49 4.73 4.73 4.73 4.73 4.73 4.73 4.73 4.73 4.73 4.73  50  Data to Determine A f Test  M.V.a  M.V.b  M.V.e  M.V.f  11 12 13 14 15 16 17 18 19 20  1.588 1.597 1.604 1.608 1.603 1.608 1.611 1.611 1.610 1.611  1.530 1.536 1.538 1.542 1.541 1.542 1.546 1.544 1.544 1.543  0.620 0.620 0.624 0.625 0.624 0.628 0.629 0.629 0.626 0.628  0.446 0.449 0.417 0.423 0.448 0.423 0.446 0.424 O.455 0.427  1.540  O.625  0.435  Average  1.605  Data from Oven Tests to Determine Test  and kj/^  Average Inside Temp.  Average Outside Temp.  170.12 170.12  139.96 140.96  151.89 148.34  0.716 0.716  182.31 181.00 184.73  148.00 147.72 151.21.  209.39 189.68 192.45  0.668 0.668 0.668  191.36 192.88  132.96 131.67  130.45 134.95  0.795 0.795  Q  dx  51 Data f o r Moisture Content and S p e c i f i c Weight of F i r and Cedar  Cedar Test  1 2 3 4 5 6 7 8 9 10 11 12  Wet Weight  Dry Weight  grams  grams  0.8248 0.8099 0.8536 0.8437 O.8502 0.8290 O.8760 1.8027 0.6860 O.5910 O.6180 0.6810  0.774  2.l6l 2.421 2.069 2.270 2.222 2.126 3.721 4.701 4.285 3.660  /o Moisture Content  Volume  S p e c i f i c Gravity  m.l.  0.802 0.794 0.799 0.776 0.814 1-677 0.659 0.567 0.591 0.655  6.59 6.44 6.48 6.30 6.38 6.83 7.62 7.51 4.10 4.23 4.57 3.97  2.426 2.413 2.433 2.407 2.420 2.433 2.407 5.190  0.319 0.315 0.330 0.330 0.330 0.319 0.338 0.323  2.028 2.268 1.936 2.111 2.075 1.984 3-469 4.389 4.006 3.4o6  6.56 6.75 6.87 7-53 7.08 7.16 7-26 7.11 6.96 7.46  3.645 4.066 3.439 3.867 3.642 3.478 6.239 7.796 6.967 5.982  0.556 0.558 0.563 0.545 0.570 0.570 O.556 O.563 0.575 O.569  O.76I  Fir  1 2 3 4 5 6 7 8 9 10  52 Specific Weight of Hardboard Sample  1 2 5 4 5 6 7 8 9 10 ll 12 15 i4 15  Thickness  Length  Width  inches  inches  inches  0.245 0.244 0.244 0.245 0.243 0.245 0.244 0.243 0.244 0.246 0.246 0.243 0.244 0.245 0.246  1.985 2.009 2.010 2.008 2.009 1.987 1.993 2.003 2.011 2.013 2.010 1.998 2.008 2.006 2.010  2.007 2.011 2.012 2.011 1.998 2.007 2.011 2.010 2.011 2.011 2.011 2.011 2.008 2.011 2.OO8  Volume 3 inches  0.976 O.986 O.987 O.989 0.975 0.977 0.978 O.978 0.987 0.996 0.994 0.976 0.984 O.988 0.993  Weight grams  14.33 14.30 14.42 14.73 lU.59 14.36 14.23 14.00 '14.35 14.76 14.81 14.32 14.46 14.55 14.54  Specific Weight lb./ft?  55.92 55.24 55.65 56.73 57-00 55.98 55.^2 5U.53 55-38 56.45 56.75 55.89 55.97 56.09 55-77  Specific Weight of Fibreglass Insulation 1 2 3 4 5 6 7 8 9 10 11 12 13 14  15 16 17 18  2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00  5-70 4.20 7.00 20.00 1.75 3.62 6.72 18.38 18.44 18.44 6.25 5.76 13.12 10.31 19.56 15.03 5.75 13.31  1.86 3.62 1.88 2.56 I.65 2.22 2.38 7-00 7.06 6.28 2.32 2.44 1.33 1.36 1.43 3.69 2.44 1.81  21.20 50 . U l  26.25 102.40 7.78 16.07 31.87 257.32 256.37 231.61 29.06 28.11 34.90 28.04 55.9h 110.92 28.06 48.18  22.04 32.31 25.24 117.12 5.74 16.26 32.56 271.64 282.39 242.55 35.99 31.05 32.18 25.74 61.68 119.18 30.68 47.87  3.96 4.04 3.66 4.36 3.9U 3-84 3-87 4.02 4.13 3-99 4.74 4.21 3.51 3.51 4.20 4.10 4.17 3.78  Thermal D i f f u s i v i t y o f H a r d b o a r d  Test 1 Time  M.V.  Test 2  M.V. - M . V . ^  M.V.  M . V . - M,  min. 0.00 1.00 2.00 3.00 4.00 4.50 5.00 5-25 5-50 5-75 6.00 6.25 6.25 6.75 7.00 7.25 7.50 7-75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 10.25 10.50 10.75 11.00  2.359 2.037 1.448 1.044 O.783 O.689 O.615 O.583 0.551 0.529 0.507 0.482 0.465 0.447 0.434 0.417 0.409 0.398 O.389 0.378 0.373 0.364 O.360 0.353 0.348 0.344 0.341 0.336 0.334 0.329 0.328  T e s t 1 , M.V.«, = 0 . 3 0 6 0.304 Test 2,  2.053 1.731 1.142 0.738 0.477 O.383 0.309 0.277 0.245 0.223 0.201 0.176 0.159 0.l4l 0.128 0.111 0.102 0.092 0.083 0.072 0.067 0.058 0.054 0.047 0.042 0.038 0.035 0.030 0.028 0.023 0.022  2.360 2.043 1.456 1.046 0.782 0.687 0.612 0.578 0.551 0.523 0.503 0.481 0.463 0.442 O.433 0.418 0.405 0.394 O.386 0.378 0.369 0.362 0.356 0.352 0.347 0.34l O.338 0.334 0.331 0.329' 0.327  2.056 1.739 1.152 0.742 0.478 0.383 0.308 0.274 0.247 0.219 0.199 0.177 0.159 0.138 0.129 0.114 0.101 0.090 0.082 0.074 0.065 0.058 0.052 0.048 0.043 0.037 0.034 . 0.030 0.027 0.025 O.023  5* I n i t i a l Temperatures Node  1 2 3  4  5 6 7 8 9 10 11 12 13  14  15 16 17 18 19 20 21 22 23 24  25  Plane a  Plane b  Plane c  Plane d  Plane e  Plane f  162.3 162.3 162.3 162.3 162.3 162.3 162.3 162.3 162.3 162.3 162.3 162.3 163.2 163.2 173.2 162.3 162.3 162.9 163.2 163.2 162.3 162.3 162.9 163.2 163.2  159.2 159.2 159.2 159.2 159.2 159.2 159.2 159-7 159-8 159.8 159.2 159.2 l6l.5 161.8 l6l.8 159.2 159.2 160.2 161 ;8 161.8 159.2 159.2 160.2 161.8 161.8  145.8 145.8 145.8 145.8 145.8 145.8  131.5 131.5 131.5 131.5 131.5 131.5 131.0 131*2 131.2 131.2 131.5 130.8 129.3  117.1 117.1 117.1 117.1 117.1 117.1 116.9 116.9 116.8 116.8 117.1 116.7 113.2 113-2 113.2  108.2 108.2 108.2 108.2 108.2 108.2 108.0 108.1 108.2 108.2 108.2 107.9 107.8 107.8 107.8 108.2 108.0 107.8 107.8 107.8 108.2 108.0 107.8 107.8 107.8  145.5 145.4 145.7 145.7 145.8 145.5 145.3  1U5.5 145.5 145.8 145.5 145.3 145.3 145.3 145.8  129.4 129.4  131.5 130.6 129.4  145.5  129.2 129.2 131.5 130.6  145.3 145.3 145.3  129.2 129.2  O u t s i d e a i r t e m p e r a t u r e , To = IO6.5  129.4  II7.I  116.3 113.3 113.0 113.0 117.1 II6.3  II3.3 113-0 113.0  Transient Temperatures from Numerical Analysis Time  Three Dimensional Analysis  hours  T25a - To  T25a - To  55.8  55-6  49.3 43.8 39-4  48.6  0.0 0.5 • 1.0 1.5 2.0 2.5 3.0  3-5  4.0 4.5 5-0  5-5 6.0 6.5 7.0 7-5 8.0 8.5 9.0 9-5 10.0 10.5 11.0  11.5 12.0 12.5 13.0  13.5 14.0  35-7 32.6 . 29.8 27.4  25.2 23.1 21.3 19.5 17.9 16.5 15.1  38.6 34.5 30.8 27.5 24.6 22.0  19-6  17.5 15.7  14.0 12.5 11.2  10.0  9.8 9.0  6.3 5-7 5-1 4.5 4.0 3.6  12.7 11.6 10.6  8.3 7.6 7.0 6.4 5.9 5-4 5.0 4.6 4.2 3.9  16.0  3.2 3.0  15.5 16.5 17.0 17.5 18.0 18.5 19.0 19-5 20.0  43-3  13.9  15.0  14.5  One Dimensional Analysis  3.5  2.7 2.5 2.3 2.1 1.9 1.8 1.6  8.9 8.0 7-1  3.2  2.9 2.6 2.3 2.1 1.8 1.6 1.5 1.3 1.2 1.0  0.9 0.8 0.7 0.7 0.6  Time  Three D i m e n s i o n a l Analysis  One D i m e n s i o n a l Analysis  T25a- To  T25a - T o  20.5 21.0 21.5 22.0 22.5  1^5 1.4 1-3  23.0  1.2 1.1 1.0  0.5 0.5 0.4 0.4  23-5  0.9  hours  24.0  0.8  0.3  0.3 0.3 0.2  Transient Temperatures from Experiment Time hours  0.0 1.0 2.0 3.0 4.0 5-0 6.0 7.0 8.0 9-0 10.0 11.0 12.0 13.0 l4.0  15-0 16.0 17-0 18.0  Test 1  T25a - To 57.1 49-3 43.6  37.1 30.9  27.1 23.1  19.6 16.9 13.5  10.9 9-3 7.8  6.1  4.9  4.0 3.3 2.6 1.8  Test 2  T25a - To 56.6 49.6  43.1 36.9 30.7  26.2 22.6  19.9  18.0 16.0 13-7 9.7 6.9 5-7 4.8 3.9 2.9 1.9 0.9  To correct f o r error i n recorder timing system, multiply times by O.96  APPENDIX G.  FIGURES  FIG. I  GENERAL  VIEW  OF  APPARATUS  FAN  CIRCUMFERENTIAL R A D I A T I O N SHIELD CIRCUMFERENTIAL HEATERS  WALL  HEATER SUPPORT GUARD  SECTION  EXIT S E C T I O N  P U L L E Y AND BELT DRIVE  TEST SECTION HEATERS W I T H SHIELDS FAN  THERMOSTAT  TEST  SECTION  TOP RADIATION SHIELD HEATER  SUPPORT  HEATER AND  SHIELD  TOP H E A T E R S WIRE S U P P O R T FOR RADIATION SHIELD  FAN  vn FIG.  2  SIDE  SECTION  OF  APPARATUS  FIG.3  FRONT  SECTION  OF  THE  APPARATUS  61  FIG. 4  FIR  FRAME  HARDBOARD  FRAME COMPONENT ( P L A N E D  FIBREGLASS INSULATION  CEDAR  5  CROSS  SECTION  A T A FRAME  COMPONENT  2 * 4 )  62  «  FIG. 6  TEMPERATURE  CONTROL  EXIT  SECTION  S Y S T E M  FOR  THE  4- OHMS HEATER 6 OHMS  HEATER  THERMOSTAT, T I  GUARD  SECTION  5 HEATERS PARALLEL  IN  10 H E A T E R S IN PARALLEL  FIG. 7  HEATER A N D C O N J R O L  CIRCUITS  FOR  THE  TEST  AND  GUARD  SECTIONS  CONDEN SER  T3  FIG.  8  SIMPLIFIED  ELECTRONIC  CONTROL  CIRCUIT  FIG. 9  RECORDER  AND  THERMOCOUPLE  SWITCH 2 0 X  66  COMPONENTS OF EXIT SECTION TEMPERATURE CONTROL SYSTEM  FIG. 10  WALL  AND  °F INSIDE  1401.  S U R F A C E OF HARDBOARD, ( P L A N E A )  ^INTERFACE OF HARDBOARD AND INSULATION, ( P L A N E B)  -EDGE OF F R A M E 130..  COMPONENT  o THERMOCOUPLE  "*  READING  CENTRELINE OF F R A M E COMPONENT A N D E D G E O F GRID N E T W O R K  120..  C E N T R E L I N E OF F R A M E  AXIS O F SYMMETRY  —AXIS -EDGE  O F GRID  COMPONENT-  OF S Y M M E T R Y  NETWORK  110..  • C E N T R E L I N E OF T E S T  SECTION  100.. I N T E R F A C E O F CEDAR ' AND I N S U L A T I O N , ( P L A N E E)  90..  -9  F I G . II  — OUTSIDE SURFACE OF CEDAR, (PLANE F )  a-  STEADY  STATE  TEMPERATURES  0\ —5  68  VERTICAL  <{; OF WALL  C E N T R E SECTION OF THE FRAME AXIS OF S Y M M E T R Y SECTION USED IN ANALYSIS  REGIONS OF 2-D HEAT FLOW REGION OF l-D HEAT FLOW AXIS OF S Y M M E T R Y ORIZONTAL <£ OF WALL  REGIONS OF 3-D HEAT FLOW  AXES OF TEMPERATURE SYMMETRY  FIG. 12  AXES OF T E M P E R A T U R E REGIONS  OF  HEAT  SYMMETRY FLOW  AND  69  a  I  o  $  2  a— <•  II  &  12  13  2X  14  15  19  20  2X  P L A N VIEW' 17  18  X= U = XT= ur= 2Z  21  23  24  -6  0.633" 0.244" 1.083" 0.750"  25  -PLANE A  u  -PLANE B -HARDBOARD  -NODE ()  -PLANE C  O  — INSULATION v  o  —PLANE  <> v  D  ELEVATION  — PLANE E  us  — CEDAR —PLANE F  •<£ OF F R A M E  COMPONENT  FIG. 13 NODE SYSTEM FOR ANALYSIS  VIEW  70  DIRECTION OF HEAT FLOW  k a - N O R M A L OR AXIAL CONDUCTIVITY k t - T A N G E N T I A L CONDUCTIVITY  GRAIN  Kt  ka  FIR  F I G . 14  |  0 DIRECTION  8  OF H E A T F L O W GRAIN  CEDAR  AXIAL  AND  T A N G E N T I A L  CONDUCTIVITIES  THERMOCOUPLES DIRECTION OF GRAIN MOTOR  HEAT T R A N S D U C E R SPECIMEN  DIRECTION OF HEAT FLOW  FIG. 15 APPARATUS FOR LONGITUDINAL CONDUCTIVITY TESTS  71  T25Q-TQ  Tzsa^-To  0  2  4  6  8  10  12  14 T I M E  F I G . 17  COOLING  CURVES  FOR  16  18  20  - (HOURS)  INSIDE  WALL  SURFACE  22  24  73  0  2  4  6  8  10  TIME  FIG. 18 THEORETICAL  12  14  16  (HOURS)  COOLING CURVES  18  20  22  & O  TEST ONE TEST TWO  EXTRAPOLATED CURVES  8  10 TIME  F I G .  19  EXPERIMENTAL  14  12  16  18  (HOURS)  C  O  O  L  I  N  G  CURVES  20  22  Tzs  a.-To  0  2  4 - 6  8  10  12  14  16  18  20  ZZ  24  T I M E - (HOURS)  FIG.20  THEORETICAL  AND CORRECTED  EXPERIMENTAL  COOLING  CURVES  

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