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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 Apri l , 1962 In presenting this thesis in partial 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 this investigation was to examine the transient heat flow through a composite wall. This wall was chosen to represent the type used in house construction. It consisted of a f i r frame, covered on one side with hardboard and on the other with cedar, and the space between the hardboard and cedar was f i l l e d with fibreglass insulation. A vapour barrier was not included as i t would offer l i t t l e resistance to heat flow. This structure, therefore, offered resistances to heat flow i n series and paral l e l . The theoretical analysis was 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 effect of the frame. The heat flow was three dimensional in the f i r s t analysis owing to the difference i n the magnitude of the parallel resistances and was one dimensional i n the approximate analysis. The two theoretical solutions both showed exponential cooling rates and agreed within five percent of each other, which shows that the effect of the frame is negligible when i t s surface area i s small as compared to the total surface area of the wall. The ratio of t o t a l wall surface area to frame area for the wall studied was 9.6 to 1.0. The wall was mounted i n a guarded hot-box apparatus and experiments were performed in order to verify the results of the theoretical analysis. The experiments consisted of establishing 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 British 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 2k Appendix B. General Physical Properties of Wood and other Wall Materials 27 Appendix C. 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 hj> Appendix F. Data h$ Appendix G. Figures 58 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 in i t ia l phase was the investigation of the transient response of heated air 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 in 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 fluid 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 luid. The results of the paper by W. A. Wolfe showed that when the fluid is a ir , 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 Air 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 in 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, in order that the heat flow would be either normal to or parallel with the grain. This is important as the grain causes 3 anisotropy in wood. Hardboard was glued and tightly screwed to the hot side of the frame ih an effort to reduce contact resistances produced by air spaces. A cedar panel was securely fastened to the cold side of the frame. 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 intact. With this arrangement, insulation could be changed, and thermocouples relocated. A fibreglass insulation was installed for the experiments performed i n this study, and was compressed into place to 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 for the cedar panel. The panel was also constructed to minimize the effects of a i r spaces between the boards. The Test Section The temperature measurements were taken i n the test section, which was the centre portion of the wall, plus an adjacent a i r enclosure (Figures 2 and 5 ) - The sides of the test section were away from the edge of the wall i n order that the heat losses at the edges would not affect the temperatures i n the test section. A control system prevented any transfer of heat across the boundaries of the a i r enclosure. Thus, a l l the heat i n the test section passed directly through the wall. The casing of the test section consisted of aluminum coated heavy kraft paper, insulated by an aluminum lined fibreglass insulation, with the l i n i n g on the outside to prevent radiation to the test section. This insulation was used to decrease the response of the test section to the surrounding guard section. Two heaters, one of four ohms resistance, and the other of six ohms, were suspended i n the test section. They could be used separately, together, i n pa r a l l e l , or series, depending on the power required. The voltage to the heaters was controlled by a variac, and this voltage determined the temperature difference across the wall. These heaters had radiation shields i n order to prevent radiation to the wall, since only conduction heat transfer was desired. A balsa wood fan was placed i n the test section to mix the a i r , and create a uniform surface coefficient over the surface of the wall. This fan was driven by an electric motor, which was on the outside of the apparatus. The Guard Section The function of the guard section was to isolate the test section. It surrounded the test section, and was enclosed by a plywood casing, with two plastic windows for viewing the inside of the apparatus (Figures 1, 2, and 3). This casing was lined with fibreglass insulation i n order to reduce heat losses. Ribbon heaters, supported by wooden rods with porcelain insulators, encircled the test section to give a uniform heat generation throughout the guard section. There were ten of these heaters i n the guard section which could be used on a continuous 110 volts, and five that were connected to 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 to the variac. Two fans, mounted in opposite corners of the apparatus, circulated the air 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 air to encircle the test section. The Exit Section The exit section was an air 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, in order to prevent radiation from the heaters being reflected to the wall. The temperature in 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 air 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 in the guard and test section. As shown in Figure 7> sensing elements of a bridge system were placed in the guard and test sections. These resistors were extremely sensitive to temperature changes. The difference in the resistances of the elements caused by a temperature difference between the test and guard s e c t i o n s , produced an unbalanced b r i d g e system. T h i s l a c k 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 t h e r e l a y ( F i g u r e 8). 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 temperature was h i g h e r , t h e r e l a y would 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 c o n t a c t s between t e r m i n a l s one and t h r e e . T h i s c l o s e d t h e c i r c u i t o f t h e 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 motor . 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 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 topped t h e motor . The s h a f t of t h e motor was connected t o t h e handle o f a v a r i a c , and as t h e motor t u r n e d , i t reduced the output v o l t a g e o f t h e v a r i a c , and decreased t h e power s u p p l i e d t o t h e guard s e c t i o n h e a t e r s . W i t h t h e output v o l t a g e o f t h e v a r i a c r e d u c e d , t h e temperature i n t h e 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 . T h i s a c t i v a t e d t h e 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 the power s u p p l i e d t o t h e h e a t e r s , and r e s t o r i n g t h e temperature b a l a n c e . As t h e motor t u r n e d from the 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 output v o l t a g e o f t h e v a r i a c ranged from 35 t o 110 v o l t s . T h i s c y c l i c c o n t r o l caused a maximum v a r i a t i o n o f 0.2 deg. F . i n t h e s teady s t a t e temperature i n t h e t e s t s e c t i o n . The Thermocouples Thermocouples were used t o measure t h e temperatures i n t h e r e g i o n s o f one, two and t h r e e d i m e n s i o n a l heat f l o w , and t o check t h e o p e r a t i o n o f t h e 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 thermocouples were i n s t a l l e d . They were connected through 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 c o n v e r t e d t h e output o f t h e 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). E a c h 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 connector 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 which c o u l d be a t t a c h e d t o any one o f t h e female connectors 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 temperatures i n any one o f t h e t h r e e r e g i o n s o f heat 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 output o f one o f t h e thermocouples every 15 seconds w i t h an a c c u r a c y o f 0.2 d e g . F . The r e l a x a t i o n method was used 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 determine t h e temperature p r o f i l e s on a k p l a n e normal t o the a x i s o f a component o f t h e frame. These temperature p r o f i l e s were used as a guide t o l o c a t e t h e thermocouples 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 wall. The thermocouple leads were taken along isothermal lines for two inches before branching away from the wall. 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 test section a i r enclosures i n order to check the operation of the temperature control system. Five thermocouples i n the guard section were connected i n para l l e l , and their signal was read on a potentiometer. This reading was compared with the output of a thermocouple located i n the test section and when the average value of each signal was equal, the controls were functioning properly. TEST PROCEDURE Preparation for the Test The fans and the control system were started and the control point adjustment was set at the maximum position. This caused the variac to supply the maximum voltage to the control heaters i n the guard section. A l l of the heater circuits were then closed to give maximum heating. One or both of the heater circuits i n the test section was closed, and the variac was set to give the voltage required for a particular temperature drop across the wall. For the tests performed i n this study, the six ohm heater was used, and the variac was set at four volts to give a temperature difference across the wall of approximately 56 deg.F. The thermostat in the exit section was set to the desired position, 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 circuits in the exit section were closed. When the temperature in the test section was near the desired value, the control point setting 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 to reach steady state, since the heat lost and gained by the test section a i r enclosure was equal during each control cycle. It was extremely d i f f i c u l t to both acquire and maintain the steady s t a t e c o n d i t i o n due t o inadequate 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 temperature d i f f e r e n c e between t h e guard s e c t i o n and t h e room, t h u s as t h e temperature i n t h e apparatus approached t h e d e s i r e d v a l u e , and as t h e room temperature changed, t h e c o n t r o l p o i n t had t o be a d j u s t e d 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 . When s teady s t a t e had been o b t a i n e d , t h e apparatus was ready f o r t h e t e s t . Performance o f t h e T e s t The temperatures were measured throughout t h e t e s t s e c t i o n w h i l e t h e s teady s t a t e e x i s t e d i n o r d e r t o determine the i n i t i a l temperatures f o r the t h e o r e t i c a l a n a l y s i s . F o l l o w i n g t h i s , s e v e r a l 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 thermocouples i n t h e two and t h r e e d i m e n s i o n a l heat f l o w r e g i o n s . The male connector was t h e n a t t a c h e d t o t h e connector w i t h t h e thermocouples I n t h e one d i m e n s i o n a l r e g i o n . T h u s , t h e t r a n s i e n t temperatures i n t h e t h r e e areas o f heat 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 throughout t h e t e s t . 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 b e g i n t h e t e s t . 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 guard 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 temperature d i f f e r e n c e o c c u r i n g between t h e hot a i r and t h e s u r f a c e o f t h e 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 t h e assumption i n t h e a n a l y s i s t h a t t h e heat c a p a c i t y o f t h e a i r . was n e g l i g i b l e . A temperature 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 t h e w a l l o f t h e t e s t s e c t i o n a i r enc losure ' i n o r d e r t o p r e v e n t heat 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 decrease i n t h e c o o l i n g r a t e . T h i s temperature 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 temperature symmetries a t t h e b o u n d a r i e s of t h e t e s t s e c t i o n would be d i s t u r b e d . T h i s temperature drop was m a i n t a i n e d by 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 heat s u p p l y . I f t h e temperature d i f f e r e n c e a c r o s s the w a l l o f t h e a i r e n c l o s u r e became t o o l a r g e , t h e temperature o f t h e a i r i n t h e t e s t s e c t i o n would become l o w e r t h a n t h e temperature i n t h e w a l l ' s s u r f a c e . T h i s would cause heat t o f l o w i n t h e wrong d i r e c t i o n and thus i n c r e a s e t h e c o o l i n g r a t e o f the w a l l . I n o r d e r 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 tems 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 temperature 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) the temperature d i f f e r e n c e between t h e guard and t e s t s e c t i o n s . F i n a l l y , when a temperature 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 t e r m i n a t e d . 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 its 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 its surrounding temperature distribution did not extend beyond four inches from the centreline. Thus, as shown in 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 girt , 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, in 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. The size of this region was chosen to make the boundaries the two dimensional heat flow regions and the corners the one dimensional regions. The Exact Analysis The small portion of the wall analysed in 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 air and apparatus components in the test section was negligible. (k) The materials were homogeneous. (5) The boundaries of the region were adiabatic. The f irst 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 air over both sides of the wall. The heat capacity of the apparatus components in the test section was minimized by using light materials with low specific heats. The effect of the heat capacity of the air was shown to be negligible in an analysis by 8 W. A. Wolfe. The fourth assumption was good in 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 in 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 in 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 specific weight throughout the wall was not known. The f i f t h assumption was based on the fact that the temperature gradients normal to the boundaries were negligible (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 a at node 25a: • ^ ^ ( T 2 0 a - T25a)<5#+ ^£2££< ( T 24a - T25a)*#"+ - ^ C ^ b - T 2 5 a ) < ^ = €h Cfl? U (T25a« - T25a) The l e f t side of the equation represents the heat flowing into and element during a time interval, dt, and the right side i s the change i n heat content during that time interval. Rearranging the above equation to solve for T 2 5 a ' . T 2 5 a » = ^ ^ ( T 2 0 a + T24a) + ZftJhdt'S25b + [ 1 - ( 2 + k b^l)°^^\ T25a 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 coefficient of T25a was termed the self influence coefficient, since i t affects i t s own future temperature. The self influence coefficient must be positive, otherwise an i n s t a b i l i t y w i l l arise in the equations. 1^ This i n s t a b i l i t y i s produced by a thermo-dynamically impossible condition, where the future temperature 9. Dusiriberre, ep. c i t . , p. 115. 10. Ibid., p. 116. a t the end of the time i n t e r v a l , w i l l become lower as the temperature at the beginning of the i n t e r v a l becomes h i g h e r . A l s o , the e o e f f i c e n t should not equal zero, 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 the heat capacty of the element and the node would then have no e f f e c t on i t s f u t u r e temperatures. The most convenient time i n t e r v a l , which made a l l of the 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 minute. The equation f o r node l a governed the s e l e c t i o n of t h i s time i n t e r v a l . The remaining 149 equations were d e r i v e d i n the same manner."^ The t r a n s i e n t temperatures were c a l c u l a t e d by s u b s t i t u t i n g the i n i t i a l temperatures i n t o the equations, and c a l c u l a t i n g the temperatures at the end of the f i r s t time i n t e r v a l . These temperatures became the i n i t i a l values f o r the next time i n t e r v a l and the c a l c u l a t i o n was repeated. T h i s procedure was continued u n t i l the temperature d i f f e r e n c e across the w a l l was n e g l i g i b l e . The c a l c u l a t i o n s were performed on the U n i v e r s i t y of B r i t i s h Columbia's computer, ALWAC I I I E. The i n i t i a l temperatures were obtained by d i r e c t measurement, and by i n t e r p o l a t i n g values from the curves drawn from the measured temperatures The Approximate A n a l y s i s The equation f o r one dimensional heat f l o w f o r node 25a 11. Appendix E. 12. Appendix F. 17 was simplified from the three dimensional case to: -^pL*(T25b - T25a)^ = ^ ^ ( 1 2 ^ ' - T25a) rearranging terms and letting M = — The maximum value for d£ was found hy letting the self influence 2 coefficient, ( l equal zero. Therefore, 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: -^^-2(T25a - T25b)^ + (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 T25df = 0.116 T25c + 0.116 T25e + O.768 T25d T25e* = 0.040 T25d + 0.130 T25f + 0.830 T25e T25f' * 0.156 T25e + 0.626 To + 0.218 T25f The in i t ia 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 irst ten hours than that shown by the theoretical results, and a faster cooling rate after the first ten hours. The determination of the exact transient period for the wall was diff icult , since the curves asymtotically approach zero, but the period was approximately 24 hours for an in i t ia l temperature difference across the wall of 55-6 deg. F. The agreement between the two tests indicated that the experimental results were consistent. OBSERVATIONS AND CONCLUSIONS 19 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 hours o f c o o l i n g was due t o heat b e i n g l o s t f rom t h e i n s i d e s u r f a c e o f t h e t e s t w a l l t o t h e guard s e c t i o n . T h i s heat 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 system, 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 temperature t o become l o w e r t h a n t h e temperature on t h e w a l l ' s s u r f a c e . T h i s poor 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 a p p a r a t u s : ( l ) a good t h e r m a l response between t h e quard and t e s t s e c t i o n s and (2) t h e l a r g e temperature 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 t h e r e l a y s . As t h e temperature i n t h e guard s e c t i o n g r a d u a l l y f e l l d u r i n g t h e c o o l i n g p o r t i o n o f a c o n t r o l c y c l e , t h e temperature i n t h e 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 . T h e r e f o r e , t h e 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 t h e c o n t r o l s o c c u r e d a f t e r a l o n g p e r i o d o f t i m e . 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 the h e a t i n g power was l a r g e enough t o cause t h e guard s e c t i o n temperature 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 t e m p e r a t u r e . T h u s , more heat was l o s t t h a n added d u r i n g a c o n t r o l c y c l e . 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 s e c t i o n 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 produced was not 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 not be added as i t would increase the heat capacty of the test section, thereby slowing down the cooling of the wall. The cooling was exponential for the f i r s t nine hours of test one, but afterwards the slope of the curve gradually increased because of the poor control (Figure 19). The straight portion of the the curve was extrapolated and new temperatures were obtained. The same was done to the second experimental curve which showed a sudden change after ten hours of cooling. The values obtained from the extrapolated portions were plotted and the resulting curves exhibited very good agreement (Figure 20). These new curves were more r e a l i s t i c as they allowed for the heat lost from the inner surface of the wall. As these curves are exponential, they may be expressed by the equation: -P T - To = (Ti - To) e *t where *?• , the time constant, depends only on the characteristics of the wall. The average value of t obtained from the slope of the logarithmic curves was 6.334 hours. Thus, the equation can be used to determine the cooling of the wall for any i n i t i a l temperature difference across the wall. The difference between the experimental and theorectical results were attributed largely to the non-homogeneity of the materials, particularily i n the case of the insulation where the variation i n the specific weight was 12.2 percent. The effect of any contact resistances and of the heat capacity of the test section would be to slow down the cooling rate i n the tests. Although this effect was indeterminate, i t was probably small owing to the careful construction of the apparatus. It was l i k e l y that the surface coefficient of heat transfer on the cool side of the wall was not uniform since the two fans were not capable of producing uniform a i r circulation over the entire surface of the wall. The surface coefficient would be lower i n the regions where the a i r movement was less over the surface. Thus, the actual mean value of the coefficient was less than the value used, since the value of the coefficient could only be measured in the region of one dimensional heat flow, where the a i r circulation was good. A l l of these factors would cause the experimental cooling rate to be less than the theoretical rate. Inaccuracies i n the thermocouples would account for some of the difference between theory and experiment. However, this error i s not l i k e l y to be more than two percent since calibration of the thermocouples showed errors of 1.5 deg. F. and less i n 100 deg. F readings. It i s possible that thermocouples may become loose after installation and cause large errors. However, no indication of this was observed on temperature records, but detection would be d i f f i c u l t i f thermocouples detached between tests. 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 in i t ia 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 t / I f , where t = 6.334 hours for the wall investigated in this thesis. (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 to Heat Transfer, New York, McGraw-H i l l Book Co. Inc., 19587 Batchelor, G. K i , "Heat Transfer by Free Convection Across a Closed Cavity Between Vertical Boundaries at Different Temperatures,,, Quarterly of Applied Mathematics, 12: No. 3, October, 1954. Campbell, W. G., Form and Style 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., 1949-Giedt, W. H., Principles of Engineering Heat Transfer, New York, D. Van Nostrand Co. Inc., 1957. Kollmann, F., Technologie Des Holzes und der Holzwerkstoffe, Berlin, Springer, Vol.1, 1951. 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 Disintegration of Wood Under Different Heating Conditions", Proceedings of American Wood-Preservers  Assoc., 47:155-68, 1951. Smith, E. G., "A Simple and Rigourous Method for the Determination of Heat Requirements of Simple. Intermittently Heated Exterior Walls", Journal of Applied Physics, 12:638-642, 194l. Tiemann, H. D., Wood Technology, New York, Pitman Publishing Corp., 1951. United States Department of Agriculture, Forest Products Laboratory, Forest Service, Wood Handbook, No. 72, 1955. United States Department of Agriculture, 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, 8lr 19-23, February, 1959• APPENDIX 2k APPENDIX A. LIST OF SYMBOLS 2 A area,ft. C specific heat of wet wood,B.T.U./lb. deg. F. Q specific heat of cedar, B .T.U. / lb . deg. F. C f specific heat of f i r , B .T.U. / lb . deg. F. specific heat of hardboard, B.T.U. / lb . deg. F. Q specific heat of fibreglass insulation,B.T.U./lb. deg. F. Cm mean specific heat of dry wood, B.T.U. / lb . deg. F. Cp specific heat of dry wood, B.T.U. / lb . 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. h0 surface coefficient of heat transfer for the cold surface, B.T.U./hr. f t . deg. F. k thermal conductivity, B .T.U. / in . /hr . f t . deg. F. kc 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. /hr . f t . deg. F. 25 kjc thermal conductivity of cedar parallel to the grain, B.T.U./hr. f t . deg. F. 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 f t . 2 S specific gravity of wood based on volume at current moisture content and weight when oven dried. t time, hr. T temperature, deg. F. T i in i t i a l hot surface temperature, deg. F. To cold air temperature, deg. F. T temperature of surrounding f luid, deg. F. W original wet weight. * cartesian coordinate p o< thermal dlffuslvity, k/ c, f t . /hr . <=<c thermal diffusivity of cedar normal and tangential to the grain, f t . /hr. thermal diffusivity of f i r normal and tangential to the grain, f t . /hr . c*^  thermal diffusivity of hardboard normal to the plane of the hardboard, f t . /hr . 2 °<i thermal diffusivity of fibreglass insulation, f t . /hr . 2 C X J C thermal diffusivity of cedar parallel to the grain,, f t . /hr. 2 o<^ thermal diffusivity of f i r parallel to the grain, f t . /hr . t h e r m a l d l f f u s i v i t y of hardboard i n the p l a n e o f t h e h a r d b o a r d , f t . / h r . m i l l i v o l t e q u i v a l e n t , m . v . / d e g . F . s p e c i f i c w e i g h t , l b . / f t . •z s p e c i f i c weight o f c e d a r , l b . / f t . s p e c i f i c weight o f f i r , l b . / f t . ^ s p e c i f i c weight o f h a r d b o a r d , l b . / f t . ^ •7. s p e c i f i c weight o f f i b r e g l a s s i n s u l a t i o n , l b . / f t . t i m e constant o f w a l l , h r . 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 b y : (a) t h e d i r e c t i o n o f t h e g r a i n ( b ) t h e s p e c i f i c g r a v i t y ( c ) t h e m o i s t u r e content and i t s d i s t r i b u t i o n ( d ) s t r u c t u r a l c h a r a c t e r i s t i c s (e) heat ( f ) temperature (a) The D i r e c t i o n o f the 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 the same, b u t i t i s g e n e r a l l y 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 t r a n s v e r s e 13 d i r e c t i o n s ( F i g u r e l 4 ) . J T h u s , wood i s an a n i s o t r o p i c m a t e r i a l . (b) The S p e c i f i c G r a v i t y 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 s p e c i f i c g r a v i t y . 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 s p e c i f i c g r a v i t y . 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 s p e c i f i c heat per u n i t volume b e i n g g r e a t e r t h a n the i n c r e a s e i n t h e t h e r m a l c o n d u c t i v i t y . T h u s , t h e t h e r m a l d i f f u s i v i t y g e n e r a l l y decreases as t h e s p e c i f i c g r a v i t y i n c r e a s e s . 1 3 . 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 P r o d u c t s 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* (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 in the wall, since the moisture content was about seven percent. It can be observed from the equation that the conductivity increases with an increase in the water content. When the heating medium is below the boiling point of water, there is no significant difference in 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, 1955-15. 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 in l 6 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 in the wood and by the magnitude of the temperature gradient between the faces of the specimen. These studies have also shown that the changes in moisture distribution were comparatively small when the average in 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 in both was under ten percent. Thus, the change in the thermal conductivity of the hardboard and cedar due to the variation in 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 Air 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 in 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 fe is desire. ( f) T emperat ure There is a slight increase in 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 . 18 . J . D. Maclean, "Rate of Disintegration of Wood Under Different Heating Conditions", Proceedings of American Wood-Preservers Assoc., 4 7 : 155-68, 1951. 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 in 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 Holz-werkstoffe, Berlin, Springer, Vol.1, 1951-of a i r at 32 deg. F. The average specific heat which now includes the moisture content i s given by: C = W'/THS T ? M M/100 + 1 Specific 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 parallel to the edge. They were also chosen free of knots and checks and were oven dried. Since the thermal conductivities i n the radial and tangential directions 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 to be homogeneous in the analysis as the varation i n the specific gravity of the cedar was h.J percent and of the f i r was 6.9 percent as 21 determined by tests. B. Hardboard The hardboard i s a fibrous material with wooden fibres randomly oriented i n the plane of the hardboard. Thus, the hardboard i s also an anisotropic material. The conductivity in any direction i n the plane of the hardboard was assumed to be equal but the conductivity through the hardboard was less since the heat flowing through i t crossed the grain. 21. Appendix C. The hardboard 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 the s p e c i f i c weight o f the samples was o n l y 2.5 p e r c e n t . The s p e c i f i c heat cannot be c a l c u l a t e d by t h e f o r m u l a used f o r wood, s i n c e the h a r d b o a r d f i b r e s a r e bonded t o g e t h e r by g l u e . C. F i b r e g l a s s I n s u l a t i o n 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 not appear t o have any d i r e c t i o n a l p r o p e r t i e s . 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 a l t h o u g h t h e v a r i a t i o n i n the s p e c i f i c weight o f t h e samples from 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 , kc, kf, ho and h The p r o p e r t i e s kh , At- , kc , and k f , and t h e s u r f a c e c o e f f i c i e n t s h0 and h were determined by a steady s t a t e temperature a n a l y s i s o f the w a l l . The heat f l u x was measured a t t h e c e n t r e o f the one d i m e n s i o n a l heat 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 rom the 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 determined by e v a l u a t i n g the a r e a under the curve w i t h a p l a n i m e t e r , and d i v i d i n g the a r e a b y the 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 heat f l u x by u s i n g a c a l i b r a t i o n curve 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 temperature d i f f e r e n c e s were measured a c r o s s t h e h a r d b o a r d , f i b r e g l a s s i n s u l a t i o n , and cedar by thermocouples . The c o n d u c t i v i t i e s kh, ki> and kc were t h e n c a l c u l a t e d from F o u r i e r ' s e q u a t i o n . U s i n g the d a t a from t e s t one and t h e f o l l o w i n g 22 e q u a t i o n s : , _ n dx dT dra-b « M-v-ar M-v-b f o r c e d a r , dT = , 6 ^ 2 " = 4.56 deg. F . • C O f o r i n s u l a t i o n , dT = 1 , 7 ^ Q ^ < 6 ^ 2 = U 5 . i 3 . d e g . F . f o r h a r d b o a r d , dT = 1 , 7 6 5 0 " ^ , T ^ = 1 . 1 4 deg. F . 2 2 . Appendix F . for cedar, dx = 0.75 inches for insulation, dx = 3-25 inches for hardboard, dx = 0.244 inches Thus, kc = 0.0633 B.T.U./hr. ft . deg. F. 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 in Appendix D. Steady State Oven Tests to Determine kih , kjc , and kjt In order to obtain , kJc , and kpf, i t was necessary to 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 air 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 in 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 in i t ia 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 lb . / 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. The specific weight of the fibreglass insulation was 4 l b . / f t . but i t was compressed to a thickness of 5'25 inches in the wall and thereby increased its specific weight to 4.92 l b . / f t . The variation in 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. 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: 90 = 0.320 B .T.U. / lb . deg. F. M/100 + Cp .0422 + .266 + .000644(105 - 52) M/100 + 1 1.0422 = 0.341 B.T.U. / lb . 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. / lb . 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 ~ T o o = e £ C " COS/TJX + C 2 sinmX] from (a) C 2 = 0 2 5 . Giedt, o£. c i t . , pp. 2 9 3 - 2 9 7 . from (c) k e~m °^Ct s i n m i = h e*"*^, cos m£ thus cotmi' = mk/^  from (b) cot ml = 0 and m = ( 2 " 2 + 1 ) - j -Upon considering (d), the solution becomes: oo f2.ru-1 ^ Tr^o? f T i - T o o " „=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 ln (T - Too) - ln (Ti - T«J = ln C 0 - t For a thermocouple, the voltage, M.V., is proportional to the temperature difference for a finite range. Thus ln (M.V. - M.V. ) = Trzo< When the natural logarithm of (M.V. - M.V^) is plotted against time, the slope of the line is equal to -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 in order to minimize the effect of the varnish on the diffusivity. The specimen was heated to a uniform temperature and immersed in a cold stream of water. The flow was made as large as possible in order to achieve assumption (b). The cooling temperatures were measured on a potentiometer and plotted. From the f irst experiment (Figure l6): / =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 = 7t- a°<  11-75 - 5-0 4(2.505)(.25)2 = 17-142 o< T h u s , = 0.00465 f t . 2 / h r . Also ,0, = = 0.255 B .T.U. / lb . 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. = 0.082 m.v. at t ='8.0 minutes Thus,«*X = 0.00465 f t . 2 / h r . and Cf, - 0.254 B .T.U. / lb . deg.'F. The average values used were: cx^  = 0.00464 f t . 2 / h r . Ch = 0.255 B .T.U. / lb . deg. F. 42 APPENDIX D. PHYSICAL PROPERTIES OF THE WALL MATERIALS = 0.541 B .T.U. / lb . deg. , F. <v = 0.520 B .T.U. / lb . deg. . F. = 0.255 B .T.U. / lb . deg. . F. Q = 0.20 B .T.U. / lb . 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. 2deg. ; F. = 4.04 B.T.U./hr. ft. 2deg. : F. - 0.00914 f t . 2 / h r . = 0.00545 f t . 2 / h r . = 0.00464 f t . 2 / h r . <*< = 0.0284 f t . 2 / h r . °<* = 0.0215 f t . 2 / h r . = 0.0268 f t . 2 / h r . = O.0229 f t . 2 / h r . = 20.22 l b . / f t . 5 = 55.07 l b . / f t . 5 ; = 55.92,lb./ft. 5 = 4.92 l b . / f t . 5 APPENDIX E. SUMMARY OF THEORETICAL EQUATIONS 43 Tla 1 .273T2a + •273T6a + •373Tlb + . .08lTla T 2 a ' = .091Tla + .046T3a + .274T7a + . ,374T2b + .215T2a T3a« = .034T2a + .034T4a + .275T8a + , •375T3b + .282T3a Tlj-a1 = .034T3a + .034T5a + .275T9a + . •375T4b + .282T4a T5a« .069T4a + • 274T10a -t • .374T5b + .283T5a T6a« = .091Tla + .046Tlla + .274T7a + •374T6b + , .215T6a 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 T l l a 1 = .03U-T6a + . 0 3 4 T l 6 a + .275T12a + .374Tllb + .283Tlla T12a ' = .03l+T7a + .034T17a + .046T13a •>. I- .091Tlla + .374T12b + .421T12a T15a* .034T8a + .034T12a + .034T14a -I- . 0 3 4 T l 8 a + .374T13b + .490T13a T14a' .03i+T9a + .034T13a + .034T15a + .034T19a + .374T14b + .49CT14a T15a* = .069T14a + .034T10a + .034T20a + .37^15b + .489T15a Tl6a' = .034Tlla + .034T21a + .274T17a + .374Tl6b + . 2 8 4 T l 6 a T17af = .034T12a + .034T22a + . 0 4 6 T l 8 a + .091Tl6a + .37^T17b + .421T17a Tl8a» = .034T17a + .034T13a + .034T19a + .034T23a + -374Tl8b + . 4 9 0 T l 8 a T19a* .03)+Tl8a + •034T14a + .034T20a + ,034T24a + .371+T19b + . 4 9 0 T l 9 a T 2 0 a ' = .069T19a + .034T15a + .034T25a + .37^T20b + ,489T20a T21a ' = •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 + . l l lTlOb + .711T5b T 6 b ' = .083T6a + . ,01TT6c + .lllT7b + .103Tlb + . 0 5 2 T l l b + .634T6b TTb' = .121TTa + . ,019TTc + .054T6b + .039T8b + .087T2b + .039T12b + .64lT7b T8b' = .158T8a + .021T8c + •038T9b + .038T7b + .071T3b + .027Tl3b + .6kTS&b T9b' = .156T9a + .021T9c + .038T10b + .038T8b + ,071T4b + .027T14b + .647T9b TlOb* = .15&T10a + .021T10c + .071T5b • +• .076T9b + .027T15b + .6U7T10b Tl lb ' = .083Tlla + .OlTTllc + •039T6b • •i- .039Tl6b + .lllT12b + .711Tllb T12b' = .15&T12a + .021T12c + .038T17b + .03&T7b + .071Tllb + .027T13b + .647T12b Tljb' = .286T13a + .027T13c + .036T8b • 4- .036T12b + .036Tl8b + .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 + •021T17C + .038T22b + .038T12b + .071Tl6b + .027Tl8b + .647T17b Tl8b* = .286Tl8a + •027Tl8c + .036T13b + .036T17b + .036T23b + .036T19b + .543Tl8b T19b' = .286T19a + .02TT19C + .036Tl4b + .036Tl8b + .036T24b + .036T2Gb + .543T19b T20bf = .286T20a + .027T20c + .072T19b + .036T15b + .036T25b + .543T20b T21bf = .083T21a + .017T21c + .076Tl6b + .lllT22b + .7HT21b T22b! = .158T22a + .021T22c + •071T21b + .072T17b + .027T23b + .647T22b T23b ' = .286T23a + .027T23C + .072Tl8b + .036T22b + .036T24b + .543T23b T2kb * = .286T24a + .027T24C + .072T19b + .036T23b + .036T25b + .543T24b T25b f = .286T25a + .027T25c + .072T24b + .072T20b + .543T25b 45 T i c 1 = • O l l T l b + • O l l T l d + .065T2c + •321T6c + .592T1C T2c' = .011T2b + .011T2d + .022Tlc + .053T3c + .151T7c + .752T2c T3c' = .011T3b + .011T3d + .040T2c + .04OT4C + .065T8c + .833T3c T4c' = .011T4b + .011T4d + .040T3c + .040T5c + .065T9C + .833T4c T5c' .OUT 5b + .011T5d + .080T4c + .065TIOC + .833T5C T6c' = .011T6b + •011T6d + •107T1C + .053T11C + .065T7C + .753T6c T7c' .Ol4T7b + .Ol4T7d + .l80T6c + .084T2c + .036T8c + .036T12c + .636T7C T8c« .0l8T8b + .Ol8T8d + .055T3C + •04lT7c + . 0 4 l T 9 c + .013T13C + .8I4T8C T9c' .Ol8T9b + .0l8T9d + .055T4c + •04IT8C + ,04lT10c + .013Tl4c + .8 l4T9c T10c» = .Ol8T10b + •Ol8T10d + •055T5C + .013T15C + .082T9C + .8l4T10c T i l e 1 = • O l l T l l b + . O l l T l l d + .040T6c + .040Ti6c + .065Tl2c + .833Tllc T12c' — .Ol8T12b + .0l8T12d + .055TIIC + .04lT7c + . 0 4 l T i 7 c + .0l3T13c + .8l4T12c T13c* = .058T13b + .058T13d + .043T8c + .043Tl4c + .043Tl8c + .043T12c + .712T13c Tl4c' .058Tl4b + . 0 5 8 T l 4 d + .043T9c • + .043T15c + .043Tl9c + .043Tl3c + .712Tl4c T15c' = .056T15b + .056Tl5d + .086Tl4c + .043T10< c + .043T20c + .712T15C Tl6c' = .011Tl6b + .011Tl6d + .c40Tllc + .040T21 c + .065T17c + .833Tl6c T17c' — .Ol8Tl7b + .Ol8T17d + .055Tl6c + .04lT12c + .04lT22c + .013Tl8c + .8l4T17c Tl8c' = .058Tl8b + • 05&Tl8d + .043T13C + .043T19^ c + .043T23c + .043Tl7c + .712Tl8c T19c' .05&Tl9b + . 0 5 8 T l 9 d + .043Tl4c + .0U3T2O c + .o43T24c + .043Tl8c + .712T19C T20c' = .05&T20b + .058T20d + .086T19C + .043T15-c + .043T25c + .712T20c T21c' = •011T21b + .011T21d + .080Tl6c + .065T22 c + .833T21C T22c' .0l5T22b + . 0 l 8 T 2 2 d + .055T21c + .013T23c + .082T17c + .8l4T22c T23C1 = •.058T23b + ,058T23d + .086Tl8c + .043T22 c + .043T24c + .712T23c T24c' = .058T24b + .058T24d + .086T19C + .043T23 c + .C43T25C + .712T24c T25c' .05&T25b + .05&T25d + .086T20C + .086T24 c + . 712T25C T l d 1 = .OllTlc + .OllTle + .065T2d + .321T6d + .592Tld 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 T8d' - .Ol8T8c + .Ol6T8e + .055T3d + .c4lT7d + .c4lT9d + .013T13d + .8l4T8d T9d* = .Ol6T9c + .01&T9e + .055T4d + .04lT8d + .O^lTlOd + .013Tl4d + .8l4T9d TlOd' = .OloTlOc + .Ol8T10e + .055T5d + .013T15d + .082T9d + .8l4T10d T l l d 1 = .OllTllc + .OllTlle + .c40T6d + .0U0Tl6d + .065T12d + .833Tlld T12df = .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 l6d ' = .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 T 2 0 d ' = . 0 5 6 T 2 0 C + .05&T20e + .086T19d + .c43T15d + .c43T25d + .712T20d T21d' = . 0 1 1 T 2 1 C + .011T21e + .080Tl6d + .065T22d + ,833T21d T 2 2 d ' = .01&T22c + .0lST22e + .055T21d + .013T23d + .082T17d + .8l4T22d T23d' = . 0 5 8 T 2 3 C + .058T23e + .086Tl8d + .c43T22d + .C43T24d + .712T23d T24d' = .058T24c + .056T24e + .086T19d + .c43T23d + .c45T25d + .712T2l+d T25d' = .056T25C + .05&T25e + .086T20d + .086T2Ud + .712T25d kl Tie' — .015Tld + .023Tlf + .122T2e + .258T6e + .582Tle T2e' = .015T2d + .023T2f + ,040Tle + .05OT3e + .138T7e + .73^ T2e T3e' ,015T3d + .023T3f + •037T2e + .OJTShe + .078T8e + .8lOT3e T 4 e ' ,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 + .771T7e T8e' — .017T8d + .040T8f + .036T7e + •036T9e + .045T3e + .0l6T13e + .8lOT8e T9e' - .017T9d + .04OT9f + .036T8e + .036T10e + .OkjSke + .0l6T14e + .8lOT9e TlOe' zz .017T10d + •040T10f + .072T9e • +• .045T5e + •Ol6T15e + .8lOT10e Ti le ' zz .015Tlld + .023Tllf + •032T6e • +  .032Tl6e + .122T12e + .776Tlle T12ef zz .017T12d + .C40T12f + .OTlTlle + .027T7e + .027T17e + .028T13e + .790T12e T15e* — .020T13d + •065T13f + .034T12e + .034T14e + .019T8e + .019Tl8e + .809T13e T14e' zz •020T14d + .065T14f + .034T13e + .034T15e + .019T9e + .019T19e + .809T14e T15e' - .020T15d + .065Tl5f + .068T14e + .019T10e + .019T20e + .809T15e Tl6e* = .015Tl6d + . 0 2 3 T l 6 f + .032Tlle + .032T21e + .122T17e + .776Tl6e T17e' .017T17d + .040T17f + .071Tl6e + .027T12e + .027T22e + .02&Tl8e + .790T17e Tl8e* = .02OTl8d + .065Tl8f + .034T17e + .034T19e + ,019T13e + .019T23e + .809Tl8e T19e' .020T19d + .065T19f + .034Tl8e + ,034T20e + .019T14e + .019T24e + .809T19e T20e ' .020T20d + .065T20f + .068T19e + .019T15e + .019T25e + .809T20e T21ef .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 T l f • = .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 T7f 1 = .078T7e + .313To + .037T2f + .0l8T12f + .085T6f + .c43T8f + .426T7f T8f 1 = .078T8e + .313To + .032T7f + .032T9f + .037T3f + .Ol8Tl3f + .490T8f T9f 1 = .078T9e + .313To + .032T8f + .032T10f + .OJfEkf + .0l8T14f + .4-9ca?9f TlOf' = .078T10e + .313To + .037T5f + .Ol8T15f + .064T9f + .490T10f T l l f ' = . 0 7 8 T l l e + .313To + .013T6f + .015Tl6f + .255T12f + . 3 2 6 T l l f T12f' = .078T12e + .313To + .085Tllf + .043T13f + .014T7f + .014T17f + .453T12f T15f* = .078T13e + .313To + .013T8f + .013Tl8f + .032T14f + .032T12f + .519T13f T14f 1 = .07&T14e + .313To + .015T9f + . 0 1 5 T 1 9 f + .032T15f + .032T13f + ,519T14f T 1 5 f* = .078T15e + .313To + .013T20f + .064T14f + .OlJTlOf + .519T15f Tl6f* = .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 1 = .078T19e + .313TO + .013T14f + .013T24f + .032T20f + .032Tl8f + ,5l9Tl9f 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 h0} h ,kl)> kc and kc-Test M.V.i M.V.a M.V.b M.V.e M.V.f M.V. Q 1 . 1.819 1.763 1-735 0.652 0.547 4.62 2 1.828 1.758 1.723 0.653 0.551 4.77 3 1.823 1.750 1.718 0.642 0.541 4.74 4 1.820 1.760 1.723 o.64i 0.542 4.69 5 1.809 I.766 1.727 0.637 0.538 4.78 6 1-771 1.706 1.675 0.623 0.530 4.52 7 1-790 1-738 1.698 0.619 0.520 4.63 8 1.675 1.643 0.578 0.487 4.54 1.667 1.627 0.554 O.457 4.54 1.650 1.617 0.540 0.4n 4.54 1.65^ 1.617 0.538 0.427 4.54 1-655 1.616 0.537 0.412 4.54 1.653 1.616 0.537 0.434 4.54 1.655 1.615 0.538 0.424 4.54 I.656 1.618 0.543 0.445 4.54 1.654 1.617 0.543 0.434 4.54 I.656 1.619 0.546 0.453 4.54 9 1.688 I.638 1.600 o.54o 4.49 1.678 1.632 1.605 0.538 4.49 1.688 1.637 1.605 0.538 4.49 1.695 I.636 1.605 0.541 4.49 1.697 1.635 1.603 0.538 4.49 1.688 I.636 1.604 0.538 4.49 10 0.330 0.301 4.73 0.329 0.303 4.73 0.328 0.302 4.73 0.326 0.300 4.73 0.326 0.299 4.73 0.326 0.296 4.73 0.323 0.296 4.73 0.321 0.293 4.73 0.315 0.289 4.73 0.309 0.280 4.73 50 Data to Determine A f Test M.V.a M.V.b M.V.e M.V.f 11 1.588 1.530 0.620 0.446 12 1.597 1.536 0.620 0.449 13 1.604 1.538 0.624 0.417 14 1.608 1.542 0.625 0.423 15 1.603 1.541 0.624 0.448 16 1.608 1.542 0.628 0.423 17 1.611 1.546 0.629 0.446 18 1.611 1.544 0.629 0.424 19 1.610 1.544 0.626 O.455 20 1.611 1.543 0.628 0.427 Average 1.605 1.540 O.625 0.435 Data from Oven Tests to Determine and kj/^ Test Average Inside Temp. Average Outside Temp. Q dx 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 5 1 Data for Moisture Content and Specific Weight of F i r and Cedar Cedar Test Wet Weight Dry Weight /o Moisture Content Volume Specific Gravity grams grams m.l. 1 0 . 8 2 4 8 0 . 7 7 4 6 . 5 9 2 . 4 2 6 0 . 3 1 9 2 0 . 8 0 9 9 O.76I 6 . 4 4 2 . 4 1 3 0 . 3 1 5 3 0 . 8 5 3 6 0 . 8 0 2 6.48 2 . 4 3 3 0 . 3 3 0 4 0.8437 0 . 7 9 4 6 . 3 0 2 . 4 0 7 0 . 3 3 0 5 O . 8 5 0 2 0 . 7 9 9 6 . 3 8 2.420 0 . 3 3 0 6 0 . 8 2 9 0 0 . 7 7 6 6 . 8 3 2 . 4 3 3 0 . 3 1 9 7 O . 8 7 6 0 0 . 8 1 4 7 . 6 2 2 . 4 0 7 0 . 3 3 8 8 1 . 8 0 2 7 1 - 6 7 7 7 . 5 1 5 . 1 9 0 0 . 3 2 3 9 0 . 6 8 6 0 0 . 6 5 9 4 . 1 0 1 0 O . 5 9 1 0 0 . 5 6 7 4 . 2 3 1 1 O . 6 1 8 0 0 . 5 9 1 4 . 5 7 1 2 0 . 6 8 1 0 0 . 6 5 5 3 . 9 7 F i r 1 2 . l 6 l 2 . 0 2 8 6 . 5 6 3.645 0 . 5 5 6 2 2.421 2 . 2 6 8 6 . 7 5 4 . 0 6 6 0 . 5 5 8 3 2 . 0 6 9 1 . 9 3 6 6 . 8 7 3 . 4 3 9 0 . 5 6 3 4 2 . 2 7 0 2 . 1 1 1 7 - 5 3 - 3 . 8 6 7 0 . 5 4 5 5 2 . 2 2 2 2 . 0 7 5 7 . 0 8 3 . 6 4 2 0 . 5 7 0 6 2 . 1 2 6 1 . 9 8 4 7 . 1 6 3 . 4 7 8 0 . 5 7 0 7 3 . 7 2 1 3 - 4 6 9 7 - 2 6 6 . 2 3 9 O . 5 5 6 8 4 . 7 0 1 4 . 3 8 9 7 . 1 1 7 . 7 9 6 O . 5 6 3 9 4 . 2 8 5 4 . 0 0 6 6 . 9 6 6 . 9 6 7 0 . 5 7 5 1 0 3 . 6 6 0 3 . 4 o 6 7.46 5 . 9 8 2 O . 5 6 9 52 Specific Weight of Hardboard Sample Thickness Length Width inches inches inches 1 0.245 1.985 2.007 2 0.244 2.009 2.011 5 0.244 2.010 2.012 4 0.245 2.008 2.011 5 0.243 2.009 1.998 6 0.245 1.987 2.007 7 0.244 1.993 2.011 8 0.243 2.003 2.010 9 0.244 2.011 2.011 10 0.246 2.013 2.011 l l 0.246 2.010 2.011 12 0.243 1.998 2.011 15 0.244 2.008 2.008 i4 0.245 2.006 2.011 15 0.246 2.010 2.OO8 Volume Weight Specific Weight 3 i n c h e s grams l b . / f t ? 0.976 14.33 55.92 O.986 14.30 55.24 O.987 14.42 55.65 O.989 14.73 56.73 0.975 lU.59 57-00 0.977 14.36 55.98 0.978 14.23 55.^ 2 O.978 14.00 5U.53 0.987 '14.35 55-38 0.996 14.76 56.45 0.994 14.81 56.75 0.976 14.32 55.89 0.984 14.46 55.97 O.988 14.55 56.09 0.993 14.54 55-77 Specific Weight of Fibreglass Insulation 1 2.00 5-70 1.86 21.20 22.04 3.96 2 2.00 4.20 3.62 50 . U l 32.31 4 .04 3 2.00 7.00 1.88 26.25 25.24 3.66 4 2.00 20.00 2.56 102.40 117.12 4.36 5 2.00 1.75 I.65 7.78 5.74 3.9U 6 2.00 3.62 2.22 16.07 16.26 3-84 7 2.00 6.72 2.38 31.87 32.56 3-87 8 2.00 18.38 7-00 257.32 271.64 4.02 9 2.00 18.44 7.06 256.37 282.39 4.13 10 2.00 18.44 6.28 231.61 242.55 3-99 11 2.00 6.25 2.32 29.06 35.99 4.74 12 2.00 5.76 2.44 28.11 31.05 4.21 13 2.00 13.12 1.33 34.90 32.18 3.51 14 2.00 10.31 1.36 28.04 25.74 3.51 15 2.00 19.56 1.43 55.9h 61.68 4.20 16 2.00 15.03 3.69 110.92 119.18 4.10 17 2.00 5.75 2.44 28.06 30.68 4.17 18 2.00 13.31 1.81 48.18 47.87 3.78 Thermal D i f f u s i v i t y o f Hardboard T e s t 1 T e s t 2 Time M . V . M . V . - M . V . ^ M . V . M . V . - M, m i n . 0 . 0 0 2.359 2.053 2.360 2.056 1 .00 2.037 1.731 2.043 1.739 2 . 0 0 1.448 1.142 1.456 1.152 3 . 0 0 1.044 0.738 1.046 0.742 4 . 0 0 O.783 0.477 0.782 0.478 4 . 5 0 O.689 O.383 0.687 0 .383 5 .00 O.615 0.309 0.612 0 .308 5-25 O.583 0.277 0.578 0.274 5-50 0.551 0.245 0.551 0.247 5-75 0.529 0.223 0.523 0.219 6 . 0 0 0.507 0.201 0.503 0.199 6.25 0 .482 0.176 0.481 0.177 6.25 0.465 0.159 0.463 0.159 6.75 0.447 0 . l 4 l 0 .442 0.138 7 . 0 0 0.434 0.128 O.433 0.129 7.25 0.417 0.111 0.418 0.114 7 .50 0.409 0.102 0.405 0.101 7-75 0.398 0 .092 0.394 0 .090 8 . 0 0 O.389 0 .083 O.386 0 .082 8.25 0.378 0.072 0.378 0.074 8 . 5 0 0.373 0.067 0.369 0.065 8.75 0.364 0.058 0.362 0.058 9 . 0 0 O.360 0.054 0.356 0.052 9.25 0.353 0.047 0.352 0.048 9 .50 0.348 0 .042 0.347 0 .043 9.75 0.344 0.038 0 . 3 4 l 0.037 1 0 . 0 0 0.341 0.035 O.338 0.034 10.25 0.336 0 .030 0.334 . 0 .030 10.50 0.334 0.028 0.331 0.027 10.75 0.329 0 .023 0.329' 0.025 11 .00 0.328 0 . 0 2 2 0.327 O.023 T e s t 1 , M.V.«, = 0 .306 T e s t 2 , 0.304 5* I n i t i a l Temperatures Node P l a n e a P lane b P l a n e c P l a n e d P l a n e e P l a n e f 1 162.3 159.2 145.8 131.5 117.1 108.2 2 162.3 159.2 145.8 131.5 117.1 108.2 3 162.3 159.2 145.8 131.5 117.1 108.2 4 162.3 159.2 145.8 131.5 117.1 108.2 5 162.3 159.2 145.8 131.5 117.1 108.2 6 162.3 159.2 145.8 131.5 117.1 108.2 7 162.3 159.2 145.5 131.0 116.9 108.0 8 162.3 159-7 145.4 131*2 116.9 108.1 9 162.3 159-8 145.7 131.2 116.8 108.2 10 162.3 159.8 145.7 131.2 116.8 108.2 11 162.3 159.2 145.8 131.5 117.1 108.2 12 162.3 159.2 145.5 130.8 116.7 107.9 13 163.2 l6l.5 145.3 129.3 113.2 107.8 14 163.2 161.8 1U5.5 129.4 113-2 107.8 15 173.2 l6l.8 145.5 129.4 113.2 107.8 16 162.3 159.2 145.8 131.5 I I 7 . I 108.2 17 162.3 159.2 145.5 130.6 116.3 108.0 18 162.9 160.2 145.3 129.4 113.3 107.8 19 163.2 161 ;8 145.3 129.2 113.0 107.8 20 163.2 161.8 145.3 129.2 113.0 107.8 21 162.3 159.2 145.8 131.5 117.1 108.2 22 162.3 159.2 145.5 130.6 II6.3 108.0 23 162.9 160.2 145.3 129.4 II3.3 107.8 24 163.2 161.8 145.3 129.2 113-0 107.8 25 163.2 161.8 145.3 129.2 113.0 107.8 O u t s i d e a i r t e m p e r a t u r e , To = IO6.5 Transient Temperatures from Numerical Analysis Time Three Dimensional One Dimensional Analysis Analysis hours T25a - To T25a - To 0.0 55.8 55-6 0.5 • 49.3 48.6 1.0 43.8 43-3 1.5 39-4 38.6 2.0 35-7 34.5 2.5 32.6 . 30.8 3.0 29.8 27.5 3-5 27.4 24.6 4.0 25.2 22.0 4.5 23.1 19-6 5-0 21.3 17.5 5-5 19.5 15.7 6.0 17.9 14.0 6.5 16.5 12.5 7.0 15.1 11.2 7-5 13.9 10.0 8.0 12.7 8.9 8.5 11.6 8.0 9.0 10.6 7-1 9-5 9.8 6.3 10.0 9.0 5-7 10.5 8.3 5-1 11.0 7.6 4.5 11.5 7.0 4.0 12.0 6.4 3.6 12.5 5.9 3.2 13.0 5-4 2.9 13.5 5.0 2.6 14.0 4.6 2.3 14.5 4.2 2.1 15.0 3.9 1.8 15.5 3.5 1.6 16.0 3.2 1.5 16.5 3.0 1.3 17.0 2.7 1.2 17.5 2.5 1.0 18.0 2.3 0.9 18.5 2.1 0.8 19.0 1.9 0.7 19-5 1.8 0.7 20.0 1.6 0.6 Time Three D i m e n s i o n a l A n a l y s i s One D i m e n s i o n a l A n a l y s i s hours T25a- To T25a - To 20.5 1^ 5 0.5 21.0 1.4 0.5 21.5 1-3 0.4 22.0 1.2 0.4 22.5 1.1 0.3 23.0 1.0 0.3 23-5 0.9 0.3 24.0 0.8 0.2 Transient Temperatures from Experiment Time Test 1 Test 2 hours T25a - To T25a - To 0.0 57.1 56.6 1.0 49-3 49.6 2.0 43.6 43.1 3.0 37.1 36.9 4.0 30.9 30.7 5-0 27.1 26.2 6.0 23.1 22.6 7.0 19.6 19.9 8.0 16.9 18.0 9-0 13.5 16.0 10.0 10.9 13-7 11.0 9-3 9.7 12.0 7.8 6.9 13.0 6.1 5-7 l 4 . 0 4.9 4.8 15-0 4.0 3.9 16.0 3.3 2.9 17-0 2.6 1.9 18.0 1.8 0.9 To correct for error i n recorder timing system, multiply times by O.96 APPENDIX G. FIGURES FIG. I G E N E R A L V I E W OF A P P A R A T U S CIRCUMFERENTIAL RADIATION SHIELD CIRCUMFERENTIAL HEATERS HEATER SUPPORT GUARD SECTION PULLEY AND BELT DRIVE T E S T SECTION TOP RADIATION SHIELD HEATER SUPPORT TOP H E A T E R S WIRE SUPPORT FOR RADIATION SHIELD FAN WALL EXIT SECTION TEST SECTION HEATERS WITH SHIELDS F A N THERMOSTAT HEATER AND SHIELD FAN vn F I G . 2 S I D E S E C T I O N O F A P P A R A T U S F I G . 3 F R O N T S E C T I O N O F T H E A P P A R A T U S 61 FIG. 4 FIR F R A M E HARDBOARD FRAME COMPONENT ( P L A N E D 2 * 4 ) FIBREGLASS INSULATION CEDAR 5 C R O S S S E C T I O N A T A F R A M E C O M P O N E N T 62 « F I G . 6 T E M P E R A T U R E C O N T R O L S Y S T E M F O R T H E E X I T S E C T I O N 4- OHMS HEATER 6 OHMS HEATER THERMOSTAT, T I GUARD SECTION 5 H E A T E R S IN P A R A L L E L 10 H E A T E R S IN P A R A L L E L FIG. 7 HEATER AND CONJROL CIRCUITS FOR T H E T E S T A N D GUARD SECTIONS CONDEN SER T3 F I G . 8 S I M P L I F I E D E L E C T R O N I C C O N T R O L C I R C U I T FIG. 9 RECORDER AND THERMOCOUPLE SWITCH 2 0 X 66 F I G . 1 0 WALL AND COMPONENTS OF EXIT SECTION TEMPERATURE CONTROL SYSTEM ° F 1401. 130.. 120.. 110.. 100.. 90.. -EDGE OF F R A M E C O M P O N E N T C E N T R E L I N E O F F R A M E C O M P O N E N T A N D E D G E OF GRID N E T W O R K INSIDE SURFACE OF HARDBOARD, ( P L A N E A) ^INTERFACE OF HARDBOARD AND INSULATION, ( P L A N E B) " * o T H E R M O C O U P L E R E A D I N G AXIS OF S Y M M E T R Y -9 a-C E N T R E L I N E OF F R A M E C O M P O N E N T -— A X I S OF S Y M M E T R Y - E D G E OF GRID N E T W O R K • C E N T R E L I N E OF T E S T S E C T I O N 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) — OUTSIDE SURFACE OF CEDAR, (PLANE F) F I G . II S T E A D Y S T A T E T E M P E R A T U R E S 0\ —5 68 VERTICAL <{; OF WALL CENTRE SECTION OF THE FRAME AXIS OF SYMMETRY SECTION USED IN ANALYSIS REGIONS OF 2-D HEAT FLOW REGION OF l-D HEAT FLOW AXIS OF SYMMETRY 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 SYMMETRY AND REGIONS OF HEAT FLOW I 2 a a— o <• II $ & 21 12 2X 17 2Z 13 2X 18 23 14 19 24 69 -6 15 20 PLAN VIEW' X= 0.633" U = 0.244" XT= 1.083" ur= 0.750" 25 () O o <> u v v us •<£ OF F R A M E C O M P O N E N T -PLANE A -PLANE B -HARDBOARD -NODE -PLANE C — INSULATION — P L A N E D E L E V A T I O N V I E W — P L A N E E — CEDAR —PLANE F FIG. 13 NODE SYSTEM FOR ANALYSIS 70 DIRECTION OF HEAT FLOW G R A I N k a - N O R M A L OR AXIAL CONDUCTIVITY k t - T A N G E N T I A L CONDUCTIVITY k a Kt | 0 DIRECTION OF HEAT FLOW GRAIN 8 FIR C E D A R F I G . 14 A X I A L A N D T A N G E N T I A L C O N D U C T I V I T I E S M O T O R T H E R M O C O U P L E S DIRECTION OF GRAIN HEAT T R A N S D U C E R S P E C I M E N DIRECTION OF HEAT FLOW FIG. 15 APPARATUS FOR LONGITUDINAL CONDUCTIVITY TESTS 71 T 2 5 Q - T Q Tzsa^-To 0 2 4 6 8 10 12 14 16 18 20 22 24 T I M E - (HOURS) F I G . 17 C O O L I N G C U R V E S F O R I N S I D E W A L L S U R F A C E 73 0 2 4 6 8 10 12 14 16 18 20 22 TIME (HOURS) FIG. 18 THEORETICAL COOLING CURVES & TEST ONE O TEST TWO EXTRAPOLATED C U R V E S 8 10 12 14 16 18 20 22 T I M E ( H O U R S ) F I G . 19 E X P E R I M E N T A L C O O L I N G CURVES Tzs a.-To 0 2 4 - 6 8 10 12 14 16 18 20 ZZ 24 T I M E - (HOURS) FIG.20 T H E O R E T I C A L A N D C O R R E C T E D E X P E R I M E N T A L COOLING C U R V E S 

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