Comparison of modes of operation for guarded hot plate apparatus with emphasis on transient characteristics

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Comparison of modes of operation for guarded hot plate apparatus

with emphasis on transient characteristics

Shirtliffe, C. J.; Orr, H. W.

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C o m p a r i s o n of M o d e s of O p e r a t i o n f o r G u a r d e d H o t P l a t e

A ~ I

A e n x r s

A p p a r a t u s with E m p h a s i s on T r a n s i e n t C h a r a c t e r i s t i c s C. J. S h i r t l i f f e and H. W. O r r D i v i s i o n of Building R e s e a r c h N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a O t t a w a , C a n a d a T h e c o n s t a n t h e a t f l u x a n d c o n s t a n t t e m p e r a t u r e m o d e s of o p e r a t i o n of t h e g u a r d e d h o t p l a t e a p p a r a t u s a r e c o m p a r e d a n d t h e e x i s t e n c e of a n o p t i m u m h e a t f l u x b o u n d a r y condition is e x a m i n e d . T h e s t e p r e s p o n s e e q u a t i o n s f o r t h e d i f f e r e n t m o d e s and f o r v a r i o u s p r e c o n d i t i o n i n g t e m p e r a t u r e s a r e t a b u l a t e d a n d p r e s e n t e d in g r a p h i c a l f o r m . O p t i m u m p r e c o n d i t i o n i n g t e m p e r a t u r e s a r e d e t e r m i n e d . E f f e c t s of d i s t u r b a n c e s a n d p l a t e c a p a c i t y a r e d i s c u s s e d a n d c o n t r o l a n d m e a s u r e m e n t p r o b l e m s c o m p a r e d . A h y b r i d m o d e of o p e r a t i o n is s u g g e s t e d . T h e e x t e n s i o n of t h e r e s u l t s t o t h e u s e of h e a t m e t e r s a n d t r a n s i e n t h e a t flow a p p a r a t u s is d i s c u s s e d . C o n s t a n t t e m p e r a t u r e o r a h y b r i d m o d e of o p e r a t i o n i s found t o b e t h e m o s t p r a c t i c a l . T h e r e s p o n s e t i m e s c a n b e a n o r d e r of m a g n i t u d e l e s s t h a n c o n s t a n t h e a t f l u x o p e r a t i o n . U s e of t h e g u a r d e d hot p l a t e a p p a r a t u s i n t h e c o n s t a n t t e m p e r a t u r e m o d e is s u g g e s t e d f o r o t h e r m e a s u r e m e n t s . K e y W o r d s : G u a r d e d h o t p l a t e , h e a t m e t e r s , m o d e s , o p t i m u m b o u n d a r y c o n d i t i o n s , r e s p o n s e t i m e , t h e r m a l c o n d u c t i v i t y , t h e r m a l diffusivity, t r a n s i e n t e r r o r s , t r a n s i e n t h e a t flow. 1. I n t r o d u c t i o n 1 . H i s t o r i c a l l y , t h e g u a r d e d h o t p l a t e a p p a r a t u s (1) i s a " S t a n d a r d s Lab1' t y p e i n s t r u m e n t w h e r e t h e e m p h a s i s is on a c c u r a c y r a t h e r t h a n on m i n i m i z i n g t h e d u r a t i o n of t h e t e s t . T h e t r a n s i e n t h e a t flow c h a r a c t e r i s t i c s and c o n t r o l p r o b l e m s h a v e t h e r e f o r e b e e n n e g l e c t e d . O t h e r t y p e s of a p p a r a t u s h a v e b e e n evolved f o r s h o r t d u r a t i o n t e s t s of l e s s e r a c c u r a c y . It w a s b e l i e v e d t h a t t h e s t a n d a r d g u a r d e d h o t p l a t e a p p a r a t u s w a s m u c h m o r e v e r s a t i l e t h a n a s s u m e d and t h a t with t h e p r o p e r c o n t r o l s y s t e m a n d m e a s u r e m e n t t e c h n i q u e s i t s u s e could b e e x t e n d e d t o r a p i d , high a c c u r a c y t e s t i n g . T h e a p p a r a t u s w a s thought t o o f f e r a d v a n t a g e s i n t h e t r a n s i e n t h e a t flow a r e a a n d f o r u s e i n conjunction with h e a t m e t e r s . A l s o , s i n c e t h e g u a r d e d h o t p l a t e a p p a r a t u s is b e i n g p r o p o s e d f o r t e s t i n g " s u p e r " i n s u l a t i o n s with e x t r e m e l y l o n g r e s p o n s e t i m e s , i t w a s b e l i e v e d t h a t t h e c o n t r o l m o d e t h a t is o p t i m u m in t e r m s of s p e e d a n d a c c u r a c y s h o u l d b e d e t e r m i n e d a n d t h e r e s p o n s e f u n c t i o n s t a b u l a t e d t o allow e s t i m a t i o n of t h e t i m e t o r e a c h a n a d e q u a t e s t e a d y s t a t e condition. T h e g u a r d b a l a n c e is a s s u m e d t o b e c o n t r o l l e d a u t o m a t i c a l l y i n t h i s a n a l y s i s s o t h a t i t d o e s not i n t r o d u c e a d d i t i o n a l d i s t u r b a n c e s i n t o t h e s y s t e m . T h i s is b e l i e v e d t o b e a r e a s o n a b l e a s s u m p t i o n a s m o s t l a b o r a t o r i e s h a v e p r o v i d e d t h i s t y p e of c o n t r o l f o r t h e i r a p p a r a t u s . T h e t e m p e r a t u r e of t h e c o l d p l a t e is a s s u m e d t o b e c o n s t a n t . F i g u r e s i n p ~ r e n t h e s e s in d i c a t e t h e l i t e r a t u r e r e f e r e n c e s a t t h e and of t h i s paper.

2. Modes of O p e r a t i o n and T r a n s i e n t C h a r a c t e r i s t i c s

T w o m o d e s of c o n t r o l of t h e hot p l a t e in t h e g u a r d e d hot p l a t e a p p a r a t u s a r e i n g e n e r a l u s e . F o r t h e s i m p l e s t and m o s t widely u s e d t h e p l a t e i s s u p p l i e d with c o n s t a n t p o w e r , t h e r e b y p r o d u c i n g a c o n s t a n t h e a t f l u x a t t h e s u r f a c e of t h e

late.

T h e s e c o n d , l e s s widely u s e d , method i s t o k e e p t h e t e m p e r a t u r e of t h e h o t p l a t e c o n s t a n t by c o n t r o l l i n g t h e p o w e r with a t e m p e r a t u r e c o n t r o l l e r . T h e l a t t e r m e t h o d w a s u s e d by S o m m e r s ( 2 ) f o r "high s p e e d t t t e s t s , but h a s not b e e n widely u s e d b e c a u s e of t h e e x t r a equipment involved and t h e l a c k of knowledge of t h e r e l a t i v e a d v a n t a g e s a n d p r o b l e m s of t h i s m o d e c o m p a r e d t o t h e c o n s t a n t h e a t f l u x m o d e . 2. 1. C o n s t a n t H e a t F l u x O p e r a t i o n T h e u s u a l s e q u e n c e of e v e n t s i n p e r f o r m i n g a t e s t with t h e g u a r d e d h o t p l a t e a p p a r a t u s with c o n s t a n t h e a t f l u x f r o m t h e hot p l a t e i s a s follows: ( i ) T h e s a m p l e i s conditioned t o t h e m e a n t e m p e r a t u r e of t h e t e s t ( i i ) T h e cold p l a t e s a r e a t t h e r e q u i r e d t e m p e r a t u r e

( i i i ) No p o w e r i s b e i n g s u p p l i e d t o t h e hot p l a t e and it i s not a t t h e r e q u i r e d t e s t t e m p e r a t u r e ( i v ) T h e t h e r m a l conductivity a n d , t h e r e f o r e , t h e p o w e r t o t h e t e s t a r e a a r e e s t i m a t e d

( v ) T h e s a m p l e i s p l a c e d i n t h e p l a t e s , t h e p o w e r t o t h e t e s t a r e a i n c r e a s e d f o r an u n s p e c i f i e d p e r i o d of t i m e , t o h e a t up t h e p l a t e and t h e s a m p l e , the"n r e d u c e d t o t h e esti-ted p o w e r . T h e conditions then a r e allowed t o r e a c h s t e a d y s t a t e

( v i ) Additional a d j u s t m e n t s of t h e p o w e r a r e u s u a l l y n e c e s s a r y and d i s t u r b a n c e s m u s t s e t t l e out.

T h e r e h a s b e e n n o e a s y way t o e s t i m a t e t h e t i m e r e q u i r e d t o r e a c h t h e s t e a d y s t a t e condition u s i n g t h i s p r o c e d u r e .

2. 2. C o n s t a n t T e m p e r a t u r e O p e r a t i o n

T h e u s u a l s e q u e n c e of o p e r a t i o n s for u s e with t h e hot p l a t e a t a c o n t r o l l e d t e m p e r a t u r e i s a s follows:

( i ) T h e s a m p l e i s conditioned t o t h e m e a n t e m p e r a t u r e of t h e t e s t ( i i ) Both t h e h o t and cold p l a t e s a r e a t t h e r e q u i r e d t e m p e r a t u r e s

( i i i ) T h e s a m p l e is p l a c e d i n t h e a p p a r a t u s a n d t h e t e s t a r e a p o w e r i n c r e a s e s a u t o m a t i c a l l y t o supply t h e r e q u i r e d h e a t a n d t h e n d r o p s t o t h e s t e a d y s t a t e value. It should b e noted t h a t n o g u e s s i n g o r s e c o n d a d j u s t m e n t s a r e r e q u i r e d . 2. 3. O p t i m u m B o u n d a r y Conditions C o n s t a n t t e m p e r a t u r e o p e r a t i o n i s e a s i e r t h a n c o n s t a n t h e a t f l u x o p e r a t i o n , a n d p r o b a b l y f a s t e r , but t h e q u e s t i o n r e m a i n s a s t o w h e t h e r t h e r e i s a b o u n d a r y condition t h a t w i l l g i v e e v e n f a s t e r r e s p o n s e . A s s u m i n g t h e r e i s a b e t t e r , n o n c o n s t a n t , h e a t f l u x b o u n d a r y condition, F ( t ) , w h e r e t is t i m e , a n d t h e a p p r o p r i a t e unit s t e p r e s p o n s e of t h e s a m p l e is V ( x , t ) , t h e s u r f a c e t e m p e r a t u r e of t h e h o t p l a t e c a n b e w r i t t e n a s given i n r e f . (3).

where t' i s t h e dummy tirne v a r i a b l e , and s u b s c r i p t t indicates t h e t i m e derivative.

By n o r m a l r e s p o n s e factor methods, F ( t ) can be found f o r any d e s i r e d r e s p o n s e . Since the a n s w e r depends, however on the s t e p r e s p o n s e of t h e sample, _{V(x, t ) , which i s a function of t h e} unknown t h e r m a l p r o p e r t i e s , t h e r e i s no p r a c t i c a l way t o find t h i s optimum condition for t h e r m a l conductivity a p p a r a t u s . The m o s t p r a c t i c a l boundary conditions would a p p e a r t o be the constant t e m p e r a t u r e boundary condition, since it i s the only boundary condition that can be specified without r e g a r d to s a m p l e p r o p e r t i e s .

2. 4. Response T i m e s

T h e t h e o r e t i c a l solutions t o the heat t r a n s f e r p r o b l e m s corresponding t o t h e constant t e m p e r a t u r e and constant heat flux m o d e s of operation a r e given in Appendix I. The solutions have been derived in nondimensional f o r m to make t h e m s e n e r a l . The t e m p e r a t u r e i s nondimensionalized a s shown in Appendix I t o be z e r o a t the cold p l a t e and 1 a t the hot plate under steady s t a t e conditions. The non- dimensional heat flux i s a l s o 1 under steady conditions. T i m e z e r o i s r e f e r r e d t o a s the t i m e at which the boundary conditions a r e applied, that i s , the t e s t i s s t a r t e d , and i s nondimensionalized by t h e t h e r m a l diffusivity "a" and t h e t h i c k n e s s , L,

T h e solutions a r e given in g e n e r a l f o r m and a l s o f o r s p e c i f i c locations in the s a m p l e a t which t h e m e a s u r e m e n t s a r e t o be m a d e for determining t h e t h e r m a l cbnductivity, s i n c e it i s t h e s e values that must approach s t e a d y s t a t e b e f o r e t h e m e a s u r e m e n t s can be made a c c u r a t e l y .

T h e r e s p o n s e t i m e of a s a m p l e i s defined h e r e t o m e a n the t i m e r e q u i r e d for t h e uncontrolled, or unspecified, v a r i a b l e at the hot p l a t e t o r e a c h an adequate steady s t a t e so a s t o give an e r r o r in t h e m e a s u r e m e n t of t h e r m a l conductivity of l e s s than the specified value. A curve of e r r o r in the

m e a s u r e m e n t v e r s u s t i m e will r e s u l t for each mode. The equations f o r e r r o r a r e given in Appendix I I. R e s p o n s e t i m e s for t h e constant t e m p e r a t u r e and constant heat flux modes with t h e initial

t e m p e r a t u r e equal to t h e s t e a d y mean t e m p e r a t u r e , W

=

0. 5, a r e given in figure 1. T h e r e s p o n s e t i m e for constant t e m p e r a t u r e mode i s s e e n t o be a l m o s t an o r d e r of magnitude l e s s than for t h e constant heat flux mode when the e r r o r s a r e s m a l l .

T h e r e s p o n s e t i m e s for s p e c i m e n s a t v a r i o u s i n i t i a l t e m p e r a t u r e s , W , a r e s h o w n i n f i g u r e s 2 and 3, for values of W x _{0 , t h e cold p l a t e t e m p e r a t u r e , t o W} = _{1 , the hot p l a t e t e m p e r a t u r e .} In each mode t h e r e i s an optimum value of W a t which the r e s p o n s e t i m e i s l e a s t . The e r r o r r e v e r s e s sign, that i s , overshoots, when t h e i n i t i a l t e m p e r a t u r e i s g r e a t e r than 0. 5 for t h e constant t e m p e r a t u r e mode and 0 . 6 3 5 for the constant heat flux m o d e , and the negative portion of t h e c u r v e can govern t h e r e s p o n s e t i m e of a t e s t . T h e optimum values of t h e initial t e m p e r a t u r e , W , a r e t h e r e f o r e approximately 0. 52 and 0. 64, r e s p e c t i v e l y , f o r the two modes. The r e s p o n s e t i m e s for the two m o d e s a t the optimum W a r e approximately the same. The r e s p o n s e t i m e for t h e constant flux mode a t optimum conditioning i s an o r d e r of magnitude l e s s than t h a t for conditioning to & e _mean

t e m p e r a t u r e ; t h i s can b e useful in c a s e s w h e r e the t h e r m a l conductivity of a specimen i s being checked and the steady s t a t e h e a t flux can b e determined beforehand. If t h e t h e r m a l conductivity i s not known then t h e final hot p l a t e t e m p e r a t u r e and t h e r e f o r e the optimum W cannot be determined.

T h e equations and f i g u r e s do not t a k e into account t h e heat s t o r a g e capacity of t h e p l a t e s , t h e t h r e e - d i m e n s i o n a l n a t u r e of the heat flow, or the t e m p e r a t u r e dependence of the p r o p e r t i e s . It should t h e r e f o r e b e recognized that the equations and f i g u r e s a r e only a guide t o the a c t u a l r e s p o n s e t i m e and that checks should be m a d e on t h e applicability of t h e f i g u r e s t o a n y p a r t i c u l a r a p p a r a t u s by t e s t i n g s a m p l e s of known t h e r m a l diffusivity.

A t e a t was made on a f o a m polystyrene s a m p l e , 2. 5 c m thick with a 20- by 20-cm guarded hot p l a t e t o c o m p a r e t h e t h e o r e t i c a l r e s p o n s e t i m e s t o the actual values. F o r constant heat flux operation, the r e s p o n s e t i m e t o

2

p e r cent e r r o r was l e s s than 1 5 s e c , when t h e hot p l a t e was p r e h e a t e d to t h e s t e a d y s t a t e t e m p e r a t u r e , and over 10 h r when t h e p l a t e was not p r e h e a t e d . In e a c h c a s e the exact constant h e a t flux w a s used to obtain t h e d e s i r e d hot p l a t e t e m p e r a t u r e . T h e t h e o r e t i c a l value was approximately 40 s e c for the specimen. In constant t e m p e r a t u r e mode t h e m e a s u r e d r e s p o n s e t i m e was approximately 70 sec. T h e r e s p o n s e c u r v e s a r e , t h e r e f o r e , better e s t i m a t i o n s of the a c t u a l r e s p o n s e t i m e s for t h e constant t e m p e r a t u r e mode than t h e constant h e a t flux mode.

2. 5. M e a s u r e m e n t P r o b l e m s E a c h m o d e of o p e r a t i o n h a s i n h e r e n t m e a s u r i n g p r o b l e m s . T h e constant h e a t f l u x m o d e of o p e r a t i o n h a s s u c h long r e s p o n s e t i m e s t h a t t h e t r u e s t e a d y s t a t e v a l u e c a n b e difficult t o r e c o g n i z e . M e a s u r e m e n t s m u s t b e m a d e o v e r a n e x t e n d e d p e r i o d t o e n s u r e t h a t it h a s b e e n r e a c h e d . T h e c o n s t a n t t e m p e r a t u r e m o d e of o p e r a t i o n r e q u i r e s t h a t t h e t e m p e r a t u r e of t h e h o t p l a t e b e c o n t r o l l e d , b y c o n t r o l l i n g t h e p o w e r t o t h e h e a t e r . T h e s t e a d y s t a t e p o w e r t o t h e h e a t e r w i l l , by d e f i n i t i o n , v a r y a s m a l l a m o u n t with t i m e . It i s n e c e s s a r y t o a v e r a g e t h e p o w e r o v e r a s u f f i c i e n t l e n g t h of t i m e t o obtain t h e m e a n v a l u e s . T h e a v e r a g i n g t i m e w i l l depend on t h e c o n t r o l l e r c h a r a c t e r i s t i c s a n d t h e t e m p e r a t u r e s e n s o r . T h e p r o b l e m of a v e r a g i n g c a n b e e a s e d c o n s i d e r a b l y by u s i n g z s t r i p c h a r t r e c o r d e r , a u t o m a t i c d a t a l o g g e r , o r a s i g n a l a v e r a g i n g c i r c u i t . 2. 6. H y b r i d M o d e T h e b e s t of both m e t h o d s c a n b e o b t a i n e d by s t a r t i n g t h e t e s t i n t h e c o n s t a n t t e m p e r a t u r e m o d e and s w i t c h i n g l a t e r t o t h e c o n s t a n t h e a t f l u x m o d e . T h i s h y b r i d m o d e of o p e r a t i o n r e q u i r e s a s i g n a l a v e r a g e r and i s a c c o m p l i s h e d b y s t a r t i n g i n t h e c o n s t a n t t e m p e r a t u r e m o d e , m o n i t o r i n g t h e a v e r a g e p o w e r s u p p l y v o l t a g e , a n d when i t b e c o m e s s u f f i c i e n t l y c o n s t a n t , s e t t i n g t h e output of t h e c o n t r o l l e r t o t h a t v a l u e , a n d s w i t c h i n g t h e c o n t r o l l e r t o m a n u a l m o d e , t h e r e b y supplying c o n s t a n t h e a t f l u x t o t h e h o t p l a t e . Depending on t h e s u c c e s s of t h e t r a n s f e r t o c o n s t a n t h e a t flux o p e r a t i o n , t h e m e a s u r e - m e n t c a n b e m a d e w i t h a s h o r t s e t t l i n g t i m e , a n d w i t h n o r m a l m e a s u r i n g i n s t r u m e n t s . 2. 7. D i s t u r b a n c e s a n d R e a d j u s t m e n t s If t h e t e m p e r a t u r e o r t h e h e a t flux of t h e h o t p l a t e m u s t b e r e a d j u s t e d s o a s t o c o n f o r m t o a s t a n d a r d t e s t condition, o r if i t i s d i s t u r b e d , t h e t i m e f o r t h e t r a n s i e n t t o s e t t l e c a n b e e s t i m a t e d f r o m t h e r e s p o n s e t i m e c u r v e s . A c h a n g e i n a p a r a m e t e r a t o n e s u r f a c e c o r r e s p o n d s t o t h e c a s e of a c h a n g e a t t h e s u r f a c e a n d z e r o i n i t i a l t e m p e r a t u r e . T h e e r r o r t h a t t h e d i s t u r b a n c e p r o d u c e s i n t h e s t e a d y s t a t e v a l u e should b e u s e d i n d e t e r m i n i n g t h e r e s p o n s e t i m e . F o r e x a m p l e , a 10 p e r c e n t c h a n g e n e e d o n l y s e t t l e t o 10 p e r c e n t of i t s f i n a l v a l u e t o p r o d u c e a 1 p e r c e n t e r r o r i n t h e m e a s u r e - m e n t . 2. 8. P l a t e C a p a c i t a n c e T h e h o t p l a t e a s s e m b l y c a n h a v e m o r e t h a n 100 t i m e s t h e t h e r m a l c a p a c i t a n c e of t h e s a m p l e and c a n t h e r e f o r e b e t h e m a j o r f a c t o r i n d e t e r m i n i n g r e s p o n s e t i m e ( 3 ) . F a i l u r e t o p r e h e a t t h e h o t - p l a t e a s s e m b l y t o t h e d e s i r e d t e m p e r a t u r e c a n g r e a t l y e x t e n d t h e t e s t , e s p e c i a l l y f o r t h e c o n s t a n t h e a t f l u x m o d e w h e r e t h e s m a l l s o u r c e of h e a t m u s t r a i s e t h e t e m p e r a t u r e of t h e l a r g e t h e r m a l m a s s . T h e i n c r e a s e i n r e s p o n s e t i m e , o r l a g , d e c r e a s e s a s t h e c o n d u c t a n c e of s a m p l e s i n c r e a s e s , and c a n v a r y f r o m a few s e c o n d s t o m a n y h o u r s . F o r c o n s t a n t t e m p e r a t u r e o p e r a t i o n , t h e hot p l a t e i s n o r m a l l y a t t h e r e q u i r e d t e m p e r a t u r e and t h e t h e r m a l m a s s c a n b e u s e d t o a d v a n t a f t e t o s u p p l e m e n t t h e h e a t e r c a p a c i t y i n h e a t i n g t h e s p e c i m e n . In t h e h y b r i d m o d e , t h e t h e r m a l c a p a c i t a n c e i s h e l p f u l d u r i n g t h e c o n s t a n t t e m p e r a t u r e i n t e r v a l a n d d e t r i m e n t a l in t h e c o n s t a n t h e a t flux i n t e r v a l . 2. 9. C h a r a c t e r i s t i c s of C o n t r o l S y s t e m s f o r Hot P l a t e s It h a s b e e n found t h a t t h e r e s p o n s e t i m e f o r t h e n o r m a l d e s i g n of h o t p l a t e s a n d n o r m a l s a m p l e s i s oo g r e a t t h a t r e s e t i s r e q u i r e d , i. e . i n t e g r a l a c t i o n with "anti-windup'' f e a t u r e a l o n g with t h e p r o p o r t i o n a l a c t i o n in t h e c o n t r o l l e r . T h e s u d d e n c h a n g e s in p o w e r output r e q u i r e d when high c a p a c i t a n c e s a m p l e s a r e p l a c e d i n t h e a p p a r a t u s m a k e i t d e s i r a b l e t o h a v e r a t e , i . e . d e r i v a t i v e a c t i o n , a s well. P r o p e r d e s i g n of t h e h o t p l a t e and p l a c e m e n t of t h e t e m p e r a t u r e s e n s o r c a n d e c r e a s e t h e l a g s i n t h e c o n t r o l s y s t e m a n d e a s e t h e s e r e q u i r e m e n t s . 2.10. F u r t h e r R e f i n e m e n t

If

t h e s h o r t e s t p o s s i b l e t e s t t i m e i s r e q u i r e d , t h e r e i s n o r e a s o n why t h e s a m p l e s c a n n o t b e p r e c o n d i t i o n e d t o t h e p r o p e r t e m p e r a t u r e g r a d i e n t in a l a r g e , s i m p l i f i e d p a i r of t e m p e r a t u r e

c o n t r o l l e d p l a t e s . T h e s a m p l e s could then b e t e s t e d r a p i d l y i n an a p p a r a t u s o p e r a t e d in t h e confitant t e m p e r a t u r e mode.

3 . Application to Heat M e t e r A p p a r a t u s

T h e g u a r d e d hot p l a t e a p p a r a t u s , when o p e r a t e d in t h e constant t e m p e r a t u r e m o d e , f o r m s t h e b a s i c c o n f i g u r a t i o n r e q u i r e d f o r m e a s u r i n g t h e r m a l c o n d u c t i v i t y with a h e a t m e t e r ; t h e a d v a n t a g e of t h i s h e a t m e t e r m e t h o d i s t h e s i m p l i c i t y of t h e m e a s u r e m e n t s .

T h e s o l u t i o n s in Appendtx I c a n be a p p l i e d t o t h e h e a t m e t e r a p p a r a t u s a s w e l l , a s long a s t h e h e a t m e t e r i s thin and h a s a low t h e r m a l c a p a c i t a n c e . E q u a t i o n s ( 5 ) , ( 6 ) , and (7) of Appendix I show t h a t t h e r e a r e t h r e e l o c a t i o n s i n a s a m p l e w h e r e o p t i m u m r e s p o n s e without o v e r s h o o t c a n b e a t t a i n e d . T h e l o c a t i o n s and r e l e v a n t c o n d i t i o n s a r e l i s t e d below. T e s t Conditions ( i ) t h e s a m p l e i s conditioned to a u n i f o r m t e m p e r a t u r e ( i i ) t h e t e m p e r a t u r e of t h e h o t and cold p l a t e s a r e c o n t r o l l e d . L o c a t i o n s and I n i t i a l T e m p e r a t u r e s ( i ) a t t h e hot s u r f a c e - s a m p l e a t t h e m e a n t e m p e r a t u r e ( i i ) a t t h e cold s u r f a c e

-

s a m p l e a t t h e m e a n t e m p e r a t u r e ( i i i ) at t h e c e n t e r of t h e s a m p l e - a n y u n i f o r m t e m p e r a t u r e .

T h e t h i r d l o c a t i o n , a t t h e c e n t e r of t h e s a m p l e , i s not only i d e a l in t e r m s of r e s p o n s e but a l s o p r o v i d e s s o m e i s o l a t i o n f r o m a n y s m a l l independent d i s t u r b a n c e s a t t h e p l a t e s and, ideally, d o e s not r e q u i r e t h e s a m p l e t o b e conditioned t o a n y s p e c i f i c t e m p e r a t u r e . T h e i n i t i a l t e m p e r a t u r e m u s t b e u n i f o r m o r a t l e a s t s y m m e t r i c about t h e c e n t e r point. If t h e r e i s t h e p o s s i b i l i t y that t h e r m a l c o n d u c t i v i t y i s a function of t e m p e r a t u r e , t h e c e n t e r p o s i t i o n a f f o r d s t h e p r o p e r location t o i n d i c a t e if t h e t e m p e r a t u r e h a s d e v i a t e d f r o m t h e m e a n value. T h e c e n t e r p o s i t i o n r e q u i r e s two i d e n t i c a l s a m p l e s . L a n g ( 4 ) h a s developed a n a p p a r a t u s with t h e h e a t m e t e r in t h i s location but i t is not c l e a r if a l l t h e a d v a n t a g e s w e r e r e c o g n i z e d . N o r r i s and F i t z r o y ( 5 ) g a v e t h e t h e o r e t i c a l j u s t i f i c a t i o n f o r t h e m e t h o d .

4. E x t e n s i o n t o T r a n s i e n t M e a s u r e m e n t C o n d i t i o n s

B e c k ( 6 ) h a s d e t e r m i n e d t h e o p t i m u m c o n f i g u r a t i o n f o r t r a n s i e n t h e a t flow a p p a r a t u s f o r s i m u l - t a n e o u s d e t e r m i n a t i o n of t h e r m a l conductivity and s p e c i f i c h e a t . When t h e t h e r m a l conductivity i s low, t h e o p t i m u m c o n f i g u r a t i o n i s a l a r g e t h e r m a l m a s s ''cold p l a t e , " p r e c o n d i t i o n e d but not continuously

c o o l e d , and a c o n s t a n t t e m p e r a t u r e w a r m s i d e with h e a t flux m e t e r e d . T h e cold p l a t e i s i n s u l a t e d on a l l but t h e a c t i v e s u r f a c e . T h e t h e r m a l p r o p e r t i e s a r e i n f e r r e d f r o m t h e cold p l a t e t e m p e r a t u r e and t h e hot s i d e h e a t flux, u s i n g a " n o n l i n e a r e s t i m a t i o n " p r o c e d u r e . T h e g u a r d e d hot p l a t e a p p a r a t u s with t e m p e r a t u r e c o n t r o l l e d h o t p l a t e c a n b e m o d i f i e d t o a t t a i n t h e r e q u i r e d configuration; t h i s a l l o w s t h e e x t e n s i o n of t h e a p p a r a t u s t o m e a s u r e m e n t of both t h e r m a l c o n d u c t i v i t y and s p e c i f i c h e a t .

With s a m p l e s of high t h e r m a l conductivity, t h e o p t i m u m c o n f i g u r a t i o n at t h e c o l d p l a t e r e m a i n s t h e s a m e w h i l e t h a t of t h e hot s i d e condition i s z e r o h e a t f1u.x. T h e r m a l p r o p e r t i e s a r e i n f e r r e d f r o m t e m p e r a t u r e s m e a s u r e d on t h e two s i d e s b y t h e s a m e p r o c e d u r e . By r e m o v i n g t h e hot p l a t e f r o m t h e g u a r d e d h o t p l a t e a p p a r a t u s , t h i s condition c a n b e a c c o m p l i s h e d .

F u r t h e r w o r k should b e c o n s i d e r e d on t h e a d a p t a t i o n of t h e g u a r d e d h o t p l a t e a p p a r a t u s t o t h i s t y p e of m e a s u r e m e n t , and a s t u d y of t h e s o u r c e s of e r r o r . Modification in d e s i g n of t h e h o t and cold p l a t e s m i g h t b e r e q u i r e d t o m a k e t h e m " u n i v e r s a l , " o r i n t e r c h a n g e a b l e p l a t e s m i g h t b e c o n s i d e r e d .

5. C o n c l u s i o n T h e c o n s t a n t t e m p e r a t u r e a n d h y b r i d m o d e s of o p e r a t i o n h a v e b e e n shown t o b e t h e m o s t p r a c t i c a l f o r t h e g u a r d e d hot p l a t e a p p a r a t u s . T h e r e s p o n s e t i m e s c a n b e a n o r d e r of m a g n i t u d e b e t t e r t h a n f o r c o n s t a n t h e a t f l u x o p e r a t i o n . T h e f i g u r e s p r e s e n t e d c a n b e u s e d a s a guide to e s t i m a t e t h e t i m e r e q u i r e d t o r e a c h a n a d e q u a t e s t e a d y s t a t e . G u a r d e d hot p l a t e a p p a r a t u s with t e m p e r a t u r e - c o n t r o l l e d p l a t e s c a n b e u s e d f o r r a p i d m e a s u r e - m e n t s , e i t h e r a l o n e o r i n conjunction with a h e a t m e t e r . T h e a p p a r a t u s a l s o m i g h t b e u s e d f o r t h e s i m u l t a n e o u s d e t e r m i n a t i o n of t h e r m a l conductivity and t h e r m a l diffusivity.

T h i s r e p o r t is a c o n t r i b u t i o n f r o m t h e Division of Building R e s e a r c h , National R e s e a r c h C o u n c i l of C a n a d a , a n d i s p u b l i s h e d with t h e a p p r o v a l of t h e D i r e c t o r of t h e Division. 6. R e f e r e n c e s (1) Method of t e s t f o r t h e r m a l conductivity of m a t e r i a l s by m e a n s of t h e g u a r d e d hot p l a t e

.

(C177 -63. B o o k of ASTM S t a n d a r d s , 1963, P a r t 3. ( 2 ) S o m e r s , E. V. , H i g h - s p e e d g u a r d e d hot p l a t e a p p a r a t u s f o r t h e r m a l conductivity of t h e r m a l insulation. T r a n s . , ASME, 7 8 , August 1956, p. 1191-6. (4 ) L a n g , D. L. , A q u i c k t h e r m a l conductivity t e s t on i n s u l a t i n g m a t e r i a l s , ASTM B u l l e t i n No. 216, S e p t e m b e r 1965, p. 58-60. (5) N o r r i s , R. H. a n d F i t z r o y , ( M r s . ) Nancy D. : A quick t h e r m a l conductivity t e s t on i n s u l a t i n g m a t e r i a l s , M a t e r i a l s R e s e a r c h and S t a n d a r d s , VI, S e p t e m b e r 1961, p. 727-729. ( 3 ) C h u r c h i l l , R. V. , O p e r a t i o n a l m a t h e m a t i c s , (6) B e c k , J. V. , T h e o p t i m u m a n a l y t i c a l S e c o n d Edition, M c G r a w - H i l l B o o k Co. I n c . , d e s i g n of t r a n s i e n t e x p e r i m e n t s f o r s i m u l - New York, 1958. t a n e o u s d e t e r m i n a t i o n s of t h e r m a l conduc

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t i v i t y and s p e c i f i c h e a t . M i c h i g a n S t a t e . u n i v e r s i t y , Ph. D. , 1 9 6 4 E n g i n e e r i n g , M e c h a n i c a l . Appendix I R e s p o n s e T i m e s f o r a n Infinite S l a b 1. B a s i c Equation T h e n o n d i m e n s i o n a l i z e d h e a t equation f o r a n i n f i n i t e s l a b with u n i f o r m b o u n d a r y c o n d i t i o n s is w h e r e

0

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[T ( x . f )

-

T ( o , ~ ) ]

/

[T ( b m )

-

T ( 0 ,

J]

X =

X/L, T a a t / L 2 , L .:t h e t h i c k n e s s of t h e s l a b , x

=

t h e s p a c e v a r i a b l e , running f r o m 0 t o L , t

=

t i m e , a

=

t h e t h e r m a l diffusivity. T h e ncx-~dimensionalized h e a t f l u x is w r i t t e n a s w h e r e q = h e a t flux, q l = t h e s t e a d y s t a t e h e a t f l u x ( - A T ( L , m)

-

T ( O , m)] /L),

X

m t h e t h e r m a l conductivity,

-

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i n d i c a t e s t h e 234

1.1. B o u n d a r y Conditions (B. C. ) 2 C o n s t a n t t e m p e r a t u r e a t X

=

0 T w o s e p a r a t e b o u n d a r y conditions a t X = 1 will b e c o n s i d e r e d : ( i ) C o n s t a n t t e m p e r a t u r e : 0(1, T)

=

1 , a n d ( i i ) C o n s t a n t h e a t flux: In e a c h c a s e t h e s t e a d y s t a t e t e m p e r a t u r e d i f f e r e n c e a n d s t e a d y s t a t e h e a t flux a r e e q u a l t o unity. 1. 2. I n i t i a l C o n d i t i o n s T h e following u n i f o r m i n i t i a l condition i s a s s u m e d : e ( x , 0)

=

W w h e r e W i s a f i n i t e constant. 1. 3. S o l u t i o n s t o B a s i c P r o b l e m S o l u t i o n s h a v e b e e n d e t e r m i n e d u s i n g b a s i c s o l u t i o n s a n d t h e "Superposition P r i n c i p l e " a s given by C h u r c h i l l (3). T h e s o l u t i o n s a r e given in t e r m s of t h e v a r i a b l e not s p e c i f i e d a t X r 1 , t h a t i s , in t e r m s of h e a t flux f o r c o n s t a n t t e m p e r a t u r e b o u n d a r y condition, a n d i n t e r m s of t e m p e r a t u r e f o r t h e c o n s t a n t h e a t f l u x b o u n d a r y condition. T h e s o l u t i o n s a r e f i r s t given i n g e n e r a l f o r m , t h e n f o r s p e c i a l c a s e s with X, o r both X a n d W, specified. T a b l e s of t h e f u n c t i o n s m a y b e obtained by w r i t i n g t h e a u t h o r s d i r e c t l y . (1) G e n e r a l F o r m of S o l u t i o n s ( a ) C o n s t a n t T e m p e r a t u r e B. C. m

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s i n (BnX/2)

.

exp (-B:T/~) n = l B o u n d a r y c o n d i t i o n s will b e a b b r e v i a t e d a s B. C.

(2) Special C a s e s

(a) Constant T e m p e r a t u r e B. C.

,

with X = I

m

"

2 T 2

ax

X , T = 1 t 2 n=

f

1 exp ( -A _n T ) - 4 W _{n = 1} exp ( -Bnr) (b) Constant Heat F l u

B.

C . , with =

(c) Constant T e m p e r a t u r e B. C. , with X

=

0. 5, W finite

(d) Constant T e m p e r a t u r e B.C.

,

with X

=

1 , W 0. 5

"

)

ax

= right -hand s i d e of eq (5) 0. 5,T ( e ) Constant T e m p e r a t u r e B. C . , with X = 0 , W

=

0. 5

a e j

-

r right-hand s i d e of eq (5) a x l o , T (f) Constant T e m p e r a t u r e , with

X

₌

1 , W = 1 2. Calculation of E r r o r s E r r o r s in determination of t h e r m a l conductivity, if m e a s u r e m e n t s a r e m a d e b e f o r e s t e a d y s t a t e conditions a r e attained, a r e derived below.

2.1. Constant T e m p e r a t u r e B. C. E r r o r a t t i m e T and position X.

s i n c e

2. 2. Constant Heat F l u x B. C. E r r o r at t i m e T and a t position X E r r o r

=

L o ( x , ~ ) - 8 ( o B r n ) J

_-

[ e x

r

- N o . T I ] 1 =

-

- 1 a t X = l 0 ( 1 , r ) s i n c e 0 ( 0 , ~ )

=

0 , all T , and 0 ( X , w )

=

X.

F i g u r e 1 R e a p o n s e T i m e s for M e a s u r i n g T h e r m a l Conductivity

-

E r r o r a t Time T _{A f t e r S t a r t of T e s t}

-

Constant T e m p . and Constant Heat Flux M o d e s

-

Initial T e m p . , W = 0. 5

C O N S T A N T H E A T F L U X M O D E

I N I T I A L T E M P E R A T U R E

=

W

O P T I M U M W

h . 6 3 5

Figure 2 Response T i m e s for Measuring Thermal Conductivity

- Error at Time 7 After Start of T e s t

-

Constant Heat Flux Mode

T =

a t / P

Figure 3 R e s p o n s e T i m e s for Measuring Thermal Conductivity

-

E r r o r at T i m e T After Start of T e s t

-

Constant Temp. Mode