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Heat transport through thermal insulation: an application of the
principles of thermodynamics of irreversible processes
Kumaran, M. K.; Stephenson, D. G.
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National Research
Conseil national
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Council Canada
de recherches Canada
Institute for
lnstitut de
Research in
recherche en
Construction
construction
Heat Tmnsport Through Thermal
Insulation: An Application of the
Principles of Thermodynamics of
Irreversible Processes
by M.K. Kumaran and D.G. Stephenson
Reprinted from
ASME Paper No. 86-WA/HT-70, p. 1-4
ASME Winter Annual Meeting
Anaheim, California, December 7
-
12, 1986
(IRC Paper No. 1445)
Price $2.00
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sston a* meellngs of Ine SI scussjon Ic pr~nted gnly ~f Irn ASME for ?siteen nontl) rntfd ~n USA ~ o l s i b l s far s 3ciery or of it. the paper l a IS a!ter the m
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raremania or oprnlonr sdv ! D.v~srons or S=c!luns. orpubl~shed In an PSME JUI ee!llg ,anted In pap vrinted In Its Jrnnl Papers ers or in d ~ s publicalrons arp ava~laille ABSTRACT
Heat Transport Through Thermal Insulation: An
Application of the Principles of Thermodynamics of
Irreversible Processes
M.
K. KUMARAN and 0. G. STEPHENSON
Building Services Section
Institute for Research in Construction
National Research Council Canada
Ottawa, Canada, K I A OR6
The p r i n c i p l e s of i r r e v e r s i b l e thermodynamics a r e u s e d t o i n t e r p r e t h e a t t r a n s f e r t h r o u g h d r y t h e r m a l i n s u l a t i o n . The h e a t f l u x i s t r e a t e d a s coupled c o n d u c t i v e and r a d i a t i v e modes of h e a t t r a n s f e r and phenomenological e q u a t i o n s a r e a c c o r d i n g l y d e r i v e d . L a b o r a t o r y measurements of h e a t f l u x t h r o u g h a specimen of g l a s s f i b r e i n s u l a t i o n a r e u s e d t o check t h e v a l i d i t y of t h e approach. An e x p r e s s i o n i s d e r i v e d f o r t h e t e m p e r a t u r e dependence of t h e a p p a r e n t t h e r m a l c o n d u c t i v i t y of d r y t h e r m a l i n s u l a t i o n . The e x p e r i m e n t a l d a t a f o r a c e r a m i c board a r e w e l l r e p r e s e n t e d by t h e e x p r e s s i o n f o r a t e m p e r a t u r e r a n g e of 800 K. An e x t e n s i o n of t h e t h e o r e t i c a l method t o d e s c r i b e s i m u l t a n e o u s h e a t and m o i s t u r e t r a n s p o r t t h r o u g h t h e r m a l i n s u l a t i o n i s s u g g e s t e d . where
(c)
i s t h e t e m p e r a t u r e g r a d i e n t i n t h e d i r e c t i o n dxof t h e h e a t t r a n s f e r and ( A ' ) and (A") a r e
p r o p o r t i o n a l i t y c o n s t a n t s t h a t d e s c r i b e c o n d u c t i v e ( s o l i d
+
g a s ) and r a d i a t i v e modes of h e a t t r a n s f e r , r e s p e c t i v e l y . The u s e of F o u r i e r r e l a t i o n and R o s s e l a n d ' s a p p r o x i m a t i o n , t o g e t h e r w i t h t h e t h e o r y of i r r e v e r s i b l e p r o c e s s e s (T.I.P.)(1-6)
c a n p r o v i d e a method t o l o o k a t t h e i n t e r a c t i o n between c o n d u c t i v e and r a d i a t i v e modes of h e a t t r a n s f e r i n t h e r m a l i n s u l a t i o n . T h i s p a p e r d e s c r i b e s s u c h a n a p p l i c a t i o n of T.I.P. t o h e a t t r a n s f e r t h r o u g h t h e r m a l i n s u l a t i o n . ' A t h r e e - t e r m e x p r e s s i o n i s d e r i v e d f o r t h e h e a t f l u x t h r o u g h t h e i n s u l a t i o n . One t e r m r e p r e s e n t s t h e c o n d u c t i v e p a r t , t h e s e c o n d , t h e r a d i a t i v e p a r t and t h e t h i r d . t h e i n t e r a c t i o n between t h e two D a r t s .KEY WORDS The b a s i c p o s t u l a t e s of T.I.P. a r e :
1 ) The r a t e (0) of e n t r o p y p r o d u c t i o n p e r u n i t volume Heat t r a n s f e r , t h e r m a l i n s u l a t i o n , thermodynamics i s g i v e n by t h e sum of t h e p r o d u c t s of f l u x e s ( J i ) of i r r e v e r s i b l e p r o c e s s e s , r a t e of e n t r o p y p r o d u c t i o n , and a s s o c i a t e d f o r c e s ( I $ ~ ) i . e . phenomenological e q u a t i o n s , m o i s t u r e . 0 =
C
Jiei
( 3 ) INTRODUCTION i Heat i s t r a n s f e r r e d t h r o u g h d r y t h e r m a l i n s u l a t i o n by a c o m b i n a t i o n of c o n d u c t i o n and r a d i a t i o n . Models d e s c r i b e d i n t h e l i t e r a t u r e i n t e r p r e t t h e h e a t t r a n s f e r a s t h e sum of s o l i d c o n d u c t i o n , g a s c o n d u c t i o n and r a d i a t i o n . I n e n g i n e e r i n g a p p l i c a t i o n s , t h e , i n t e r a c t i o n between c o n d u c t i o n and r a d i a t i o n i s u s u a l l y n e g l e c t e d and t h e a p p a r e n t t h e r m a l c o n d u c t i v i t y (Aapparent) of a t h e r m a l i n s u l a t i o n i s w r i t t e n aswhere (As) i s t h e c o n t r i b u t i o n from s o l i d c o n d u c t i o n , (A ) from g a s c o n d u c t i o n and (Ar) from r a d i a t i o n .
g
These models u s e F o u r i e r r e l a t i o n and Rosseland a p p r o x i m a t i o n t o w r i t e e x p r e s s i o n s f o r t h e h e a t f l u x (9) a s 2 ) The components of t h e f l u x a r e l i n e a r f u n c t i o n s of t h e components of f o r c e s and a r e g i v e n by phenomenological e q u a t i o n s as where t h e c o e f f i c i e n t s Lik a r e c a l l e d phenomenological c o e f f i c i e n t s . 3 ) The c r o s s phenomenological c o e f f i c i e n t s , s u b j e c t t o c e r t a i n m a t h e m a t i c a l r e s t r i c t i o n s , obey t h e Onsager r e c i p r o c a l r e l a t i o n s
(g)
i . e . L . . = L . 1 J ~i ( 5 )Presented at the Winter Annual Meeting Anaheim, California - December 7-12, 1986
Thus f o r a n i r r e v e r s i b l e p r o c e s s , a s s o c i a t e d w i t h
two s i m u l t a n e o u s f l u x e s , J1 and J2
where and @ 2 a r e t h e f o r c e s c o r r e s p o n d i n g t o J1
and J 2 , r e s p e c t i v e l y and
Heat T r a n s p o r t and E n t r o p y P r o d u c t i o n
The e n t r o p y f l u x t h r o u g h any m a t e r i a l is:
The d i r e c t i o n of t h e S v e c t o r is t h e same a s t h e d i r e c t i o n of t h e h e a t f l u x ( q ) , s i n c e t h e t e m p e r a t u r e (T) i s a s c a l a r . I f we make t h e x a x i s p a r a l l e l t o t h e e n t r o p y and h e a t f l u x v e c t o r s , t h e d e r i v a t i v e i s dx d S = l . * - q
.dT.
-
dx T dxT2
d x ( 9 )The second t e r m of Eq. ( 9 ) i s t h e r a t e a t which
e n t r o p y i s b e i n g produced i n u n i t volume of t h e
m a t e r i a l ; i.e.,
When h e a t i s t r a n s f e r r e d by c o n d u c t i o n ,
Thus f o r h e a t c o n d u c t i o n , from Eqs. ( 3 ) , ( 4 ) , ( 1 0 ) and ( 1 1 ) I f t h e h e a t i s t r a n s f e r r e d by r a d i a t i o n qr
-
T3dT
dx ( 1 4 ) ( t h i s i s t h e Rosseland a p p r o x i m a t i o n f o r t h e r m a l r a d i a t i o n ) . I n t h i s c a s e When h e a t c o n d u c t i o n and h e a t t r a n s f e r by r a d i a t i o n o c c u r t o g e t h e r , t h e f l u x and f o r c e v e c t o r s h a v e two components, and t h e phenomenological e q u a t i o n t h a t i n t e r r e l a t e s t h e s e components becomes:where a , b and c a r e t h e phenomenological c o e f f i c i e n t s . S i n c e q c and q r a r e i n t h e same d i r e c t i o n , t h e y may b e added a l g e b r a i c a l l y t o o b t a i n t h e t o t a l h e a t f l u x The f i r s t t e r m i n t h e b r a c k e t g i v e s t h e c o n d u c t i v e p a r t of t h e h e a t t r a n s f e r , t h e t h i r d term, t h e r a d i a t i v e p a r t and t h e second t e r m , t h e i n t e r a c t i v e p a r t . T h i s i n d i c a t e s t h a t t h e a p p a r e n t fv$rmal c o n d u c t i v i t y s h o u l d b e a q u a d r a t i c f u n c t i o n of T
.
The o b j e c t i v e of t h i s p a p e r i s t o check t h i s h y p o t h e s i s .A TEST OF THE HYPOTHESIS
The form of t h e t e m p e r a t u r e dependence of t h e
a p p a r e n t t h e r m a l c o n d u c t i v i t y t h a t i s i n d i c a t e d by
Eq. ( 1 8 ) c a n b e checked by measuring t h e t o t a l r a t e of
h e a t f l o w t h r o u g h s e m i - t r a n s p a r e n t m a t e r i a l s . It i s
c o n v e n i e n t t o d e f i n e a q u a n t i t y Q a s
Then d x = ( a
+
2 b e ~ l . '+
c.T3)dT
dx (20)Hence from ( 1 8 ) and ( 2 0 ) .
t h e n q = - - . dQ dx (21) I f one measures t h e , t o t a l h e a t f l u x (q1-2) t h r o u g h a sample of t h i c k n e s s t , when t h e s u r f a c e s a r e k e p t a t T1 and T2, S i m i l a r e q u a t i o n s c a n b e w r i t t e n when t h e s u r f a c e s a r e
m a i n t a i n e d a t T3 and T,,, T5 and T6, e t c . These
e q u a t i o n s c a n b e combined i n t o a m a t r i x form
b
and s o l v e d f o r
5
,
-
and2
.
T e s t Specimens l a b o r a t o r i e s i n Canada and t h e United S t a t e s
(10).
Two t e s t specimens were prepared from a s h e e t of These d a t a a r e i n t h e form of t h e a p p a r e n t thermal medium d e n s i t y g l a s s f i b e r board produced by Johns c o n d u c t i v i t y a t f o u r mean t e m p e r a t u r e s r a n g i n g from Manville Co. f o r t h e N a t i o n a l Bureau of S t a n d a r d s , 533 K t o 1363 K. These d a t a were f i t t e d by a q u a d r a t i c Washington, D.C. The average d e n s i t y of t h e sample was i n ~ 1 . ~ . Table 2 g i v e s t h e s e d a t a and t h e v a l u e s of 51 kg*m-j. Specimens were c u t and placed i n wooden a , b and c t h a t were d e r i v e d from them, and shows t h e frames. The dimensions of t h e specimens were 58.42 x c a l c u l a t e d v a l u e s of t h e a p p a r e n t thermal c o n d u c t i v i t y . 58.57 x 2.64 cm. The two specimens weighed 461.2 g and The q u a d r a t i c i n T ~r e p r e s e n t s t h e d a t a v e r y w e l l * ~461.3 g, and were i d e n t i c a l i n a l l o t h e r r e s p e c t s . over t h i s r e l a t i v e l y wide range of temperatures. The These specimens were d r i e d i n an oven a t 323 K f o r one v a l u e s of t h e c o e f f i c i e n t s a , b, and c depend on t h e week and t h e n t h e y were mounted on a 60 cm guarded h o t n a t u r e of t h e m a t e r i a l , and may a l s o depend on t h e p l a t e a p p a r a t u s b u i l t according t o ASTM s p e c i f i c a t i o n s e m i s s i v i t y of t h e bounding s u r f a c e s and t h e t h i c k n e s s
(2).
Steady s t a t e measurements were made a t t e n of t h e sample. Measurements on s i m i l a r samples of d i f f e r e n t s e t s of boundary temperatures. d i f f e r e n t t h i c k n e s s e s a r e needed t o check on t h e s ep o s s i b i l i t i e s . These r e s u l t s show, however, t h a t t h e s e EXPERIMENTAL RESULTS c h a r a c t e r i s t i c c o e f f i c i e n t s a r e n o t f u n c t i o n s of
temperature o r temperature g r a d i e n t , and t h a t t h e The t e s t r e s u l t s and t h e d e r i v e d v a l u e s of a , b a n a l y s i s based on T.I.P. i s v a l i d f o r t h i s t y p e of and c a r e g i v e n i n Table 1, a l o n g w i t h v a l u e s of h e a t i n s u l a t i o n m a t e r i a l . The i n t e r a c t i o n between t h e h e a t f l u x c a l c u l a t e d u s i n g t h e s e v a l u e s of a , b and c and conduction and r a d i a t i o n i s t a k e n i n t o account w i t h o u t t h e boundary t e m p e r a t u r e s f o r t h e v a r i o u s t e s t s . The having t o e s t a b l i s h t h e temperature d i s t r i b u t i o n s t a n d a r d d e v i a t i o n between t h e measured and c a l c u l a t e d through t h e m a t e r i a l .
v a l u e s of h e a t f l u x i s 0.04 W/m2 when t h e f l u x ranged
from 23.4 t o 52.6 w/m2. Table 2
Table 1 The apparent thermal c o n d u c t i v i t y (happarent) of a ceramic board a t d i f f e r e n t temperatures. Heat flow through a dry sample of medium d e n s i t y g l a s s
f i b e r i n s u l a t i o n (51 kg*m-3); T1 and T2 a r e s u r f a c e
t e m p e r t u r e s and q1-2 i s t h e h e a t f l u x T 'apparent (W/meK) (K) Measured Calculated* 41-2 533 0.073 0.073 Measured Calculated*
2
)2)
w/m2 w/m2 813 0.114 0.113 293.92 273.97 23.47 23.43 1088 0.189 0.189 303.11 283.16 24.24 24.29 1363 0.312 0.311 312.63 293.48 24.21 24.24 a = 5.40 x W/m-K b = 1.92 x w / ~ * K ~ . ~ 322.08 303.25 24.75 24.78 c = 9.40 x 10-I W/m*K4 331.61 313.62 24.72 24.66 *'apparent = a+
2 b ~ l . ~+
c T~343.13 323.3,, 28.47 28.43 An Extension t o Heat T r a n s f e r Through Moist M a t e r i a l s The p r i n c i p a l advantage of t h e TIP approach i s 354.02 333.88 30.21 30.24 t h a t i t can be extended t o i n c l u d e a d d i t i o n a l
components i n t h e f l u x v e c t o r . An e n c l o s e d porous 363.46 343.21 31.65 31.66 system t h a t h a s moist m a t e r i a l a d j a c e n t t o a warm
boundary w i l l have vapour flowing from t h e warm r e g i o n 314.22 286.33 34.90 34.93 t o t h e c o o l e r r e g i o n . The h e a t f l u x ( q v ) a s s o c i a t e d
w i t h t h i s vapour flow can be r e p r e s e n t e d a s , 333.61 293.86 52.61 52.60
dPV dT
a = 1.896 x W/m-K q v = - k ( h * - ) - dT dx
b = 1.764 x 10-7 ~ / r n * K ~ . ~
c = 4.520 x 10-lo w / ~ * K ~ where k = vapour p e r m e a b i l i t y of t h e m a t e r i a l ,
t = 26.4 mm h = e n t h a l p y of s a t u r a t e d vapour a t T, Pv = s a t u r a t i o n vapour p r e s s u r e of water a t T. 2b (T2.5- T$.5)
*ql-2 =
5
(Ti-T2) +-
t 2.5t dPv
Around 300 K t h e product h c a n be approximated by
+
5
(T:-
T;) A.~15*4. 1 dT4 t
As
t h e NRC guarded h o t p l a t e a p p a r a t u s i s l i m i t e d l T h i s approximation h a s been d e r i v e d from t a b u l a t e d t o t e m p e r a t u r e s between 273 K and 365 K, t h e v a l i d i t y v a l u e s of e n t h a l p y and vapour p r e s s u r e of water a t of Eq. (16) could only be e s t a b l i s h e d over t h i s l i m i t e d s a t u r a t i o n v s temperature. These d a t a a r e p u b l i s h e d r a n g e of temperature. However, some h i g h t e m p e r a t u r e i n ASHRAE Handbook of Fundamentals, 1985, Ch. 6, d a t a were a v a i l a b l e from a s e t of round-robin Table 2: p. 6.6.T h i s means t h a t t h e f l u x ( J v ) s h o u l d be
The phenomenological e q u a t i o n f o r h e a t t r a n s p o r t by c o n d u c t i o n , r a d i a t i o n and m o i s t u r e m i g r a t i o n becomes
The c o e f f i c i e n t s a , b and c can be d e t e r m i n e d from t e s t s on a d r y sample and t h e c o e f f i c i e n t s d, e and f from a d d i t i o n a l t e s t s on a m o i s t sample of t h e same m a t e r i a l . Of c o u r s e , t h e c o q o n e n t qv of h e a t f l u x w i l l o n l y o c c u r when t h e r e i s m o i s t u r e i n t h e warm p a r t s , o f t h e m a t e r i a l . As soon a s a l l t h e m o i s t u r e h a s m i g r a t e d t o t h e c o l d e s t r e g i o n , t h e m o i s t u r e f l u x and a s s o c i a t e d h e a t f l u x w i l l s t o p ; t h e c o n d u c t i o n and r a d i a t i o n w i l l c o n t i n u e a s l o n g a s a t e m p e r a t u r e g r a d i e n t e x i s t s . e - CONCLUSION The p o s t u l a t e s of t h e Thermodynamics of I r r e v e r s i b l e P r o c e s s e s s i m p l i f y t h e a n a l y s i s of h e a t t r a n s f e r i n s i t u a t i o n s where more t h a n one mode of h e a t t r a n s f e r o c c u r s . T h i s a p p r o a c h t a k e s a c c o u n t of energy s h i f t i n g from one mode of t r a n s p o r t t o a n o t h e r w i t h i n t h e m a t e r i a l w i t h o u t h a v i n g t o d e t e r m i n e t h e
t e m p e r a t u r e d i s t r i b u t i o n through t h e m a t e r i a l . The approach, which h a s been shown t o be a p p l i c a b l e t o h e a t t r a n s f e r t h r o u g h t h e r m a l i n s u l a t i o n , may a l s o be a p p l i c a b l e t o h e a t t r a n s p o r t t h r o u g h s e m i - t r a n s p a r e n t g a s e s i n t h e combustion chambers of f u r n a c e s and i n t e r n a l combustion e n g i n e s .
It a p p e a r s t h a t t h e i n t e r a c t i o n between c o n d u c t i v e and r a d i a t i v e modes of h e a t t r a n s f e r amounts t o
a p p r o x i m a t e l y 6% of t h e h e a t f l u x t h r o u g h t h e two specimens, f o r t h e r a n g e of t e m p e r a t u r e r e p o r t e d i n t h i s paper. F u r t h e r t e s t i n g i s needed t o check t h e v a l i d i t y of Eq. ( 2 7 ) when m o i s t u r e m i g r a t i o n c o n t r i b u t e s t o t h e h e a t t r a n s p o r t t h r o u g h porous m a t e r i a l s . The c u r r e n t s t u d y h a s d e m o n s t r a t e d t h a t h e a t t r a n s p o r t t h r o u g h a s e m i - t r a n s p a r e n t medium l i k e g l a s s f i b e r i n s u l a t i o n can be c a l c u l a t e d by u s i n g t h r e e c o e f f i c i e n t s t h a t c h a r a c t e r i z e t h e m a t e r i a l . The v a l u e s of t h e s e c h a r a c t e r i s t i c c o e f f i c i e n t s c a n be d e t e r m i n e d from measurements o f t h e t o t a l h e a t f l o w t h r o u g h a sample of t h e m a t e r i a l w i t h v a r i o u s c o m b i n a t i o n s of t h e boundary t e m p e r a t u r e s . These c h a r a c t e r i s t i c c o e f f i c i e n t s a p p e a r t o be independent of t e m p e r a t u r e o v e r a wide range. I n t h i s r e s p e c t , t h e y a r e p r e f e r a b l e t o t h e v a l u e of t h e a p p a r e n t t h e r m a l c o n d u c t i v i t y , which i s a f u n c t i o n of t e m p e r a t u r e . The a p p a r e n t t h e r m a l c o n d u c t i v i t y c a n be c a l c u l a t e d f o r any v a l u e of mean t e m p e r a t u r e when t h e v a l u e s of t h e c h a r a c t e r i s t i c c o e f f i c i e n t s a r e known. a b d b c e d e f ACKNOWLEDGEMENT The a u t h o r s g r a t e f u l l y acknowledge t h e t e c h n i c a l a s s i s t a n c e o f M r . J . G . T h s r i a u l t . T h i s p a p e r i s a c o n t r i b u t i o n of t h e I n s t i t u t e f o r Research i n C o n s t r u c t i o n , N a t i o n a l R e s e a r c h C o u n c i l o f Canada. REFERENCES T-l
dT
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