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Calculation of transient heat flow through walls and roofs
Mitalas, G. P.
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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 C A N A D A C O N S E I L N A T I O N A L D E R E C H E R C H E S DU C A N A D AA h!
A y x ,:g,
C A L C U L A T I O N O F T R A N S I E N T H E A T FLOW T H R O U G H W A L L S AND R O O F S b Y G . P . M i t a l a s R e p r i n t e d from A S H R A E T R A N S A C T I O N S P r i c e 1 0 c e n t s R e s e a r c h P a p e r No. 3 9 3 of t h e D i v i s i o n of B u i l d i n g R e s e a r c h OTTAWA F e b r u a r y , 1 9 6 9 N R C 1 0 6 2 9CALCUL DES ECOULEMENTS DE CHALEUR TRANSITOIRES AU TRAVERS DES MURS ET DES TOITURES
SOMMAIRE
L'auteur utilise la me'thode de d6termination du facteur de re'ponse pour le calcul des 6coulements transitoires de chaleur au travers des murs e t des toitures, dans le c a s o; les coefficients d e transmission superficielle de la chaleur ne sont pas constants. Cette m6thode e s t valable dans des conditions ambiantes pe'riodiques e t non p6riodiques, e t convient pour les calculs des transmissions superficielles de chaleur principalement par radiation e t Gvaporation
No.
2086
G. P. MITALAS
Member ASHRAE
Calculation of Transient Heat Flow Through Walls and Roofs
An a c c u r a t e c a l c u l a t i o n of h e a t f l o w a n d t e m p e r a - t u r e a t t h e o u t e r a n d i n n e r s u r f a c e s of w a l l s a n d r o o f s i s a n i m p o r t a n t s t e p i n t h e d e s i g n of a b u i l d i n g a n d i t s a i r - c o n d i t i o n i n g s y s t e m . T h e commonly u s e d m e t h o d s ' v 2 of c a l c u l a t i n g t e m p e r a t u r e a n d h e a t f l o w through b u i l d i n g e n c l o s u r e s a r e b a s e d o n t h e a s - s u m p t i o n s t h a t t h e a m b i e n t c o n d i t i o n s v a r y i n e x - a c t l y t h e s a m e w a y o n s u c c e s s i v e d a y s , a n d t h a t t h e s u r f a c e h e a t t r a n s f e r c o e f f i c i e n t s d o n o t v a r y a t a l l . T h e s e a s s u m p t i o n s h a v e b e e n n e c e s s a r y b e c a u s e n o p r a c t i c a l t e c h n i q u e s w e r e a v a i l a b l e t o c o p e w i t h t h e a c t u a l c o n d i t i o n s . With t h e d e v e l o p m e n t of d i g i t a l c o m p u t e r s , h o w e v e r , i t is n o w q u i t e p r a c t i c a b l e t o t a k e a c c o u n t of h e a t t r a n s f e r c o e f f i c i e n t s t h a t d e p e n d o n t h e s u r f a c e t e m p e r a t u r e , a m b i e n t a i r t e m p e r a t u r e a n d w i n d c o n d i t i o n s . I t i s a l s o n e a r l y a s s i m p l e t o d e a l w i t h n o n - p e r i o d i c a m b i e n t t e m p e r a t u r e a n d r a d i - a t i o n c o n d i t i o n s a s w i t h p e r i o d i c c o n d i t i o n s . C u r r e n t m e t h o d s ' 1 2 a r e n o t a d e q u a t e f o r m a k i n g e c o n o m i c a s s e s s m e n t s of m o d e r n c o m p l e x a i r - c o n d i - t i o n i n g s y s t e m s u t i l i z i n g v a r i o u s e n e r g y s o u r c e s , b e c a u s e t h e y c a n n o t a c c o u n t for t h e r e a l v a r i a t i o n s of w e a t h e r c o n d i t i o n s w i t h t i m e . A n e w m e t h o d of p r e d i c t i n g a n n u a l e n e r g y r e q u i r e m e n t s for h e a t i n g a n d c o o l i n g , b e i n g c o n s i d e r e d by t h e A S H R A E T a s k G r o u p o n E n e r g y R e q u i r e m e n t s , u t i l i z e s h o u r l y v a l - u e s of w e a t h e r p a r a m e t e r s f o r a w h o l e y e a r . T h e G. P . M i t a l a s is a R e s e a r c h O f f i c e r , B u i l d i n g S e r v i c e s S e c t i o n , D i v i s i o n of B u i l d i n g 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 o d a , O t t a w a . T h i s paper i s a c o n t r i b u t i o n of t h e D i v i s i o n o f B u i l d i n g R e s e a r c h of t h e 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 o d a and i s w i t h the a p p r o v a l of t h e D i r e c t o r of t h e D i v i s i o n . T h i s paper w a s prepared for p r e s e n t a t i o n a t t h e A S H R A E A n n u a l M e e t i n g , L a k e P l a c i d , N e w Y o r k , June 2 4 - 2 6 , 1 9 6 8 . r e s p o n s e f a c t o r m e t h o d p r e s e n t e d i n t h i s p a p e r per- m i t s t h i s t y p e of a p p r o a c h ; i t c a n h a n d l e p e r i o d i c o r n o n - p e r i o d i c a m b i e n t c o n d i t i o n s a n d v a r i a b l e s u r f a c e h e a t t r a n s f e r c o e f f i c i e n t s , a n d is s u i t a b l e for c a s e s w h e r e r a d i a t i o n a n d e v a p o r a t i o n a r e t h e p r e d o m i n a n t m e c h a n i s m s o f h e a t t r a n s f e r a t a s u r f a c e . I t is of i n t e r e s t t o n o t e t h a t t h e v a l u e s of t h e T o t a l E q u i v a l e n t T e m p e r a t u r e D i f f e r e n t i a l s i n t h e A S H R A E H A N D B O O K O F F U N D A M E N T A L S ( 1 9 6 7 ) w e r e c a l c u l a t e d by c o m p u t e r u s i n g t h e m e t h o d d e - s c r i b e d i n t h i s p a p e r . T h e c o m p u t a t i o n c o u l d h a v e b e e n d o n e by t h e c o n v e n t i o n a l m e t h o d s , b u t i t w a s m u c h e a s i e r to u s e t h e t i m e s e r i e s t e c h n i q u e s . T H E BASIS O F T H E M E T H O D T h e h e a t flux t h r o u g h t h e s u r f a c e s of a w a l l c a n b e e x p r e s s e d i n t e r m s of t h e s u r f a c e t e m p e r a t u r e a n d t h e r m a l c h a r a c t e r i s t i c s of t h e w a l l u s i n g t h e time s e r i e s t e c h n i q u e s d e s c r i b e d i n R e f e r e n c e s
3, 4
a n d5 .
T h u s , a l l u n k n o w n h e a t f l u x e s a t t h e s u r f a c e c a n b e r e l a t e d t o t h e s u r f a c e t e m p e r a t u r e s a n d , if t h e t e m p e r a t u r e h i s t o r y of t h e s u r f a c e i s k n o w n , t h e h e a t b a l a n c e e q u a t i o n s c a n b e s o l v e d t o find t h e c u r r e n t v a l u e of s u r f a c e t e m p e r a t u r e . When t h e h e a t t r a n s f e r c o e f f i c i e n t d e p e n d s o n t h e s u r f a c e t e m p e r a t u r e i t i s n e c e s s a r y t o s o l v e t h e e q u a t i o n s by a n i t e r a t i v e pro- c e d u r e . O n c e t h e s u r f a c e t e m p e r a t u r e h a s b e e n f o u n d i t b e c o m e s p a r t of t h e t e m p e r a t u r e h i s t o r y ts b e used' for f u t u r e c a l c u l a t i o n s . Dry S u r f a c e T h e h e a t t r a n s f e r a t a d r y s u r f a c e of a s l a b i s by c o n d u c t i o n , c o n v e c t i o n a n d r a d i a t i o n ( F i g . 1 ) a n dS u r f
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A v 4 t h e a l g e b r a i c s u m of t h e h e a t t r a n s f e r c o m p o n e n t s by t h e s e m e c h a n i s m s m u s t e q u a l z e r o . T h i s is t h e b a s i s of t h e h e a t b a l a n c e e q u a t i o n f o r a s u r f a c e . F o r o n e - d i m e n s i o n a l a n d l i n e a r h e a t c o n d u c t i o n ( i e : h e a t c o n d u c t i o n t h a t c a n b e d e s c r i b e d by l i n e a r d i f f e r e n t i a l e q u a t i o n s ) t h e s u r f a c e h e a t f l u x e s a n d t e m p e r a t u r e s c a n b e r e l a t e d by t h e t h r e e s e t s of r e s p o n s e f a c t o r s X , Y a n dZ.
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w h e r e q l V n , q ~ , ~ , 01,,, d 2 , , a r e t h e t e r m s i n t h e t i m e s e r i e s f o r h e a t f l u x a n d t e m p e r a t u r e r e s p e c t i v e l y . T h e s u b s c r i p t n i s a n i n t e g e r t h a t i n d i c a t e s t i m e ; f o r s i m p l i c i t y n is u s e d i n s t e a d of nA w h e r e A i s t h e q 1 q 2>
1 1 v t i m e i n t e r v a l . S u b s c r i p t s 1 a n d 2 d e n o t e t h e t w o s i d e s of t h e s l a b , a n d by d e f i n i t i o n t h e h e a t f l u x i n t h e d i r e c t i o n from s u r f a c e 1 t o 2 is p o s i t i v e . T h e u p p e r l i m i t s of t h e s u m m a t i o n s c a n b e r e p l a c e d by a f i n i t e i n t e g e r J s i n c e x i , y , a n d z j t e n d t o z e r o a s j b e c o m e s l a r g e . T h e r e s p o n s e f a c t o r s e t s X , Y a n dZ
c a n b e c o m - p u t e d f o r a n y m u l t i - l a y e r s l a b p r o v i d e d t h e h e a t f l o w is o n e - d i m e n s i o n a l 6 . T h e f a c t o r s f o r w a l l s a n d r o o f s w h e r e h e a t f l o w is n o t s t r i c t l y o n e - d i m e n s i o n a l (e.g.: h o l l o w t i l e , b r i c k o r c o n c r e t e b l o c k w a l l s ) m u s t b e d e t e r m i n e d e x p e r i m e n t a l l y . H e a t t r a n s f e r by c o n v e c t i o n b e t w e e n t h e s l a b s u r - f a c e 1 a n d a d j a c e n t a i r is g i v e n by C l , n = h l , n ( a l , n-
61,n)(3)
w h e r e , h l , , = f i l m c o n d u c t a n c e a l , , = a m b i e n t a i r t e m p e r a t u r e .In g e n e r a l , th'e film c o n d u c t a n c e d e p e n d s on t h e a m b i e n t t e m p e r a t u r e , s l a b s u r f a c e t e m p e r a t u r e a n d a i r motion; e.g.: for n a t u r a l c o n v e c t i o n a t a v e r t i c a l s u r f a c e 1 T h e r a d i a n t h e a t t r a n s f e r a t s u r f a c e 1 c o n s i s t s of the long-wave r a d i a t i o n e m i t t e d by t h e s u r f a c e r l , n = ~ 1 , ~ 0 ( 4 6 O
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a n d r a d i a t i o n a b s o r b e d by t h e s u r f a c e 1 . n = a l , n I l , n + c l , n L l , n(6)
w h e r e , u = S t e f a n - B o l t z m a n n c o n s t a n t (= 0.1712 x c l P n = a b s o r p t i v i t y o r e m i s s i v i t y of . t h e s u r f a c e 1 for l o n g - w a v e r a d i a t i o n 11,, = s h o r t - w a v e radiacion i n c i d e n t on s u r f a c e 1 a l , n = a b s o r p t i v i t y of s u r f a c e 1 for s h o r t - w a v e r a d i a t i o n L1,, = l o n g - w a v e r a d i a t i o n i n c i d e n t on s u r f a c e 1. E q s 3 , 4, 5 a n d6
a p p l y for s u r f a c e 2 if t h e s u b s c r i p t i s c h a n g e d from 1 t o 2 . U s i n g t h e a b o v e n o t a t i o n , t h e h e a t b a l a n c e e q u a - t i o n s for t h e t w o s u r f a c e s a r e S l , n - r1.n + C l , n - 41,n = 0 (7) S ~ , n - r2.n + C 2 . n + q2.n = 0 - (8) S u b s t i t u t i o n of t h e e x p r e s s i o n s for q l , , , q 2 , n , c ~ , ~ a n d ~ 2a n d c o l l e c t i o n of t h e unknowns on t h e l e f t - , ~ hand s i d e g i v e s (-hl,, - xo)Oi,n + ~ o 0 z . n - r l , n + h l , n a l , n = T h e a b o v e e q u a t i o n s , a n d t h e o t h e r s r e l a t i n g r l , , , r2,n, h l , , a n d h 2 + t o t h e s u r f a c e t e m p e r a t u r e s , c a nbe s o l v e d for
el,,
a n d 0 2 , n u s i n g a n i t e r a t i v e pro---- - - * T h i s i s just a s a m p l e of t h e f a m i l y of e q u a t i o n s t h a t r e l a t e s u r f a c e f i l m c o n d u c t a n c e , t e m p e r a t u r e s , s u r f a c e o r i e n t a t i o n and v e l o c i t y of forced a i r m o t i o n . c e d u r e . T h e s i m p l e s t , but n o t n e c e s s a r i l y t h e f a s t e s t m e t h o d i s : 1. G u e s s d l , , a n d 0 2 , n v a l u e s . ( T h e s i m p l e s t g u e s s i s to t a k e t h e v a l u e a t n e q u a l t o i t s p r e v i o u s v a l u e . ) 2. U s i n g t h e s e v a l u e s c a l c u l a t e h l , n , h 2 , n , r l , , a n d r 2 , n by E q s 4 a n d 5.
3.
S u b s t i t u t e a b o v e v a l u e s i n E q s9
a n d 1 0 a n d s o l v e f o r d l , , a n d 0 2 , n . 4. R e p e a t t h e c a l c u l a t i o n s . S t o p w h e n t h e d i f f e r e n c e b e t w e e n t h e v a l u e s of t h e t e m p e r a t u r e s o b t a i n e d i n s u c c e s s i v e i t e r a t i o n s i s n e g l i g i b l e . Normally t h e s l a b s u r f a c e t e m p e r a t u r e h i s t o r i e s a r e n o t known a t t h e s t a r t of c o m p u t a t i o n , but they a r e r e q u i r e d for t h e c o m p u t a t i o n of t h e right-hand s i d e s of E q s9
a n d 10. T h e r e f o r e , t h e c a l c u l a t i o n s s h o u l d b e s t a r t e d w e l l i n a d v a n c e of t h e t i m e for which t h e s o l u t i o n i s required. T h i s m a k e s t h e re- s u l t s t h a t a r e of i n t e r e s t i n d e p e n d e n t of t h e a s s u m e d s u r f a c e t e m p e r a t u r e v a l u e s for t h e time prior t o t h e s t a r t of c a l c u l a t i o n . Wet S u r f a c e T h e h e a t t r a n s f e r a t a w e t s u r f a c e d e p e n d s a l s o on t h e e v a p o r a t i o n r a t e of t h e w a t e r . T h e tt.eory of s i - m u l t a n e o u s h e a t a n d m a s s t r a n s f e r b e t w e e n w a t e r - w e t t e d s u r f a c e a n d a i r g i v e s (Ref 1 , p. 7 4 , E q 45) w l , n = K l , n ( E o , n - E l , , ) ( 1 1 ) w h e r e w l , , = h e a t t r a n s f e r t o s u r f a c e 1 by c o n v e c t i o n a n d e v a p o r a t i o n h1,n K1,n =- c o e f f i c i e n t t h a t r e l a t e s h e a t t r a n s f e r a n d e n t h a l p y p o t e n t i a l h l , , = c o n v e c t i o n h e a t t r a n s f e r c o e f f i c i e n t y = s p e c i f i c h e a t of a i r - w a t e r v a p o r mixture i n f r e e s t r e a m E l , , = e n t h a l p y of a i r i n c l o s e c o n t a c t with w e t s u r f a c e E,,, = e n t h a l p y of a i r i n f r e e s t r 6 a m . T h e h e a t b a l a n c e e q u a t i o n for a w e t o u t e r s u r - f a c e , b a s e d o n t h e a b o v e e q u a t i o n i sunknown t e r m s on t h e left-hand s i d e g i v e s e q u a t i o n s s i m i l a r to E q s
9
o r 1 0 . T h e s i g n i f i c a n t d i f f e r e n c e is t h a t t h e e n t h a l p y El:, is a n e x t r a unknown; h o w e v e r , i t c a n b e r e l a t e d to w e t s u r f a c e t e m p e r a t u r e a s - s u m i n g t h a t t h e a i r i n c l o s e c o n t a c t with t h e w e t s u r - f a c e is s a t u r a t e d and is a t w e t s u r f a c e t e m p e r a t u r e . F o r e x a m p l e , a n e m p i r i c a l r e l a t i o n b e t w e e n e n t h a l p y , E , [ B t u / l b of dry air] and s a t u r a t e d a i r t e m p e r a t u r e , 8 , [ d e g F] for s t a n d a r d a t m o s p h e r i c p r e s s u r e is E l , , = 1 . 5 3 1 3+
0.2077881,n+
0.44496 x - 0.48448 x 1 0 - 4 8 3 I , ,+
0 . 5 3 3 4 8 x 1 0 - ~ 8 ~ (13) T h i s t h e n g i v e s a n o t h e r i n d e p e n d e n t e q u a t i o n , s o t h a t t h e s e t of e q u a t i o n s c a n b e s o l v e d for s u r f a c e t e m p e r a t u r e s by a n i t e r a t i v e p r o c e d u r e .*In t h e c a s e of a roof flooded with w a t e r t h e h e a t s t o r a g e of t h e w a t e r l a y e r must be a c c o u n t e d for. A s s u m i n g t h a t t h e w a t e r t e m p e r a t u r e w i l l b e n e a r l y uniform throughout t h e l a y e r b e c a u s e of t h e rela- t i v e l y h i g h w a t e r c o n d u c t i v i t y a n d mixing d u e t o c o n v e c t i o n c u r r e n t s , t h e h e a t s t o r a g e , Q,, of t h e w a t e r l a y e r is g i v e n by w h e r e B n is t h e h e a t s t o r a g e c a p a c i t y of t h e w a t e r den . l a y e r p e r u n i t a r e a a n d i s t h e r a t e of w a t e r tem- d t p e r a t u r e r i s e . T h i s time d e r i v a t i v e c a n b e approxi- m a t e d by T h e a d d i t i o n of - Q n term i n w e t s u r f a c e h e a t b a l - a n c e e q u a t i o n p e r m i t s c a l c u l a t i o n s of h e a t t r a n s f e r through f l o o d e d roofs. S I M P L I F I C A T I O N S T h e n e e d for i t e r a t i o n is e l i m i n a t e d if t h e c o n v e c - tion h e a t t r a n s f e r c o e f f i c i e n t s a t t h e s u r f a c e s a r e a s s u m e d to b e i n d e p e n d e n t of s u r f a c e t e m p e r a t u r e s , a n d t h e l o n g w a v e r a d i a t i o n e m i s s i o n i s approxi- m a t e d by a l i n e a r e x p r e s s i o n . In t h i s c a s e t h e s o l - a i r t e m p e r a t u r e c o n c e p t c a n be u s e d a n d t h e s u r f a c e * T h e m a x i m u m e r r o r i s 5 0.05 B t u / l b o f d r y a i r for t h e t e m p e r a - t u r e r a n g e 20 t o 100 F. t e m p e r a t u r e s a r e g i v e n by w h e r e , A l , , a n d A 2 , n = s o l - a i r t e m p e r a t u r e for the o u t s i d e s u r f a c e a n d its e q u i v a l e n t for t h e i n s i d e s u r f a c e H I , , a n d H 2 , n = a p p r o p r i a t e h e a t t r a n s f e r coeffi- c i e n t s t o b e u s e d with A l , , a n d A Z , n r e s p e c t i v e l y
.
If t h e s u r f a c e h e a t t r a n s f e r c o e f f i c i e n t s a r e a s - s u m e d t o be c o n s t a n t they c a n be i n c o r p o r a t e d i n t o t h e o v e r a l l r e s p o n s e f a c t o r s e t s , which r e l a t e h e a t flux d i r e c t l y t o t h e s o l - a i r temperature. T H E A C C U R A C Y O F T H E M E T H O D T h e t i m e s e r i e s r e p r e s e n t a t i o n b a s e d on triangular p u l s e s a p p r o x i m a t e s t h e f u n c t i o n o v e r e a c h time i n t e r v a l by a s t r a i g h t l i n e . T h e e r r o r s i n c a l c u l a t e d h e a t f l u x e s a n d s u r f a c e t e m p e r a t u r e s , b e c a u s e of t h e time s e r i e s r e p r e s e n t a t i o n , a r e t h e r e f o r e f u n c t i o n s of t h e t'ime i n t e r v a l A a n d t h e s e c o n d a n d h i g h e r time d e r i v a t i v e s of t h e f u n c t i o n s . An a p p r o p r i a t e t i m e i n t e r v a l A for p a r t i c u l a r prob- l e m s c a n b e s e l e c t e d by i n s p e c t i o n of input vari- a b l e s . T h e a d e q u a c y of t h i s A c a n b e c h e c k e d by performing t h e c a l c u l a t i o n s u s i n g A a n d r e p e a t e d A using-. T h e d i f f e r e n c e b e t w e e n t h e two r e s u l t s i n - 2 d i c a t e s t h e m a g n i t u d e of t h e error. A time i n t e r v a l of 1 hr i s u s u a l l y a d e q u a t e for building h e a t t r a n s f e r c a l c u l a t i o n s 5 . CONCLUSION O n e - d i m e n s i o n a l n o n - s t e a d y h e a t flow through w a l l s a n d r o o f s c a n be c a l c u l a t e d by t h e time s e r i e s tech- n i q u e s . T h e d a t a n e e d e d for t h e s e c a l c u l a t i o n s a r ema1 r e s p o n s e factor s e t s and t h e s u r f a c e h e a t trans- fer c o e f f i c i e n t s that may or may not be c o n s t a n t with time a n d temperatures. T h e error introduced by the time s e r i e s r e p r e s e n t a t i o n of t h e v a r i a b l e s c a n be p r a c t i c a l l y e l i m i n a t e d by u s i n g s u f f i c i e n t l y s m a l l time i n t e r v a l s for time s e r i e s r e p r e s e n t a t i o n . the i n s i d e a n d o u t s i d e ambient c o n d i t i o n s , the w a l l or roof thermal c h a r a c t e r i s t i c s d e f i n e d by the ther-
ACKNOWLEDGEMENT
T h e author gratefully a c k n o w l e d g e s t h e many helpful s u g g e s t i o n s made by Dr. D. G. S t e p h e n s o n .
2 . The Institution of Heating and Ventilating Engineers
Guide, London, 1965.
R E F E R E N C E S
1. ASHRAE HANDBOOK OF FUNDAMENTALS, 1967.
3. A. Tustin, A Method of Analyzing the Behaviour of Linear Systems in Terms of Time Series, Jour. Inst. Elec. Eng., Vol. 94, Part I
-
A, No. 1 , 1947, pp. 130- 142.4. D. G. Stephenson and G. P. Mitalas, Cooling Load Calculations by Thermal Response Factor Method, ASHRAE TRANS., Vol. 73, Part I, 1967.
5. G. P . Mitalas and D. G. Stephenson, RoomThermal
Response Factors, ASHRAE TRANS., Vol. 73, Part I,
1967.
6. G. P . Mitalas and J . G. Arseneault, Fortran IV Program to Calculate Heat Flux Response Factors for Multi-Layer Slab, Division of Building Research, National Research Council of Canada, Computer Program 26, 1967.
DISCUSSION
J.D. McALLISTER (Trenton, N.J.): It would be of i n t e r e s t , if Mr. Mitalas h a s made s u c h comparisons, to h a v e a n i n d i c a t i o n of t h e difference in t h e order of magnitude in a p p a r e n t d e s i g n energy require- ments for a t y p i c a l i n s t a l l a t i o n when c a l c u l a t e d c o n v e n t i o n a l l y a n d when c a l c u l a t e d u s i n g i t e r a t i v e time s e r i e s t e c h n i q u e s .
MR. MITALAS: We d i d not make t h i s comparison. L E S L I E A. BARRON (Chicago, Ill.): T h e paper i s very good in i t s orientation. T h i s i s c e r t a i n l y s o m e t h i n g t h a t t h e p r o f e s s i o n n e e d s to know. How- e v e r , t h e w e i g h t of the math involved l e a d s me t o wonder w h e t h e r o r not i t c o u l d be handled i n a day- to-day office procedure.
Sometime a g o , our industry w a s a s k e d a s i m i l a r q u e s t i o n a s t o t h e d i u r n a l h e a t flow i n masonry c a v i t y w a l l s i n s u l a t e d with vermiculite masonry
fill. T h e only w a y w e could r e a l l y come up with
a n y a n s w e r s t o them w a s t o very c a r e f u l l y instru- ment t h e w a l l , produce t h e ambient c o n d i t i o n s d e s i r e d , r e v e r s e t h e h e a t flow o n a diurnal b a s i s , a n d p i c k off "on t h e fly," s o to s p e a k , instru- ment r e a d i n g s from which w e predicated t h e w a l l
performance. A c t u a l l y , i t might h a v e b e e n very b a s e work, but t h e C o r p s of E n g i n e e r s s e e m e d to be s a t i s f i e d a n d a c c e p t e d the information. T h i s w a s not p u b l i s h e d in a p a p e r but w a s a n industry bulletin only.
W.F. S T O E C K E R (Urbana, 111.) T o c l a r i f y h i s s t a t e m e n t , I b e l i e v e the q u e s t i o n i s : I s there a p o s s i b i l i t y of a n office using the approach which you h a v e p r e s e n t e d i n your p a p e r ?
MR.
MITALAS: I think i t i s f e a s i b l e t o d o t h e s e c a l c u l a t i o n s for routine jobs.R.H. T U L L (Morristown, N. J.): I would l i k e t o s p e a k a s a Chairman of t h e T a s k Group on E n e r g y Requirements. In regard t o t h e f i r s t comment, w e h a v e i n c l u d e d in t h e program of the t a s k group a project w e c a l l "Operation Cross-Check." T h i s i s t o a n s w e r the p a r t i c u l a r q u e s t i o n a s t o how t h i s c a l c u l a t i o n c o m p a r e s with t h e p r e s e n t c a l c u - lation methods.
We a r e comparing a t y p i c a l building by a b o u t s i x d i f f e r e n t programs of c a l c u l a t i o n procedures,
r e s p o n s e f a c t o r t e c h n i q u e . We h o p e t h a t w e w i l l h a v e s o m e t h i n g t o r e p o r t on t h i s a f t e r a p e r i o d of s o m e months. Z U L F I K A R CUMALI ( S a n F r a n c i s c o , Cal.): My q u e s t i o n is c o n c e r n e d w i t h t h e s o l u t i o n of t h e m a t r i x w h i c h r e l a t e s t h e p r e s e n t v a l u e s of t h e t e m p e r a t u r e s t o t h o s e a t t h e p r e c e e d i n g t i m e in- t e r v a l . When w e a s s u m e c o n s t a n t s u r f a c e h e a t t r a n s f e r c o e f f i c i e n t s , all w e n e e d d o is i n v e r t t h i s matrix a n d u s e i t for t h e r e s t of t h e c a l c u l a t i o n s , i e , i t i s i n v e r t e d o n l y o n c e . T h e r e is n o p r o b l e m h e r e . H o w e v e r , w h e n t h e h e a t t r a n s f e r c o e f f i c i e n t s a r e n o t c o n s t a n t , t h e m a t r i x m u s t b e i n v e r t e d a t e a c h t i m e i n t e r v a l . If t h e te'mperature d e p e n d e n c e of t h e c o e f f i c i e n t s a r e s i g n i f i c a n t , t h e i n v e r s i o n m u s t b e d o n e i t e r a t i v e l y , if n o t w e c a n t a k e t h e i r v a l u e s a t t h e l a s t t i m e i n t e r v a l as a n a p p r o x i m a t i o n . T h e q u e s t i o n r e f e r s t o t h e a m o u n t o f time re- q u i r e d t o d o t h i s as c o m p a r e d t o s o l v i n g i t d i r e c t l y w i t h o n e inversion. T h e o t h e r q u e s t i o n i s , if w e a s s u m e t h a t t h e h e a t t r a n s f e r c o e f f i c i e n t s a r e c o n s t a n t h a v e w e u s e d t h i s m e t h o d t o i t s f u l l e s t e x t e n t ? We c a n d o t h e s a m e t h i n g by u s i n g T r a n s - form m e t h o d s , w h i c h w a s t h e w a y t h e p r o b l e m w a s s o l v e d o r i g i n a l l y . If y o u a s s u m e t h a t t h e h e a t t r a n s f e r c o e f f i c i e n t s c a n b e e x p r e s s e d as a h a l f - i n t e g e r f u n c t i o n of tem- p e r a t u r e o r a s f u n c t i o n s of t i m e o n l y o r if t h e y a r e c o n s t a n t , t h e n y o u c a n s i m p l y m a k e a f r e q u e n c y r e p r e s e n t a t i o n o f y o u r i n p u t o r y o u r b o u n d a r y c o n - d i t i o n s a s c o n v o l u t i o n s w i t h y o u r i n p u t . T h e r e f o r e , s i n c e y o u c a n u s e t h e s e t of r e g u l a r c o n d u c t i o n h e a t t r a n s f e r e q u a t i o n s for t h i s form of t h e i n p u t s , y o u c a n a u t o m a t i c a l l y s o l v e them. T h e s o l u t i o n is a n a l y t i c , a l t h o u g h n o t a s i m p l e a n a l y t i c o n e i n t h i s case. A l l I a m s a y i n g is t h a t if d u e t o c o m p u t e r time c o n s i d e r a t i o n s w e d o n ' t t a k e i n t o a c c o u n t t h e d a y s w h e n t h e c o e f f i c i e n t s a r e d e p e n d e n t o n e i t h e r t i m e a n d t e m p e r a t u r e by a s s u m i n g t h e m c o n s t a n t , t h e n t h e o l d m e t h o d s s e e m t o h a n d l e t h e s a m e p r o b l e m a l r e a d y . MR. MITALAS: I d o n o t h a v e t h e i n f o r m a t i o n c o n c e r n i n g c o m p u t e r t i m e r e q u i r e m e n t s ; t h e r e - f o r e , I c a n n o t a n s w e r t h e s e q u e s t i o n s . T h e p r o b l e m of c a l c u l a t i n g s u r f a c e h e a t f l u x e s b e c o m e s more t i m e c o n s u m i n g o n c e t h e s u r f a c e h e a t t r a n s f e r c o e f f i c i e n t s a r e f u n c t i o n s of temper- a t u r e s . H o w e v e r , t h e c a l c u l a t i o n s of s u r f a c e h e a t f l u x e s w h e n s u r f a c e h e a t t r a n s f e r c o e f f i c i e n t s a r e c o n s t a n t a r e v e r y s i m p l e u s i n g t h e r m a l r e s p o n s e f a c t o r s s i n c e c o n s t a n t s u r f a c e c o e f f i c i e n t s c a n b e c o n s i d e r e d as e x t r a l a y e r s o f a w a l l o r roof. MR. CUMALI: T h e p o i n t I w a s g o i n g t o m a k e w a s t h a t by u s i n g t h e F a s t F o u r i e r T r a n s f o r m t e c h n i q u e , I c a n transform 8000 h r s d a t a i n s o m e t h i n g l i k e
5
s e c o n d s , w h i c h I t h i n k w o u l d b e c o m p a r a b l e t o t h e t i m e f o r c o m p u t i n g t h e r e s p o n s e f a c t o r s . C.M. A S H L E Y ( M a n l i u s , N.Y.): I w o u l d l i k e t o r a i s e a q u e s t i o n of a s o m e w h a t d i f f e r e n t c h a r a c t e r . In s t u d y i n g s o m e of t h e c h a n g e s w h i c h w e r e m a d e b e t w e e n t h e e a r l i e r e d i t i o n s o f t h e A S H R A E G U I D E AND D A T A BOOK a n d t h e1967
e d i t i o n , o n e w h i c h p r o v i d e d t h e g r e a t e s t i n f l u e n c e o n t h e c h a n g e i n h e a t t r a n s f e r t h r o u g h w a l l s a n d r o o f s w a s t h a t i n t h e c o m b i n e d r a d i a t i o n a n d c o n v e c t i o n c o e f f i c i e n t on t h e o u t s i d e of t h e w a l l . I t w a s c h a n g e d , I be- l i e v e , from4
~ t u / f t ~ F p e r h r t o3.
I t r i e d t o find w h a t t h e f a c t u a l b a s i s w a s for e i t h e r o n e o r t h e o t h e r a n d I c a m e t o t h e c o n c l u s i o n t h a t t h e r e w a s a v e r y p o o r b a s e f o r both. A l s o , I think t h a t t h e r e is a r e a l q u e s t i o n a s t o w h e t h e r t h e d a t a w h i c h d o e x i s t j u s t i f y t h e c h a n g e from4
t o3.
P e r h a p s i t might b e a r g u e d t h a t u n d e r s o m e l o w w i n d c o n d i t i o n t h e"3"
is j u s t i f i e d , but I s e r i o u s l y q u e s t i o n i t . I w o u l d l i k e t o a s k Mr. M i t a l a s w h e t h e r h e k n o w s of a d d i t i o n a l d a t a on w h i c h y o u c a n b a s e t h e m e t h o d w h i c h h e is d e s - c r i b i n g , p a r t i c u l a r l y w i t h r e s p e c t to t h i s p a r t i c u l a r s u b j e c t , a n d a l s o w i t h r e s p e c t t o t h e i n s i d e c o e f - f i c i e n t ? MR. MITALAS: N o , w e d o not. MR. A S H L E Y : If y o u c h e c k b a c k through t h e d a t a o n w h i c h t h e o u t s i d e s u r f a c e c o e f f i c i e n t is b a s e d , y o u w i l l f i n d t h a t t h e c u r v e s a r e b a s e d upon wind v e l o c i t y a n d c o n v e c t i o n c o e f f i c i e n t s o n s m a l l l a b o - r a t o r y s a m p l e s . A l s o , t h e t e m p e r a t u r e c o n d i t i o n s a r e s u c h t h a t t h e r a d i a t i o n f a c t o r is q u i t e d i f f e r e n t from t h a t w h i c h w o u l d b e u s e d i n t h e s u m m e r time. A s n e a r l y as I c a n d e t e r m i n e , t h e v a l ~ e of4
w o u l d b e j u s t i f i e d a t s o m e w h e r e a r o u n d5
mph.I would s a y t h a t t h e a v a i l a b l e d a t a a r e s o poor that a l m o s t n o information c a n be a c c e p t e d a s re- a l l y b e i n g s a t i s f a c t o r y .
W.F. S T O E C K E R : I t would s e e m a p p r o p r i a t e t h a t t h i s m a t t e r b e referred t o t h e T e c h n i c a l Committee on L o a d C a l c u l a t i o n s , w h i c h i s r e s p o n s i b l e for t h a t c h a p t e r in t h e ASHRAE GUIDE AND DATA BOOK.
T. KUSUDA (Washington, D.C.): A s t h e c h a i r m a n of t h e s u b c o m m i t t e e for h e a t i n g and c o o l i n g l o a d s of t h e ASHRAE T a s k Group For E n e r g y R e q u i r e - m e n t s , I must e m p h a s i z e t h e s i g n i f i c a n c e of t h e r e s p o n s e f a c t o r c o n c e p t d e m o n s t r a t e d i n t h i s p a p e r a n d t h e paper t h a t Follows in t h i s s e s s i o n . U v a l u e a n d t h e e f f e c t i v e t e m p e r a t u r e d i f f e r e n c e ( E D T ) , t h e c o n c e p t s o w i d e l y known a n d u s e d i n the p a s t , a r e s t i l l u s e f u l i n e v a l u a t i n g t h e d e s i g n load For s e l e c t i n g t h e h e a t i n g a n d c o o l i n g equipment. B u t t h e c o n c e p t i s u t t e r l y inabdequate for t h e hour by hour e v a l u a t i o n of i n s t a n t a n e o u s h e a t g a i n a n d h e a t l o s s , r e s p o n d i n g to t h e c h a n g i n g c l i m a t i c c o n d i t i o n s a n d e n e r g y u s e s c h e d u l e s . Although i t s c o n c e p t i s n o t n e w , t h e r e s p o n s e f a c t o r t e c h n i q u e h a s b e e n r e c e n t l y r e v i t a l i z e d by M i t a l a s and S t e p h e n s o n mainly on a c c o u n t o f t h e w i d e s p r e a d a c c e s s i b i l i t y to h i g h - s p e e d large- memory c o m p u t e r s . I may s a y t h a t t h e r e s p o n s e f a c t o r c o n c e p t a c t u a l l y r e v o l u t i o n i z e d h e a t t r a n s - fer a n a l y s e s : many problems h i t h e r t o c o n s i d e r e d u n s o l v a b l e or e x t r e m e l y c u m b e r s o m e to s o l v e now c a n r e a d i l y b e a n a l y z e d .
T h e f i n i t e d i f f e r e n c e method, t h e Schmidt Method, a s t a n d a r d p r o c e d u r e for many t r a n s i e n t h e a t c o n d u c t i o n p r o b l e m s ; very time-consuming a n d y e t i n a c c u r a t e , h a s b e c o m e o b s o l e t e in t h e p r e s e n c e of t h e r e s p o n s e f a c t o r method, w h o s e a c c u r a c y i s p r a c t i c a l l y t h e s a m e a s t h e e x a c t s o l u t i o n . T h e h e a t c o n d u c t i o n problems a n a l y z e d i n t h i s p a p e r a r e only the beginning of t h e e x t e n s i v e ap- p l i c a t i o n of t h e r e s p o n s e Factor method which i s e x p e c t e d t o follow in t h e n e a r future. It i s very f o r t u n a t e t h a t t h e ASHRAE T a s k Group For E n e r g y R e q u i r e m e n t s h a s b e e n a b l e to t a k e a d v a n t a g e of t h i s t e c h n i q u e i n d e v e l o p i n g i t s " P r o p o s e d P r o c e d u r e for Determining H e a t i n g a n d C o o l i n g L o a d s for E n e r g y C a l c u l a t i o n s " . S i n c e t h e r e s p o n s e f a c t o r method i s e x p e c t e d t o b e i n c r e a s i n g l y m e n t i o n e d in the f u t u r e , I would l i k e t o urge a l l of t h e ASHRAE members who a r e e n g a g e d in h e a t t r a n s f e r a n a l y s i s to become fa- miliar with t h i s r e m a r k a b l e t e c h n i q u e .