Experimental determination of the Z-transfer function coefficients for houses

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ASHRAE Transactions, 93, 1, pp. 146-160, 1987

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Experimental determination of the Z-transfer function coefficients for

houses

Barakat, S. A.

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Experimental Determination of the

Z- Transfer Function Coefficients

for Houses

by S.A. Barakat

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L I B R A R Y

Appeared in

ASHRAE Transactions 1987

Part

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p. 146-160

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Reprinted by permission of the

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Des experiences conpes @ialement en vue de d6-er

les coefficients de fonction de

transfert

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charge cdorifique dus awc apports solaires, ainsi que les coefficients de fonction

de transfert air de pi&ce, ont

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r6alistks sur un site d'essai. Trois series de coefficients ont

CtC

obtenus pour differents niveaux de masse de maisons B ossature de bois. Ces niveaux,

qui couvrent une plage inferieure

B

celle du

ASHRAE

Handbook, sont particuli6rement

utiles pour les calculs CnergCtiques visant les maisons. Les niveaux de masse examinks

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46, 130

et

535 kg/m2 (9,4,26,6 et

110 IbIpi2) d'aire de plancher.

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I

EXPERIMENTAL DETERMINATION OF THE Z-TRANSFER

FUNCTION COEFFICIENTS FOR HOUSES

!

S.A.

B a r a k a t , Ph.D.,

P.E.

I

S p e c i a l l y designed e x p e r i m e n t s were performed a t an o u t d o o r t e s t f a c i l i t y t o determi.ne t h e c o o l t n g - l o a d z-trii11sEer f u n c t i o n c o e f f i c i e n t s due t o s o l a r g a i n i n p u t , as w e l l a s t h e room-air t r a n s f e r f u n c t i o n c o e f f i c i e n t s . Three s e t s of c o e f f i c i e n t s were o b t a i n e d f o r d i f f e r e n t mass l e v e l s i n wood-f rame h o u s e s . ' h e s e l e v e l s , which c o v e r a r a n g e lower t h a n t h o s e c o v e r e d i n t h e ASHRAE Handbook, a r e p a r t i c u l a r l y u s e f u l f o r e n e r g y c a l c u l a t i o n s f o r h o u s e s . The mass l e v e l s examined a r e based on 4 6 , 130, and 535 kg/& (9.4, 26.6, and 110 l b / f t 2 ) of f l o o r a r e a .

I

INTRODUCTION

The z - t r a n s f e r f u n c t i o n t e c h n i q u e ( a l s o c a l l e d t h e t h e r m a l r e s p o n s e f a c t o r o r t h e w e i g h t i n g f a c t o r method), i n t r o d u c e d by S t e p h e n s o n and M i t a l a s (1967; M i t a l a s 1972, 1978; ASHRAE 1985), i s a well-known method f o r e n e r g y a n a l y s i s of b u i l d i n g s . With t h i s t e c h n i q u e , h o u r l y t h e r m a l l o a d s can be c a l c u l a t e d based on t h e p h y s i c a l d e s c r i p t i o n of t h e b u i l d i n g and t h e ambient w e a t h e r c o n d i t i o n . These l o a d s a r e t h e n u s e d , t o g e t h e r w i t h i n f o r m a t i o n a b o u t t h e

h e a t i n g / c o o l i n g s y s t e m , t o c a l c u l a t e t h e s p a c e a i r t e m p e r a t u r e and h e a t a d d i t i o n / e x t r a c t i o n r a t e s . To perform t h e s e c a l c u l a t i o n s , t h e method u t i l i z e s s e t s of p r e c a l c u l a t e d t r a n s f e r f u n c t i o n c o e f f i c i e n t s . These a r e g i v e n i n t h e ASHRAE p r o c e d u r e s (ASHRAE 1975) f o r t h r e e room mass l e v e l s ; namely, l i g h t , medium, and heavy, c h a r a c t e r i z e d r e s p e c t i v e l y by 146, 341, and 6 3 5 kg/m2 ( 3 0 , 70, and 130 l b / f t 2 ) of f l o o r a r e a .

An e x a m i n a t i o n of c o n v e n t i o n a l wood-frame h o u s e s y i e l d s a room s p e c i f i c mass v a l u e as low a s 50-60 kg/& (10-12 l b / f t 2 ), which i s much lower t h a n ASHRAE v a l u e s f o r l i g h t w e i g h t

b u i l d i n g s . The p r e c a l c u l a t e d r e s p o n s e f a c t o r s g i v e n i n ASHRAE (1975) a r e , t h e r e f o r e , n o t a p p r o p r i a t e f o r most h o u s e s . T h i s is c o n f i r m e d by t h e l a c k of agreement between measured and p r e d i c t e d e n e r g y consumption when ASHRAE v a l u e s were used i n a n e n e r g y a n a l y s i s program

(Konrad and L a r s e n 1 9 7 8 ) t o s i m u l a t e t h e p e r f o r m a n c e of a wood-f rame s t r u c t u r e . F i g u r e 1 shows t h e r e s u l t s of one s u c h c a l c u l a t i o n . T h i s l a c k of agreement i n d i c a t e s t h a t t h e s i m u l a t e d p e r f o r m a n c e c o n s i s t e n t l y i m p l i e s a much h e a v i e r b u i l d i n g .

I n t h i s s t u d y , e x p e r i m e n t s were c a r r i e d o u t t o d e t e r m i n e t h e c o o l i n g - l o a d t r a n s f e r f u n c t i o n c o e f f i c i e n t s f o r s o l a r h e a t g a i n i n p u t and t h e room-air t r a n s f e r f u n c t i o i l

c o e f f i c i e n t s f o r t h r e e d i f f e r e n t mass l e v e l s of houses c o r r e s p o n d i n g t o 46, 130, and 535 kg/m2 (9.4, 26.6, and 110 l b / f t 2 ) of f l o o r a r e a .

I

THE Z-TRANSFER FUNC'PION METHOD

I

For a dynamic h e a t t r a n s f e r s y s t e m , dependent and i n d e p e n d e n t v a r i a b l e s can be r e l a t e d by

I d i f f e r e n t i a l e q u a t i o n s . The s o l u t i o n of t h e s e e q u a t i o n s may t a k e s e v e r a l forms. I n t h i s

1

c a s e , t h e i n p r ~ t and o u t p u t z - t r a n s f orms ( i .e., t r a n s f orms of i n d e p e n d e n t and d e p e n d e n t v a r i a b l e s ) a r e r e l a t e d by a z - t r a n s f e r f u n c t i o n as f o l l o w s :

S.A. Harakat is a R e s e a r c h O f f i c e r , I n s t i t u t e f o r R e s e a t c h 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 of Canada, Ottawa, Canada, K1A OR6.

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where O(z) = z-transform of t h e o u t p u t I ( z ) = z - t r a n s f o r m of t h e i n p u t

G(z) = system t r a n s f e r f u n c t i o n

The z - t r a n s f o r m a t i o n of a time-dependent f u n c t i o n (e.g., s o l a r r a d i a t i o n , o u t s i d e a i r t e m p e r a t u r e ) is a sequence of v a l u e s of t h e f u n c t i o n a t e q u a l time i n t e r v a l s ( u s u a l l y one h o u r ) ; t h a t is, i t i s a 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 f u n c t i o n .

The most i m p o r t a n t c h a r a c t e r i s t i c of t h e z - t r a n s f e r f u n c t i o n method, a s compared w i t h o t h e r c a l c u l a t i o n methods, is t h a t t h e i n p u t s and t h e o u t p u t s a r e sequences of v a l u e s e q u a l l y spaced i n time. Thus, t h e weather r e c o r d s of o u t s i d e a i r t e m p e r a t u r e and s o l a r r a d i a t i o n , which a r e given on ar? h o u r l y b a s i s , can be used a s i n p u t w i t h l i t t l e o r no p r e p r o c e s s i n g . The main l i m i t a t i o n of t h e z - t r a n s f e r method i s t h a t t h e system under c o n s i d e r a t i o n must be

c o n s t a n t with time and l i n e a r ; ( i n o t h e r words, t h e thermal p r o p e r t i e s and h e a t t r a n s f e r c o e f f i c i e n t s used i n t h e system must be c o n s t a n t , and independent of t e m p e r a t u r e ) .

The z - t r a n s f e r f u n c t i o n method f o r energy c a l c u l a t i o n s is e x p l a i n e d i n C h a p t e r 26 of t h e 1985 e d i t i o n of ASHRAE Fundamentals (ASHRAE 1985). The c a l c u l a t i o n c o n s i s t s of two s t e p s . I n t h e f i r s t s t e p , t h e space a i r t e m p e r a t u r e is assumed t o be c o n s t a n t a t some r e f e r e n c e v a l u e and t h e i n s t a n t a n e o u s h e a t g a i n s / l o s s e s f o r t h e room a r e c a l c u l a t e d . These i n c l u d e s o l a r and i n t e r n a l h e a t g a i n s and envelope conduction and i n f i l t r a t i o n l o s s e s . A c o o l i n g l h e a t i n g l o a d f o r t h e room ( d e f i n e d a s t h e r a t e a t which h e a t must be removedladded t o maintain t h e a i r t e m p e r a t u r e i n the room a t t h e c o n s t a n t r e f e r e n c e v a l u e ) is c a l c u l a t e d f o r each component of i n s t a n t a n e o u s h e a t g a i n l l o s s .

The z - t r a n s f e r f u n c t i o n t h a t r e l a t e s t h e z-transforms of t h e room h e a t g a i n , I ( z ) , t o t h e room c o o l i n g + l o a d , Q ( z ) , may be e x p r e s s e d a s

where v o , vl

,

and wl a r e t h e z - t r a n s f e r f u n c t i o n c o e f f i c i e n t s . While v,, and vl a r e d i f f e r e n t f o r each type of heat g a i n , wl is t h e same f o r a l l types and depends only on t h e thermal s t o r a g e c h a r a c t e r i s t i c s of t h e room. I n t h e time domain, t h e room c o o l i n g load is given by

where t i s t h e t i m e and A i s t h e t i m e i n t e r v a l between s u c c e s s i v e v a l u e s , u s u a l l y one hour.

A t t h e end of t h e f i r s t s t e p , t h e c o o l i n g l o a d s from v a r i o u s h e a t g a i n s a r e added t o g e t t h e t o t a l c o o l i n g load f o r t h e room.

I n t h e second s t e p , t h e n e t h e a t i n p u t t o t h e room ( t o t a l c o o l i n g load l e s s any h e a t e x t r a c t i o n o r a d d i t i o n provided by t h e HVAC system) i s used t o c a l c u l a t e t h e d e v i a t i o n of t h e space a i r t e m p e r a t u r e from t h e r e f e r e n c e v a l u e s e t d u r i n g t h e c o o l i n g load c a l c u l a t i o n .

' 1 ' 1 1 ~ z-Lt-;i~l?tLer l r ~ t ~ c t i o r i tlliit r e l a t e s ttlc z-transfotmtl of t h e ~ p t l c e tt?~nl)t!rature d e v i n t i o n from t h e r e f e r e n c e , 8 ( z ) , t o t h e n e t h e a t i n p u t t o t h e s p a c e , QN(z), is termed t h e room-air ( o r a i r - t e m p e r a t u r e ) t r a n s f e r f u n c t i o n and is given by

where g o , g l , g2 and pl a r e t h e t r a n s f e r f u n c t i o n c o e f f i c i e n t s . The v a l u e of t h e c o e f f i c i e n t

p, is e q u a l t o w1 f o r t h e same room mass. The n e t h e a t i n p u t and space t e m p e r a t u r e a r e r e l a t e d by

+TO avoid r e p e t i t i o n , r e f e r e n c e s t o h e a t i n g l o a d s w i l l be e l i m i n a t e d ; t h e s e loads can always be c o n s i d e r e d a s n e g a t i v e c o o l i n g l o a d s .

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Some i n t e r e s t i n g p r o p e r t i e s of t h e t r a n s f e r f u n c t i o n c o e f f i c i e n t s can be o b t a i n e d by a p p l y i n g t h e s t e a d y - s t a t e c o n d i t i o n s . For t h e cooling-load t r a n s f e r f u n c t i o n c o e f f i c i e n t s , i t can be shown t h a t

where f r e p r e s e n t s t h e f r a c t i o n of t h e s t e a d y - s t a t e h e a t g a i n t h a t a p p e a r s a s a c o o l i n g l o a d . The c o e f f i c i e n t s given i n ASHKAE Fundamentals a r e such t h a t f = 1. An e s t i m a t e of f can be made u s i n g t h e procedure i n C h a p t e r 26 of t h e Handbook. The v c o e f f i c i e n t s can be a d j u s t e d a c c o r d i n g l y .

S i m i l a r l y , under s t e a d y - s t a t e c o n d i t i o n s , E q u a t i o n 4 can be r e a r r a n g e d t o g i v e

where KT i s t h e thermal conductance of t h e room, i n c l u d i n g a l l h e a t flow p a t h s .

I n ASHRAE l i t e r a t u r e , t h e a i r - t e m p e r a t u r e c o e f f i c i e n t s a r e p r e s e n t e d a s normalized

f a c t o r s t o remove t h e e f f e c t of v a r i a t i o n s i n room thermal conductance, and t h e e f f e c t of room s i z e . The normalized t r a n s f e r f u n c t i o n c o e f f i c i e n t s a r e c a l c u l a t e d a s

where A f is t h e f l o o r a r e a . E q u a t i o n 7 becomes

Another p r o p e r t y i n h e r e n t i n t h e a i r t e m p e r a t u r e c o e f f i c i e n t s is a measure of t h e h e a t c a p a c i t y of t h e room. It can be shown t h a t t h e h e a t c a p a c i t y , C , i s g i v e n by

THE OUTDOOR TEST FACILITY

The t e s t f a c i l i t y , d e s c r i b e d i n d e t a i l i n Barakat (19841, c o n s i s t s of t h r e e one-story

i n s u l a t e d wood-frame b u i l d i n g s w i t h basements. They a r e d i v i d e d i n t o f o u r two-room u n i t s and f o u r single-room u n i t s . Three of t h e f o u r two-room u n i t s ( U n i t s 1 t o 3 ) were used f o r t h e experiments. C o n s t r u c t i o n d e t a i l s and thermal r e s i s t a n c e v a l u e s of t h e w a l l s and c e i l i n g s a r e given i n T a b l e s 1 and 2.

Each of t h e two-room u n i t s c o n s i s t s of a s o u t h and a n o r t h room w i t h a c o n n e c t i n g door. Each s o u t h room has a s o u t h - f a c i n g window with a n e t g l a s s a r e a of 2.6 m 2 ; each n o r t h room has a 1

d

window f a c i n g n o r t h . The t h r e e u n i t s d i f f e r only i n t h e type and amount of thermal mass used a s i n t e r i o r f i n i s h of t h e w a l l s , a s given i n Table 2. A l l rooms i n t h e f a c i l i t y a r e h e a t e d t o 20°C w i t h an e l e c t r i c baseboard h e a t e r c o n t r o l l e d by a p r e c i s i o n c o n t r o l l e r t o w i t h i n O.l°C. To e l i m i n a t e any h e a t i n g load due t o outdoor a i r i n f i l t r a t i o n , a s l i g h t p o s i t i v e p r e s s u r e is maintained i n a l l rooms, u s i n g a i r p r e h e a t e d t o t h e same t e m p e r a t u r e a s

t h e a i r t n t h e t e s t room.

! Measurements were t a k e n of t h e f o l l o w i n g : t h e a v e r a g e i n d o o r a i r t e m p e r a t u r e a t mid-height of each room (1.2 m above t h e f l o o r ) ; t h e t e m p e r a t u r e of t h e a t t i c a i r and t h e c o r r i d o r a i r ; t h e s o l a r r a d i a t i o n i n c i d e n t on a h o r i z o n t a l s u r f a c e , and on a s o u t h - and a n o r t h - f a c i n g v e r t i c a l s u r f a c e ; t h e d i r e c t normal component of t h e s o l a r r a d i a t i o n ; t h e e l e c t r i c space-heating energy consumption of each room; and t h e wind speed and d i r e c t i o n .

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A d a t a logger was used t o s c a n and r e c o r d t h e d a t a on magnetic t a p e . Wind d a t a ,

r a d i a t i o n rind t e ~ n p e r a t ~ r r e v a l u e s were scanned e v e r y minute, averaged over n s p e c i f i c p e r i o d , and recorded. Energy consumption v a l u e s were accumulated and recorded f o r t h e same p e r i o d .

EXPERIMENTAL PROCEDURE

111 p r i n c i p l e , the experiments were based on measuring t h e room response (heating-system power o u t p u t o r room-air t e m p e r a t u r e ) t o a known i n p u t ( s o l a r g a i n o r p e r t u r b a t i o n i n t h e system i n p u t t o t h e room). A p p l i c a t i o n of t h e t r a n s f e r f u n c t i o n r e l a t i o n s h i p s of E q u a t i o n 2 o r 4

u s i n g t h e measured i n p u t and o u t p u t d a t a would t h e n y i e l d t h e v a l u e s of t h e c o e f f i c i e n t s . Since t h e t e s t u n i t s a r e l o c a t e d o u t d o o r s , t h e o v e r a l l measured o u t p u t c o n t a i n e d a second s i g n i f i c a n t component, which is t h e o v e r a l l r e s p o n s e of t h e room t o t h e outdoor weather parameters. This component had t o be s u b t r a c t e d b e f o r e t h e t r a n s f e r f u n c t i o n could be o b t a i n e d .

To e l i m i n a t e t h e dynamic e f f e c t of s o l a r g a i n through windows and t h e a s s o c i a t e d o v e r h e a t i n g of t h e space on sunny days, both windows of each u n i t were shaded from s o l a r r a d i a t i o n by an e x t e r n a l , v e r t i c a l shade a b o u t 40 cm i n f r o n t of t h e window.

For t h e s o l a r t r a n s f e r f u n c t i o n e x p e r i m e n t s , t h e shade on t h e s o u t h window was t o t a l l y o r p a r t i a l l y removed on a sunny day t o a l l o w s o l a r g a i n through t h e window f o r 1 hour. T h i s c r e a t e d a 1-hour s o l a r i n p u t p u l s e on t h e room. The magnitude of t h e p u l s e ( s i z e of o p e n i n g ) was chosen s o t h a t t h e room-air t e m p e r a t u r e would not exceed t h e s e t p o i n t ( r e f e r e n c e )

t e m p e r a t u r e . A l l t h e p a r a m e t e r s , i n c l u d i n g t h e e l e c t r i c h e a t i n g power consumption f o r t h e room, were measured e v e r y f i v e minutes. Mid-hour v a l u e s were then used f o r t h e a n a l y s i s .

For t h e room-air t r a n s f e r f u n c t i o n e x p e r i m e n t s , t h e window remained f u l l y shaded and an a d d i t i o n a l e l e c t r i c h e a t e r , placed i n t h e c e n t e r of t h e u n i t , was s t a r t e d a t a p r e s c r i b e d time driring the n i g h t , and k e p t on f o r t h r e e o r f o u r h o u r s , c a u s i n g t h e room t o o v e r h e a t . The t e m p e r a t u r e r i s e above t h e r e f e r e n c e v a l u e ( s e t p o i n t ) is t h e o u t p u t response f o r Equation 5 ( o r i n p u t for E q u a t i o n 4 ) .

F i g u r e s 2 and 3 r e p r e s e n t examples of the measured parameters and outdoor weather c o n d i t i o n s d u r i n g a s o l a r p u l s e experiment, and d u r i n g a h e a t - a d d i t i o n p u l s e experiment, r e s p e c t i v e l y . F u r t h e r d e t a i l s of both t y p e s of experiments a r e given i n a companion p u b l i c a t i o n (Barakat 1985).

DATA ANALYSIS

I n both p u l s e e x p e r i m e n t s , t h e t o t a l o u t p u t , t h a t is, t h e change i n power consumption f o r t h e s o l a r g a i n e x p e r i m e n t s , and t h e r i s e i n room temperature f o r t h e h e a t a d d i t i o n e x p e r i m e n t s , was the r e s u l t of two simultaneous i n p u t s : t h e i n t e n t i o n a l l y g e n e r a t e d p e r t u r b a t i o n and t h e outdoor weather parameters (ambient t e m p e r a t u r e , wind and s o l a r r a d i a t i o n on t h e b u i l d i n g s u r f a c e s ) . To s e p a r a t e t h e n e t o u t p u t due t o t h e p u l s e , t h e component due t o outdoor w e a t h e r v a r i a t i o n was f i r s t s u b t r a c t e d from t h e t o t a l o u t p u t . A t r a n s f e r f u n c t i o n r e l a t i o n s h i p was then e s t a b l i s h e d between t h e i n p u t p u l s e and t h e c o r r e s p o n d i n g n e t o u t p u t , and t h e

c o e f f i c i e n t s were determined. The procedure is d e s c r i b e d i n d e t a i l i n t h e f o l l o w i n g paragraphs.

S t e p 1. Measuring Room Response t o Outdoor Weather P a r a m e t e r s

A number of p e r i o d s were chosen d u r i n g which no p u l s e s were a p p l i e d and t h e room

t e m p e r a t u r e remained a t a sei: p o i n t . The measured power consumption, t h e r e f o r e , r e p r e s e n t e d the response t o t h e outdoor weather parameters. Using m u l t i p l e l i n e a r r e g r e s s i o n , a t r a n s f e r f u n c t i o n was o b t a i n e d l i n k i n g t h e i n p u t parameters and t h e room power c o n s u m ~ t i o n , Ea* T h i s f u n c t i o n has t h e form

E,(z) a. + a l z m l

=, ( 1 1 )

-

1

+

bl

z-'

I n t h e time domain t h i s t r a n s l a t e s i n t o

E a ( t ) = a,,AT*(t)

+

alAT*(t-1)

-

blEa(t-1) ( 1 2 )

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where A P i s a weighted t e m p e r a t u r e d i f f e r e n c e d e f i n e d a s and UA = o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t f o r a p a r t i c u l a r w a l l , windaw o r c e i l i n g AT a Ts

-

Tr, f o r o u t s i d e w a l l s = Ta

-

Tr, f o r p a r t i t i o n w a l l s and c e i l i n g = To

-

Tr, f o r windows Ts = s o l - a i r t e m p e r a t u r e of t h e o u t s i d e s u r f a c e Ta = a i r temperature of t h e a d j a c e n t room o r a t t i c To = o u t d o o r a i r t e m p e r a t u r e Tr = room t e m p e r a t u r e ( s e t a t 20°C).

The o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t s , UA, were c a l c u l a t e d based on t h e c o n s t r u c t i o n d e t a i l s . S o l - a i r t e m p e r a t u r e s were c a l c u l a t e d u s i n g t h e f o l l o w i n g d a t a : s u r f a c e a b s o r p t a n c e ; s o l a r r a d i a t i o n i n c i d e n t on t h e s u r f a c e (measured f o r s o u t h and n o r t h w a l l s and c a l c u l a t e d from h o r i z o n t a l measurements For e a s t and west w a l l s [Barakat 19801); and t h e outdoor f i l m heat t r a n s f e r c o e f f i c i e n t c a l c u l a t e d h o u r l y a s a f u n c t i o n of wind speed (ASHRAE 1985).

The c o e f f i c i e n t s ( a o , al

,

and b l ) of E q u a t i o n 12 were determined u s i n g m u l t i p l e l i n e a r r e g r e s s i o n . The adequacy of t h e s e c o e f f i c i e n t s was checked by u s i n g them t o p r e d i c t t h e power consumption d u r i n g o t h e r p e r i o d s . F i g u r e 4a shows measured power consumption f o r a four-day p e r i o d i n J a n u a r y 1983; i t a l s o shows t h e p r e d i c t e d power u s i n g t h e t r a n s f e r f u n c t i o n

c o e f f i c i e n t s c a l c u l a t e d f o r t h e same d a t a , w i t h t h e f i r s t v a l u e a s an i n i t i a l c o n d i t i o n . F i g u r e 4b shows t h e measured power v a l u e s f o r a d i f f e r e n t p e r i o d i n March 1983 compared w i t h t h e v a l u e s p r e d i c t e d u s i n g the c o e f f i c i e n t s produced f o r t h e d a t a i n F i g u r e 4a. The

p r e d i c t l o n , which has an rms e r r o r of l e a s than 25 W (compared with an a v e r a g e powor of 380 W over t h e p e r i o d ) , is good.

S t e p 2. G e n e r a t i n g an I n p u t P u l s e

For a p e r i o d w i t h an i n p u t p u l s e , t h e c o e f f i c i e n t s g e n e r a t e d i n S t e p 1 were used t o p r e d i c t t h e room response due t o ambient weather, E a ( t ) , u s i n g E q u a t i o n 12. The n e t o u t p u t response due t o t h e p u l s e , E ( t ) was then c a l c u l a t e d a s

P

where E ( t ) r e p r e s e n t s Q f o r t h e s o l a r g a i n experiments ( E q u a t i o n 2) and QNt f o r t h e room-air temperaeure e x p e r i m e n t s [ ~ ~ u a t i o n 4 ) , and E ( t ) is t h e t o t a l power consumption of t h e room ( i n c l u d i n g a d d i t i o n a l h e a t e r i n p u t i n t h e room-air t e m p e r a t u r e e x p e r i m e n t s ) . S t e p 2 i s

i l l u s t r a t e d i n F i g u r e 5 f o r a s o l a r p u l s e a t noon i n t h e s o u t h room of U n i t 2 and i n F i g u r e 6 f o r a h e a t - a d d i t i o n p u l s e i n Unit 1. I n t h e s e f i g u r e s and a l l t h o s e t h a t f o l l o w , t h e h o u r l y d i s c r e t e d a t a p o i n t s a r e j o i n e d by s t r a i g h t l i n e s f o r both measured and p r e d i c t e d r e s u l t s . Data used t o produce t h e response f a c t o r s t a r t only a t t h e p u l s e time.

For t h e s o l a r p u l s e , t h e i n p u t i s c a l c u l a t e d a s

where I is t h e a v e r a g e s o l a r r a d i a t i o n i n t e n s i t y over t h e 1-hour p u l s e p e r i o d , F i s t h e s o l a r g a i n f a c t o r f o r double g l a z i n g c a l c u l a t e d f o r t h a t hour (Barakat 1980); and A i s t h e n e t g l a z i n g a r e a exposed d u r i n g t h e p e r i o d .

S t e p 3. C a l c u l a t i n g t h e T r a n s f e r Function C o e f f i c i e n t s

M u l t i p l e l i n e a r r e g r e s s i o n a n a l y s i s was used t o produce a f i r s t approximation t o t h e t r a n s f e r f u n c t i o n c o e f f i c i e n t s i n Equations 2 and 4. These c o e f f i c i e n t s were t h e n used a s i n i t i a l assumptions f o r a s o p h i s t i c a t e d p r e d i c t i o n program t o g e n e r a t e more a c c u r a t e v a l u e s of t h e c o e f f i c i e n t s . A p r e d i c t i o n program was used (DFMF'P 1968), which c a l c u l a t e s t h e l o c a l minimum of a f u n c t i o n ( t h e e r r o r v a l u e ) of s e v e r a l v a r i a b l e s ( t h e c o e f f i c i e n t s ) u s i n g t h e

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method of F l e t c h e r and Powell (1963). This is t h e same method commonly used f o r t h e d e s i g n of r e c u r s i v e d i g i t a l f i l t e r s ( S t e i g l i t z 1970). The minimized f u n c t i o n was t h e square of t h e d i f f e r e n c e between t h e measured and p r e d i c t e d responses. The method s u b s t i t u t e s t h e p r e d i c t e d value f o r t h e previous hour i n Equation 2 o r 4 t o c a l c u l a t e t h e v a l u e and e r r o r f o r t h e

p r e s e n t hour, r a t h e r than t h e measured v a l u e f o r t h e previous hour, as i n r e g r e s s i o n

c a l c u l a t i o n s . a

RESULTS

S o l a r Responae F a c t o r s

A t o t a l . of 11 experiments were performed during t h e 1980181 h e a t i n g season i n t h e s o u t h rooms of IJnlts 1 , 2 , and 3 of t h e t e s t F a c i l i t y . Table 3 g i v e s t h e average c a l c u l a t e d values of the t r a n s f e r f u n c t i o n c o e f f i c i e n t s f o r a l l experiments and t h e average f o r each room

( l i g h t , medium, heavy). The t a b l e a l s o g i v e s values of t h e nus d i f f e r e n c e between t h e measured and p r e d i c t e d responses. I n a d d i t i o n , F i g u r e 7a t o c shows examples of both t h e measured and p r e d i c t e d c o o l i n g load response t o a s o l a r p u l s e f o r each of t h e l i g h t , medium and heavy rooms.

A s with t h e response f a c t o r s i n t h e t h e c o e f f i c i e n t s p r e s e n t e d i n Table 3 a r e normalized, a s explained e a r l i e r i n t h i s Note, t o g i v e an f value ( i n Equation 6 ) of 1. Before n o r m a l i z a t i o n , average c a l c u l a t e d values of f were 0.7, 0.89, and 0.86 f o r t h e t h r e e room mass l e v e l s , r e s p e c t i v e l y .

Room-Air Response F a c t o r s

Experiments were performed i n two h e a t i n g seasons, 1980181 and 1982183. Between t h e two p e r i o d s , an a d d i t i o n a l e x t e r n a l l a y e r of i n s u l a t i o n was a p p l i e d t o t h e w a l l s and c e i l i n g s of a l l t e s t u n i t s . The change i n thermal c h a r a c t e r i s t i c s i s r e f l e c t e d i n t h e d a t a i n Table 1. I n 1980181, experiments were performed i n a l l t h r e e 2-room u n i t s f o r both rooms simultaneously ( p a r t i t i o n door open). I n 1982183, the medium-weight u n i t was not a v a i l a b l e f o r f u r t h e r experiments. D e t a i l s of t h e experiments performed during both h e a t i n g seasons (19 i n a l l ) a r e given i n Table 4. F i g u r e 8a t o c shows examples of the measured and p r e d i c t e d net h e a t i n p u t t o t h e room ( o u t p u t ) corresponding t o t h e measured v a r i a t i o n i n room temperature ( i n p u t ) f o r each of t h e t h r e e room mass values. The r e s u l t i n g average response f a c t o r s f o r a l l t h e experiments a r e l i s t e d i n Table 4. R e s u l t s of i n d i v i d u a l t e s t s a r e a l s o given by Barakat

( 1985).

Due t o measurement e r r o r s , the t r a n s f e r f u n c t i o n c o e f f i c i e n t s o b t a i n e d d i d not e x a c t l y s a t i s f y t h e s t e a d y - s t a t e c o n d i t i o n given by Equation 7. To s a t i s f y t h i s c o n d i t i o n , t h e g2

coef € i c i e n t was c a l c u l a t e d from t h i s equation. The o v e r a l l thermal conductance of t h e u n i t ,

KT, was obtained by adding t h e c o r r i d o r - w a l l conductance t o t h e indoor-outdoor UA v a l u e of t h e u n i t obtained i n a previous i n v e s t i g a t i o n (Barakat 1984). The maximum change i n any v a l u e was l e s s t h a n 10%.

In a d d i t i o n , t o be c o n s i s t e n t with t h e ASHRAE p r e s e n t a t i o n , t h e room-air response f a c t o r s were normalized u s i n g Equation 8. These a d j u s t e d and normalized f a c t o r s a r e p r e s e n t e d i n Table 5 f o r t h e t h r e e mass l e v e l s .

A s i n d i c a t e d e a r l i e r , t h e thermal c a p a c i t y of a b u i l d i n g can be determined u s i n g Equation 10. Table 5 g i v e s t h e r e s u l t i n g values f o r t h e t h r e e u n i t s , a s w e l l a s t h e values c a l c u l a t e d from the c o n s t r u c t i o n d e t a i l s , t h e b u i l d i n g elements, and t h e s p e c i f i c h e a t of each element ( t a k e n from Table 2). The d a t a i n t h e t a b l e show good agreement between t h e two s e t s of thermal c a p a c i t y values.

DISCUSSION

The response f a c t o r s obtained i n t h i s s t u d y , t o g e t h e r with t h e values i n t h e ASHRAE Handbook, a r e summarized i n Table 6. Examination of t h e decay c o e f f i c i e n t s (wl and p l ) shows t h a t t h e new values a r e c o n s i s t e n t with t h e mass l e v e l s of t h e b u i l d i n g . The new response f a c t o r v a l u e s , t h e r e f o r e , f i l l t h e gap t h a t e x i s t e d f o r wood-frame houses, A d d i t i o n a l work i s

needed, however, t o produce t h e remaining s e t s of c o o l i n g load response f a c t o r s (conduction, l i g h t s , e t c . ) .

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Work is underway t o c a l c u l a t e t h e z - t r a n s f e r f u n c t i o n c o e f f i c i e n t s a n a l y t i c a l l y , f o r t h e same mass l e v e l s r e p o r t e d i n t h i s s t u d y , u s i n g t h e procedure developed by M i t a l a s (1983).

P r e l i m i n a r y r e s u l t s of t h e s e c a l c u l a t i o n s (Sander 1985) a r e r e p o r t e d i n Table 7. C a l c u l a t e d

v a l u e s f o r t h e s o l a r c o o l i n g load and t h e room-air t r a n s f e r f u n c t i o n c o e f f i c i e n t s show f a i r l y

good agreement with e x p e r i m e n t a l v a l u e s . The l a r g e v a l u e s of t h e new g c o e f f i c i e n t s of t h e

room-air t r a n s f e r f u n c t i o n a r e probably due t o t h e l o c a t i o n s of h e a t i n p u t and t e m p e r a t u r e

measurement. A s mentioned p r e v i o u s l y , t e m p e r a t u r e was measured a s t h e a v e r a g e (of 12

l o c a t i o n s ) a t room mid-height. However, i f t h e measured a v e r a g e t e m p e r a t u r e is l e s s t h a n t h e

t r u e mixed average assumed i n any c a l c u l a t i o n , t h i s w i l l r e s u l t i n higher values of t h e g

coef F i c i e n t s compared with t h o s e c a l c u l a t e d a n a l y t i c a l l y . I n a d d i t i o n , euch d i f Eerencea may

account f o r t h e d i s c r e p a n c y between t h e assumed l i n e a r indoor c o n v e c t i v e f i l m c o e f f i c i e n t s used i n t h e c a l c u l a t i o n , and t h e a c t u a l c o n v e c t i v e c o e f f i c i e n t s f o r t h e w a l l s of t h e t e s t u n i t s .

The response f a c t o r s i n T a b l e 7 were used i n t h e ENCORE Program (Konrad and L a r s e n 1978) t o s i m u l a t e t h e performance of Unit 1 of t h e t e s t f a c i l i t y f o r t h e same p e r i o d shown i n

F i g u r e 1. Room t e m p e r a t u r e and power consumption a r e compared w i t h measured v a l u e s i n

F i g u r e 9. The new response f a c t o r v a l u e s gave r e s u l t s t h a t corresponded much b e t t e r t o a c t u a l

monitored performance f o r such l i g h t w e i g h t b u i l d i n g s . F u r t h e r improvement i n t h e s i m u l a t i o n

r e s u l t s , p a r t i c u l a r l y f o r heavyweight b u i l d i n g s , can be expected i f more c o e f f i c i e n t s a r e used t o e x p r e s s t h e z - t r a n s f e r f u n c t i o n .

SUMMARY

New s e t s of z - t r a n s f e r f u n c t i o n c o e f f i c i e n t s were g e n e r a t e d e x p e r i m e n t a l l y f o r t h e s o l a r

c o o l i n g l o a d and t h e room-air temperature. The c o e f f i c i e n t s were o b t a i n e d f o r wood-frame

houses with t h r e e d i f f e r e n t s p e c i f i c mass l e v e l s : 46, 130, and 535 kg/m2 of f l o o r a r e a .

These c o e f f i c i e n t s improved t h e thermal performance p r e d i c t i o n c a p a b i l i t y of energy a n a l y s i s

programs based on the response f a c t o r p r i n c i p l e . The response f a c t o r s r e p o r t e d i n t h i s Note

complement t h o s e i n t h e ASHRAE Handbook and i n c r e a s e t h e scope of t h e response f a c t o r method.

1

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I

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M i t a l a s , G.P. 1972. "Transfer f u n c t i o n m t h o d of c a l c u l a t i n g c o o l i n g l o a d s , h e a t e x t r a c t i o n and space temperature". ASHRAE J o u r n a l , Vol. 14, No. 12, (Dec.) pp. 54-56.

M i t a l a s , G.P. 1978. "Comments on t h e z - t r a n s f e r f u n c t i o n method f o r c a l c u l a t i n g h e a t t r a n s f e r i n b u i l d i n g s " . ASHRAE T r a n s a c t i o n s , Vol, 84, P a r t 1 , pp. 667-674.

M i t a l a s , G.P. 1983. "Room dynamic thermal response, Proceedings of t h e 4th I n t e r n a t i o n a l Symposium on t h e Use of Computers f o r Environmental Engineering R e l a t e d t o Buildings", pp. 25-31, Tokyo, Japan.

Sander, D.M. 1985. 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 Research Council of Canada, p r i v a t e communication.

S t e i g l i t z , K. 1970. Computer-aided d e s i g n of r e c u r s i v e d i g i t a l f i l t e r s " . IEEE T r a n s a c t i o n s on Audio and E l e c t r o a c o u s t i c s , Vol. AU-18, No. 2, pp. 123-129.

Stephenson, D.G., and M i t a l a s , G.P. 1967. "Cooling load c a l c u l a t i o n by t h e thermal response f a c t o r method". ASHRAE T r a n s a c t i o n s , Vol. 73, P a r t 1.

TABLE 1

C h a r a c t e r i s t i c s of Test U n i t s

Wall thermal r e s i s t a n c e (m2 OC/W) C e i l i n g thermal r e s i s t a n c e (m2 OC/W) Floor thermal r e s i s t a n c e (n? OC/W) South-window g l a s s a r e a (m2 )

North-window g l a s s a r e a (m2 )

Window thermal r e s i s t a n c e (m2 OC/W) Floor a r e a per room

(d

)

Room h e i g h t (m) TABLE 2 Thermal S t o r a g e C h a r a c t e r i s t i c s of Test U n i t s T e s t Thermal c a p a c i t y a u n i t (MJ/K) C o n s t r u c t i o n L i g h t

-

Standard wood-frame, 12.7 mu

gypsum board f i n i s h on w a l l s and c e i l i n g s ,

c a r p e t over wooden f l o o r .

i

Medium

-

A s above, but 50.8 mm gypsum board f i n i s h on w a l l s and 25.4 mm f i n i s h on c e i l i n g . Heavy

-

I n t e r i o r wall f i n i s h of 101.6 mm b r i c k , 12.7 mm gypsum board f i n i s h on c e i l i n g , c a r p e t over wooden f l o o r . a I n c l u d e s framing i n w a l l s and c e i l f n g TABLE 3 S o l a r T r a n s f e r Function C o e f f i c i e n t s Weight v~ W1 RMS f i t e r r o r (W) Light 0.635 -0.240 -0.605 11.7 Medium 0.400 -0.140 -0.740 9.2 Heavy 0.335 -0.205 -0.870 9.2 153

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TABLE 4

Room-Air Response Factors

- -- - RMS f i t Number of Date _go _gl ₈₂ _P1 e r r o r (W) t e s t s Light Average 1981 334.4 -398.8 91.2 -0.601 33.5 3 Corrected g2 334.4 -398.8 82.8 -0.601 Average 1983 368.9 -508.4 153.7 -0.693 25.9 b Corrected g2 368.9 -508.4 151.0 -0.693 Medium

-

Average 1981 446.3 -547.7 120.2 -0.724 34.0 Corrected g2 446.3 -547.7 113.5 -0.724, Heavy Average i981 453.7 -529.2 81.5 -0.901 30.2 Corrected g2 453.7 -529.2 80.1 -0.901 Average 1983 400.3 -452.2 51.4 -0.895 24.0 4 Corrected g2 400.3 -452.2 56.3 -0.895 TABLE 5

Normalized Room-Air Response Factors

Thermal

Unit Data _go

_*

_gl

_*

_g2* _P1 _capacity

( J / K ) - - - Light 198 1 10.40 -13.37 2.97 -0.601 1983 11.92 -17.35 5.43 -0.693 Average 11.16 -15.36 4.20 -0.647 1.98(1.96)a Medium 1981 14.50 -18.58 4.08 -0.724 3.78(4.21)a Heavy 1981 14.67 -17.55 2.88 -0.901 1983 13.00 -15.02 2.02 -0.895 Average 13.84 -16.29 2.45 -0.898 11.18(11.62)a aValues i n brackets a r e from Table 2

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TABLE 6

Experimental and ASHRAE Response Factors

- - . . - - - - -

Exptl. Exptl. ASHRAE ASHRAE Expt 1. ASHRAE light medium light medium heavy heavy Specific mass

(kg/m2 )

of floor area 4 6 130 146

34

1 535 6 35

Solar

TABLE 7

Calculated Response Factors

Light Medium Heavy

Specific mass ( kg/m2 of floor area Solar Conduction Lighting (incandescent ) Room-air temperature

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0 0 0 0 0 0 0 0 0 0 0

m N d 4 cuo 0 0

I I l n 0 ln

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(17)

(18)

a )

R O O M T E M P E R A T U R E

MEASURED 20.5"C AVERAGE

---

PREDICTED 20.6OC AVERAGE

-

I I I I I 1 I I I I I I I I

C

MEASURED 342.0 k w h TOTAL

b l

H E A T I N G P O W E R

_---

PREDICTED 328.9 k w h TOTAL

C O N S U M P T I O N

-I

D E C 1 9 8 0

D A Y O F M O N T H

J A N 1 9 8 1

F i g u r e 9 . ENCORE s i m u l a t i o n r e s u l t s f o r l i g h t u n i t s u s i n g new r e s p o n s e f a c t o r s ( T a b l e 7)