A procedure for deriving thermal transfer functions for walls from hot-box test results

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A procedure for deriving thermal transfer functions for walls from

hot-box test results

arch Consell national uncll Canada de recherches Canada

lnstitut de recherche en construction

A

Procedure for Deriving Thermal

Transfer Functions for

Walls

from

Hot-Box Test Results

D.G.

Stephenson,

K.

Ouyang

and

W.C.

Brown

internal Report

No.

568

Date

of

issue: May

1988

~ N A L Y Z E D

This is an internal report of the Institute for Research in Construction. Although not intended for general distribution, it may be cited as a reference in other publications.

A Procedure f o r D e r i v i n g Thermal T r a n s f e r F u n c t i o n s f o r Walls from Hot-Box T e s t R e s u l t s : P a r t I

by

D.G. Stephenson,

K.

Ouyang, W.C. Brown

IRC/NRC 1988 A b s t r a c t

A guarded hot-box

i s

used t o determine t h e U-value of a w a l l

specimen a t two d i f f e r e n t mean temperatures. With t h e h o t s i d e t e m p e r a t u r e h e l d c o n s t a n t , t h e c o l d s i d e t e m p e r a t u r e i s v a r i e d a t a c o n s t a n t r a t e d u r i n g t h e t r a n s i t i o n between t h e two s t e a d y - s t a t e c o n d i t i o n s . The r e s u l t s of t h i s t e s t a r e used t o determine t h r e e time c o n s t a n t s of t h e w a l l , which a r e t h e n used t o c a l c u l a t e t h e c o e f f i c i e n t s of t h e z - t r a n s f e r f u n c t i o n f o r t h a t w a l l .

I n t r o d u c t i o n

The t r a n s f e r f u n c t i o n approach f o r c a l c u l a t i n n h e a t flow t h r o u g h w a l l s and c e i l i n g s h a s been endbrsed by ASHRAE f o r th; p a s t 20 y e a r s , and i s widely used throughout North America. The procedure r e q u i r e s , a s d a t a , t h e c o e f f i c i e n t s of t h e wall's t h e r m a l t r a n s f e r f u n c t i o n s , and h o u r l y v a l u e s of t h e t e m p e r a t u r e f o r t h e o u t e r f a c e of t h e w a l l . 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 many w a l l s have been p u b l i s h e d i n t h e ASHRAE Handbook of Fundamentals ( 1).

U n f o r t u n a t e l y , t h e s e d a t a a r e a l l based on t h e assumption t h a t t h e h e a t flow through t h e s e w a l l s i s one dimensional. There

i s

no allowance f o r framing members t h a t a c t a s h e a t b r i d g e s through i n s u l a t i o n , n o r f o r such common b u i l d i n g m a t e r i a l s a s hollow c o n c r e t e blocks. T h e r e f o r e , t h e t r a n s f e r f u n c t i o n d a t a i n t h e ASHRAE Handbook of Fundamentals may n o t a c c u r a t e l y r e p r e s e n t t h e t h e r m a l performance of r e a l w a l l s . Consequently t h e r e

i s

a growing i n t e r e s t i n being a b l e t o d e r i v e t h e s e d a t a from t h e r e s u l t s of t e s t s on f u l l s c a l e w a l l specimens. T h i s paper p r e s e n t s a

procedure f o r doing t h i s . It i n v o l v e s u s i n g a guarded hot-box w a l l t e s t i n g f a c i l i t y ; t h e IRC f a c i l i t y is shown s c h e m a t i c a l l y i n F i g u r e 1. A t e s t c o n s i s t s of t h r e e phases:

I ) An i n i t i a l s t e a d y - s t a t e U-value t e s t w i t h t h e room-side

environmental t e m p e r a t u r e a t Th and t h e c l i m a t e s i d e t e m p e r a t u r e a t Tci.

1 1 ) A t r a n s i t i o n phase d u r i n g which t h e c l i m a t e chamber t e m p e r a t u r e i s changed from Tci t o Tcf. The t e m p e r a t u r e i s changed a t a c o n s t a n t r a t e d u r i n g t h i s t r a n s i t i o n .

111) A f i n a l s t e a d y - s t a t e U-value t e s t w i t h t h e room-side t e m p e r a t u r e a t Th and t h e c l i m a t e s i d e t e m p e r a t u r e a t Tcf.

F i g u r e 2 shows t h e temperature v s time curve o b t a i n e d f o r a t y p i c a l t e s t , and t h e c o r r e s p o n d i n g v a l u e s f o r t h e h e a t f l u x i n t o t h e room-side f a c e of t h e t e s t w a l l . These r e s u l t s can b e used t o determine t h r e e

t i m e - c o n s t a n t s f o r t h e w a l l , p l u s t h e i r a s s o c i a t e d r e s i d u e s . These t i m e - c o n s t a n t s and r e s i d u e s p l u s t h e U-value a r e used a s t h e d a t a t o

c a l c u l a t e t h e c o e f f i c i e n t s f o r a r a t i o n a l 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 h e a t f l u x through t h e room-side f a c e of t h e w a l l t o t h e t e m p e r a t u r e a t t h e o u t s i d e s u r f a c e . The procedure f o r t h i s l a t t e r c a l c u l a t i o n

i s

b a s i c a l l y

t h e same a s i s d e s c r i b e d i n Stephenson and M i t a l a s ( 2 ) . The d i f f e r e n c e i s t h a t t h e time-constants and r e s i d u e s a r e d e r i v e d from t e s t r e s u l t s r a t h e r t h a n from t h e dimensions and t h e r m a l p r o p e r t i e s of t h e m a t e r i a l s t h a t make up t h e w a l l .

Theory

The Laplace t r a n s f o r m s of Th and Tc a r e r e s p e c t i v e l y

el

and

e2,

and t h e t r a n s f o r m s of t h e h e a t f l u x e s Qh and Qc a r e and

4

r e s p e c t i v e l y . These a r e r e l a t e d by

where t h e elements A, B, C and D of t h e t r a n s m i s s i o n m a t r i x a r e f u n c t i o n s of t h e thermal p r o p e r t i e s and dimensions of t h e m a t e r i a l s i n t h e w a l l and t h e h e a t t r a n s f e r r e s i s t a n c e a t t h e s u r f a c e s . T h i s f o r m u l a t i o n , which is t a k e n from Carslaw and J a e g e r ( 3 ) . assumes t h a t t h e thermal p r o p e r t i e s of t h e w a l l a r e c o n s t a n t . Equation 1 can

be

r e c a s t a s

The element C h a s been e l i m i n a t e d by u s i n g t h e f a c t t h a t A-D

-

B*C

-

1. The f u n c t i o n s D/B, 1 / B and A/B a r e r e f e r r e d t o a s t h e Laplace t r a n s f e r f u n c t i o n s of t h e w a l l .

The t r a n s f e r f u n c t i o n l / B ( s ) can be r e p r e s e n t e d a s

where Tn a r e t h e t i m e c o n s t a n t s of t h e w a l l (i.e., t h e p o l e s of 1/B(s) a r e

a t s = - I / T ~ )

A l l t h e t r a n s f e r f u n c t i o n s of Equation ( 2 ) have t h e i r poles a t

s

=

s i n c e B(s) i s t h e common denominator.

When c a l c u l a t i n g t h e h e a t f l u x through a w a l l o r roof i t

i s

more convenient t o u s e z-transforms of t h e t e m p e r a t u r e s and f l u x e s r a t h e r t h a n t h e Laplace t r a n s f o r m s . (The z-transforms a r e sometimes r e f e r r e d t o a s 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 s of t h e s e q u a n t i t i e s ) . An e q u a t i o n s i m i l a r t o e q u a t i o n 2 r e l a t e s t h e z-transforms of h e a t f l u x t o t h e z-transforms of t e m p e r a t u r e s : v i z ,

where

and

The z - t r a n s f e r f u n c t i o n s can be approximated by

and

where U = t h o v e r a l l conductance of t h e wall

s%

z = e

,

6 = 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 terms i n t h e t i m e s e r i e s f o r t h e t e m p e r a t u r e s and h e a t f l u x e s ,

s

= t h e parameter i n Laplace t r a n s f o r m s ,

N

-

t h e number of terms i n t h e v a r i o u s sums (N may be d i f f e r e n t f o r each summation)

an,bn,dn = 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 .

Equation 4 is u s u a l l y w r i t t e n i n t h e time domain a s

where T

i s

t h e e n v i r o n m e n t a l t e m p e r a t u r e i n t h e b u i l d i n g

a t

t i m e t , h , t

Tc,t i s t h e environmental temperature o u t s i d e

a t

t i m e t.

The approach used i n t h i s paper i s t o approximate l / B ( s ) by t h e sum of o n l y t h r e e ( o r i n some c a s e s f o u r ) terms, and t h e n t o u s e t h e s e v a l u e s of

a and T t o c a l c u l a t e bn and dn. When v a l u e s of a a r e r e q u i r e d ( i . e . ,

wgen ins?de t e m p e r a t u r e i s changing), t h e y a r e obta?ned from e x p e r i m e n t a l l y determined v a l u e s of t h e w a l l ' s f r e q u e n c y r e s p o n s e and t h e t i m e c o n s t a n t s , a s d e s c r i b e d i n P a r t I1 of t h i s paper.

S t e p #1 _{Determination of t h e a,,}

&

T~ of a

wall

If t h e s t a r t of phase I1 ( s e e F i g u r e

2)

i s

t a k e n a s t = 0 , Qi a s t h e v a l u e of h e a t f l u x d u r i n g t h e s t e a d y - s t a t e a t t h e end of p h a s e I, and Qf a t t h e s t e a d y - s t a t e i n phase 111, t h e n t h e h e a t f l u x , Q t , a t any t i m e t d u r i n g phase I1 can be r e p r e s e n t e d ( 3 ) by:

where t* i s t h e d u r a t i o n of phase 11. T h i s can be i n v e r t e d t o g i v e

where

and _Qt + Qi + Qf

-

tx Qi ( t

-

r )

Qf

-

Qi

This asymptote i s a s t r a i g h t l i n e with a s l o p e of

t,

,

and i t e q u a l s Qi when t =

r.

Thus t h e v a l u e of

r

can be o b t a i n e d from t h e i n t e r s e c t i o n of t h i s asymptote and t h e v a l u e of Q

i *

I n phase 111,

where t ' = t-t*

S i n c e ~ ~ ( O < t < t * ) is e q u a l t o E t t ( t * < t ) , Equation 12 can be combined w i t h

Equation 9 t o g i v e t h e f o l l o w i n g e x p r e s s i o n f o r

r

A v a l u e of 'l can be o b t a i n e d i n t h i s way from e a c h v a l u e of Q t , and t h e c o r r e s p o n d i n g Qt*+t, f o r 0

<

t

<

t*.

Equation 1 2 i n d i c a t e s t h a t a s t ' becomes l a r g e

where T i s t h e l a r g e s t time-constant. 1

Thus

,

and i t s v a l u e a t This asymptote i s a s t r a i g h t l i n e with a s l o p e of

--

T,

l

a T 1 1

t = O ( i . e . , t

-

t * )

i s

Thus t h e asymptote g i v e s v a l u e s f o r T and 1 al' I f e q u a t i o n 12 i s i n t e g r a t e d from t* t o i t g i v e s Thus DI t*

I

( Q ~

-

Q t ) d t where A 5 t* Qf

-

Qi

One more r e l a t i o n s h i p can be d e r i v e d from t h e f a c t t h a t

When e a u a t i o n s 9 and 11 a r e d i f f e r e n t i a t e d t h e y g i v e

Theref o r e

E q u a t i o n s 10, 16 and 18 a r e used t o de ermine t h e time c o n s t a n t s and

₁

r e s i d u e s . However, i f t h e t r a n s f e r f u n c t i o n

n)

i s t o be approximated by a f i n i t e number of terms r a t h e r t h a n a n i n f i n i t e number, t h e sums of t h e

v a r i o u s f i n i t e s e r i e s must be t h e same a s t h e sums of t h e c o r r e s p o n d i n g i n f i n i t e s e r i e s , i . e .

Values of al and r can be d e r i v e d d i r e c t l y from t h e test r e s u l t s u s i n g e q u a t i o n 15. Thus, f f N=2 t h e r e would be only two undetermined c o n s t a n t s , a and T 2 , s o i t would n o t be p o s s i b l e t o s a t i s f y a l l t h r e e

8

r e l a t i o n s . n t h e o t h e r hand, i f N=3 t h e r e would be f o u r undetermined c o n s t a n t s , and t h e t h r e e r e l a t i o n s a r e n o t s u f f i c i e n t t o determine a l l f o u r c o n s t a n t s . One of t h e c o n s t a n t s must be used a s a n undetermined c o n s t a n t t o be determined by some o t h e r c o n s t r a i n t . It i s convenient t o l e t

3

be t h e undetermined c o n s t a n t .

Let a2

+

a = F = 1-a

3 I

where

r

and A a r e determined from Equation 13 and 17, r e s p e c t i v e l y . These c a n be solved t o g i v e

There is t h e a d d i t i o n a l c o n s t r a i n t t h a t r2 and r3 must be r e a l and p o s i t i v e . T h i s l i m i t s t h e p o s s i b l e range f o r

3

a s f o l l o w s :

a ) i f G~

>

FH

b) i f G 2 = FH

a3 must be z e r o and T2

-

--&

C) i f G 2

<

FH

g3

must be between z e r o and

F.

The b e s t v a l u e f o r a3 i s determined by t r y i n g s e v e r a l v a l u e s w i t h i n t h e a l l o w a b l e range, and f o r each of t h e s e determining a c o n s i s t e n t set of

v a l u e s f o r a2, T2 and T3. Then u s i n g t h e s e v a r i o u s s e t s of v a l u e s a v a r i a n c e

nt

i s c a l c u l a t e d a s :

and Et

i s

determined from experiment ( c t = Qf

-

Qt f o r t * < t )

The b e s t v a l u e f o r a3 is t h e one t h a t g i v e s t h e s m a l l e s t v a l u e f o r Gnt2.dt. This i s a " l e a s t squares" curve f i t t i n g procedure t h a t s a t i s f i e s t h e c o n s t r a i n t s on Qtand i t s f i r s t d e r i v a t i v e .

Example 1

T h i s procedure

i s

i l l u s t r a t e d i n example 1. I n t h i s example,

however, t h e v a l u e s of Q t have been c a l c u l a t e d from t h e e x a c t s o l u t i o n f o r a homogeneous s l a b r a t h e r t h a n being expe i m e n t a l r e s u l t s .

_f

Hence, t h e v a l u e s of

"

a r e due t o t h e approximation of I n a r e a l t e s t t h e r e would a l s o be some e x p e r i m e n t a l e r r o r i n t h e measured v a l u e s t h a t would a l s o c o n t r i b u t e t o

nt.

For a homogeneous s l a b of t h e f o l l o w i n g p r o p e r t i e s

under t h e f o l l o w i n g test c o n d i t i o n s ,

A

= 179.19 h 2

The r e s u l t i n g h e a t f l u x , Q and Et a r e g i v e n i n Table I.

t '

Table 1

For t h e remainder of phase 11, Qt

i s

given by

96 where a = 2.000 and

rl

=

-

= 9.7268

h

1 112 (These a r e t h e v a l u e s t h a t would be i n d i c a t e d by t h e r e s u l t s of a p e r f e c t ramp t e s t ) . From e q u a t i o n 20

Thus a3 must have t h e o p p o s i t e s i g n of F, i.e. p o s i t i v e .

Table I1 g i v e s t h e v a l u e s of

9 ,

T2 and T3 t h a t a r e c o n s i s t e n t with v a r i o u s v a l u e s of a3. The lower p a r t of t h e t a b l e g i v e s t h e v a l u e s of rlt

f o r t h r e e of t h e cases. These v a l u e s show t h a t t h e approximation g e t s b e t t e r a s a3 i n c r e a s e s , b u t beyond

4

= 3 t h e improvement i s v e r y s l i g h t .

Table I1

S t e p

112

Determining 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

The problem a t t h i s s t e p i s t o determine t h e c o e f f i c i e n t s b- and d-

I, LL

t h a t make

e q u i v a l e n t t o

Values of an and T,, were determined i n s t e p

81.

Equivalence of t h e

two t r a n s f e r f u n c t i o n s r e q u i r e s t h a t t h e p o l e s of t h e two c o r r e s p o n d , and t h a t t h e y have t h e same frequency r e s p o n s e a t f r e q u e n c i e s t h a t a r e of p a r t i c u l a r importance i n t h e i n t e n d e d a p p l i c a t i o n .

The p o l e s of t h e Laplace t r a n s f e r f u n c t i o n a r e a t s = - 1 / ~ - . Hence

n t h e p o l e s of t h e z - t r a n s f e r f u n c t i o n must be a t

z

= e -61 Tn. 3 3 - 6 / ~ ~ T h e r e f o r e

c

dnz-" =

n

(I-e z - l ) n=O n= 1 which g i v e s t h e c o e f f i c i e n t s dn a s :

where Let where

-

Vf =

C

dn cos (2rmsf) n=O 3

C

dn ' Z -n n-0 3 Wf

-

-

d n s i n (2rm6f) n=O 1 2 6

6

= Vf

+

i Wf z-e

Then matching t h e frequency r e s p o n s e of R/B(z) t o R/B(s) f o r a harmonic d r i v i n g f u n c t i o n w i t h a frequency of f r e q u i r e s and

-

C

bn s i n (2rm6f) = XfWf

+

YfVf n-0 A t s t e a d y - s t a t e (i.e., f 0) 3

Theref o r e

Equations 31, 32 and 33 can be s o l v e d t o f i n d t h r e e bn c o e f f i c i e n t s . I n m a t r i x form

cos ( 2 n 6 f ) (34)

- s i n ( 4 n 6 f )

I f it is d e s i r e d t o match t h e frequency response a t a second frequency, t h e r e would have t o be f i v e bn c o e f f i c i e n t s . Two more rows and two more columns would be added t o t h e s q u a r e m a t r i x , and two more terms i n t h e column m a t r i x on t h e r i g h t s i d e of t h e e q u a t i o n .

I f t h e s q u a r e m a t r i x i n e q u a t i o n 34 i s M , t h e n bn a r e g i v e n by

Example I1

With 6

-

1 h , and f = 1 c y c l e / 2 4 hours

Thus

Table

I11

compares the frequency responses of R/B(s) and R/B(z) with

the correct" values obtained from an analytical solution for this

homogeneou 2sl

b.

These values show that R/B(z) matches the correct values

f:

B

well when

<

4.

But when this number is larger, the difference between

R/B(Z) and the correct frequency response is substantial. Whether this

difference will lead to a significant error in Qt depends on the frequency

content of the driving function Tc.

is a dimensionless parameter

where

L

-

thickness of slab

P =

density

c

=

specific heat

A

-

thermal conductivity

f

-

frequency

Example I11

-

R e s u l t s of a Ramp Test on a Heavy Wall w i t h Heat Bridges

A ramp t e s t was c a r r i e d o u t on a w a l l with a 127 ram l a y e r of foam p o l y s t y r e n e sandwiched between two 89 mm wythes of c o n c r e t e . The sample was brought t o a s t e a d y - s t a t e w i t h Th = 21°C and Tci

-

-7.loC. Then t h e

t e m p e r a t u r e on t h e c o l d s i d e was reduced t o Tcf = -30.0°C a t a c o n s t a n t r a t e over a p e r i o d of t* = 60 hours. Table I V g i v e s t h e v a l u e s of t e m p e r a t u r e and t o t a l h e a t flow a t 3-hour i n t e r v a l s o v e r t h e 7-day test p e r i o d .

The U-valueof t h e specimen can be determined from t h e i n i t i a l and f i n a l s t e a d y - s t a t e r e s u l t s . The v a r i a t i o n of U w i t h mean t e m p e r a t u r e can t h e n be c a l c u l a t e d from For t h i s t e s t Qi = 86.6 W, Qf = 156.6 W , and t h e t e s t a r e a A = 5.946 m2. These v a l u e s l e a d t o : Uo = 0.517

w m - k l

and dU

-

= 1.66 x lo-' w ~ - ~ K - ~ dT m A s

-

i s v e r y s m a l l it is v a l i d t o analyze t h e r e s u l t s from t h e t r a n s i e n t dTm

Table I V

-

Phase I Ramp Started a t 0.0 Phase I1 Ramp Finished a t 60.0

Phase 111

Table I V (cont'd)

t T~ Qt

€t

'

t

'

=t-t*

Values of

r

a r e g i v e n i n Table I V f o r each v a l u e of Qt i n Phase X I . These were c a l c u l a t e d u s i n g Equation 13, i.e.

The v a l u e s i n t a b l e I V show t h a t t h e b e s t v a l u e f o r

r

i s 13.5 h o u r s .

Table I V shows t h a t

st,

has n o t decayed t o z e r o when t ' r e a c h e s 60.0 h ( a p e r i o d t h a t e q u a l s t * ) . Thus Et w i l l a l s o n o t be z e r o when t = t* ( i . e . a t t h e end of phase 1 1 ) . T h i s must be t a k e n i n t o account when c a l c u l a t i n g c t , f o r t h e e a r l y p a r t of phase 111. S u b t r a c t i n g E ~ from ~ +Qt ( f o r ~ t

e*)

c o r r e c t s f o r t h e r e s i d u a l e f f e c t of phase 11, t h a t is For example: e t c . E 60.3

=

156.6

-

156.4 = 0.2 E 63.3 = 156.6

-

156.5 = 0.1

The v a l u e s of E ~ , a r e t h e n used t o determine

4

and TI, ( o r a s i n t h i s example ao, To, a and TI). F i g u r e 3 shows v a l u e s of i n et, v s t v . A t

l a r g e v a l u e s of t f t h e p o i n t s l i e a l o n g a c u r v e t h a t i s s l i g h t l y concave upward and t h i s c u r v a t u r e i s due t o t h e h e a t b r i d g e s , which conduct some h e a t through t h e i n s u l a t i o n i n p a r a l l e l w i t h t h e main h e a t flow path. To account f o r t h i s e f f e c t , v a l u e s o f Et, f o r t'X2 hours can be r e p r e s e n t e d by t h e sum of two e x p o n e n t i a l t e r m s : A r e g r e s s i o n f i t of t h e d a t a t o t h i s e q u a t i o n g i v e s : a = 0.056 0 ? 0 = 25.0 h a = 1.670 1 T~ = 8.33 h

I n t e g r a t i n g E from t ' = 12 t o t ' = OD y i e l d s a v a l u e of 57.3 W-h.

The i n t e g r a l from t f

-

0 t o t ' 12 can be e v a l u a t e d n u m e r i c a l l y . It e q u a l s 114.3 W*h. ( T h i s v a l u e was o b t a i n e d u s i n g h o u r l y v a l u e s r a t h e r t h a n t h e 3-hour v a l u e s i n Table IV.)

Thus, from Equation 17

Also from Equation 20

T h e r e f o r e G'

-

FH = 0.48 h 2

A s 'G FH and F

<

0, a2 must be l e s s t h a n z e r o and

9

must be g r e a t e r t h a n z e r o i n o r d e r f o r r2 and r3 t o be r e a l and p o s i t i v e . Various p a i r s of v a l u e s f o r a2, a3 were t r i e d and

gave r e s u l t s i n good agreement w i t h t h e e x p e r i m e n t a l v a l u e s f o r t h e e a r l y p a r t of t h e t e s t . From Equation 21 t h e corresponding v a l u e s f o r T2 and T3

a r e :

These v a l u e s c a n be used t o d e t e r m i n e t h e c o e f f i c i e n t s of R/B(z) j u s t a s was done i n Example 11.

Table V

Summary of R e s u l t s

D i s c u s s i o n

F i g u r e 3 shows t h e importance of o b t a i n i n g v a l u e s of €+ o v e r a s l o n g a p e r i o d a s p o s s i b l e . When t h e r e a r e some h e a t b r i d g e s throGgh l a y e r s of i n s u l a t i o n t h e y g i v e r i s e t o a long time-constant w i t h a s m a l l a s s o c i a t e d r e s i d u e . T h i s means t h a t t h e w a l l may appear t o have reached a

s t e a d y - s t a t e , b u t t h e h e a t f l u x w i l l continue t o change v e r y slowly f o r many hours. Thus it i s good p r a c t i c e t o maintain t h e c o n s t a n t t e m p e r a t u r e c o n d i t i o n s f o r a t l e a s t a day a f t e r i t seems t h a t t h e s t e a d y - s t a t e has been reached.

The concept of a t r a n s f e r f u n c t i o n t h a t embodies t h e t h e r m a l

c h a r a c t e r i s t i c s of a w a l l i s v a l i d o n l y i f t h e t h e r m a l p r o p e r t i e s of t h e w a l l a r e independent of t h e t e m p e r a t u r e and t i m e . The i n i t i a l and f i n a l s t e a d y - s t a t e r e s u l t s can be used t o check whether t h i s assumption i s v a l i d . T h i s check should be made b e f o r e making a d e t a i l e d a n a l y s i s of t h e r e s u l t s from t h e t r a n s i e n t p a r t of t h e t e s t .

dU

The o u t p u t from a ramp t e s t should be t h e v a l u e s of Uo, 'rn and =n

.

These v a l u e s w i l l e n a b l e a u s e r t o c a l c u l a t e t h e v a l u e s ofubn and dn f o r any time-step. When v a l u e s of a n a r e r e q u i r e d it

is

n e c e s s a r y t o have v a l u e s of t h e A / B and D / B t r a n s f e r f u n c t i o n f o r t h e f r e q u e n c i e s of

p a r t i c u l a r i n t e r e s t . This procedure i s explained i n P a r t 11. Conclusion

A ramp t e s t s t a r t i n g from a s t e a d y - s t a t e and ending w i t h a n o t h e r s t e a d y - s t a t e y i e l d s enough i n f o r m a t i o n t o determine t h r e e time c o n s t a n t s of t h e w a l l , which a r e t h e n used t o c a l c u l a t e t h e z - t r a n s f e r j u n c t i o n

c o e f f i c i e n t l / B ( z ) f o r t h e w a l l . Acknowledgement

The a u t h o r s wish t o e x p r e s s t h e i r a p p r e c i a t i o n t o M r . J. Richardson and

Mr. G.

K e a t l e y f o r t h e i r a s s i s t a n c e i n p r e p a r i n g t h e w a l l specimen f o r t e s t and i n c a r r y i n g o u t t h e ramp test. The q u a l i t y of t h e r e s u l t s i s due t o t h e i r s k i l l i n o p e r a t i n g t h i s complex f a c i l i t y .

References

1. ASHRAE Handbook of Fundamentals (1977).

2. Stephenson & Mitalas, Calculation of Heat Conduction Transfer Functions

for Multi-Layer Slabs. ASHRAE Trans. Vol. 77, Part 11, 1971, p.

117-126.

3. Carslaw & Jaeger, Conduction of Heat in Solids, Oxford University

WEATHER SlDE CHAMBER

ROOM SIDE CHAMBER

CALORIMETER BOX

TEST SPECIMEN

-

(2.44 m X 2.44 m)

CONVECTIVE HEATERS

POLYSTYRENE TEST FRAME

-

INSIDE AIR TEMPERATURE THERMOCOUPLES

ITh)

@ BAFFLE TEMPERATURE THERMOCWPLES q b )

@ OUTSIDE AIR TEMPERATURE THERMOCOUPLES Q)

-

+ INDICATES AIR FLOW

FIGURE 1

A Procedure f o r Deriving Thermal T r a n s f e r Functions f o r Walls from Hot-Box T e s t R e s u l t s : P a r t I1

BY

D.G. Stephenson,

K.

Ouyang, W.C. Brown

P a r t I of t h i s paper p r e s e n t s a procedure f o r d e t e r m i n i n g t h e t i m e - c o n s t a n t s (and a s s o c i a t e d r e s i d u e s ) f o r a w a l l from t h e r e s u l t s of a hot-box w a l l test. This second p a r t p r e s e n t s a procedure f o r o b t a i n i n g t h e f r e q u e n c y r e s p o n s e of a w a l l from t h e r e s u l t s of a second s e t of hot-box t e s t s . Both t h e time-constants and t h e f r e q u e n c y response a r e r e q u i r e d when c a l c u l a t i n g t h e c o e f f i c i e n t s of t h e z - t r a n s f e r f u n c t i o n s .

This second p a r t of t h e paper a l s o p r e s e n t s a procedure f o r t h e c a l i b r a t i o n , o r o b t a i n i n g t h e t r a n s f e r f u n c t i o n s , f o r t h e hot-box.

Determining t h e Frequency Response of a Wall

A s e c t i o n of a w a l l i s i n s t a l l e d i n a hot-box w a l l t e s t i n g a p p a r a t u s _{- . -} j u s t

as

i t would be f o r a U-value t e s t . The arrangement is shown

D s c h e m a t i c a l l y i n f i g u r e 1. To d e t e r m i n e t h e f r e q u e n c y response of t h e

r

t r a n s f e r f u n c t i o n t h e t e m p e r a t u r e i n t h e c l i m a t e chamber is kept c o n s t a n t w h i l e t h e power t o t h e m e t e r i n g box i s v a r i e d s i n u s o i d a l l y a t v a r i o u s

f r e q u e n c i e s . The r e s u l t i n g v a r i a t i o n i n t h e t e m p e r a t u r e i n t h e m e r i n g box

i s

measured a t r e g u l a r i n t e r v a l s , and from t h e s e d a t a a v a l u e of _spi2rrf

7

$1

can be d e r i v d f o r t h e p a r t i c u l a r f r e q u e n c y , f , used f o r t h e t e s t . The

r

t r a n s f e r f u n c t i o n can a l s o be o b t a i n e d from c y c l i c t e s t r e s u l t s . I n t h i s c a s e t h e c l i m a t e chamber t e m p e r a t u r e i s v a r i e d s i n u s o i d a l l y and r h e power t o t h e m e t e r i n g box i s k e p t c o n s t a n t .

The m e t e r i n g box i s assumed t o be a l i n e a r system with t h r e e

t r a n s f e r f u n c t i o n s E, I and J. The E f u n c t i o n a l l o w s f o r t h e h e a t c a p a c i t y and t r a n s p o r t l a g of t h e h e a t i n g system, and I and J account f o r t h e h e a t c a p a c i t y of t h e a i r and b a f f l e i n t h e m e t e r i n g box and f o r t h e h e a t t h a t f l o w s i n t o ( o r o u t o f ) t h e s h e l l of t h e m e t e r i n g box, t h a t

is

@ = Laplace t r a n s f o r m of t h e power, P, s u p p l i e d t o t h e metering box

A = a r e a of t h e t e s t w a l l

Qw = Laplace t r a n s f o r m of t h e h e a t f l u x i n t o t h e t e s t w a l l from t h e m e t e r i n g

box, r e p r e s e n t e d by:

D 1

ow

=

{,J.ol

-

y'02} ( 2 )

Q1 = Laplace t r a n s f o r m of Th, t h e t e m p e r a t u r e i n t h e metering box Q2 = L a p l a c e t r a n s f o r m of Tc, t h e t e m p e r a t u r e i n t h e c l i m a t e chamber

0 = Laplace t r a n s f o r m of T t h e t e m p e r a t u r e i n t h e guard space s u r r o u n d i n g t h e m e t e r i n g chamber g'

I f T is c o n s t a n t it can be s u b t r a c t e d from Th and Tc, and t h e n J - 0

g

can be dropped from e q u a t i o n 1 (i.e., T i s t h e r e f e r e n c e o r z e r o l e v e l f o r g t h e t e m p e r a t u r e s ) . Thus, from ~ q u a t i o n g 1 and 2

This is t h e b a s i c r e l a t i o n t h a t can be used t o determine t h e value of

-

1 E s-iw

and when and a r e known f o r s - i w ( u = 2nf where f h t h e

s - i w A

I

D f r e q u e n c y ) . Conversely when a w a l l i s t e s t e d t h a t has known v a l u e s of and

1 E

- t h e r e s u l t s c a n be used t o determine

_B

and

2

*

T h i s is t h e way t h e m e t e r i n g box

i s

c a l i b r a t e d .

D e t e r m i n a t i o n of

chamber t e m p e r a t u r e i s kept a t a c o n s t a n t v a l u e , Tc, and t h e power t o t h e metering-box i s v a r i e d s i n u s o i d a l l y a t a n a n g u l a r v e l o c i t y w, t h a t i s T h i s c y c l i c a l v a r i a t i o n of t h e power produces a c o r r e s p o n d i n g c y c l i c a l v a r i a t i o n of Th Th =

Th

+

Th

e i ( w t + Y) Thus w i t h Tc c o n s t a n t

The v a l u e s of t h e r e a l and imaginary p a r t s i n t h i s complex

Example I

Data f o r t h e m e t e r i n g box o b t a i n e d from c a l i b r a t i o n f o r

e x p r e s s i o n depend on t h e v a l u e of w. M e n

i

as-iw

and

f

(

a r e known

s=i

w from a c a l i b r a t i o n of t h e h o t box ( a s e x p l a i n e l a t e r ) , t h e v a l u e of D

-

_{can be o b t a i n e d from t h e r e s u l t s of a c y c l i c a l t e s t u s i n g} s=iw e q u a t i o n 6.

Test conditions and measured quantities:

Derived quantities :

I

Theref o r e

D e t e r m i n a t i o n of

For t h i s t e s t t h e c l i m a t e chamber t e m p e r a t u r e , Tc, i s v a r i e d

s i n u s o i d a l l y and t h e power t o t h e m e t e r i n g chamber i s k e p t c o n s t a n t . T h i s c a u s e s a s i n u s o i d a l v a r i a t i o n of Th.

I n t h i s c a s e

and from Equation 3:

Example I1

T e s t c o n d i t i o n s and measured q u a n t i t i e s :

Derived q u a n t i t i e s : From t h e p r e v i o u s t e s t

0.22 1-211.1

Thus

_B

_{= 7 - 8 7}

k)

_{( )}

-I

These v a l u e s f o r t h e r e a l and imaginary p a r t s of

"

Is=iw

can be compared w i t h t h e v a l u e s d e r i v e d from t h e t i m e - c o n s t a n t s an r e s i d u e s . It should be n o t e d , however, t h a t t h e v a l u e s d e r i v e d from t h e r e s u l t s of t h e c y c l i c a l t e s t s may have a r e l a t i v e l y l a r g e u n c e r t a i n t y because t h e a m p l i t u d e

?

can be q u i t e s m a l l . The v a l u e of

-

h

,

on t h e o t h e r hand, w i l l have a

s - i W

I

...

much s m a l l e r probable e r r o r because of a l a r g e r magnitude of T,,.

C a l i b r a t i o n of t h e

meter in^

Box The t r a n s f e r f u n c t i o n s

and f o r t h e m e t e r i n g box can b e determined f o r v a r i o u s f r e q u e n c i e s i n t h e same way a s t h e f r e q u e n c y response of a w a l l i s determined. But f o r t h i s c a l i b r a t i o n t h e t e s t w a l l must be one whose t r a n s f e r f u n c t i o n s a r e known. This i s b e s t obtained with a w a l l t h a t i s made of a homogeneous m a t e r i a l whose thermal p r o p e r t i e s a r e known. I n t h i s c a s e t h e t r a n s f e r f u n c t i o n s

-

and 1

1

can be

*

I

s - i w =i w

d e r i v e d from t h e dimensions and t h e r m a l p r o p e r t y v a l u e s . Thus f o r a c a l i b r a t i o n t e s t

-

and

I

a r e t h e unknowns. A 6 ' i W

*

I

s = i

W I n t h i s c a s e t h e f i r s t t e s t has P h e l d c o n s t a n t w h i l e Tp i s v a r i e d 1 s i n u s o i d a l l y , i.e. t h e same t e s t c o n d i t i o n s a s f o r jj.. E q u a t i o n 7 can be r e a r r a n g e d t o

D

I

1

Cz

+

z)

=

'z)

(

1u!

) ( 8 )

D 1 Yh

IY

A s

-

_B

and a r e known f o r t h e w a l l t h a t

-

is used f o r t h e c a l i b r a t i o n ,

L

e q u a t i o n 8 g i v e s t h e v a l u e f o r

2

For t h i s c a s e a 100 mm expanded p o l y s t y r e n e w a l l was used and t h e

I

E

v a l u e s of

-

_A and - w e r e found t o be _A

and : = 0 . 1 6 8 m

D

I

Then having determined t h e v a l u e of

5

+

x,

i t can be used i n

E

WEATHER SlDE CHAMBER

ROOM SlDE CHAMBER

CALORIMETER BOX TEST SPECIMEN

CONVECTIVE HEATERS

INSIDE AIR TEMPERATURE THERMOCOUPLES (T, )

@ BAFFLE TEMPERATURE THERMOCOUPLES (Th)

-

@ OUTSIDE AIR TEMPERATURE THERMOCOUPLES (T,)

INDICATES AIR FLOW

FIGURE 1