NBSIR 81-2353

Documentation of Program for

Determination of Conduction

Transfer Functions

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

November 1 981

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 81-2353

DOCUMENTATION OF PROGRAM FOR

DETERMINATION OF CONDUCTION

TRANSFER FUNCTIONS

B. A. Peavy

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

November 1 981

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

DOCUMENTATION OF PROGRAM FOR DETERMINATION OF HEAT

CONDUCTION TRANSFER FUNCTIONS

by

B.A. Peavy

ABSTRACT

Conduction transfer functions are used to predict the time-dependent onedimensional conduction heat transfer at surfaces of single- or multi-layer

building constructions based on heat flux and temperature history at each

surface. By the use of conduction transfer functions, heat transfer problems

employing non-linear boundary conditions such as thermal radiation and timedependent changes in the surface film resistances can be solved.

Because conduction transfer functions are analtyically derived with an initial

time condition of zero temperature potential throughout the solid materials, it

becomes necessary to initialize the computation by including exposure to a number of outdoor weather cycles such that satisfactory initial conditions of

temperature and heat flux exist at the inside and outside surfaces.

The program is set up for the use of 1-, 2-, or 3-hour time intervals, depending

on the thickness of the building construction. The program allows for the

combination of two building constructions, e.g., the parallel heat flow paths

found in wood-frame walls.

Keywords: conduction heat transfer; conduction transfer functions;

initialization of heat transfer problem; parallel heat flow;

response factors; thick building construction.

iii
TABLE OF CONTENTS

Page

ABSTRACT ill

1. INTRODUCTION 1 2. RESPONSE FACTOR ANALYSIS 3

3. CONDUCTION TRANSFER FUNCTIONS 14

4. INITIALIZATION OF SURFACE TEMPERATURES AND HEAT FLUXES 16

5. CONCLUSIONS 18

6. DESCRIPTION OF SUBROUTINES FOR DETERMINING CONDUCTION TRANSFER

FUNCTIONS 19

6.1 SUBROUTINE PC 19

6.2 SUBROUTINE ABC 22

6.3 SUBROUTINE ROOTS 24

7. REFERENCES 28

APPENDIX 29

LIST OF TABLES

Table 1. Definitions for Jm , Lm and A^ m 6

Table 2. Number of Cycles Needed to Obtain Initialization 17

LIST OF FIGURES

Figure 1. Cross section through an n-layer building construction 4

Figure 2. Heat flux necessary to attain triangular temperature

pulse at surface of a solid 9

Figure 3. Set of overlapping triangular temperature pulses 13

iv
1 . INTRODUCTION

To evaluate the dynamic performance of a building, it is necessary to include

calculations for the conduction heat transfer through all the components of the

building envelope.

Conduction transfer functions are used to predict the time-dependent one- dimensional conduction heat transfer at surfaces of single or multi-layer building constructions based on the heat flux temperature history at each surface.

For example, the heat flux values at time t and at the outside and inside

surfaces of a particular construction are [1]:

k 1 Qo,t = E SjjQo^-m + £ (Yn) k;Ti $ t-n+1 ~ ^njk^o , t-n+l) (1)

m=l n=l

Qi.t

k 1

E smQi,t-m + 1 ( xn,kT i, t-n+1 " Yn,kTo , t-n+1)

m=l n=l

( 2 )

where Xn |c ,Yn and Z n ^

are k-th order conduction transfer functions, Sm

are coefficients of past heat flux history, and T-^ and T0 are the inside and outside temperatures for the present and past time intervals. If surface film

resistances are defined as constants at the surfaces, T^ and TQ are temperatures of the ambient air near the respective surfaces. If the surface film

resistances are set equal to zero, Tj[ and T0 are surface temperatures, and

it is therefore possible by use of appropriate heat balances at the surfaces

to employ non-linear boundary conditions, such as thermal radiation and timedependent changes in the surface film resistances.

Conduction transfer functions of zeroth order are thermal response factors and

are calculated by the program subroutine PC (see appendix) using the methodology

outlined in reference [2]. Conduction transfer functions of higher order are determined in the same subroutine using the methodology outlined in reference

[1]. The required thermal and physical properties of each building construction are input as layer-by-layer data by the user in a format as shown by the

comment card listing for subroutine PC. A maximum of seven layers is possible

excluding the outside and inside surface film thermal resistances when used.

Necessary properties for each layer are thickness, thermal conductivity, density, specific heat, and air space or surface film thermal resistance when

applicable. A set of conduction transfer functions is then generated.

The output for computation purposes is for each building construction,

1 < IR < 2°

a) conduction transfer functions

X(IR,N) , Y ( IR , N ) , Z(IR,N) 1 < N < 20

b) coefficients of past heat flux history

S ( IR,M) 0 < M £ 5

c) defined building construction conductance U(IR)

1
d) integers: NTR(IR,1)=L, NTR(IR,2)=K, NTR(IR,3)=J

L is the number of conduction transfer functions

K is the number of heat flux coefficients, and also the order of

the conduction transfer functions

J is the time intervals, 1, 2, or 3 hours.

The program subroutine PC allows for the combination of two building

constructions, such as parallel heat flow paths as found in wood-frame walls

with both cavity and wood stud construction. Both constructions are entered

with the lighter (smaller weight per unit are) construction entered first and

the percent area of both inserted as shown in the examples.

For thick building constructions (usually thicknesses greater than 3 feet), a 1-hour time interval is not a sufficiently large value for allowing the effects

of the temperature history on one surface to be transmitted to the heat flux

on the other surface. For this reason time intervals of 2 and 3 hours are used for the thicker constructions. Algorithms for the use of 1- , 2-, and

3-hour time intervals in the determination of heat flux quantities are contained

in subroutines PI, PR an PQ. These subroutines are used to initialize the

surface temperatures and heat fluxes for the various constructions. Subroutine

PQ performs the summation on known past surface temperatures and heat fluxes,

namely

k 1 ^o , t

^ SmQ0jt _m + £ (^n,k^i , t-n+1 “ ^n,k^o , t-n+1) (3)

m=l n=2

W

k 1 ^ SmQi,t-m + ^ '^n,k^i , t-n+1 ” ^n,k^o , t-n+1

)

m=l n=2

(4)

Also, it is assumed for each time interval:

Qo,t - Ho[G(t)-T0it J (5)

Qi,t “ Hj.[Ti >t-Ta .] (6)

where G(t) is temperature variation of the outdoor temperature and Ta is the

indoor temperature. Setting (5) equal to (1), and (6) equal to (2), two simultaneous equations are derived from which values for the surface temperatures

T0}t and T^ jt can be found, and the heat fluxes are computed from (1) and (2), or (5) and (6). For each diurnal (24-hour) cycle and each building construction, the sums of the outside and inside heat fluxes and the surface temperatures are printed. When the sums of the heat fluxes become closely equal to

each other and change very little from the previous cycles, it can be assumed

that the initialization process has been completed for a particular building

construction.

2
2. RESPONSE FACTOR ANALYSIS

For one-dimensional heat flow in an individual layer of one ormore parallel

layers of a building construction (see figure 1), the partial differential

equation for conduction heat flow is given by

3x2

m _ JL

9v_m

3t

( 7 )

where vm is the temperature potential with respect to a datum plane temperature,

x is a dimension along which heat is flowing, is the thermal diffusivity of

the layer material, and t is the time. For continuity of temperature and heat

flow at the interface of the layers, perfect contact is assumed, so that at

x=bm the following conditions apply

vm-l

Vi

'm

dv.m-1

dx

dvm

dx

( 8 )

(9)

where K is the thermal conductivity of the respective layer material. At the

exposed surfaces x=o and x=bn , the heat fluxes are assumed proportional to the

temperature difference between the fluid (air, gas or liquid) and the surfaces,

and are represented by the temperature relationships

dv-,

-R

oKi s

1 - at x = 0 ( 10 )

and

"Vn

dv

dx

n = v - g(t) at x = bn (ID

where R0 and R^ are the surface film resistances, and f(t) and g(t) are the

fluid temperatures as a function of time. When RQ and/or R^ is zero, the

time temperature function(s) f(t) and/or g(t) represent temperature(s) at the

respective surface(s). Applying the Laplace transform to (7), a solution for

the transform of the temperature becomes

m

- ^

q(x-b) , nm s nr + B. -qnm (x-bm ) nr nr m ( 12 )

2 where p is the Laplace parameter, qm

= p/a , and A^ and B

ffl are constants to

be determined from conditions at the two surfaces of an individual layer.

Applying the Laplace transform to (8), (9), (10), an (11), expressions for the

transforms of the temperatures at the exposed surfaces of layer 1 and layer n (figure 1) are given by

vj. =• [P 1+Q 1 + V 2 /p(S]_+Tj_)] + ~ — [sinh x /p/aj_ +V]i /p cosh x /p/a^] (13)

3
1

Figure 1. Cross section through an n-layer building construction

4

x=o

x=bi

x=b

2

x=b

n

-2

x=b

n

-i
Vn

Gf (p)

W

+

g(p)

W

[sinh (bn

-x) / p/an + V 2 /p cosh

[P 2 + Q 2 + V 1 /p(S 2 -T2 )]

(b n

-x) /p/an ]

where

( 14 )

Pi = I Jmsinh[(Nm-x//ai> /p]

5 1 = S J^oshCCN^x/Za!) / p]

P 2 = E Jmsinh[(Nm-(bn-x)//an )/p]

5 2 = S Jmcosh[(Nm

-(bn-x)//an ) /p]

P = E Jmsinh Nm /

p

S = E Jm

cosh Nm / p

(The above summations are over m=l to 2 n_^,

Ql = E Lmsinh[(Em+x//ai) / p]

Ti = E Lm

cosh[(Em+x/a][) /p]

Q2 = E Lmsinh[(Em

-(bn-x)//an ) /p]

T2 = E Lmcosh[(Em-(bn-x)//an ) /p]

Q

= E Lm

sinh Em /p

T = E Lm

cosh Em /p

n being the number of layers.)

am

Knri-1 ^ am/am+l

G = 2 n

" 1 /(l+o 1 )(l+a2 )...(l+an

_1)

( l~om ) / ( 1+Oni)

Kn

H = G— /a^/

a

n

V 1 = RoKl/^ai v2 = RiKn/^an

n Nm - E A^ m( ki“ki-l) /^ a i , Em = Nm ~ 2b]_//a]_, A^ m = 1 1=1

(Jm , Lm and are defined in table 1.)

W = P + Q + V 1V2 p(P-Q) + /p [V 2 (S+T) + V 1 (S-T)]

The transforms of the heat flux at x=0 and at x=b n are found by differentiating

(13) and (14) with respect to x and multiplying by minus one and the thermal

conductivity of layer 1 and n, respectively:

Fi = KL ^p

W /a.

([S-T + V 2 /p (P-Q)] f(p) - Hg ( p )

)

(15)

Kn _

F„ = — t Gf (P> - 1 S+T + Vi /p (P-Q)] g(p)}

W /a,

(16)

5
m

12

3

4

56

78

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

Table 1. Definitions

km A2,m

for Jm ,

A3 ,m

Ljjj and

A4 ,m

Ai ,m

A5 ,m A6 ,m A7,m

1 k l

1 1 1 1 1 1 kik2 k2 -1 1 1 1 1 1 kl k3 k3 -1 -1 1 1 1 1 k2 k3 kl k2 k3

1 -1 1 1 1 1 kl k4 k4 -1 -1 -1 1 1 1 k2 k4 kl k2 k4

1 -1 -1 1 1 1 k3 k4 kl k3 k4

1 1 -1 1 1 1 kl k2 k3 k4 k2 k3 k4 -1 1 -1 1 1 1 k4 k5 kl k4k5

1 1 1 -1 1 1 k3 k5 kl k3 k5

1 1 -1 -1 1 1 k2 k5 kl k2 k5

1 -1 -1 -1 1 1 kl k5 k5 -1 -1 -1 -1 1 1 kl k2 k3k5 k2 k3 k5 -1 1 -1 -1 1 1 klk2 k4k5 k2 k4 k5 -1 1 1 -1 1 1 kxk3k4k5 k3 k4 k5 -1 -1 1 -1 1 1 k2 k3 k4k5 kl k2 k3 k4k5

1 -1 1 -1 1 1 k5 k6 kl k5 k6

1 1 1 1 -1 1 k4 k6 kl k4 k6

1 1 1 -1 -1 1 k3 k6 kl k3 k6

1 1 -1 -1 -1 1 k2 k6 kl k2 k6

1 -1 -1 -1 -1 1 kl k6 k6 -1 -1 -1 -1 -1 1 kl k2 k3 k6 k2 k3 k6

-1 1 -1 -1 -1 1 k1 k2 k4 k6 k2 k4 k6 -1 1 1 -1 -1 1 kl k2 k5 k6 k2 k5 k6 -1 1 1 1 -1 1 k!k3 k4k6 k3 k4 k6

-1 -1 1 -1 -1 1 kl k3 k5 k6 k3 k5 k6

-1 -1 1 1 -1 1 klk4k5 ke k4k5 k6 -1 -1 -1 1 -1 1 k2 k3 k4 k6 kl k2 k3 k4k6

1 -1 1 -1 -1 1 k2 k3 k5 k6 kl k2 k3 k5 k6

1 -1 1 1 -1 1 k2 k4 k5 k6 kxk2 k4k5 k6

1 -1 -1 1 -1 1 k3 k4 k5 k6 k1 k3 k4k5 k6

1 1 -1 1 -1 1 kl k2 k3 k4 k5 k6 k2 k3 k4 k5 k6

-1 1 -1 1 -1 1

6
The inversion of (15) and (16), as shown in reference [3], is performed by

evaluating the residues at the poles of the denominator W = 0, where p = -0 2 or /p = i 0 , which gives the relationship

W(3) = (l-V^S 2

) E Jm

sin Nm

6 + (l+V^g2

) E Lm

sin Em

0

"b 3 [ (V2+V 1 ) E Jm cos Nm8 + (^ 2 —Vj_) E L^cos E^E] = 0

and the differentiation of W with respect to p evaluated at p = -0 2 gives

U = (1-VjV2

B 2

) Z Jm

Nmcos NJ5 +(1+V1V

2

B 2

) Z L

fflEm

cos E

ffl B

+ (V 2+V!)[Z Jm

cos »,( -(I JmNmsin NmB]

+ (

V

2 ~V ^ )

[ E Lmcos Em3 - 3 E Lm

Emsin E m3]

- 23V1V2

[E J msin Nm3 - E Lm

sin Em0]

= U

(17)

dW

where 210 (t~) 0 d P p=-6 2

The residues at the poles p = -0 2 are

2Ki B? -

'13

2 — E [f(-3f)(D1 -V 2D

2 3) ~ Hg(-0?)] e'

6it

and

2K 3: n0

2 —n " i — o — 0 -3 -i t

E — [Gf(-3|) " (D3 -ViD2 3) gC-3f )] e

1 /an Ui

where satisfy W( 8 ) = 0, and

D 2. — E (

J

mcos Nm8 -Lmcos Euj8 ), B 3 — E (JjjjCOS Njjj8 "KLjjjCOS Ejq3),

D 2 = E(Jm

sin Nm3 -Lmsin E m0).

(18)

(19)

( 20 )

Of particular concern is the evaluation of the first root of (17). This can be

done expeditiously by expanding the sines and cosines in their series and considering only the first two terms, in order to obtain an initial estimate of the

first root

W

B * A

l 3-A2 3 3 (21)

or 3 2

2 « A-|yA2 with

Al = 2 (Jm

Nm + LmEm) + (V]+V 2 ) E Jm + (V 2 -V X ) E Lm

7
and

Next, it is expedient to define the temperature functions f(t) and g(t). First,

these functions are defined as triangular temperature pulse functions (see

figure 2), and solutions obtained for the heat flux at subsequent time intervals

may, by superposition, be used as shown in equation (1). The time function is

defined by:

which when substituted for f(p) and g(p) in (15) and (16) gives double poles at

p=0 (reference 3, p. 78). The following expressions are limits of the necessary

functions of (15) and (16) for evaluating the residues at the double poles.

f(t) = 0 = t/6 = 2-t/6

= 0

f(p) = 0 t < 0 ( 22 ) = 1/Sp2 = (l-2e-P 5 )/6p 2 = (1-e-pG ) 2 /6p2

0 < t < 6 6 < t < 26

t > 26

Lim w_ = Al + a2 p

p+o v p

Lim (S - T) + V

2/p(P-Q) = Bj^ + B2 p

p+o

Lim (S + T) + V-,/p(P-Q) = C

± + C

2 p.

and A2 are defined following (21) and

Bi E ( Jjjj-Ljjj) , Cj E (Jni ^m)

For the first term in (15), the residue for 0<t<5 is

( 23 )

8
1VIJ.N3J.0d 3dniVd3dW31

^-CMOGOCO^-CMOCNI^-CO O CD CD cb CD CD CD

wnwixvw 01 ouvd - xnu ivbh sovjdns

Figure 2. Heat flux necessary to attain triangular temperature pulse

at surface of a solid.

9

TIME
for the last term in (16), the residue is

z t "_ ^ ( tC l

, A1C2-A2 C

1

) ( 24 )

and for the first term in (16) and the last term in (15), the residue is

Yt- -_ Kn

G

/t - _f)

.2

1

6 /a "1 A' (25)

where

K1B

1 K

n

C

l

^l/a, Al/a~

Kn

G

= _1

^l/oT ^

n

and R is the total thermal resistance of the n-layers plus R3 and R2. The

response factors X]_, and Z\ from (23), (24), (25), (19) and (20) evaluated

at t=<5, become

_ K1 x 1>0 = X 6 Z (D X - V 2D2 6i)

^

Yl,0 = Y 6

Kn

G

Z (D 3 ) ^

z l,0 = z 6 —j=- z ( d3 " v lD2 B i) n where

2 e_e \

s

4»i

For X2j Q> Y 2> q and z 2,0 evaluated at t=26,

K

/

a

Vs

2

1 ^1 -6-t6

X

2,0 -"i

- Y 6 E (®1”V2D2

3

i

) 4’ i (e - 2)

Y2,0 "1 ‘ Y

6 -

2

z ^(e"Bi - 2) (27) n

10
z 2,0 - z 6 -

1

E (D 3 -V 1 D2

3 1 )^i (e -2),

and for t>2 <5 ,

(28)

/a.

K‘n

n Equations 7, m=l,2,...n, are linear partial differential equations which, for

the continuity conditions (8) and (9), and the boundary conditions (10), (11),

and (22), give solutions for the heat flux at discrete time, t=j6, by equations

26, 27, and 28 in a generalized form (refer to equation 7):

where the temperature amplitudes of the triangular pulses in equations 22,

initiated at time equal zero, are and Tn

at the exposed surfaces. Typical

response to a unit triangular pulse at an exposed surface is shown in figure 2.

For a continuous set of triangular temperature pulses of aribitrary amplitudes

and placed at a time interval t=5 apart, the heat flux at a time t=T becomes a linear combination of present and past temperature pulses and the respective

solutions from (26), (27) and (28). Using this principle of superposition, the

heat flux relations become:

*lU«> = TlX jj0 - Tn

Y J>0

l’n(j^) = TiY j>0 ~ Y n

z j,o

(29)

(30)

L

F l,x 1 x j,o T l,T-j+l Y

j ,o Tn, t- j+1 (31)

L

F n,t = 1 Y j,o T l,T-j+l “ z j,oTn,T-j+l

3=1

(32)

11
This can best be illustrated by referring to figure 3, which shows a set of

overlapping isosceles triangles connected at their peaks by straight lines. The

rationale is to imagine that the effect of each separate temperature pulse in

past time will have some contribution to the heat flux at the present time.

The nature of the resultant temperature distribution is assumed linear between

each pulse. This seems intuitively obvious when one observes that the area of

overlap is identically equal to the open area under the straight line drawn

between two adjacent pulses. A verification of response factors was performed

in reference 2 where f ( f ) and g(t) in (17) are defined by trigonometric series

and the results were compared to confirm the linear variation between

temperature pulses.

12
IVIlNBiOd 3aniVd3dW31

Figure 3. Set of overlapping triangular temperature pulses.

13

t- 56

t4(5

r- 3

<5

t-28

TIME
3. CONDUCTION TRANSFER FUNCTIONS

For some types of construction, the number of terms in equations 31 and 32

needed to obtain accurate results is inordinately large for computational purposes. For this reason, it is necessary to reduce the number of terms. One

way to do this is to define a response factor or a zeroth order conduction

transfer function for j greater than three as follows:

W. = P, e

1

2 -jM + P

2 e

2 “ j&2 6 + P

3 e

2 -3336

from (28) such that Si < & 1+1

* To eliminate the first term, perform the

operation

-S 2 5 w. -1 = w. - e 1 w. - 3,1 3,o 3-1,0

, 2 2 . = P

2 e

0 A

[ 1-e 21 ] +

2 -jS 2

<S

Continuing one step further

W

d,2

= P

3,1

" 6

2 “j33 6

2 -3 2

6 W3-1,1

2 2 2 2 (S3-8 2 )<5 (S3-Si)6

!-e J z

] [ 1-e

J 1

] +

This process can be continued to give Wj, 3 , Wj

} 4 , etc. From the above, it can

be seen that the modified response factors become smaller and smaller in magnitude to a point where they can be ignored for computational purposes. Performing the operation Fi^-RiF^t-i, (Fl, t ”RlF l,t-l) ~ R2 ( F1 , t-l“Rl> F l, t-2 > etc *» (31) and (32) can be transformed to the following form

k

E m=l

k

F l,t - Z (-'f'V.t-. + Z ( X j,k tJ

+

1

j = l

? n,t

= S (-l)m+F

SmFn ^ t _m + l (Yj>k T 1;t.j+1 m=l 3=1

Y j,k T o,t-j+l )

Z j,k To,t-j+lwhere

Rm "

2 “3m5

and

S x = Ri+R2 +R3+R4+R5

S 2 = Ri(R2+R3+R4+R5 ) + R2 (R3 +R4+R5 ) + R3 (R4+R5 ) + R4R5

(33)

(34)

14
s3 = R1R2( R3+R4+R5) + R1R3( R4+R5) + R4 R5( Rl+R2+R3> + R2 R3( R4+R5)

S 4 = RiR2 (R3 R4+R3 R5+R4 R5 ) + R3R4R5(Ri+R2)

s 5 = RlR2R3 R4R5

In equation 33, for the case when the temperature T-^ is constant and T0 is zero

for all times, the heat fluxes are equal and constant due to the steady-state

condition, then the conductance of the construction is solved for by the

relationship

C

Ej=l

1 +

X

k

E m=l

(-1

)

m

Sm

(35)

Comparison of the thermal conductance of a construction with values computed

from (35) is needed as a check on the accuracy of the computed conduction

transfer functions.

15
4. INITIALIZATION OF SURFACE TEMPERATURES AND HEAT FLUXES

Conduction transfer functions are analytically derived with an initial time

condition of zero temperature potential throughout the solid(s). This is not

a realistic initial condition for the use of conduction transfer functions to an existing building that has been exposed to outdoor weather cycles before the

need to determine heat flow through the building construction. For this reason

it becomes necessary for the building construction to be exposed to a number of

outdoor weather cycles such that satisfactory initial conditions of temperature

and heat flux history exist at the inside and outside surfaces before subjecting

the building construction to the actual outdoor weather variation.

Presently the initialization is performed on an arbitrary basis and is seriously

in need of quantification, especially in regard to very thick or heavy-weight

constructions. For lightweight constructions such as wood frame walls, only

two or three diurnal weather cycles are necessary for initialization. But with

an increase in thickness or weight per unit surface area, the necessary number

of diurnal cycles can be considerably increased.

There are many ways by which initialization may be accomplished, but one way to

substantiate that initialization has been completed is to use repeated identical

diurnal weather cycles. Here, for a diurnal cycle, the average heat flux at the

inside and outside surfaces will be approximately equal after exposure to a sufficient number of cycles.

An example of initialization is given in table 2, which gives the ratio of the

average heat flux at both the outside and inside surface for a diurnal cycle to

the average cyclic heat flux which would occur under steady periodic conditions.

The tabular values were computed for 1/2, 1, 2, and 3 feet of heavyweight con- crete (120 lbs/ft^) and indicate that as the thickness increases, the number of

cycles necessary to achieve initialization also increases.

Initialization is accomplished in subroutine PI, which in turn picks up the

outdoor and indoor temperature variations and the inside and outside surface

coefficient of heat transfer for a 24-hour period from a calling program. This

subroutine calls subroutine PQ to determine W0 t and W-^ t (3) and (4), to be

used in the simultaneous solution involving ( 1 5 , (5), (2) and (6). Solution

for the present surface temperature is found in subroutine PR as well as the

solution for Q 0 t Qi t (1) an<^ (2). This subroutine also updates the temperatures and heat fluxes. For every 24-hour period, the subroutine PI sums the

heat flux and temperatures at the outside and inside surfaces for the separate

building constructions, and prints out the values.

The subroutine PI was developed to illustrate initialization and is not

necessarily recommended for use in working computer programs. This is because

there are so many applications for conduction transfer functions as to types of

heat balance involving the surface temperature that it would be impossible to

devise a subroutine to be all-encompassing. The subroutines PI, PR and PQ are mainly to give as an example the use of conduction transfer functions in

conduction heat transfer problems.

16
12

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Table 2. Number of Cycles Needed to Obtain Initialization

Value of Ratio = average heat flux for cycle

average heat flux steady periodic cycle

at outside and inside surface of heavyweight concrete

of various thicknesses (K=l Btu h“-*-f t”^F- I

, p = 120 lbft -

^,

C=0 . 2 Btu lb_1F_1

)

Thickness of concrete, ft.

1/2 1

outside inside

.769 .368 1.062 .218

1.095 .950 1.297 .820

1.000 1.000 1.001 .993

1.000 1.000

2 3

1.625 .049 2.179 .008

1.902 .445 2.554 .167

1.218 .820 1.602 .491

1.064 .947 1.305 .730

1.019 .985 1.159 .858

1.005 .996 1.083 .925

1.002 .999 1.044 .960

1.000 1.000 1.024 .978

1.014 .987

1.008 .992

1.005 .995

1.004 .996

1.004 .997

1.003 .997

1.003 .997

17
5. CONCLUSIONS

Conduction transfer functions are closed-form analytical solutions for

one-dimensional conduction heat flow in single or multi-layer building con- structions based on a knowledge of temperature and heat flux history occuring

at constant time intervals. Values of heat flux obtained by this method are based on the assumption that the temperature varied linearly over the time

interval (ref. 2). The use of the principle of superposition or use of a set

of overlapping triangular temperature pulses is possible for the solutions to

linear, homogeneous, partial differential equations (ref. 4), when the solution

becomes a linear combination of the separate solutions.

Conduction transfer functions of kth-order are generated from the zeroth order

by the following recursion formula:

W

n,k Wn,k-1 - e

2 -M w.n-l,k-l

where 3^ is the kth roof of (17), n are integers greater than one, and values

for k-l=0 are defined by (26), (27), and (28). The heat fluxes. at the inside

and outside surfaces are then given by (33) and (34).

The output of subroutine PC provides conduction transfer functions and

coefficients of past heat flux history by which the heat flux (eqs. 33 and 34)

may be evaluated. The routine needs two additional subroutines, ABC and ROOTS.

Particular features of the program subroutine are:

1. When an enclosed air space occurs within a building construction, the

thermal diffusivity is assumed to be 0.75 ft^/h. If the thickness is not

defined for the air space, it is assigned a value 0.08333 foot (1 inch).

The thermal conductivity is then defined as the thickness divided by the

thermal resistance.

2. The program allows for the combination of two building constructions that

would act as parallel heat flow paths in a wall, roof or floor. An example

would be wood-frame walls with the cavity and wood studs as two separate

constructions through which heat must pass. The resulting conduction

transfer functions are linear combinations of the ratio of area to the

total area and the conduction transfer functions for each construction.

3.

The present program is intended for the use of 1-, 2-, or 3-hour

time intervals. For thick building constructions (usually thicknesses

greater than 3 feet), a 1-hour time interval is not a sufficiently

large value for allowing the effects of temperature history on one surface

to be transmitted to the heat flux on the other surface. By allowing the

2- or 3-hour time interval, the effects are more noticeable, as evidenced in values for Yn Q , which transmits the temperature potential on one surface to the other surface.

18
6. DESCRIPTION OF SUBROUTINES FOR DETERMINING CONDUCTION TRANSFER FUNCTIONS

6.1 SUBROUTINE PC

SUBROUTINE PC(IR)

Description

This subroutine calculates the conduction transfer functions of order k, Xn

Yn k> z n kthat are necessary to determine the dynamic conduction heat transfer

through building constructions.

The approach taken by subroutine PC is presented in reference 2 for determining

response factors and in reference 1 for determining conduction transfer functions. Equations 26, 27, and 28 are closed-form solutions for heat flow at the

two exposed surfaces of single and multi-layer building constructions. Determination of the roots of equation 17 for multi-layer slabs presents the most

difficulty due to the unpredictable nature of the equation.

Subroutine Calling This Routine

MAIN PROGRAM

Subroutines Called by This Routine

SUBROUTINE ABC

SUBROUTINE ROOTS

Common Blocks

Variables Placed

in Common Blocks

/ CBA/

/CZ/

S,C

XX, YY, ZZ, R, NTR, U

Declarations

DIMENSION S(64) , B(7), C(6), E(6), L(8), K(8), SP(8), RE(8),

1 RM(8 , 60) , D(8) , AL(7 ) , Y(50), X(50), W(50), T(50), Z(5), A(20, 6), P(20, 6),

2 V(20 , 6)

DATA ST/1H*/

REAL K,L

19
Input

Source of Data Name Description

Card Images

Card Image

L, K, D, SP,

RE, RM

Card Image JA, IN, JB JA = total number of layers and may

include surface film resistances; if

JA=0 , RETURN

IN = time increment, may be 1, 2, or 3

hours

JB=0, no parallel heat flow paths

JB=1, initiates combination of this

construction with following

unformatted layer properties and

description

L = layer thickness

K = thermal conductivity

D = density

SP = specific heat

RE = air space or surface film

resistance

RM = description of layer

Read only if JB=1

JC = percent area in construction

1 already read in

JD = percent area in construction 2 to

be read following

Output for Computation (labelled common)

Name

*-U order conduction transfer functions for construction

JA, IN, JC, JD

XX(IR,L) X k k

th

n » order i IR=1,2

YY(IR,L) Yn,k eq. 33,34

ZZ(IR,L) Z n , k

eq. 34

R(IR,K) Jm heat flux CO'

U(IR) U building construction conductance

NTR NTR(IR, 1)

NTR(IR,2)

NTR(IR, 3)

L, number of conduction transfer functions

K, number of heat flux coefficients

J, time interval, 6

20
Calculation Procedure

1. Set IR=0, a counter for describing the number for a building construction.

2. For each building construction, the total number of layers, time interval,

and parallel heat flow indicators are read in followed by a separate card

image for each layer which includes on each card the thickness, thermal

conductivity, density, specific heat, thermal resistance, and the layer

description. If the first and last layer densities are each less than

. 001 , the thermal resistance(s) are defined for the inside and outside surface film resistances, and the number of layers is reduced by the number of

surface film resistances. If the resulting value for the number of layers

is greater than seven, the subroutine is exited by a return to the calling

routine.

3. The thermal diffusivity for each layer is calculated from the ratio of the

thermal conductivity to the product of the density and specific heat. For an air space, the thermal diffustivity is set equal to 0.75. If no air

space thickness is defined, the thickness is set equal to 0.08333 and the

thermal conductivity is set equal to the thickness divided by the air space

thermal resistance.

For each layer, the thickness divided by the square root of the thermal

diffusivity is defined (loop 20). Values for km

(from eq. 14) are defined

(loop 21). Values for Nm and Em

are defined (see definitions following

eq. 14 and table 1 by statements 1 to 7, according to the value J for the

number of layers).

4. The summation terms for evaluation of A]_, A2 , B]_, B2 , C]_, and C2 are determined in subroutine ABC (1=1) for evaluation of equations 23, 24, and

25, t= 6 . Values of A^ and A2 are used to give an initial estimate to the

first root of (17). Subroutine ROOTS determines the roots of (17) and the

individual terms in the series of (26) for Xi q, Y]_ q» anc* Z 1 0 which are defined in loop 50, as well as X2 q» y 2 0 * an<^ z 2 O*^^* 27).’ Values for

Xj 0 , Y j,0> anc* z i,0 ( j-3,4, . . . ,2d) are’defined in loop 56 (eq. 28). For

less* than 0 . 12 ’it has been shown that Y]_ 0

an<* Y2 0 are negligibly

small and that the values are sometimes the result of loss of significance

in their determination and are therefore set equal to zero.

5. If the initial estimate for 81 (eq. 21) is less than 0.15, the time interval

is set equal to 2. If the estimate is less than 0.10, the time interval is

set equal to 3.

6 . A criterion for the order of the conduction transfer functions is:

[ l X

20 ,k 1 + I Y 20 ,k I + I Z 20 ,k

] < 4 x 10"6

.

if the relationship is not satisfied, the next greater order is computed

(B s k £. 5) f r °m the recursion relationship.

21
W

n,k " W

n,k-1 e ^k

6 Wn-l,k-l

Once the criterion is satisfied, the coefficients of past heat flux history

are computed and the computed thermal conductanes (35) are compared to con- ductance computed from the input properties of the building construction.

If the denomination of (35) is less than 15^, either this time interval is

measured (but not greater than three hours) or the building construction

thickness is reduced by a factor of 0.95 and the conduction transfer

functions are recomputed.

7. The number of conduction transfer functions (1 < L < 20), to be used in

heat flux calculations is determined from the criterion:

+ l YL,kl + K.kM < - 7 * 10

-7

L,k I

8. When the parallel heat flow indicator, JB equals 1, the zeroth order

conduction transfer functions are computed for both building constructions.

The ratio of the areas of the constructions to the total area, JC and JD, are input, before the second building construction. A linear combination

of the respective areas and the conduction transfer functions gives a resultant conduction transfer function for the parallel heat flow path.

Computation then continues at step 6 above.

6.2 SUBROUTINE ABC

SUBROUTINE ABC(X,Z,J,I)

Description

This subroutine computes summation elements needed for (1=0) determining roots

of the characteristic equation 17, and values for (18). It also computes summation elements needed for (1=1) determining A]_, A2 , B^, B2 , C]_, and C 2 for

substitution in equations 21, 23, 24, and 25.

Subroutine Calling This Subroutine

PC

ROOTS

Subroutine Called by This Routine

None

Common Blocks

/ CBA/

Variables Obtained

from Common Blocks

S(64) , C(6)

Variables Placed

in Common Blocks

None

22
Declarations

DIMENSION

Input

Source of

Data

/ CBA/

/ CBA/

ROOTS

PC, ROOTS

PC, ROOTS

Output

Name

Z(I=0)

Z(I=1)

Z(l) , T(64) , V(64 , 4)

Name Description

S ( 64 ) Nm , m=l ,2 , ... ,32

Em , M+32=33 , 34 , . .

.

, 64

equations 13 and 14

C ( 6 ) k.£, i=l,2,...,6

equations 13 and 14

X Values of g for computing summation

elements of equations 17 and 18

J Number of layers in building

construction

I 1=0, gate for determining summation

elements for equations 17 and 18.

1=1, gate for determining summation

elements for equations 21, 23, 24

and 25.

Description

Z(l) = E Jm

sin Nmg Z(2) = 2 Jm cos Nmg Z ( 3 ) = Z Jm

Nmcos Nmg Z(4) = Z JmNm sin Nmg Z ( 5 ) = Z Lm sin Emg Z(6) = Z Lm cos Emg Z ( 7 ) = E LmEm cos Emg Z(8) = E LmEm sin Emg

Z(l)

Z(2)

Z(3)

Z(4)

Z(5)

Z(6)

Z(7)

Z(8)

'-'m ^m^m0 J N

z m m-i JN„ m m

V^mi

Lm

Ems

LmEm

23
Calculation Procedure

1. Set K=l, if 1=1, go to 20; if 1=0, refer to calculation procedure 3.

for K = 1 T(N)

for K = 2 T(N)

for K = 3 T(N)

for K = 4 T(N)

= 1 = S (N) = S(N) 2 = S(N) 3 N = 1 to 64

2. For each value of K with the assigned values for T(N), from statement 40,

Y = W = 0, an assigned GO TO statement based on the number of layers sends

control to the appropriate location to give the summations.

Y = E Jm T(N), W = E Ln T(N), J n and Ln

's are defined in table 1. When K

is greater than 4, a return is made to the calling routine.

3. for K = 1

K = 2

K = 3

K = 4

T(N) = sin M S (N)

T(N) = cos 8 * S (N)

T(N) = S (N) cos 6 * S(N)

T(N) = S(N) sin 8 * S(N)

Use calculation procedure 2 above.

6.3 SUBROUTINE ROOTS

SUBROUTINE ROOTS (AB, AC, AD, VI, V2

,

DEL, B, Z, P, R, J, M)

Description

This subroutine calculates the positive roots of equation 17. It also

computes equation 18 and common portions of the summation terms of the

response factors (see equations 26, 27, and 28). It is essential that all

the positive roots of (18) be computed up to a value of about 5.5. If any

roots are missed erroneous values for the response factors will result.

Any algorithm for finding the roots of (18) must be used with caution

because the nature of some building constructions will give unpredictable

behavior to the equation. Because equation 18 is the derivative of equation

17, the Newton-Raphson method for finding roots is used when in the vicinity

of the root.

Subroutine Calling This Routine

PC

Subroutine Called by This Routine

ABC

Common Blocks

None

24
Declarations

DIMENSIONS B(l), Z(l), P(l), R( 1) , Y(8)

Input

Source of

Data Name Description

PC AB k]//ai

PC AC ^n/^an

PC AD G, see eq. (14)

PC VI RlKi//ai, eq. (14)

PC V2 RLKn//an » eq* ( 14 )

PC DEL Time interval hr.

PC J Number of layers

PC Z(l) Approximation to first root

Output

Name Description

M Number of roots computed

B Roots

,

g, (M values)

Z Kl

^i

/ ai

(Di~V2D23i) , eq. 25 (M values)

P

KnG

Vi (D3 -V]D2 Bi) (M values)

R

Kn

/ot n ^-VjE^gi) (M values)

25
Calculation Procedure

1.

Initialize

1=0, gate for use in subroutine ABC

M=l, count on number of roots computed

K=0, iteration counter

U=0.1, tentative increment

V=Vi*V 2 , see W(3), eq. 17

G=V 2_V^, see W(3), eq. 17

H=V 2 “V^, see W(3), eq. 17

A=3= .005

D=W( .005)

X=Z(1), initial approximation to first root, eq. 21

E=.15X, initial increment used for finding first root.

2. Root finding loop begins with statement 10. Call subroutine ABC to

calculate A=W(3) and Q=W'(3). If A is less than .2 x 10“^, the value X=3

is considered to be the root and exits from the loop to statement 50. If

A/Q is less than 5x10“^, computation goes to statement 40. If the iteration

counter K is greater than 2, computation goes to statement 40. If the product A*D is positive, go to statement 30 where the present increment is

added to X. If the product A*D is negative, incrementing has gone past the

value for root go to statement 20, which subtracts the present increment;

reduce the increment by one-half. If M=l, go to statement 10; otherwise the

iteration counter K is increased by 1 and go to statement 10. In statement

30, the value of X is increased by the present value for the increment E; in

statement 40, the value of X is changed using the Newton-Raphson Method,

X=X-A/Q [$ n+i = 3 n

- W(3 n)/W’ (3) ] , iteration counter K is increased by one.

If K is greater than 9, exit from loop to statement 50; otherwise go to

statement 10.

-3^6 ? 3. Statement 50, store root 3(M)=X=3m , compute ¥m

=2e /3^<SW'(3), Z(M)

P(M), and R(M) for output. If 3 m

is greater than 5.5, RETURN to subroutine

PC, if M is greater than 50, RETURN to subroutine PC. Set iteration count

to zero.

4. The initial increment for the determination of each root is based on the

magnitude of the first root, or

26
U = .005

U = .002

U = .001

U = 0.1 3i < 1

•5 < &i < 1

.25 < 0i < .5

3i < ,25

5. Initialize

C = X = 3 m

E = U

M = M+l

N = 1

Initialize increment

Indexing for next root

Index for initializing loop

6. Initializing root finding loop starts with statement 70. Increment C by the

value of E, call subroutine ABC in order to calculate A=W(8), Q=W'(8). If

N=l, set D=A and S=Q. At statement 80, increase index N by 1, increase

increment E by the initial increment U and go to statement 70. If N is

greater than 1, go to statement 90; if A*D is less than zero, go to statement 100; if A/Q or W(B)/W’(8) is greater than zero, go to statement 80

above. If A/Q is less than zero, the increment E is set equal to 2U, X=C+E

and go to statement 10 (see item 2 above). Statement 100, the present E is

divided by 2, then X=C-E and go to statement 10.

27
7. REFERENCES

1. B. Peavy, "A Note on Response Factors and Conduction Transfer Functions,"

ASHRAE Transactions , Volume 84, Part 1, 1978.

2. B. Peavy, "Determination and Verification of Thermal Response Factors for

Thermal Conduction Applications," NBSIR 77-1405, National Bureau of

Standards, April 1978.

3. H. S. Carslaw and J. C. Jaeger, "Operational Methods in Applied Mathematics,

Second Edition, Oxford University Press, 1948.

4. M. Lokmanhekim, "Convolution Principle as Applied to the Heat Transfer

Problems of Buildings and Fundamentals of Its Efficient Use," LBL-6866,

Lawrence Berkeley Laboratory, University of California, August 1977.

28
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01. Summary date

Yr. Mo. Day

8 1 1 1 1 0

04. Software date

Yr. Mo. Day

FEDERAL INFORMATION PROCESSING STANDARD SOFTWARE SUMMARY

02. Summary prepared by (Name and Phone)

Bradley A. Peavy 301 921 3532

05. Software title Computation for Conduction Transfer Function

Initialization of Heat Conduction Problem

06. Short title

03. Summary action New Replacement Deletion a

Previous Internal Software ID

07. Internal Software ID

08. Software type 09. Processing mode 10. Application area General Specific Automated Data Computer Systems Management/

] | System 1 1 Interactive f] Support/Utility p Business

[X] Computer Program pj Batch p| Scientific/Engineering p Process Control

| | Subroutine/Module

| | Combination | | Bibliographic/Textual p Other

11. Submitting organization and address National Bureau of Standards

Washington, DC 20234

12. Technical contact(s) and phone

B. A. Peavy

Bldg. 226, Room B 124

NBS 742 Tel 301 921 3532

13. Narrative Program (subroutine PC) computer conduction transfer functions of multi-layer building

constructions for thermal conduction applications. Provision is made for including or excluding surface film thermal resistances, for the combination of two building con- structions acting as parallel heat flow paths, and for the use of 1-, 2-, or 3-hour

time intervals, depending on the thickness of the building construction. Subroutine PC

may be easily modified to accommodate any time interval necessary.

Subroutines PI, PQ, and PR are included to define the initialization of the heat con- duction from an initial datum plane temperature to a steady periodic condition. The

subroutines use conduction transfer functions from subroutine PC, the inside and outside

surface film coefficient, and a 24-hour history of inside and outside air temperatures

The inside and outside surface temperatures are found from the heat balances at the

surfaces

14. Key words: conduction heat transfer; conduction transfer functions; initialization of

heat transfer problem; parallel heat flow; response factors; thick building construction

15. Computer manufr and model UNIVAC 1108

16. Computer operating system

Time sharing syste

17. Programing language(s) m FORTRAN V

18. Number of source program state- ments

19. Computer memory requirements 20. Tape drives None

21. Disk/Drum units None

22. Terminals

23. Other operational requirements

24. Software availability

Available B

Limited In-house only

25. Documentation availability

Available Inadequate In-house only

26. FOR SUBMITTING ORGANIZATION USE

L

185-101 Standard Form 185

1974 July

U.S. Dept, of Commerce—NBS

(FIPS. Pub. 30)
NBS-114A IREV. 2-80

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publ ication Date

SHEET (See instructions) NBSIR 81 2353 November 1981

4. TITLE AND SUBTITLE

DOCUMENTATION OF PROGRAM FOR DETERMINATION OF HEAT CONDUCTION TRANSFER FUNCTIONS

5. AUTHOR(S)

Bradley A. Peavy

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

10.

SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here) Conduction transfer functions are used to predict the time-dependent one-dimensional

conduction heat transfer at surfaces of single- or multi-layer building constructions

based on heat flux and temperature history at each surface. By the use of conduction

transfer functions, heat transfer problems employing non-linear boundary conditions

such as thermal radiation and time-dependent changes in the surface film resistances

can be solved.

Because conduction transfer functions are analytically derived with an initial time

condition of zero temperature potential throughout the solid materials, it becomes

necessary to initialize the computation by including exposure to a number of outdoor

weather cycles such that satisfactory initial conditions of temperature and heat flux

exist at the inside and outside surfaces.

The program is set up for the use of 1-, 2-, or 3-hour time intervals, depending on the thickness of the building construction. The program allows for the combination of

two building constructions, e.g., the parallel heat flow paths found in wood-frame

walls

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

conduction heat transfer; conduction transfer functions

transfer problem; parallel heat flow; response factors;

13.

AVAILABILITY

fcxJ Unlimited

initialization of heat

thick building construction.

14. NO. OF

PRINTED PAGES

| | For Official Distribution. Do Not Release to NTIS

J Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

47

15. Price

^^Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $6.50

USCOMM-DC 6043-P80
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