NAT L INST. OF STAND & TECH NIST

PUBLICATIONS

•IBSIR 77-1405

REFERENCE

Jetermination and Verification

of Thermal Response Factors

. for Thermal Conduction

Applications

Bradley A. Peavy

Building Thermal and Service Systems Division

Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, D C. 20234

Final Report

April 1978

U.S. DEPARTMENT OF COMMERCE 100

NATIONAL BUREAU OF STANDARDS

NBSIR 77-1405

DETERMINATION AND VERIFICATION

OF THERMAL RESPONSE FACTORS

FOR THERMAL CONDUCTION

APPLICATIONS

Bradley A. Peavy

Building Thermal and Service Systems Division

Center for Building Technology

National Engineering Laboratory

National Bureau of Standards

Washington, D.C. 20234

Final Report

April 1978

U.S. DEPARTMENT OF COMMERCE. Juanita M. Kreps, Secretary

Dr. Sidney Herman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

DETERMINATION AND VERIFICATION OF THERMAL RESPONSE FACTORS

FOR THERMAL CONDUCTION APPLICATIONS

by

Bradley A. Peavy

ABSTRACT

New formulas for calculating thermal response factors for multiple-layer

construction have been developed by a rigorous derivation. A comparison

was made of the time for computation between the presently used matrix

algebra method and the method given in this paper. Results were obtained

using the new method in one-fiftieth to one-half of the computational

time necessary to obtain solutions from the matrix algebra method.

Comparisons with another analytical method were performed to verify the

accuracy of the response-factor technique.

Key Words: Dynamic conduction heat transfer; heat transfer; thermal

response factor; verification.

i

DETERMINATION AND VERIFICATION OF THERMAL RESPONSE FACTORS

FOR THERMAL CONDUCTION APPLICATIONS

by

Bradley A. Peavy

1. INTRODUCTION

The analytical solutions needed to describe temperature or heat flow for

steady periodic or transient conduction heat transfer in multi-layer walls,

ceilings and floors are quite complicated and are not available for nonlinear conditions such as thermal radiation at surfaces and time-dependent

changes in surface film coefficients. It has therefore been expedient

to employ "approximation" methods by which non-linear conditions may be

satisfied. One such method, which is the result of an analytical formulation, is termed the "response-factor method," and solves for temperatures

or heat flows of multi-layer constructions based on the past temperature

history.

A response-factor method defined by Kusuda [1]* uses overlapping triangular

pulses to compute response factors for a particular construction. These

response factors are then used to determine temperatures and heat flows

in response to previously occurring events. In order to handle multi-layer

constructions for solution of response factors by computers, a matrix

algebra was developed for an arbitrary number of layers. There are some

inherent difficulties in using this approach, in that the calculation

of response factors can be quite time consuming because certain functions

* See references cited at end of text

1
are not well suited for efficient calculation. A portion of this paper

will be devoted to the development of equations which allow for increased

computational efficiency- The algorithms presented are sufficient for the

determination of up to a seven layer composite, which is probably more

than sufficient for possible building constructions.

Unfortunately, the accuracy of the temperatures and heat flows values calculated by response-factor methods has received very little verification.

Although it is well known that the thermal properties of building materials

are not as well defined as they should be, this is no excuse for not knowing the error incurred by the use of an approximate method for whatever

thermal properties may be assigned to a particular program for solution.

A portion of this paper will be devoted to a comparison between numerical

results from an analytical solution and results from calculations using

the response-factor technique.

2. ANALYSIS

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

layers, the partial differential equation is given by

m ^ 1 m

3x2 a 9t

m

where v^ is the temperature potential above a datum plane, 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

2
between layers, perfect contact is assumed, i.e..

V = V

m-1 m

and

dv ^ dv

K , m-1 _ K m_

where K is the thermal conductivity of the respective layer. Applying

m

the Laplace transform to (1) gives

d V „ ^ m z - z p

dx^ %

for which a solution is:

-

q (x-b J -q (x-b J , m m-1 , _ ra m-1

Vmm= A e + B e m

„ dv r <L_ (x-b 1 ) -q (x-b , ) -i K m „ I . nn m-1 „ ^m m-1

m — = Kq Ae -Be dx m m L m m J

where b , is the distance from x=0 (surface of first layer) to the m-1

nearest face of the m-th layer. Applying the continuity conditions (2)

gives the following relations

2 A T -y 1 -y 1 m-1 . -^m-l ^ , ^

•'m-l = A e + k , B e (1+0,) m m-1 m

m-i

2 B , y T y , m-1 ,

. 'm-l , ^ 'm-l

= k , A e + B e

(1 + 0 , ) m-1 m m

m-1

3
where

a K

m

a ,

/ m-1

m-1 K - ./ a m-1 V m

k

1-0

m-1, m-l 1 + a . m-1

Vi m-1 m-2 V ^ m-1 v ^ fa T /a T

/

m-1 y m-1

and is the thickness of the (m-l)-th layer. This process is continued

until the constants and B^^ (pertaining to the layer with face exposed

at x=0) are found in terms of and (pertaining to the n-th layer with

face exposed at x=b^) . At x=0, the heat flux is proportional to the

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

surface, and is represented by the relationship

where is the surface film resistance and f(t) is fluid temperature

as a function of time. Similarly, the boundary condition at x=b^ is

"'2'" ' \ - (5)

where R2 is the surface film resistance and g(t) is a fluid temperature

as a function of time. When either or R2 is zero, the time temperature

function represents temperatures at the respective surfaces. The resulting

expressions for the transform of the temperatures in layer 1 and layer n

are given by
(6)

and

n iEl[sl„h (b^-x) + 7p cosh (tj^-x)l +

(7)

where

^

= z J

m

slnh /p (N - ^

m

•x/yo^) ^1 = Z L

m

sinh /p (E^+x/^^)

z J

m

cosh /p (N -x/JaT^) ^1 = Z L

m

cosh /p (E^+x/Za^

)

^2

= z J

m

sinh /p (N^-.(b^-x)//a;) ^2 = z L

m

sinh »/p m n n

^2

= z J

m

cosh /p (N - a -x)//^^)

n n

^2 = z L

m

cosh x/p (E -a -x) /Ja) m n n

P = z J

m

sinh v/p N Q

= z L

m

sinh \/p E

m

S = z J

m

cosh v/p N T = z I'm cosh \/p E

m

.n-2

(The above summations are over m=l,2,... 2 , n being the number of layers.)

W = P + Q + V^V2P(P-Q) + v/p [^V2(S+T) + V^(S-T)]

G = 2''"V(l+aj^)(l+c^)- • 'd+Vl^' " = "T"

GK

n

N = m

i.

1

m

i=l

E = N

m m

a.a

n

and J , L and A. are defined in Table 1. m m x,m

5
TABLE 1. Definition For J , m

L and

m

A.

x,m

m J

m

L

m

A

1 1 ki 1

2 kik2 k2 -1

3 kik3 k3 -1

4 k2k3 kik2k3 1

5 kik^ k4 -1

6 kaki, kik2ki^ 1

7 kskit kiksktt 1

8 kik2k3ki^ ^2^3 kit -1

9 ki+ks kiki^ks • 1

10 kiksks 1

11 k2k5 , kik2k5 1

12 kiks k5 -1

13 kik2k3k5 1^2k3k5 -1

14 kik2k4k5 k2kttk5 -1

15 klk3k^k5 kski+ks -1

16 k2k3k^k5 kxk2k3ki+k5 1

17 kikske 1

18 ki+ke kiki+ke 1

19 k3k6 ki kskg 1

20 k2k6 kik2k6 1

21 kike kg -1

22 kik2k3k5 k2k3k6 -1

23 kik2 ki+ kg k2kitk6 -1

24 kik2k5k6 k2k5k6 -1

25 kikski+ke k3ki+k6 -1

26 kik3k5k6 k3k5k6 -1

27 kiki+kske kifkske -1

28 k2k3ki+k6 kik2k3ki+ke 1

29 k2k3k5k6 kik2k3k5k6 1

30 k2kitk5k6 kik2kitk5k6 1 31 kski+kskg ki k3 kj+ ks kg 1

32 kik2k3ki+k5k6 k2k3ki,k5kg -1

A

3 ,ni A A

c; ,m

A A

1 1 1 1 1

1 1 1 1 1

-1 1 1 1 1

-1 -1 1 1 1

-1 -1 1 1 1

-1 -1 1 1 1

1 -1 1 1 1

1 -1 1 1 1

1 1 -1 1 1

1 -1 -1 1 1

-1 -1 -1 1 1

-1 -1 -1 1 1

1 -1 -1 1 1

1 1 -1 1 1

-1 1 -1 1 1

-1 1 -1 1 1

1 1 1 -1 1

1 1 -1 -1 1

1 -1 -1 -1 1

-1 -1 -1 -1 1

-1 -1 -1 -1 1

1 -1 -1 -1 1 1 1 -1 -1 1

1 1 1 -1 1

-1 1 -1 -1 1

-1 1 1 -1 1

-1 -1 1 -1 1

-1 1 -1 -1 1

-1 1 1 -1 1

-1 -1 1 -1 1

1 -1 1 -1 1

1 -1 1 -1 1

6
The transforms of the heat flux at x=0 and at x=b are found by differenn

tiating (6) and (7) with respect to x and multiplying by minus one and the

respective thermal conductivity:

n/p

F, =

"' W /a - n

= [cf (p) - is+T + v/F (P-Q)}g(p)l • (9)

W v/a L ^ •'J n

The inversion of (8) and (9) is performed by evaluating the residues at

the poles of the denominator W=0, where p= -3 or \/p = iB, which gives

the relationship

Wq = (1-V.V_B^) E J sin N 3 + (1+V,V^3^) sin E 3

p IZ m m L Z m m

|s-T + V2 x/p (P-Q)} f(p) - Hi(p)j

+ 3 KV^+V, |_/lm) E J cos N 3 + (Vo-V, ) E L cos E 3 m zim m

(8)

= 0 (10)

2

and the differentiation of W with respect to p evaluated at p= -3 gives

U = (1-V,V„3^) E J N cos N 3 + (1+V,V_3^) E L E cos E 3 11 mmm 11 mmm '

+ (V^+Vt zi|_m ) 1 E J cos N 3 - 3 E J N sin N 3I m mm m

+ (V^-VjIe zij^mL cos E 3 - 3 E L E sin E 3] m mm ml

- 23V,V„rEJ sin N^3 - E L sin E^3l (11)

1 m m m m
2

The residues at the poles p= -3 are

and

^n3

XD.-V^D^B) - Hi(-3,^

)J

e

2K 'ST' 3.^ r _ „ _ „ -, -3.^ t ^ / 7r"|Gf(-3. ) - (D3-V1D23) g(-3p e n 1

-'

where

D = E(J cos N 3 - L cos E 3), D„ = S (J cos N 3 + L cos E 3), i mm mm 2. mmmm

D„ = Z(J sin N 3 - L sin E 3).

J mmmm

Of particular concern is the evaluation of the first root of (10) . 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 m .,

^ A^3 - A23^ = 0 ; , ; ; ^

2 or 3-j^ ~ A^/A2

with

= Z (J N + L E ) + (Vt+V„) Z J + (V,-V^) E L

1 mmmm LI m Zi m

and

V +V V -V

= V^V^ Z (J N -L E ) + — Z J N + — — Z L E

2 12 mmmm 2 mm 2 mm

1 + -Z-Z(JN +LE ) . 0 mm mm

8
Consider the triangular pulse function of Kusuda [1], where

f(t) = 0

= t/6

= 2-t/6

= 0

f(p) = 0

= l/6p^

= (l-2e-P^)/6p2

= (l-e-P'^)2/6p2

t £ 0

0 < t £ 6

6 < t £ 26

t > 26

which when substituted for f(p) and g(p) in (8) and (9) gives double

poles at p=0. Following are limits of the necessary functions of (8)

and (9) for evaluating the residues at the double poles.

Lim

iw) = ^ + A2P

Lim

p-)-o S - T + V2 v/p (P-Q) = +

^ S + T + /F (P=Q) = + C2P

and A2 are defined by (14) and

B, = E (J - L ) , C, = S (J + L ) 1 m m * 1 m m'

B = E (J N 2 - L E ^) + V„ Z (J N - L E ) z z. mm mm mm mm

= E(JN^ + LE^)+V, Z(JN - LE) / z mm mm 1 mm mm

For the first term in (8) , the residue for 0<t<6 is

X. = K^rtB3^

_^ V2:Vl

(16)

9
for the last term in (9) , the residue is

K

Z. = n ^

^1^2 ^2*^1 1

.1 J

(17)

and for the first term in (9) and the last term in (8) , the residue is

K G A. n 6ja

tA,

1

->

(18)

where

KG,

n 1

At /a

1 n

K G

n A^/a

1 n

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

The response factors X^, and evaluated at t=6, become

Xo -

VOL,

^

Y, = Y, - ^n^ V

\/an

(19)

'6 v/a

^1(D, n

where

-3.

"i

10
For Y2 and evaluated at t=26,

1 - ^1 V- '^i ^

2

2 K G —3

1^2 = 4--^6-7^I (20) n

2

and for t>26,

For larger values of j , the response factors decrease with increase in

j by a connnon ratio — i.e., for N sufficiently large,

-3? 6 X.+l Y.+l Z.+l

3 j 3

for all 3>N, where is the first root of (10) . Thus for j greater than

N, the response factors can be computed by the relationship

X, = X. • CR . 3+1 3

The temperatures f(t) and g(t) may be constructed from a series of triangular

pulse functions (15), overlapping in time by an amount 6. By the principle

of superposition, the heat flux at time t is determined from the sum of the

products of the response factors and the temperatures from the present time

to preceding time intervals of duration 6. The heat flow at time t=T and

11
at the x=0 face is then

F, = (X. - Y. V .^,) (23) 1,T i=l 1 l,T-i+l 1 n,T-i+l

and the heat flux at x=b is n

F = (Y. V, - Z. V .^.) (24) n,T i=l 1 l,T-i+l 1 n,T-i+l

where V is the temperature potential in relation to a fixed datum plane

temperature, which is referenced to time, t=T, t-1, t-2, etc.

For certain types of constructions, the number of required response

factors can become quite large to give a reasonable accuracy to the heat

fluxes of (23) and (24). Conduction transfer functions can reduce this

number considerably and are defined from the relationship, F^ ^

- CR F^

of (23) and similarly from ^24). The conduction transfer functions are

then defined as follows:

^1 " ^1 ^1 "

'^l ^1 " ^1

I I I Xj = Xj-CR Xj_^ Yj = Yj-CR Yj_^ = Zj-CR (25)

Using these functions, the heat flux at t=T and x=0 becomes

F, = "CR . Ft 1 + Z (X-V, - Y-V ._^,) 1,T 1,T-1 . J l,T-j+l J n,T-j+l'

From (21) and (25), it can be seen that the conduction transfer functions

are numerically very small numbers as j approaches N, and the number of

functions needed for computation purposes is considerably reduced. However,

f 12
initially it is necessary to know the value of F-^ t_1 • This must be

determined from several iterations over the past temperature history

of Wi and •

3. RESPONSE FACTORS

Using the algorithms of the previous section, the response factors of

equations (19), (20), and (21) are calculated from the computer program

found in Appendix A. To show time savings in response-factor calculations,

identical multi-layer constructions were input into both the program of

Kusuda [1] and the program of Appendix A. Six multi-layer constructions

are shown in Table 2, and the computation times for the two programs are

shown in Table 3. As can be seen, considerable time saving is available

from the program of Appendix A.

In the example of Table 2, there is an evident lack of enclosed air spaces

in the building construction. Kusuda [2] assumed only a purely thermal

resistance effect of air spaces during dynamic temperature changes. It

is the supposition of the author that both the air space thickness and

the heat capacity of the air should be considered for heat transfer calculations. From literature values, the thermal diffusivity of dry air

varies from 0.639 ft^/h at 0 °F to .977 ft^/h at 120 °F, and these values

are reduced somewhat by the presence of water vapor. For the program of

Appendix A, the thermal diffusivity of air in air spaces is assumed to

be 0.75 ft^/h, the air space thickness is assumed to be one inch if not

specified by the input card, and the thermal resistance is as defined for

steady-state air-space values.

13
Table 2. Multi-Layer Constructions

Layer

Descript ion

Layer

Thickness

ft

Thermal

Conductivity

Btu/h ft F Density

Ib/ft^

Speci f ic

Heat

Btu/lb F

Case 1 Two Layers

1 1-IN MINERAL FIBERBOARD

2 4-IN LIGHT-WEIGHT CONCRETE

0.833

.3333

,035

,100

23.0

40.0

.140

.200

Case 2 Three Layers

1 5/8-IN PLASTER BOARD

2 3-1/2 IN BATT INSULATION

3 3/4-IN WOOD SIDING

,0521

,2917

,0625

.094

.026

.800

50.0

2.0

36.0

,260

,220

,280

Case 3 Four Layers

1 1/2- IN PLASTER BOARD .0417

2 3/4-IN POLYSTYRENE INSULATION .0625

3 4-IN COMMON BRICK .3333

4 4-IN FACE BRICK ' .3333

.094

.094

.417

.750

50.0

50.0

120.0

130.0

,260

,260

,190

,190

Case 4 Five Layers

1 1/2-IN PLASTER BOARD

2 1-IN BATT INSULATION

3 4-IN L.W. CONCRETE

4 3/4-IN BOARD INSULATION

5 1-IN STUCCO

,0417

,0833

.3333

,0625

,0833

.094

.026

.100

.017

.400

50.0

2.0

40.0

2.2

116.0

,260

,220

,200

, 290

,200

Case 5 Six Layers

1 3/4-IN ACOUSTIC TILE

2 4-IN HEAVY WEIGHT CONCRETE

3 2-IN ROOF INSULATION

4 2-IN H.W. CONCRETE

5 3/8-IN FELT

6 1/2-IN SLAG

,0625

,3333

,1667

,1667

,0313

,0417

.035

1 .000

.025

1 .000

.110

.830

30.0

140.0

5.7

140.0

70.0

55.0

.200

.200

.200

.200

.400

.400

Case 6 Seven Layers

1 1/2-IN PLASTER BOARD

2 1-IN BATT INSULATION . 3 1/2-IN PLYWOOD

4 3-IN H.W. CONCRETE

5 1/2-IN PLYWOOD

6 3/4-IN BOARD INSULATION

7 1-IN STUCCO

,0417

,0833

,0417

,2500

,0417

,0625

,0833

.094

.026

.067

1.000

.067

.017

.400

50.0

2.0

34.0

140.0

34.0

2.2

116.0

,260

,220

,290

200

,290

,29 0

,200

14
Table 3. Execution Time for Computation of Response Factors, Seconds

Kusuda [1] Appendix A Time Savings

Two layers

Three layers

Four layers

Five layers

Six layers

Seven layers

2.037

8.885

1.902

3.018

1.609

9. 786

.187

.174

.378

.625

.846

1 .681

1.850

8.711

1.524

2.39 3 .763

8.105

Table 4. Heat Flux at Inside Surface of Wood Frame

2 Construction with Air Space, Btu/h ft

Time (hr) No heat capacity 3 1/2" air space 5 1/2" air space

1 1.,742 1,.021 .954

2 1

,

.262 1 .242 1 .203

3 1,,359 1,.342 1.319

4 1

,

.548 1,.525 1.494

5 1., 707 1..691 1 .666

6 .964 1

,

.109 1.233

7 - 1.,966 - 1 .,598 - 1.154

8 - 6..113 - 5,.684 - 5.024

9 -10..410 - 9,.924 - 9.214

10 -14,.248 -13,.809 -13.144

11 -17,.427 -17,.065 -16.501

12 -19.. 720 -19,.463 -19.042

13 -20,.908 -20,.783 -20. 544

14 -20..827 -20,.849 -20.822

15 -19..497 -19,.660 -19.846

16 -17,.038 -17,.334 -17.718

17 -13,.592 -13,.996 -14.552

18 - 9,.416 - 9,.908 -10.599

19 - 4,.960 - 5,.459 - 6.201

20 - 1,.785 - 2,.088 - 2.639

21 .746 .870 - 1.128

22 .214 .281 - .414

23 + .239 + .185 + .093

24 .680 .628 .547

15
The introduction of these changes does alter the response factors and

heat flux amplitudes in response to external temperature variations.

Table 4 gives heat flux values at the inside surface for a wood-frame

construction, with no heat capacity [2], and 3 1/2 and 5 1/2 inch air

spaces. The external temperature variation is taken from Figure 1.

A. VERIFICATION OF RESPONSE FACTORS

Response factors as defined in this paper are the result of an analytical

formulation which employs a past time history of overlapping pulses for

the temperatures at or adjacent to surfaces of composite solids to determine the heat flux or temperature at present time. A linear variation

in temperature over the time period, (usually one hour) is assumed. The

ability of response factors to give accurate values for heat flux has

been questioned. For this reason, it is appropriate to compare the numerical results from the response factor calculation with those from another

analytical calculation.

Analytical solutions can be found from equations (6) and (7), where the

temperature -time functions can be defined by the trigonometric series.

f(t) = Z (A„ cos W„t + B„ sin W„t)

n n n n

f(p) = Z An P ^ BnW,n

(27)

2 2 p + W ^ n

and

g(t) = Cq + Z (C^ cos W^t + sin W^t ) ; g(p) = — + Z C„p + D„W„

n*^ n n

2 2

p + W ^ n

(28)

16

Considering only the steady periodic condition, the residues at the poles

at p=0 and p=+iw^ can be found for the surface temperatures. Heat flow

at the surfaces can then be obtained from (4) and (5).

The six examples of Table 2 were used in determination of heat flux for

both the response factor and analytical solution. The temperature variation,' g(t) shown in Figure 1 was used for the test cases, with f(t)=0

and Ri = 0.68 and Ro = .333 hft^F/Btu. The coefficients C„ , C and D

i z o ' n n were determined from 96 points where there was linear interpolation

between the hourly points. This was necessary to assure a nearly linear

variation in temperature between the hourly points for the function g(t).

When coefficents were determined for 24 and 48 points the agreement between

the heat flux for the analytical and response factor was not as good.

Heat flux was computed both from the response factors [(23) and (24)] and

the conduction transfer function (26). For all cases examined the heat

flux at the inside surface was less than one percent different from the

values computed for the analytical solution. At the outside surface, the

agreement was not as good, particularly around 5 and 19 hours, where

there are sudden changes in the slope of the temperature curve (as

shown on Figure 1). The detail figures show that the function g(t) used

in the analytical solution varies considerably from the linear form

required by the response - factor method in the time period 4 to 5 and 19

to 20, It is expected that if more points were used, there would be

a better agreement at the outside surface.

18
Table 5. Comparison of Heat Flux Values Calculated from

Response-Factor and Analytical Methods Btu/h ft

Case 2. Case 3.

Oil ^ c 1 H P fillT fcJUX 'Fflf* X. PC In s 1 d p Sur face Out side Sur far p In s i. cl € Si 1 T* (JUL J-f* ^5ar*\^ pC

I\c 9 p • An A

1

Rg s p • Anal

.

Re s p

.

An3

1

Rp Q n nil An ad 1X •

1L 1 1 28 1 . 103 .358 .358 39 .450 39 .437 -5 468

0

i. • HO / 503 .442 .442 34 . 607 34 609 -4H • Rft7 0 U / -AH • Rftft 0 U U

•JJ 1 177 1X • 152 X ^ 4L. 479 .478 32 679 32 668 H • 7 -/ H . 7 L 1 . 268 1 .195 .542 .542 30.798 30. 768 -3 . 763 -3 . 763

652 304 .604 .604 27 . 1 71 27 .036 -3 271 -3 2 70

u - Q 947 - 9 338 .430 .434 - 2.856 2 920 -2^ • 810 X ft0 -9z Rl ft • 0 X u

7 «,1 S A7Q -15 366 _ Ago - .486 - 38.739 - 38 726 -9 408 -9 AOR

QO -1X U • JS'^l .J X -1 fi 41

9

X U . *tX ^ -1X • 933 7 -J -J -1 933 - 65 508 - ftS \J J • 4fll HO X -9Z . 110 X X u -9 1 OQ

Q 1 O • OO

J

X \J t J J i J • H J 0 -3 ASft OH 7 .JX OH • 7 0 X — 9 ni 9 — 9 m 1

L \J -1 5 951 -1 5 807 -4 861 -4 . 862 - 96 946 J \J • U 7 H -2^ . 1X 70/ V/ -9Z • 1X ftQ u 7

1 1 1 1 i. H • ^ OX -14X *+ . 1X Oft V/O -6 047 -ft 048 -1X 01vf X . 1X «JO 3ft -1X 01U X * 0\} 7ft 1 \J -9 SQO -9 SRQ

-11 -1XX1 • J17Q1

y

-ft Q '^O -ft Q 19 - Qft yyj •yQ7S1

J

- Qft QOA -^J . Z9 JOTR — J . Z9TJ 7/

1

'\ - 7 701 - 7/ . ^SfiS u ^ -7 439 -7 4 34 - 83 ft9ft - 830^.0ft9fi zo -Zi 0 SR -A nS7

1 A _ "I 070 J , J f y — J . ZOO -7/ . ARR too -7 AQO - ftl 9RQ - ft'^ 9"^"^ —AH .7/0 Q 7R —AH .7/0 Q7R

15 .709 .783 -7.091 -7.093 - 37.984 - 37.933 -5.916 -5.916

16 5.175 5.183 -6.284 -6 . 284 - 8.151 - 8 . 1 24 -6.786 -6.787

17 8.621 8.604 -5.110 -5.111 21.992 22.007 -7.509 -7.510

18 12.274 12.076 -3.661 -3.662 53.149 53.093 -8.015 -8 . 01

6

19 11.223 10.820 -2.065 -2.064 71.912 71.763 -8.254 -8.255

20 1.296 1.332 - .811 - .806 58.711 58.703 -8.195 -8.197

21 1.226 1.242 - .333 - .332 52.360 52.354 -7.857 -7.858

22 1.366 1.371 - .113 - .113 48.294 48.290 -7.332 -7 .332

23 1.520 1.517 .055 .055 45.399 45.394 -6.722 -6.722

24 1 .676 1.648 .214 .214 43.178 43.164 -6.901 -6.091

19
Comparison of calculated heat flux values from the analytical and responsefactor methods is shown in Table 5 for Cases 2 and 3.

5. CONCLUSIONS

Formulas for calculating thermal response factors for plane multi-layer

constructions have been developed as given by equations (19), (20), and

(21). A computer program to obtain response factors based on these

formulas is found in the Appendix. A comparison was made of the time

for computation of response factors between the matrix algebra method

of Kusuda [1] and the method given in this paper. A considerable saving

in computation time is realized by the methods of this paper.

Presently, the response- factor calculations for constructions with air

spaces assume only a purely thermal resistance effect of air spaces

during dynamic temperature variations [2]. The heat transfer across an

air space involves the nature of the bounding surfaces, the intervening

air, orientation of the space and the direction of heat flow: and hence

the three modes of heat transfer —radiation, convection, and conduction —

influence heat flow in an air space. It would be impractical to simulate the three modes of heat transfer for constantly changing air space

surface temperatures, but it is felt that a reasonable approximation

should include both the heat capacity and air space thickness foi dynamic

heat transfer calculations. These were included in the computer program

of the Appendix. The introduction of these changes gives different

values for the response factors and resulting heat-flux amplitudes.

20
Response factors are analytical formulations from the partial differential

equations for heat conduction. When properly applied, the responsefactor method gives correct values for temperature and heat flow for

conduction heat transfer problems.

6. REFERENCES

1. T. Kusuda, Thermal Response Factors for Multi-Layer Structures of

Various Conduction Systems, ASHRAE Transactions, Vol. 75, 1969.

2. T. Kusuda, NBSLD, The Computer Program for Heating and Cooling

Loads in Buildings, NBS-BSS-69, U.S. Government Printing Office,

Wash. , D.C. 20402.

7. CONVERSION FACTORS TO METRIC (S.I.) UNITS

Physical Quantity

Length

Area

Temperature

Density

Thermal Conductivity

Thermal Resistance

Thermal Diffusivity

Heat Flux

Specific Heat

Time

To Convert

From To Multiply By

ft m 3.0480 E-1

ft^ 9.2903 E-2

F C (F-32)/1.8

Ibm/ft^ kg/m^ 1.6018 E+1

Btu/h ftF W/mK 1.7296 E+0

h ft^F/Btu m^K/W 1.7623 E-1

ft^/h m^/s 2.9900 E-3

Btu/h ft^ W/m^ 3.1525 E+0

Btu/lbmF j/kgK 4.1840 E+3

h s 3.6000 E+3

21
APPENDIX A

Appendix A gives a Fortran listing of computations for the response

factors and includes input of data identical to that of Kusuda [1].

:

• J .

;

22
C PBOGBAM pec COMPOTING KESP0NSE FACT6RS «P MULTI-LATEB BUILDING

C CeNSTSDCTIONS (ONE TO SEVEN LATEKS ) F0K THEBNAL CeNDOCTI0N

C AFPLICATIflNS

C INPUT

C DEL- TIME INCREMENT ( EQU 15)

C JA- NUHBEB 0F LATEBS IN CGNSTBUCTION (MAT INCLUDE SUBFACE FILMS)

C THEBM0PBTSICAL PfiOPEBTIES 0F EACH LAYEK (ONE CABD PEB LAT£fi« 1 0F7 . O

)

C SUBFACE FILM RESISTANCES MAT BE INCLUDED AS LATEBS- IN 'BE' 0NLT

J C IF INCLUDED FeB BOTB SIDES - 'JA* SHOULD NOT EXCEED 9

C eRDEBING OF LATEBS FBOM INSIDE TO OUTSIDE

C L - THICEMESS OF EACH LATEB

C K - THERMAL CONDUCT ITITT OF LATER

, CD- DENSITT OF LATER

C SP- SPECIFIC HEAT OF LATEB

C BE- TEEBMAL RESISTANCE (FOB AIB SPACES OB SUBFACE FILMS ONLT)

C DESCBIFTIflN OF EACH LATEB (ONE CABD PEB LATER- 6A6 ) C KM- CONTAINS DESCRIPTIONS FOB EACH LATEB (HOLLERITH)

C BLANE CABD INDICATES MO MOBE CONSTBUCTIONS TO BE INPUTTED

C

c

C SUBROUTINE BOOTS - COMPUTES BOOTS OF BQU 10

C

C SUBBOUTINE ABC - COMPUTES ( I-l ) TERMS FOB EQUS 16.17.18

C (I-O) SUMMATION TEBMS FOB EQUS 10.11

CcC OUTPUT

C THEBMOPHYSICAL FBOPEBTIES OF EACH LATER AND DESCBIPTIONS

C U N).X( N).D( N).SP( N).B£( N).BM( I.N).I-1.6 (EXCLUDING SUBFACE FILMS)

C STATEMENT - SUBFACE FILMS INCLUDED OR EXCLUDED

C X - PBINTOUT OF TEBMS USED IN COMPUTING EQOS 16.17.18

C BB.BC.BD.BE.BF.bg - TEBMS OF EQUS 16.17.18

C T - BeOTS OF EQU 10

C CB- COMMON RATIO EQU 22

C RESPONSE FACTORS

C XX( N ).TT( N ).ZZ( N), N-l.IB IB DEFINED BT EQU 22

PARAMETER F-60.G-100

DIMENSION S(64 ).B( 7).C(6 ).B( 6 ).L( 8 ).K(e ). D( B).SP( e).KE( 8). BMC 8.6).

lAU 7 ).XC F ),T( F ).W( F ),T( F ).XX( G ).TT(G ).ZZ( G ) COMMON/CBA/S.C

KEAL K.L

100 F06MAT ( 1017)

102 FORMAT (10F7.0)

105 FOBILAT ( 6A6 ) 201 F0BMAT ( 10F12.6 ) 202 POBMAT (3P16.8)

203 FOBMAT ( II O.Fl 1 .4. Fl 0. 3, FIO. 1 . FIO .3. FS. 2. 2X.6A6 ) 205 FOBMAT ( 26B BUILDING CONSTBUCTION NO. 14 ) 206 FOBMAT (33B ROOTS OF CBABACTEB ISTIC EQUATION)

207 FOBMAT ( 24E BESFONSE FACTOBS X.T.Z)

208 FOBMAT( IHl ) 209 P0BMAT( IB ) 210 FORMAT (56S RESPONSE FACTORS ARE CALCULATED FBOM SURFACE TO SURFAC

AB ) 211 FORMAT (56B RESPONSE FACTORS INCLUDE SUBFACE FILM RESISTANCES - SI

A«F6.3.eH AND S2-F6.3

)

212 FOBMAT (14B COMMON BATI0-F9.6)

BEAD < 5.102 ) DEL

IB-0

23
97 WRITE <6,20e)

99 K£AO < 5,&00> JA

J "J

A

BJ-.O

.^ K2-.0

IF (j.EQ.oi oe xe zoo BEAD (5,102) U 1 ).K( 1 ).!)( 1 ).EP(1 ).KE( 1 ) IA-2

IF (D( 1 ).GT..001 > Ge TO 96

Kl -BB( 1 ) J-J-1

IA-1

98 Ce 101 M-IA.J

101 BEAD (5.102) U N ).K( M ). S( N ). SP( M ).BE( N ) Ifi-IB»1

JUEAC <S.105» (BM( l.I ).I-1.6)

103 DO 104 M-IA,J

104 BEAD (5.105) ( BM( N« I ), I «1 . 6 ) IF (D(J).GT. .001) OC TO 106

B2-SE( J ) ' J-J-l

106 IF ( J.GX.7.0B. J.EQ.O ) oe TO 200

PC 110 N-1 ,J

IF (WN)) 109.107.109

107 AUN)-,75

IF ( L( M ).LT. .001 ) L(N)-. 083333

E( N )-I.( M )/B£( N ) Ge TO 110

109 AU M)-K( M)/(D(10*SF( M))

110 CeMTINOE

A-.O

19 De 20 N^l.7

B(N)-.0

IF (M.LEeJ) AA-AA*L( M )/X( N ) 20 IF (M.LE.J) B(N)-U M)/SQBT( AUN) ) AD-1,

IF ( j.EQ.i ) oe Te 1 I-J-1

DO 21 N-1 ,1

E(M)-£( V*l )*SQBT( AL( M )/AL( M*l ) )/K( K ) C( N >-( 1 . -E( M ) )/( 1 . *E( M ) ) 21 AD-2.»AD/( 1, E(N) ) OO TO < 1 ,2.3.4,5.6.7 ), J

7 S(32)-B(7)-B(6)»B(5)-B(4)*B(3)-B(2)*B( 1 ) 8( 31 »-B( 7)-B( 6 )«B( 5 )-B(4 ) B( 3 )»B( 2 )*B( 1 ) S(30)'>B( 7)-B( 6 )«B( 5)<-B( 4 )>B( 3 )*B( 2 )*B( 1 ) 8( 29 )-B( 7 »-B< 6 )*B( 5 )*B( 4 )-B( 3 )»B( 2 )»B( 1 ) S(28)-B( 7)-B( 6 )-fi( 5)«B(4 >-B( 3 )*B(2 )«B( 1 ) S( 27 )-B< 7 )-B( 6 ) B( 5 )-B( 4 )-B< 3 »-B( 2 )»B( 1 ) 8(26 »-B( 7 )-B( 6 )*B( S ) *B( 4 )-B( 3 )-B( 2 )«B( 1 ) S(25)-B< 7)-B(6)-B(5)*B(4)«B(3)>B(2)*B( I ) S(24)-B( 7 )-B( 6 )»B( 5 )»B( 4 ) B( 3 )-B( 2 >*B( 1 ) S(23 )-B( 7 )-B( 6 )-B( 5)«B(4 )*B( 3)-B(2 )*B( 1 ) 8(22 )-B( 7 )-B< 6 )-B( 5 )-B( 4 )«B( 3 )-B( 2 )*B( 1 ) S( 21 )-B( 7 )-B( 6 )-B( 5 )-B( 4 >-B( 3 )-B( 2 )*B( 1 ) S(2 0)-B( 7)-B( 6 )-B( 5)<-B( 4)-B( 3 )«B( 2 )*B( 1 ) S( 19)-B< 7)-B( 6 )-B( 5 )-B(4 )«B( 3 )*B( 2 )»B( 1 ) S( 1 8 )''B( 7 )-B( 6 )»B( 5 )*B( 4 )«B( 3 )*B( 2 )«B( 1 ) S( 17)-B( 7)-B( 6 )*B( 5)*B(4 »^B( 3 )*B( 2 )»B( I ) 6 S(16)-B( 7 )*B( 6 )-B( 5 )«B( 4 )-B( 3 )*B(2 >*B( 1 )

24
S( 15 )-B( 7 )«B< 6 )-B( 5 >*B( 4 )-B( 3 )*B( 2 >*B( 1 ) SC1« )-B( 7 )«B( 6 >-B( 5 »»B( 4 )«B( 3 )-B( 2)*B( 1 I S( 13 >-B( 7 )*B( 6 )-B( 5 )-B( 4 )*B( 3 »-B< 2 )*B( 1 > S( 1 2 ) -B( 7 I •BC 6 )-BC 5 )-B( 4 )-B( 3 >-B( 2 ) •Bt 1 ) Sdl >-B<7 l*B(6l-B(5l-BC4)-B(3)»B(2l*BC 1 > S( 10 l-B( 7 »»B( 6 )-B< S )-B<4 )*B( 3 )*B( 2)«B( 1 ) 6( «|-B(7J»B(6)-B(«)«B(4»*B(3»*B(2)^BC 1 ) S SC e }-B(7)*B(6)*B(5>-B(4)«B(3)-B(2)*BC 1 > 6( 7 1-B( 7 )*B( 6 )*B( 5 )•&( 4 )*B( 3)*B( 2 )«B( 1 ) 8( 6 }-B( 7 )*B( 6 )*B( 5 >-B( 4 )-B( 3 )*B( 2 > *B( 1 » S( S )-B( 7 1*B( 6 )«B( 5 >-B( 4)-B( 3 )-B< 2 >*B( 1 > 4 S( 4 )-B( 7 )«B( 6 )»B< 5 »»B(4 )-B( 3 )»B( 2>»B< 1 > S( 3)«B(7)*B(6)*B(SI*B(4)-B(3)*B(2)*B( 1 I 3 S( 2)-B(7)*B(6)«B(S)*B(4)*B(3)<-B(2)*B( 1 ) 2 8( 1 »-B(7)»B(6)*B(5)»B<4)*B(3»»B(2»»B( I » oe TC 9

X S( 1 )-B( 1 ) 9 D0 e M-1.32

I»N*32

8 S( I )-S( M )-2.*B( 1 ) AB-E( 1 >/SCBTt ALC 1 1 ) AC-K( J )/SQBT( ALi J ) ) 1-B1*AB

V2-B2*AC

1*1

CALL ABC( CA.X. J.I

)

CA"^1«V2

CB-V2*V1

CC-V2-T1

BA-X( 2 )*X( 6)«CB«Z< 1 >«CC«Z<S ) BB«AB«( X( 1 )-Z( 5 ) )/SA

BC-AC«( X( 1 )*Z( 5 ) )/BA

BI>-AC*AS/BA

BH-( X( 4 )*X( e ) )/6. €*•( X( 2 )-X( 6 } )( CB»X< 3 >*CC»X( 7 ) )/2.

BI -( X( 3 )-X( 7 ) )/2, V2»( X( 2 )-X< 6 ) ) BJ-t X( 3)*X(7I )/2. X( 2 )-X( 6) ) BE-AB*< BA«BI-BB*< X( 1 )-X( 5 I ) l/( DEL*BA»BA ) BP-ACX BA*BJ*BB*( X( 1 )*X( 5 )! )/( DEL*BA*BA ) BG--AC«AC«BB/ ( DEL«BA*BA ) X( 1 )-SQHT( BA/BB)

CALL Be0TS( AB,AC.A£.V1,T2.DEL«T«X,T.W.J.M ) XX(1>-BB*BE

ZZ( 1 )-BC»BF

TT(1)-BD»B0

XX( 2 )—BE

ZZi 2 )—BP

TT( 2 )"-BG

DO 50 M-l.M

XX( 1 >-XX(l )-X(M)

ZZ( 1 )-ZZ(l>-«(M)

TY(1 )-TT(X)-T(M)

CA -EXP( -DEL»T( N »»«2 )-2.

XX( 2 >-XX( 2 >-X( M >*CA

ZZ( 2 >*ZZ( 2 )•«( N >*CA

50 T«2>-TY(2)-T(M>«CA

CB-EXP( •OELvTC 1 >••£ ) 90 56 1-3,

G

XX< I )-.0

T« I )-.0

ZZ< I )-.0

25
XB-1

CC-I-3

CA-CC«T( 1 )»«2»»E1

IF CCA. 01. 20.) Oe TO S9

DO 55 M-l,M

CA»DEL«TCNK«2

CB-CC«CA

IF <CB.GT.40.) oe TO 56

CI>-BXP( -CA ) CD-EXP( -CB)»( I, -CD )»«2

XX( Z )-ZX( I )-X( N )*CI>

ZZ( I )-ZZ( I )-( M )*C0

55 TXC 1 )-TX( I )-T( N)»CD

52 CA-ABS( XX( I )/XX( X-1 )-CB )*ABS(TT( I )/TY( X-1 HCB ) CA-CA*ABS( ZZ( I )/ZZ( I-l )-CB ) 56 IF ( CA. LI. .00003 ) GO TQ 59

59 VBITE f6.205) IB

De 204 I-1,J

204 VEITE (6.203) I . U I ). K( I ).D( I ). SP( I ). B£( I ). ( Bkl( I.N ).N- J . 6 ) VBITE (6.209)

IF (J.EC.JA) VBITE (6.210)

IF (J.NE.JA) VBITE (6,211) K1.K2

VBITE (6.206)

VBITE (6.201) (T(M).N-1.5)

VBITE (6.209)

VBITE (6,207)

VBITE (6,202) ( XX( M ),TT( N ).ZZ( M ),N-1 . IB ) VBITE (6.212) CB oe Te 97

200 STOP

END

SCBBeOTINE AfiC(X.Z.J.I)

DIMENSieN Ul ).T(64).V(64.4)

Cei<UeN/CfiA/S( 64 ).C( 6 ) K-1

IF ( i.BQ.i > oe Te 20

M-l

IF (J.GT.l) ll-2»*J/4

i>e e M-i.N

De e 1-1.2

L-N

IF ( I.EQ.2 ) L-N*32

B-X«S(L)

A-SIN(B) . B<>ces(B)

V(L,1 )-A

(L,2)-B

V(I,,3)-B«S( L)

e (L,4)-A«S(L)

12 Ve 14 N-1,64

14 T(M)-V(N.K)

15 T-.O

V-.O

10 oe TO ( 1 ,2,3.4,5,6,7 ),

J

7 A-C( 1 »•< C( 2)»T( 24 i*Cl 3 )«T( 26 )« 4 )•( T( 27)*C( 2 )»C( 3)«T( 32 ) » )*T( 17)

A-C( 5 )•( A«C(2 )•(€( 3 )»T( 29 )€( 4 )«T( 30))*C(3 )«C( 4 )»T(31 ) )*C( 2)«T( 20)

26
B-C(4 )«<T< ie)*C( 2 )*(C< 1 >«T( 23 )*C< 3 )*T( 28) > •a 1 >«C( 3 )*T( 25) ) A»A*B»C( 3 )»<T( 19 )*CC 1 )»C( 2 )»T< 22 ) )*C< 1 )»T( 21 ) Y-T*C<

A-C( 2 )»(C( 3 )«C< )«T( 60)»C< 3)»a 5 )»T< 61 )*C(4 )•€< 5)«T( 62) ) B-C( 2 )«T( 52 »*C1 3 >«T( 51 )»C( )»T( 50)*a 5 )<K T( 49 )»C( 3)»a 4 )«T( 63 ) ) B-T( 53 )»C( I )•( A*B )»C( 3 )•( C( 4 )«T( 57)*C< 5 )»T( 58))*C(4 )*C< 5 )»T( 59 ) A-C(2 )«(C( 3)*T( 54 )*C( 4 )*T( 55)»a 5 )•( T< 56)»C( 3 )*C( 4)«T( 64 ) ) ) «-ff*C( 6)*( A*B ) 6 A-Cf 4 )*(TC 9 )*C( 1 )*(C( 2 )*T( 14)*C< 3 )*T( 15)) )*0( 2 )«T( 11 )*C< 1 )*T( 12 > T-T»C< 5 )»( A*C( 3 )•( T( 10 )»C( 2 )*( C< 1 )»T( 13)*C< 4)»TC 16) ) ) ) A-T(44 i*C( 1 )»( C(4 )»T( 41 )*C( 3 )*T( 42 )*C( 2 )»T( 43 ) )*C<3 )•« 4 )»T< 47 ) W«W*C( 5)»( A*C< 2 )»(C( 3)»TC 45 )*C( 4 )( T( 46)»a I )»C( 3 )*T<4e ) )) ) 5 T -T 4 )»( C( 1 )»T( 5 ) »C( 2 )»T< 6 ) «3 )•< T( 7 )*C< 1 )»C« 2 )»T( 8 ) ) ) W-W«C< 4 )»C T( 37 )*Cl 1 )•< C( 2 )»T( 38 )« 3 )»T( 39 > )*C( 2 )•« 3 )»T( 40 » » 4 T-Y*C( 3 )*( C( 1 )*T{ 3 )*C( 2 )»T<4) ) W-W»CC3)»( T( 35)*C( 1 )»C<2)»T« 36))

3 T "Y *Cl 1 )»C( 2 )«T< 2 ) U-W €( 2)«T(34 ) 2 Y-Y*T< 1 ) »-W»C( 1 )»T(33 ) «K)-Y

ZCK*4)-«

16 X'X*l

IF ( X.EQ.l ) Oe 70 22

IF (K.LE.4) oe 10 12

KETDBM

1 Z< E ) -T( 1 ) W K«4 )-,0 oe xe 16

20 De 21 K-1,64

21 T(M)-1. oe Ye 15

22 DO 23 N«1.64

23 T( N )-T( M )«S( M ) IF ( E.LE.4 ) oe re 15

BETOBM

END

SDBfieOTINE KeOTS( AB . AC. AD. VI , V2, DEL.B.Z.P, B. J, M

)

DIUEMSieM B( 1 ),Z< 1 ).P( 1 ).B( 1 ).Y( 8)

I-O

V-Vl«Y2

0-V2»Vl

H-V2-V1 I F"2.

V«.C2

M-l

K-0

A-,005

CALL ABC( A. Y.J.I ) D -Y< 1 )»Y( 5 )»A»( 0«Y( 2 )*B»Y( 6 )<-A*V*< Y< 1 )-Y< 5 ) ) ) X-Z<1)

E".15«X

4 CALL ABCtZ.Y.J.I)

A -Yd )*Y(5)*X»C G»Y(2)*E*Y(6>-S*V*( Y( 1 )-Y( 5 ) ) ) T-( 1 .•Y»X»2 )»YC 3 )€ 1 . V«X*X )»Y( 7 )-2. •X^Vl Y( 1 )-Y( S ) ) IF f ABS( A).LT.l .E-6) Oe TO 7

27
IP ( K.GT.2 ) 00 T0 6

C INCKEUEMTING Te eSTAXM FIfiST ZEB0 - AND PPB0XZ1UTE P0B 0THESS

IF <A*I>) s.s.e

5 X-X-E

E-B/2.

IF <M.E0.1 ) 00 T« 4

X"X»1

00 TO 4

a x-x*E oe T0 4

C KEWTeM-EAFHSflN ICETEOD FOB SetVING BQU 10

6 Q-T ©•i T< 2 >-I»Y( 4 ) )*B*( T( 6 )-X«T( 6 ) ) X-X-A/Q

C M0T Te EXCEED 8-TB ITESATI0N

IF f K.GT.e I 00 TO 7 oe le 4

7 B( M )-X

A-£ZP( •X'X^DEL)

Q-T*0*C T( 2 l-X»T<4 } >»B»( Y< 6 >-X»T( 8) > E"2.«A/( X*1»DEL«Q ) Z< M )•£•< T( 2)-Y( 6)-X»V2»( TC 1 >-T( 5 » ) >*AB

P01)*E«AC«AD

B( U )-E«AC*( T( 2 )*T( 6 )-X*Vl*( T< 1 }-T( SI) ) IF ( X«X«DEL.GT.30. ) KETCBM

IF (M.EQ.60) BETOBM

X-0

IF I M.GT.l I Ge Te t IF ( X.GT. .5 ) GO TO 1 F-3,5

0-.002

1 x-x*c

CALL ABC( X. T.J.I

)

D-T<1 »*Y(5)»X*( G*T(2 )*fi*T(6 )-X*V*f TCI >-T(S)))

IF f U.EQ.l ) E-X/4.

IF (M.GT.l ) E-( X<-B(ll-1 ) )/F

X-X»E

00 Te 4

END

28
FEDERAL INFORMATION PROCESSING STANDARD SOFTWARE SUMMARY

01. Summary date 02. Summary prepared by (Name and Phone) Bradley A. Peavy (301) 921-3503

03. Summnry action New Replacement Deletion

Previous EDInternal Software ID

Yr. Mo. Day 7I7 lb ?ll 05. Software title Determination of Thermal Response Factors 04. Software date

Vr. Mo. Day

07. Internal Software ID

06. Short title 08. Software type

Automated Data Q System

Computer Program

(~) Subroutine/Module

09. Processing mode

1

[ Interactive S Batch

1 I Combination

10. General Computer Systems

Q Support/Utility ^

Scientific/Engineering

1

1 Bibliographic/Textual

Application are

Management/

Business

Process Control Other

a

Specific

11. Submitting organization and address National Bureau of Standards

Washington, D.C. 20234

12. Technical contact(s) and phone

B.A. Peavy

Building 226, Room B104

(301) 921-3503

13. Narrative Program computes thermal response factors of multi-layer building constructions (one to seven layers) for thermal conduction applications. Provision is made for including or excluding surface film thermal resistances .Input includes time increment (1 hour, 1/2

hour, etc.), number of layers, thermal properties and descriptions of each layer. Output

includes the thermal response factors, X, Y and Z. Main program is amendable to use

as a subprogram in a larger program for use in thermal conduction applications.

14. Keywords

Dynamic conduction heat transfer, thermal response factors

15. Computer mdnufr and model

UNIVAC 1108

16. Computer operating system 17. Programing language(s)

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 Limited In-house only B

25. Documentation availability

Available Inadequate In-house only

26. FOR SUBMITTING ORGANIZATION USE

185-101 29

standard Form 185 1974 July

U.S. Dept. of Commerce—NBS

(FIPS, Pub. 30)
NBS-1 MA (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUHl.K ATION OK RI-ll^ORTNO.

NBSIR 77-1405

2. Gov't Accession

No.

3. Recipient's Accession No.

A nrinri am p\ ci i d t" i *t i i

" 4, 1 i 1 I, r. ANU bU b 1 1 1 L t*. DETERMINATION AND VERIFICATION OF THERMAL RESPONSE FACTORS

FOR THERMAL CONDUCTION APPLICATIONS

5. Publication Date April 1978

6. Performing Organization Gode

7. AUTHOR(S) Bradley A. Peavy

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 462 6528 ^742 6528)

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report & Period Lovered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

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

bibliography or literature survey, mention it here.)

New formulas for calculating thermal response factors for multiple-layer construction

have been developed by a rigorous derivation. A comparison was made of the time for

computation between the presently used matrix algebra method and the method given

in this paper. Results were obtained using the new method in one-fiftieth to one-half

of the computational time necessary to obtain solutions from the matrix algebra

method

.

Comparisons with another analytical method were performed to verify the accuracy

of the response-factor technique.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by sem.coions) Dynamic Conduction heat transfer; heat transfer; thermal

response factor; verification.

18. AVAILABILITY Unlimited

Q_ For Official Distribution. Do Not Release to NTIS

1 Order From Sup. of Doc, U.S. Government Printing Office Washington. D.C. 20402. SD Cat. No. C13

1 ! Order I-rom National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF PAGES

20. SEC:UR1TY CLASS

(THIS PAGE)

UNCLASSIFIED

22. Price

30 USCOMM-DC 29042-P74