Lumping errors of analog circuits for heat flow through a homogeneous slab

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Lumping errors of analog circuits for heat flow through a

homogeneous slab

Stephenson, D. G.; Mitalas, G. P.

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Ser

TH1

N21r

2 no. 139

c.

2

I

BLDG

NATIONAL

RESEARCH

COUNCIL

CANADA

DlVlSlON OF BUILDING RESEARCH

LUMPING ERRORS O F ANALOG CIRCUITS FOR

HEAT FLOW THROUGH A HOMOGENEOUS SLAB

BY

D. G. S T E P H E N S O N AND G. P . MITALAS

REPRINTED FROM

INTERNATIONAL DEVELOPMENTS I N HEAT TRANSFER AMERICAN SOCIETY O F MECHANICAL ENGINEERS

PART I, SECTION A, 1961, P. 28 - 3 8

RESEARCH PAPER NO. 139

OF T H E

DIVISION OF BUILDING RESEARCH

PRICE 25 CENTS

OTTAWA

OCTOBER 1961

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This publication

is

being d i s t r i b u t e d by the Division

of Building R e s e a r c h of the National R e s e a r c h Council.

It

should not be reproduced in whole or in p a r t , without p e r m i s

-

sion of the o r i g i n a l publisher. The Division would be glad to

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able a t p a r in Ottawa, to the R e c e i v e r ~ e n k r a l

of Canada,

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Circuits for Heat Flow Through

I

National Research Council Div. of Building Research Ottawa, Canoda

a Homogeneous slab"

ABSTRACT

D. G. Stephenson

G. P. Mitalas

T h i s paper compares t h e transfer functions for s e v e r a l electrical analog circuits, both a c t i v e and p a s s i v e , with the theoretical functions for heat conduction through a homogeneous s l a b . T h e results in- d i c a t e t h e magnitude of t h e errors due to s p a c e lumping a s a function of non-dimensional frequency for e a c h circuit. T h i s permits t h e s e l e c t i o n of the s i m p l e s t circuit which will a c h i e v e a specified accuracy over a specified frequency range.

1. INTRODUCTION

An accurate calculation of t h e heat flux and temperature distribution through the walls, ceiling or floor of a room requires t h e simultaneous calculation of the room s i d e surface heat flux of a l l the elements enclosing the room. Reference

[l]

d i s c u s s e s the u s e of an analog computer for t h i s type of bujlding heat transfer problem. It i s pointed out in t h a t paper that one of t h e important considerations in s e t t i n g up a computer for a room h e a t transfer calculation is:

6 c

Iiow t o simulate accurately t h e wall, roof and floor s e c t i o n s with a s few computer elements a s possible." T h i s paper p r e s e n t s a method of designing analog circuits to c a l s u l a t e one-dimensional h e a t conduction through a homogeneous s l a b with a s p e c i f i e d accuracy.

T h e a n a l y s i s differs from earlier s t u d i e s [ 2 , 3 , 4 , 5 ] in two respects: it compares the frequency r e s p o n s e of the analog c i r c u i t s with t h e theoretical frequency re- s p o n s e of a homogeneous s l a b ; and it i n c l u d e s elec- tronic analog circuits with their p o s s i b i l i t i e s f o r u s i n g higher accuracy difference expressions, a s well a s the p a s s i v e resistance-capacitance ladder networks.

Previous s t u d i e s [ 2 , 3 , 4 , 5 ] have considered the transient r e s p o n s e of p a s s i v e analog circuits a n d have concluded that many elements are. required if a n analog i s to represent accurately t h e response of a s l a b to a sudden change in the driving function. Only the low frequency components of t h e o u t s i d e surface temperature, however, have any s i g n i f i c a n t effect on t h e conditions i n s i d e buildings. T h u s any analog that is accurate for frequencies up to t h e third or fourth harmonic of a diurnal driving function i s quite s a t i s - factory for calculating conditions inside a building.

Designing an analog circuit for a limited frequency range l e a d s to simpler circuits than a r e needed for an a c c u r a t e response t o a s t e p change in t h e driving function. T h u s the frequency response approach i s u s e d in t h i s investigation.

Carslaw and Jaeger [6] s h o w that when t h e tern- peratures a t each surface of a homogeneous s l a b vary sinusoidally, the surface temperatures and h e a t flows a r e related by linear equations which c a n be ex- p r e s s e d as:

* T a b l e s I - I V h a v e been deposited a s Document 6 6 7 9 w i t h the American Documentation Institute P u b l i c a t i o n s Project. A copy may be secured by citing the Document number and by remitting 8 1 . 2 5 for photoprints, or $ 1 . 2 5 for 35 mm microfilm.

(5)

(6)

T h i s error can be reduced by choosing t h e value of

( A ~ ) ~ ( ~ )

d x 2

=

a-l

- ? a +

O i + ,

-r*-

( A x ) 4

(~1:)

-

(7)

a s o that t h e coefficients of t h e neglected d e r i v a - i t i v e s a r e a s s m a l l a s p o s s i b l e . F o r i n s t a n c e , when for

1 <

i

<

N-1

d 3 8

a =

1 ,

the c o e f f i c i e n t of

-

in

( 6 )

i s z e r o s o t h e

d x 3

a2e

E q u a t i o n s

(5)

a n d

( 6 )

also apply a t points

N

and

N-1

first term in the for

7 _ax

is

r e s p e c t i v e l y when

Ax

i s replaced by

- A x

and the

m2

(2).

This

is the same for

all

t h e internal

s u b s c r i p t s

12

1 ,

0 by

N ,

points. T h e error c a n be reduced by d e c r e a s i n g

1

by

N-1,

and

Ax,

i.e. i n c r e a s i n g t h e number of lumps. T h e f i r s t

2

by

N-2.

two terms which a r e neglected in t h e e x p r e s s i o n

Ax

i s t h e d i s t a n c e between the e q u a l l y s p a c e d inter- f o r the temperature gradient a t t h e s u r f a c e , however,

. - n a l p o i n t s a n d X 1 - X o X

-

- N

-

X ~ - l a =

Ax

( A x )

( A x ) 3

(5

)

a r e

-

3 (5)

1

-

12

₁

T h e terms including t h e third and a l l higher When a =

0.5,

the third derivative term i s r e d u c e d to d e r i v a t i v e s a r e u s u a l l y neglected. T h i s c o n s t i t u t e s

t h e error in t h e finite difference approximations. 8

(s),

a n d the fourth derivative term d i s -

INTEGRATOR

FIG.20 ANALOG CIRCUIT T O COMPUTE SURFACE H E A T F L U X AND I N T E R N A L TEMPERATURES FOR A HOMOGENEOUS SLAB WHEN SURFACE T E M P E R A T U R E IS GIVEN.

FIG.2b ANALOG CIRCUIT T O COMPUTE SURFACE TEMPERATURE AND TEMPERATURES THROUGH A HOMOGENEOUS SLAB WHEN SURFACE H E A T F L U X IS GIVEN.

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a p p e a r s . T h i s improvement in t h e a c c u r a c y of t h e e x p r e s s i o n for t h e s u r f a c e g a d i e n t i s p a r t i a l l y o f f s e t by t h e r e a p p e a r a n c e of a

"

6

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in t h e e x p r e s s i o n for

1

(

e)

1

T h e c i r c u i t s for a n operational amplifier type of a n a l o g s h o w n in F i g s . 2 a and 2 b a r e b a s e d on t h e difference e q u a t i o n s for a = 1.0

voltage a t a n y internal point i i s d e s c r i b e d by

T h u s if l*R =

-

( A x )

t h i s t y p e of c i r c u i t w i l l s o l v e the s e t of ordinary dif- f e r e n t i a l e q u a t i o n s for the temperatures through a s l a b .

With t h i s t y p e of a n a l o g t h e s u r f a c e h e a t flux i s r e p r e s e n t e d b y current; t h u s

for 1

<

i

<

N-1 _{T h i s i s e q u i v a l e n t to u s i n g}

Ax = L / N

T h e c i r c u i t s for a = 0.5 a r e only s l i g h t l y more com- p l i c a t e d .

T h e v a l u e s of t h e t r a n s m i s s i o n matrix coeffi- c i e n t s for t h i s type of a n a l o g circuit h a v e b e e n c a l c u l a t e d a n d t h e r e s u l t s for a = 1.0 a r e g i v e n i n T a b l e I, a n d for a = 0.5 in T a b l e

11.

T h e e x p r e s - s i o n s u s e d for c a l c u l a t i n g t h e s e c o e f f i c i e n t s a r e derived i n Appendix

11.

It i s p o s s i b l e t o o b t a i n t h e matrix c o e f f i c i e n t s u s i n g a n a n a l o g computer but t h e r e s u l t s o b t a i n e d in t h i s way i n c l u d e t h e errors d u e t o imperfections of t h e computer components. T h e r e s u l t s c a l c u l a t e d from t h e e x p r e s s i o n s in Appendix

I1

r e p r e s e n t the performance of a n a n a l o g with p e r f e c t components. T h e s i g n i f i c a n c e of t h e d i f f e r e n c e s between t h e matrix c o e f f i c i e n t s for an a n a l o g circuit a n d t h e t h e o r e t i c a l coeffi- c i e n t s for a homogeneous s l a b a r e d i s c u s s e d later.

A p a s s i v e network of r e s i s t a n c e s a n d c a p a c i - t a n c e s can a l s o b e u s e d t o s i m u l a t e t h e h e a t flow in a s l a b . In t h i s c a s e t h e temperatures a r e repre- s e n t e d by v o l t a g e s and t h e h e a t f l o w s by c u r r e n t s . F i g u r e 3 i s a t y p i c a l p a s s i v e a n a l o g c i r c u i t . T h e A T a y l o r ' s s e r i e s e x p a n s i o n about x = O g i v e s T h u s t a k i n g 1, a s proportional t o

%

i n v o l v e s neglect- a . A x d28 ing

7 (=),

a n d a l l t h e h i d e r i v a t i v e s . T h i s higher e r r o r i n t h e s u r f a c e h e a t flux i s a n important d i s a d v a n t a g e of the p a s s i v e networks compared with t h e o p e r a t i o n a l amplifier networks.

T h e

AN

a n d

BN

t r a n s m i s s i o n matrix c o e f f i c i e n t s have b e e n e v a l u a t e d for r e s i s t a n c e - c a p a c i t a n c e networks with a = 0.5 for

N

= 3 ( 1 ) 14. (When a = 0.5 an N lump circuit c o n s i s t s of N-1

"T"

networks in s e r i e s . ) T h e r e s u l t s a r e g i v e n in T a b l e

111.

The

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method u s e d w a s similar to t h e one outlined in T h e error in t h e output h e a t flow i s o b t a i n e d by Appendix I1 for t h e amplifier networks except that interchanging UN a n d

VN

in (20). Thus, t h e maximum the expression for the temperature gradient a t t h e _{possible error in t h e boundary h e a t flows i n d i c a t e d by} s u r f a c e w a s _{t h e analog c a n be c a l c u l a t e d e a s i l y when UN a n d}

_VN

'N

-

'N-I a r e known.

T h e r e s u l t s of t h e matrix coefficient calculations for t h e operational amplifier and t h e resistance- where

Ax

=

L

( N + 2 a - 2 )

3. DISCUSSION OF RESULTS

capacitance c i r c u i t s are plotted in F i g s . 4 a , 4b, and A.. 7 1

N 1

5

in the form

-

and

.

T h e theoretical BN

'

AN

BN

v a l u e s for a homogeneous s l a b taken from reference

7

are a l s o plotted s o the v a l u e s of UN and

VN

c a n be T h e h e a t flows indicated by a n analog circuit measured directly from the graphs. Values of UN a n d are given by

VN are given in T a b l e IV a s functions of $ a n d N for

-

1

40

-

₈₀

t h e various c i r c u i t s . T h e s e data were measured off large s c a l e graphs similar t o F i g s . 4 and

5. [.I=+[;

:I.[.]

(I4) tional accurate, T h e amplifier r e s u l t s of t h e t y p e c i r c u i t s in T a b l e analog s t u d i e d , IV with s h o w a for that = 0.5 the the i s computation opera- the most of surface h e a t flows.

(The s u b s c r i p t s indicate t h a t the quantity i s a s - T h e followinE example i l l u s t r a t e s how the charts

_-

s o c i a t e d with a lumped analog circuit.) If it i s a s - and t a b l e s c a n b e u s e d to c a l c u l a t e the number of sumed that do = din arid

ON

= OOut, equation (14) can lumps needed for a particular problem.

b e subtracted from

(2)

t o give

Example

1.

Problem: How many lumps a r e required in a n operational amplifier a n a l o g circuit ( a = 0.5) to c a l c u l a t e ( 1 5) the heat flow into a 6-in. c o n c r e t e s l a b with a n error

of not more than 2 Btu/ft2 hr when the s u r f a c e tem- 'out perature h a s a n amplitude of 5

F

a t a frequency of

4 cycles/day, and t h e other s u r f a c e i s perfectly in- sulated.

T h e lefthand column matrix e l e m e n t s a r e t h e errors in

a

= 0.04 f t 2 / h r the h e a t flux a s c a l c u l a t e d b y the analog circuit.

k = 1.0 Btu/ft hr

F

A

A N L e t - -

-

= U N e j 61 B N

1

1 - - -

= VNei

6

B N ,j (wt+6,) 'out =

I

'out

I

(17)

R

= L / k = 0.5 f t 2 hr F / B t u when gout = 0

1

'out =

- din

A

T h e error in the analog h e a t flow will have i t s maxi- T h u s l'outl

-

f

0.3

mum p o s s i b l e value of I'in

I

'N

I

'in

I +

VN l'out

I

T h e maximum error in t h e s u r f a c e h e a t flux i s -

-

I

gin

- go

lmax

R

(20)

loin

I

P o u t

I

(9)

FIG.40 POLAR COORDINATE P L O T OF l / A N AND I / B ~ FOR AN OPERATIONAL AMPLIFIER CIRCUIT WITH o = 0.5 AND a = 1.0.

T h u s if t h e error i s t o be l e s s t h a n

2

B t u / f t 2 hr where t h e matrix e l e m e n t s with s u b s c r i p t x a r e for

=

x

6

a n d t h e q u a n t i t i e s without s u b s c r i p t s a r e

T h e following v a l u e s of

UN

_{a n d} a r e taken for

4 =

LJS

.

F o r a n a n a l o g circuit with a = 1.0 from T a b l e IV for

4 =

2.0. V a l u e s for

4 =

1.8

c a n be

obtained by interpolation but the n e a r e s t plotted

v a l u e of 4 u s u a l l y w i l l s u f f i c e t o find the required

N

_BN

T h u s a 4-lump c i r c u i t m e e t s t h e requirement. If the important r e s u l t s of a n a n a l o g computation a r e t h e temperatures a t different p o i n t s through a s l a b rather than s u r f a c e h e a t flux t h e error in t h e tempera- t u r e s may be e s t i m a t e d a s follows:

T h e temperature a t p l a n e x in F i g .

1

i s :

where

Ai

_{a n d}

Bi

a r e matrix e l e m e n t s for a n i lump a n a l o g c i r c u i t with

4i

= I

4.

_{F o r v a l u e s of}_a,_other

N

than unity, t h e

Ai

a n d

Bi

v a l u e s cannot b e t a k e n from t h e t a b l e s but c a n be c a l c u l a t e d by t h e method given in Appendix

11.

T h u s t h e error i n the a n a l o g temperature for the p l a n e xi _{i s}

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Since t h e s e c o e f f i c i e n t s a r e functions of

x / L

a s well a s

N

and

4,

it s e e m s impractical t o prepare . -

t a b l e s of t h e type given for the h e a t flux errors. T h e v a l u e s of the s e p a r a t e f a c t o r s can be obtained from T a b l e

I.

T h e s e c o n d example i l l u s t r a t e s t h e pro- cedure.

Example

2.

Problem: F i n d the maximum p o s s i b l e error in the temperature a t

x

= 0.75

L

for a 6-in. concrete s l a b which i s represented by a four lump operational amplifier analog circuit, a = 1.0. P r o p e r t i e s and boundary conditions s a m e a s for example

1.

Solution:

4

= 1.8 but u s e 2.0 t o avoid need for interpolation of t h e t a b l e s .

1

Data: A . 7 5 L = _.4694

L!??

T h e s e g i v e p . 7 5 L -

03

1

= 0.01

F

MAX and / = 0 . 7 8 F Figure

5

s h o w s that, in g e n e r a l , a r e s i s t a n c e - c a p a c i t a n c e a n a l o g circuit r e q u i r e s more lumps t h a n an operational amplifier a n a l o g circuit to a c h i e v e t h e s a m e a c c u r a c y for heat flow c a l c u l a t i o n s . An

FIG.4b P O L A R C O O R D I N A T E P L O T O F I / A ~ A N D I / B ~ F O R A R E S I S T A N C E - C A P A C I T A N C E L A D D E R N E T W O R K WITH o = 0.5.

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FlG.5 P O L A R COORDINATE P L O T OF AN/BN FOR A RESISTANCE-CAPACITANCE L A D D E R NETWORK WITH a = 0.5 AND AN O P E R A T I O N A L A M P L I F I E R CURCUIT WITH a = 0.5 A N D a = 1.0.

N

lump c i r c u i t o f e i t h e r t y p e , h o w e v e r , w i l l g i v e D. C . B a x t e r of t h e D i v i s i o n of h l e c h a n i c a l E n g i - e x a c t l y t h e s a m e t e m p e r a t u r e s a t t h e i n t e r n a l p o i n t s . n e e r i n g of N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a . I t s h o u l d b e p o s s i b l e , t h e r e f o r e , to u s e a hybrid T h i s p a p e r i s a c o n t r i b u t i o n from t h e D i v i s i o n of a n a l o g which w i l l h a v e t h e s a m e a c c u r a c y a s t h e B u i l d i n g R e s e a r c h of N a t i o n a l R e s e a r c h C o u n c i l of o p e r a t i o n a l a m p l i f i e r t y p e . I t w o u l d u s e a m p l i f i e r C a n a d a a n d i s p u b l i s h e d with t h e a p p r o v a l of t h e a d d e r s t o c a l c u l a t e t h e h e a t f l u x e s a n d a l e s s e x - D i r e c t o r of t h e D i v i s i o n . p e n s i v e r e s i s t a n c e - c a p a c i t a n c e 'network t o corn- p u t e t h e t e m p e r a t u r e s .

REFERENCES

ACKNOWLEDGMENTS

_1.

_{M i t a l a s , G.P., S t e p h e n s o n , D.G., a n d} T h e a u t h o r s g a t e f u l l y a c k n o w l e d g e t h e a s s i s - B a x t e r , D.C., "Use of a n a n a l o g c o m p u t e r f o r room t a n c e t h e y h a v e r e c e i v e d from

J.

H.

Milsum a n d a i r c o n d i t i o n i n g c a l c u l a t i o n s

."

P r o c e e d i n g s of t h e

(12)

Second Conference of the Computing a n d D a t a P r o c e s s i n g Society of C a n a d a , held in Toronto, .June 1 9 6 0 , pp. 175-192 (NRC6099).

2. P a s c h k i s , V. and Heisler,

M.P.,

"The accuracy of measurements in lumped R-C c a b l e cir- c u i t s a s used in the study of t r a n s i e n t h e a t flow," Trans. Amer. Inst. Elect. E n g i n e e r s , Vol. 63, 1944, p. 165.

3. Klein, E.O.P., Touloukian,

Y.S.,

a n d E a t o n ,

J.R.,

"Limits of accuracy of e l e c t r i c a l a n a l o g c i r c u i t s u s e d in the solution of t r a n s i e n t h e a t conduction problems," presented a t the Annual Rleeting of the Amer. Soc. of Mech. E n g i n e e r s , 1952 (ASME P a p e r No. 52-A-65).

4.

Lawson, D.I. and illcGuire, J.H., "The solution of transient h e a t flow problems by analogous e l e c t r i c a l networks," P r o c e e d i n g s , I n s t i t u t w n of Mechanical Engineers,

(A),

Vol. 167, 1953, p. 275.

5. Friedmann, N.E., "The truncation error in

a semi-discrete analog of t h e h e a t equation," J o u r n a l of Math. a n d P h y s , Vol. 35, No. 3, 1956, p. 299.

6. Carslaw, H.S. a n d J.C. Jaeger, Conduction of h e a t i n solids, Oxford University P r e s s , Second Edition, 1959.

7 .

Shirtliffe, C . J. a n d Stephenson, D.G., T a b l e s of c o s h ( l + i ) X, sinh ( l + i ) X

(1+ i) X a n d ( 1

+

i)

X

s i n h ( 1 + i )

X.

National R e s e a r c h Council, Division of ~ u i l d i n ~ ' ~ e s e a r c h , Ottawa, January 1961, 7 1 pp. (NRC6159).

APPENDIX

I:

FINITE DIFFERENCE EXPRESSIONS

FOR SPACE DERIVATIVES

L e t s u b s c r i p t 0 indicate x = 0, s u b s c r i p t

1

indi- c a t e x = a Ax, a n d s u b s c r i p t 2 indicate x=(l+a)Ax. Roman numeral superscript i n d i c a t e s t h e order of derivative with r e s p e c t t o x.

A T a y l o r ' s s e r i e s expansion about the point x = a Ax g i v e s I (Ax)" I1 (AX)' 111 + and

8 , = 8 ,

+ A x e ,

+

-

2!

8,

+----

3 !

8,

(&)4 e l IV + 4!

. . .

(1.2)

Multiplying 1.2 by a and adding t o

1.1

g i v e s a ( l c a )

B,

+

a e 2 = ( l + a ) 0 ,

+

-

(AX)'

8,"

+

2

F o r t h e o t h e r internal p o i n t s ( 1

<

i

<

N

-

1 ) t h e s e c o n d derivative expression i s similar t o (1.4) ex- c e p t t h a t a =

1,

i.e.

Differentiating (1.1) g i v e s

Equations (1.1) and (1.2) give

I

1

l-a a A x e l =

-

a(l+a) a' +

7

'1 + l + a

--'

2

-

a I11 a(1-a)

AX)"^,

- -

2 4 (Ax),

8,

IV (1.7) Multiplying (1.6) by A x a n d substituting (1.7) a n d (1.4) for Ax

8,

and (Ax)"

8,

I' respectively g i v e s

when a = 1.0

11

8 , - 2 8 , + 8 ,

8,

= + 0 8 , I11

--

(Ax)' IV ( I . ~ ) (Ax)

"

1 2

(13)

w h e n a =

0.5 Ax

=

L / N

(11.5)

H e n c e

%

-

12 el

+

4 %

_-

A x

₀

111 _--

( A d 2 IV

_{( I .}

_-30N

₊

3(Ax)'

6

16

2 (11.6)

d

a

a n d

dt

-

(Oi)

= -

( A x ) 2

( O i - ! -

28i

+

eitl)

(11.8)

f o r 1

L

i

L

N - 1

APPENDIX 11: CALCULATION OF TRANSFER

FUNCTIONS FOR AN OPERATIONAL AMPLIFIER

_{Expression for}

_{A N .}

TYPE ANALOG CIRCUIT

T h e s i n u s o i d a l t e m p e r a t u r e a n d h e a t f l o w a t t h e t w o s u r f a c e s of a s l a b a r e g i v e n by e q u a t i o n

(1).

A

s i m i l a r e x p r e s s i o n r e l a t e s t h e v o l t a g e s i n a n a n a l o g c i r c u i t w h i c h s i m u l a t e s t h e h e a t f l o w a n d ternpera- t u r e in a s l a b . T h e s u b s c r i p t on t h e V's i n d i c a t e s t h e t h e r m a l q u a n t i t y w h i c h t h e v o l t a g e r e p r e s e n t s . F o r s i m p l i c i t y in n o t a t i o n

8,

i s s u b s e q u e n t l y u s e d i n p l a c e of

V0

0 a n d s i m i l a r l y for t h e o t h e r q u a n t i t i e s w i t h t h e under- s t a n d i n g t h a t i t r e f e r s t o t h e v o l t a g e w h e n u s e d i n c o n n e c t i o n with a n a n a l o g c i r c u i t .

N

i s t h e t o t a l n u m b e r of l u m p s t h a t a r e u s e d i n t h e c i r c u i t s o

AN

a n d

BN

a r e t h e t r a n s m i s s i o n m a t r i x c o e f f i c i e n t s for a n

N

lump a n a l o g c i r c u i t .

A

s u b s c r i p t i on t h e tern- p e r a t u r e s a n d h e a t f l o w s i n d i c a t e s t h a t t h e q u a n t i t y p e r t a i n s t o t h e o u t p u t of t h e i t h lump. T h u s t h e s u b s c r i p t s

0

a n d

N

r e p r e s e n t t h e t w o s u r f a c e s . C a s e

I . A//

L u m p s t h e Same. T h i s i s t h e c a s e w h e r e a =

1.0.

E q u a t i o n s

( 5 )

a n d

( 6 )

b e c o m e a n d

40N-1

-

30N

-

- ( E ) ~ =

2 A x

When

qN

=

0 _8,

₌

_AN

_ON

a n d

4 0 N - ! = 30N

+

a

L e t

-

= 1

( A d 2

T i = P { ~ ~ I =

1 e-st

t)i

dt

T r a n s f o r m i n g e q u a t i o n s

(11.8)

f o r i = N-1 a n d

(11.9)

g i v e s

("+a

T N - !

=

TN

+ T N - 2

(11.10)

L e t

s + 2

= y

TN

-

1

2

T h e n

-

=

-

TN

' - Y

T h e f o l l o w i n g g e n e r a l r e c u r r e n c e r e l a t i o n s h i p h o l d s for a l l o t h e r p o i n t s T ~ t l - i

T~

t z - i = y

(T) -(T)

(11.14)

T N

f o r 2 r i

N

F o r e a c h v a l u e of

N

t h e r a t i o

(14)

T o g e t t h e s t e a d y periodic r e s p o n s e of a s y s t e m to a s i n u s o i d a l driving function only r e q u i r e s sub- s t i t u t i n g j w for s everywhere in t h e e x p r e s s i o n for t h e L a p l a c e transform of the r e s p o n s e ,

where w = 2 1 7 / P . S i n c e a / ( A x ) ' h a s b e e n a s s u m e d F o r e a c h

N

t h e ratio

_77,

_/

S { R ~ ,

)

= f { B N ) e q u a l t o unity F o r example: for

N

= 3 F o r example: for

N

= 3 T h e i n v e r s i o n of t h i s i s j u s t t h e s a m e a s for t h e AN terms i.e. where

Z

= 2 [ 1 + i ( $ ) 3 3 2 ' - 4 2 - 2 T h u s A, = 4 - 2 Where

2

= 2 + j w = = 2

[

1 + j

@)!

-

(11.19)

Case

2.

H a l f Lumps at the Surfaces. T h i s i s t h e s i t u a t i o n where a = 0 . 5 . T h e finite difference ex- p r e s s i o n s become

E x p r e s s i o n for B N .

When

ON

= 0 _do₌_{BN ( R q N )}

a n d

3 (Ax)'

T h e differential e q u a t i o n s for t h e o t h e r tempera- t u r e s a r e given by 11.8 for I & i

I

N

-

2 .

Again l e t d A x ) ' = 1 and t a k e L a p l a c e t r a n s - forms of 11.20 and 11.21. T h i s g i v e s : a n d 8 do

-

12

0,

+

4 0,

(2)

= 3 ( A x ) Hence

T h e development of the e x p r e s s i o n s for AN

a n d BN for t h i s c a s e i s similar t o c a s e 1 e x c e p t that t h e f i n a l s t e p i s

and

T h e transform of equation (11.8) g i v e s t h e general recurrence r e l a t i o n s h i p for

3 1 i I N

(15)

A U T H O R S R E P R I N T *

* P r e s e n t e d a t the 1 9 6 1 I n t e r n a t i o n a l H e a t T r a n s f e r Conference h e l d A u g u s t 28-September 1, 1961, University of Colorado, Boulder, Colorado. U.S.A. P a p e r s p r e s e n t e d for d i s c u s s i o n a t t h i s m e e t i n g h a v e b e e n p u b I i s h e d in I n t e r n a t i o n a l Developmencs in l l e a t Transfer by T h e American S o c i e t y of Mechanical. E n g i n e e r s . 29 West 39th Street. New York 18. N.Y.