Stability of Numerical Heat Flow Calculations

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Technical Note (National Research Council of Canada. Division of Building Research), 1961-07-01

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Stability of Numerical Heat Flow Calculations

Harmathy, T. Z.

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DIVISION OF BUILDING RESEARCH

NATIONAL RESEARCH COUNCIL OF CANADA

'fE C1HI!if ][

C!

AlL

NOTlE

No.

339

NOT FOR PUBblCATION FOR INTERNAL USE

PREPARED BY T. Z. Harmathy CHECKED BY _GWS APPROVED BY _NBH

PREPARED FOR General Distribution

DATE _{July 1961}

ｾｾ

SUBJECT _{STABILITY OF NUMERICAL HEAT FLOW CALCULATIONS}

In a paper (1) the author showed that in numerical

(step -wise) calculations of one -dimensional heat flow the temperature at some i-th reference point can be computed by means of the

following cor relation

("+1) " .

J 'I. - FJ TJ

+

F TJ.·

+

F TJ

ｾｩ - i (i-I) (i-I) i i 1 i (HI) (HI) (1)

where the expressions for iF{i_l) ,

FI,

and F{Hl) depend on

>:C

the location of the point , and on the "capacity number" and "conductivity number" (or "transfer number II) at the appropriate temperature or temperature difference.

Attention was called to the fact that in all cases

*

The reference point may be inside, interfacial or surface point.

- l

-It was mentioned that major error s can be eliminated from the calculations by the satisfaction of a so -called "criterion of stability" which was given as

Ｎｆｾ

>

O.

1 1

The validity of this inequality remains, however, to be proven. The "stability" of an equation means that errors existing in the solutions at t

=

j.6..t will not increase but will be "smoothed out" during subs equent calculations.

Let

ＮＶＮＮｔｾｩ｟ｬＩＧ

.6..Ti t

｡ｮ､ＮＶＮＮｔｾｩＫｬＩ

be small errors in

ｔｾｩ｟ｬＩ

J T{, and

ｔｾｩＫｬＩＧ

By virtue of equations (1) and (2) the

resulting error in

ｔｾｪＫｬＩ

+ (HI) A . is

1 1

( 3)

.6..

ﾷＨｔｾｪＫｬＩ

+ (j+l) A .) =

ＮｾＮ

.6..Tj. +

(1

1 1 1 (1-1) (1-1)

If <> is a numerical value such that

ｾ ｾｩＩ AT J." - i (i-I) - i.1'(Hl) ｾ 1 (4) then <>

>

ｉＮＶＮＮｔｾｩ｟ｬＩＱ

I.6..T{1 1.6..

ｔｾｩＫｬＩ

I

1.6.. (Tl j +

1)

+

ＨｪＫＱＩａｩＩｉｾｬｬ

-

ｩｆｾｩ｟ｬＩ

-

ｩｆｾｈｬＩｬ＼＾

ＫＨｩｆｾｩ｟ｬＩ

+ i F

{i+l»)

<> ( 5)

"

3

-The condition of the errors to be "smoothed out" is

I

ｾ ＨｔｾｪＫＱＩ

+

U+

1)A_i ))

<

1

6

which is fulfilled if

11

-

i F{i -1)

-

iF{i+1) + ( iF{i -1) + iFt+l)

)

<

1 i. e. , if

iF{i_l) + iF{i+l)

<

1

.

Thus, by virtue of equation (2) inequality (3) follows.

Nomenclatur e

(6)

(7) F j t ｾ t T ｾ

T

''temperature coefficient", dimensionless l, 2, 3, . . . . time, hr time step, hr Temperature, F error in temperature, F Gr eek Letters

maximum error in temperature, F

temperature equivalent of the latent heat abso.rbed or evolved by unit volume of the solid, F

,

4

-Subscripts

(i-I), i, (HI) if placed after the symbol: at the (i-1)-th, i-th, (Hl)-th reference point, respectively; if placed

before

ｴｨｾ

sYmbol: in the expression written for the (i-1) -th, i..;th, (i+l)-th reference point, respectively.

Superscripts

j, (j+l) if placed after the symbol: at t = j At, (j+l)A t, respectively; if placed before the symbol: during the

(j-1)&

<

t

<

j At, j At

<

t ｾ (j+l) A t interval, respectively.

Reference

1. _{Harmathy, T. Z. A Treatise on Theoretical Fire Endurance}