Transient temperatures in a slab heated or cooled on one side

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Transient temperatures in a slab heated or cooled on one side

Harmathy, T. Z.

NATIONAL RESEARCH COUNCIL

CANADA

DIVISION OF BUILDING RESEARCH

TRANSIENT TEMPERATURES IN A SLAB

HEATED OR COOLED ON ONE SIDE

by

T. Z. Harmathy

ANALYZED

Internal Report No. 206

of the

Division of Building Research

OTTAWA

PREFACE

The Fire Research Section of the Division has as one of its

responsibilities the study of the fire endurance of building e lernerita,

Many cases arise in which it is desirable to atternpt to calculate

the rise of ternpe r atur e in a panel or slab which is exposed to fire

on one side. An analytical solution is available for the case in

which one face is suddenly exposed to a new temper atur e, This

equation has been rearranged and used as the basis for a graphical

presentation which facilitates the solution of such pr oblerns ,

This work which is now reported deals also with the dete eminatton

of heat flow in the slab.

The author is a research officer with the Fire Research

Section of the Division and is responsible for fire endurance studies.

Ottawa

October 1960

N. B. Hutcheon

TRANSIENT TEMPERATURES IN A SLAB HEATED OR COOLED ON ONE SIDE

by

T. Z. Harmathy':'

This paper discusses the case of a large slab, one face of which is suddenly exposed to a fluid of temperature T., while the

1

other continues to be in contact with a fluid of constant temperature, T .

o

The need for determining the transient temperatures in such slabs has arisen in the estimation of the "fire endurance" of building elements (the length of time a building member can function under

fire exposure). The solution which is presented here in graphical

and tabulated forms can be used whenever the thermal resistance of the fluid film on the T. side is at least one order of magnitude lower

1

than the resistance of both the slab and the film on the T side. The

o

resistance of the film on the T. side is , as a rule, much lower than

1

that on the other side whenever

l. The fluid at T. is liquid and the fluid at T is gas.

1 0

2.

The fluid at T. is in forced motion, while the fluid at T

1 0

is stagnant.

3. T. is much higher than T

.

1 0

If the thermal resistance of the film on the T. side is

com-1

pletely neglected, the following equations result:

ｾＬ Fire Research Section, Division of Building Research, National·

2 -T

=

T. 1

aT

h(T-T ) + k - = 0 o

ax

T=T o

o

< x

<L

x = 0 t

=

0 (1 ) (2) ( 3) (4)

If the wall is heated, T.

>

T ; if it is cooled, T. < T .

1 0 1 0

The analytical solution of the problem has been presented

*

by Carslaw and Jaeger (Ref. (1), p.126). By rearranging their

equation into the form

T - T 00 T - T 00 0 1

+

h i = 2 k

1

+

h; (1 -

J)

(5 ) a correlation is obtained which can very conveniently be represented on semi-logarithm paper (Fig. 1) by families of curves similar to those

obtained by Gurney (2) and Gurney and Lurie (3, 4) for various cases

in which T

=

T.. Here, of course, the steady- state temperature is not

00 1

constant, but is a function of x:

T =T + ( T . - T )

00 0 1 0

The a are positive

n

h.J

a cot a + - - = 0 k

hi (

x)

1

+k

1

-1

l+h..t k

roots of the equation

( 6)

( 7)

t" Numbers in parenthesis refer to similarly nurrrbe r e d references in

3

-(Some numerical values are given in Appendix IV of Ref. (1).)

Since Fig. 1 is a condensed representation of a function of

three independent variables, it cannot be detailed enough without the

danger of intricacy. In many cases, especially if the group

'tHI/,2

is

small, it might be necessary to use the numerical values listed in Table 1.

Interpolation can best be done on semi-logarithm paper. If

higher accuracy is required, a few values of the ratio (T - T)

I

(T - T )

00 00 0

2

might be calculated, either in the range

(Ktll. )

>

0.3, where only the first

term of the series is irnpor tant, or in the range 0

<

(Ktlt

2)

<

0.05, where

the following approximation can be used:

- T 1

+

h..€ ( - e rf x

)

T

..t-oo _k

=

1 -h l (8) T - T x 2

ｴＮ｛ｾ

00 _{0 1}

+ -

_{( 1}

-r)

k ).2

In practice the temperature of the slab surface in contact with the

fluid of temperature T is of primary interest. These temperatures can

o

more conveniently be taken from Fig. 2, which is a detailed plot for

xli

=1

2

and for lower

ktl..1

values. From eq. (6) the steady-state temperature of

this surface is T = T

+

(T. - T ) oow 0 1 0 1 1

+ hL

k

(9)

4

-Another point of interest is the flow of heat in the slab.

From the equation of Carslaw and Jaeger the following equation can

be derived: 2

tt

t - a -n

1.2

(10) where

aT

q= - k -

_ax

h q = (T. - T ) h D 00 1 0 ｾ 1

+T

and

By substituting ;

=

0 in eq. (10) an equation is obtained for the

heat flow from the fluid of temperature T i in to the slab. A detailed

plot of this equation is seen in Fig. 3.

If there are significant fluid film resistances on both sides of the

slab, the graphs in Figs. 1 to 3 and Table I cannot be used for calculating

the temperature distribution. However, the analytical solution (5). and

numerical values (6) for this case are also available.

( 11)

(12)

5

-Nomenclature

h = coefficient of heat transfer between the wall surface and

2

the fluid at T temperature, Btu/hr ft OF

o

k = thermal conductivity of solid, Btu/hr ft 0 F

1

=

thickness of wall, ft

2 q :: heat flow, Btu/hr ft

T = temper ature, 0 F 2 ｾ = thermal diffusivity, ft /hr Subscripts o

=

at t

=

0, bulk for x ..

.L

i

=

bulk for x 4: 0 m

=

maximum w

=

surface at x

=1

00

=

steady state Bibliography

1. Conduction of Heat in Solids. H. S. Carslaw and J. C. Jaeger. Clarendon

Press, Oxford, 2nd ed , , 1959. 5l0p.

2. Heating and Cooling of Solids of Special Shapes. H. P. Gurney.

Unpub-lished monograph, M.1. T. Library. 53p.

3. Charts for estimating temperature distributions in heating or cooling

solid shapes. H. P. Gurney and J. Lurie. Ind. Eng. Chern; ,

6

-4. Heat Transmission. W. H. McAdams. McGraw-Hill, New York, 2nd ed. ,

1942, p. 32.

5. A Method of Computing the Transient Temperature of Thick Walls.

P. R. Hi l.l, National Advisory Committee for Aeronautics,

Technical Note 4105, Washington, 1957.

6. Temperature Tables Part 1. One-layer Plate, One-space Variable, Linear.

C. J. Thorne. U. S. Naval Ordnance Test Station. Rept , No. 5562,

TABLE 1

Numerical Values of Equation (5)

(T

- T)/(T

- T )

00 00 0 2

ttt/i

h.l

x 0 0.05 O. 1 0.2 0.4 0.8 l.6 k

I

0 1.0 0 0 0 0 0 0 0.2 1.0 0.4729 0.3452 0.2443 O. 1467 0.0547 0.0076 0 0.4 1.0 0.7941 0.6286 0.4616

o.

2790 0.1040 0.0144 0.6 1.0 0.9422 0.8185 0.6304 0.3839 0.1431 0.0199 0.8 l.0 0.9884 0.9191 0.7363 0.4513 O. 1682 0.0234 1.0 l.0 0.9969 0.9493 O. 7723 0.4745

o.

1769 0.0246 0 1.0 0 0 0 0 0 0 0.2 l.0 0.4353

o.

2985 0.1910 0.0956 0.0248 0.0017 0.5 0.4 l.0 0.7624 0.5714 0.3812 0.1921 0.0498 0.0034 0.6 l.0 0.9277 0.7734 O. 5447 0.2768 0.0718 0.0048 0.8 1.0 0.9842 0.8909 0.6577 0.3368 0.0874 0.0059 l.0 l.0 0.9955 0.9295

o.

6996 0.3593 0.0932 0.0063 0 l.0 0 0 0 0 0 0 0.2 1.0 0.4143 0.2724 0.1616 0.0697 0.0134 0.0005 1 0.4 l.0 0.7425 0.5357 0.3318 O. 1441 0.0278 0.0010 0.6 l.0 0.9174 0.7412 0.4860 0.2130 O. 0411 0.0015 0.8 l.0 0.9807 O. 8680 O. 5988 0.2645 0.0510 0.0019 l.0 l.0 0.9941 0.9123 0.6433 0.2851 0.0550 0.0020 0 l.0 0 0 0 0 0 0 0.2 l.0 0.3918 0.2445 O. 1305 0.0451 0.0055 0.0001 2 0.4 l.0 0.7193 0.4937 0.2749 0.0955 0.0117 0.0002 0.6 l.0 0.9037 0.6988 0.4120 O. 1444 0.0178 0.0003 0.8 1.0 0.9754 0.8335 0.5181 0.1829 0.0225 0.0003 1.0 l.0 0.9921 0.8846 0.5633

o.

1996 0.0245 0.0004 0 l.0 0 0 0 0 0 0 0.2 1.0 0.3675 0.2143 0.0978 0.0237 0.0014 0.0000 5 0.4 l.0 0.6911 0.4432 O. 2098 0.0511 0.0031 0.0000 0.6 1.0 0.8844 0.6398 0.3185 0.0779 0.0047 0.0000 " 0.8 l.0 0.9655 0.7753 0.4047 0.0996 0.0060 0.0000 l.0 l.0 0.9869 0.8319 0.4445 O. 1096 0.0066 0.0000 0 l.0 0 0 0 0 0 0 0.2 l.0 0.3411 0.1817 0.0651 0.0090 0.0002 0.0000 0.4 l.0 0.6568 O. 3821 O. 1403 0.0195 0.0004 0.0000 00 0.6 l.0 0.8556 0.5551 0.2101 0.0292 0.0006 0.0000 0.8 l.0 0.9437 0.6683 0.2593 0.0361 0.0007 0.0000 1.0 l.0 0.9660 0.7071 O. 2771 0.0386 0.0007 0.0000

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