T he B aker C astor Oil C om pan y, N ew Y ork, N. Y.
H
EAT flow through a conducting material is unsteady or transient when the temperature at any given point in the material changes with time. Conditions of unsteady heat conduction are often encountered in engineering work— for example, in the molding and extruding of plastics, thermal processing of food, control of temperature in massive concrete structures, heat treatment of metals, annealing of glass, vulcanization of rubber, flow of heat through walls, etc. Hence, not infrequently the engineer is confronted with the problem of computing the temperature distribution within a material as a function of time.
Several methods are available, and each has advantages as well as disadvantages and limitations. These methods are usually limited in application to some simple geometrical shape such as a plane slab, a cylinder, a sphere, a box-shaped figure, etc. How
ever, most shapes can be considered as approximating one or more of these simple forms. Table I lists several of the methods employed in solving problems of unsteady heat conduction.
Thi3 article deals exclusively with the graphical method of ob
taining temperature distributions. It extends the methods al
ready in use to include several new situations which have not been discussed before in the manner presented here. Credit for the establishment of the graphical method is usually given to Schmidt (19) although the discussion of a paper by Emmons (S) brought out that priority for the method should be assigned to L. Binder.
Although the graphical method is approximate, it circumvents many of the difficulties imposed by the purely mathematical ap
proach. For example, it is readily applicable to more or less com
plex situations which cannot be simply resolved by a rigorous mathematical analysis. Essentially, the graphical method con
sists in replacing the differential equation by an equation of finite differences. In keeping with this idea, the method will be pre
sented in terms of algebra and finite increments of change.
U N ITS
The rate of heat diffusion through a solid conducting ma
terial depends on the factors of temperature difference, time, and
the thermal conductivity, specific heat, and density of the con
ducting material. In working with heat conduction problems, it is necessary that these factors "be expressed in a set of consistent units.
Unless stated otherwise, the units used here are given in Table II. The diffusivity of a material is defined by the equation,
a = k/(cp)
The rate of heat transfer through a unit cross section o f area is as
sumed to follow the basic heat transfer equation, q = k(A t/ Ax)
T E M P E R A T U R E D IS T R IB U T IO N A C R O S S B O U N D A R Y O F C O M P O S IT E W A L L
Schmidt (19) and Sherwood and Reed (SO) outlined graphical methods for obtaining the temperature distribution across the boundary interface of a composite wall. Both methods, however, are rather involved. A third method, considerably simpler, is given here which requires the use of only a single auxiliary con
struction line.
All three procedures require the time period AS to be the same for both of the materials making up the composite wall. Let primes be used to identify the quantities referring to one side of the composite wall:
a s = a o '
IxJ CL D h
-<
cr
L J
Q_
2 ÜJ
I
-Dl STANCE
Figure 1. Method for Obtaining Temperature Distribution across a Composite Wall Interface 990
Ta b l e I. Me t h o d s Em p l o y e d i n Pr o b l e m s o f Un s t e a d y He a t Co n d u c t i o n
Exactness; basic equations lend themselves to. study of general
ized interrelations existing among several factors contrib
uting to unsteady heat conduc
tion
Disadvantages and Limitations
Basic equations in general are highly in
volved and require extensive computa
tion work in getting results. Complex
ity of problem may defy mathematical analysis. Function of initial temp, distribution required. In general, limited to one material and constant k, c, and h
References
e, 8, 12, 18)
Numerical results eval- Quite exact;
uated by prepared tained charts and tables
results quickly ob- Limited to coverage of charts and tables (1,6 ,1 3 ,1 4,1 8 ,2 2 ) extant in literature. Considerable
work necessary to obtain sufficient points for over-all temp, distribution Numerical procedure: ap
idly obtained. Method widely applicable, especially to right- angled shapes. Gives over-all temp, distribution
Good approx. results can be quickly obtained. Applicable to complex temp, distributions (two materials). Gives in visual form the over-all temp, distribu variety of problems, especially to periodic temp, fluctuations over long periods
Fair accuracy
Approx. solution, although precision of method is probably better than data available for problem. Cylindrical and spherical shapes call for somewhat more involved procedure than right- angled shapes
Approx. solution, although accuracy of method is probably better than data available for problem. Construction work requires care and is somewhat tedious
(3)
(4, 5, 7, 9, 11, 16, 17, 19, SO, t l )
Expensive and elaborate equipment re- (16) quired. Training necessary to master technique of setting up apparatus and operating controls. Limitation of use is largely matter of skill with which operator can adjust controls to simu
late unsteady-state conditions
Considerable equipment required. Many (10) inherent faults and imperfoctions in proposed apparatus appear to limit method to rough and approx. results
This means that the lamina thickness Ax and Ax' are inter
dependent. Hence in selecting a thickness value for Ax, it must be borne in mind that this selection arbitrarily fixes A x' for the other side. A judicious choice can usually be made which gives convenient lamina thickness values for both sides.
In the proposed new method, lamina half sections of thickness Ax/2 and A x'¡2 are laid off to either side of the interface bound
ary to give a mixed but full lamina section altogether. The inter
face surface is considered as the midplane of this composite lam
ina section. T o the side of the interface having the larger Ax, an auxiliary construction line is drawn parallel to the boundary line and at a distance Ax (k'/k) from it, or at a distance A x’ (k/k') if
A x' is larger than A x (Figure 1).
In proceeding with the graphical process, temperature steps or temperature increments through the walls to either side of the interface boundary are obtained by standard graphical procedures as described by previous writers (9, 19, SO, SI).
Briefly, the graphical procedure for obtaining successive tem
perature distributions through a homogeneous solid as a function of time consists in progressively connecting alternate temperature values, as plotted on the midplane lines of the several laminae sections into which the solid is conveniently divided, by straight lines. The intersection of each connecting straight line with the intervening midplane which it crosses gives the desired new tem
perature value for the intervening midplane. By proceeding across the solid in some regular manner and by dealing with three adjacent midplanes at a time, a series of temperature increments or jumps are obtained and, when interconnected, result in a new temperature distribution. A succeeding distribution is obtained
by building on a previously completed distribution. The time interval elapsing between successive distributions is given by the relation AB — (A i)* /2a, an expression which involves the lamina thickness value and the diffusivity of the solid under considera
tion.
T o provide for the heat flow contribution of a fluid at a solid- fluid interface, a point corresponding to the temperature of the fluid is located at a distance (k/h) beyond the surface of the solid (a pole center). Any subsequent graphical construction which gives the temperature distribution is then made to tie in with this pole center point. If the value of half the lamina thickness (Ax/2) is less than (k/h), a Active midplane is also to be drawn at a distance (Ax/2) outside of the solid-fluid interface to aid in the graphical construction. This is a purely mechanical device to aid
Ta b l e II. Co n s i s t e n t Un i t s
Rate of heat transfer Q B.t.u./hr.
Total heat transferred Q B.t.u.
Radius r Ft.
Vol. 36, No. 11
- — 4.5 h- 3öft(msutatinq brick)
DISTANCE
-Figure 2. Temperature Distribution across a Composite Brick Wall in closely approximating the temperature distribution actually
occurring. For details of these graphical procedures reference should be made to the sources already noted.
The temperature jumps for the interface midplane, however, are obtained by considering the auxiliary line as one of the adja
cent midplanes and using it in conjunction with the midplane to the other side of the interface boundary to obtain a temperature increment. At all times the temperature of the auxiliary line assumes the temperature value of the midplane it replaces. In Figure 1 the method is shown in detail; U, im, and U give initial temperatures, lm being that o f the interface midplane. T o obtain a new value for the midplane temperature A0 time units later, . temperature U is projected horizontally to intersect the auxiliary line. A straight line connecting U, with this point of intersection cuts the interface midplane to give in, the new temperature value required. The following problem gives an illustrative application of the method.
Pr o b l e m I. A 13.5-inch firebrick wall (9.00-inch firebrick, 4.50-inch insulating brick), initially at a uniform' temperature of 100° F., is subjected to a furnace temperature of 2900° F. for 12 hours. Assuming the values tabulated below, determine the approximate temperature distribution through the wall at the end of this 12-hour period:
W all Material k p e a h
Firebrick 1.02 125 0 .2 6 0.0314 2 0 .0
Insulating brick 0.103 30 0 .2 3 0.0149 5 .0
B y successive trials it is found .that a division of the 0.75-foot firebrick "wall into 3.5 laminae results in a division of the
0.375-foot insulating brick wall into 2.5 laminae. Using these values, the Ad time period for either side of the wall is approximately 0.74 hour:
Thickness, No. o%
W all Material Ft. Laminae Ax At
Firebrick 0 .7 5 3 .6 0 .2 1 4 0.73
Insulating brick 0 .3 7 5 2 .5 0 .1 5 0 0 .7 5 The graphical solution to the problem is given in Figure 2.
The temperature distribution at the end of the 12-hour period is given at the end of the (12/0.74 = ) 16th AS time period.
D E R IV A T IO N O F C O M P O S IT E W A L L R E L A T IO N Consider the three laminae C, M, and H in Figure 1. During a relatively short time period AS, the heat which flows into the composite lamina M (from II) per unit of area is:
Qin — k(lh im) ^
During the same time period the heat which flows out of lamina M (into C) is:
Q ^ = k'(tm - Ic) ~
If AO' is made equal to A0, the net gain of heat by lamina M be
comes:
Qgain — Qin Qout
2500’
2000T
Ld cr
Z ) 1---- I50U
<
cr
L d CL
Ld
I
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C O M P O S IT E W A LL
November, 1944 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y
C
t i n fnrh
A
C O M P O S IT E WALL£ ) numbering -su rfa ce conductance ■
V '
|rs w
-Figurc 3.
num bering-surface conductance finite
Modified Spacing Arrangements to Be Used in Obtaining Temperature Distributions
This heat which is gained by lamina M goes to increase its av
erage temperature from tm to t * ; hence the heat absorbed by the lamina (per unit of area) is:
latedto A rb y the equation A0 = ( Ar)*/2a. How
ever, in laying off the midshell distances, it is necessary to modify the distance Ar between the midradii of adjacent shells.
One spacing schedule for modifying the mid
radius separation distances when the surface conductance is involved is given in column A, Table III, where R is the outside radius of the cylinder and r0, n, r,, etc., are the midshell radii, the numbering starting from the outside (Figure 3E). If the surface conductance is infinite— i.e., if the surface and ambient fluid temperatures are always the same— the spacing schedule in column B of Table III is suggested (Figure 3D).
As done in the case of the plane wall situation, the temperature distribution across a solid-fluid boundary is accomplished by locating a pole center at a distance k/hbeyond the surface of the cylinder and at a temperature corresponding to the temperature of the ambient fluid. An illus
trative problem is given to demonstrate the method.
Pr o b l e m II. A steam pipe (1.25-inch nominal diameter) is covered with cork insulation 1.4 inches thick. The inside of the pipe is suddenly raised from 88° to 211° F. by admitting steam.
Assuming an initial pipe and insulation tempera
ture of 88° F. and neglecting the thermal capacity of the pipe, determine the temperature- distribution through the insulation at the end of 12 minutes,
IB Vo
T
v Q( , j = ( i „ _ g p W + ^ Cork insulationMaterial
k
0 .0 2 5 0.485 8 .1 5 0.00633
h 3 .9
Equating these two expressions for the heat gain, solving for (in — tm ), and arbitrarily setting
(Ax)* , , (A x ')’
A0 = c - and A 9 — AO = • ,
2a 2a'
it is possible to obtain the following expression for ln:
1
(l + — ^
V k A x ' /
_
, ( l k\AX” V + A x ' ) + ‘ k A x ' + tm
At first glance it would appear that U in this expression could not be easily obtained by graphical means. However, a graphical solution can be simply obtained (Figure 3A). If the distance be
tween midplanes C and M is made equal to Ax', and the distance between midplanes M and H is made equal to Ax (k '/ k ), a straight line connecting i* and Uon midplanes H and C,respec
tively, automatically cuts midplane M at a point corresponding to a value of U, the temperature of the midplane M at the end of Ad time units in accordance with the requirements of the equa
tion.
T E M P E R A T U R E D IS T R IB U T IO N T H R O U G H A C Y L IN D E R Trinks (21) outlines a graphical procedure for obtaining the
■temperature distribution across a solid cylinder. The method assumes the cylinder to be made up of a series of concentric cylindrical shells fitted inside of each other, each of thickness Ar.
The temperature at the midradius of any shell is taken as a meas
ure of the average temperature of the shell, and a plot of these midshell temperatures against their radial distances gives the temperature distribution across the cylinder.
The method of obtaining increments of temperature change for given increments of change in time is similar to the method for the plane slab or wall, and as before, time interval AB is
re-A solution is given in Figure 4. The cork insulation is divided into 5>A concentric shells, giving a A r value of 0.0212 foot and a A0 time period of 0.0355 hour. The schedule outlined in column A of Table III was followed. The temperature distribution at the end of 12 minutes is nearly the distribution given at the end of the 3.5th AStime period.
Actual measurements have been made on this particular setup.
The graphical solution gives values which are about 5° F. higher on the average than the measured values. This is probably due to the fact that the thermal capacity of the steel pipe was neg
lected in the graphical procedure and hence a temperature lag at the inner surface of the cork insulation was not taken into ac
count.
D E R IV A T IO N O F C Y L IN D E R R E LA TIO N
Consider the three concentric shells H, M, and C in Figure 5, each of thickness A r and having midradius temperatures of Ik, tm,and le at midshell radii of rk, rm,and r c, respectively. During a relatively short time period AO,the heat which flows into shell M (from H) per foot of linear length is:
( i k - l „ ) k -A O -2 * + y )
Ar
T a b l e III. S p a c i n g S c h e d u l e s f o b C y l i n d e r s _______ M odified Spacing Separation R adii Involved
R and ro R and n
n and ri n and n n and n
A , finite surface conductance
A r/2 Ar/2 Let Y - 2 Ar-R
Y / (n + n ) Y / ln + r,) Y/(u + u)
(etc.)
B, infinite surface conductance
Ar Let X - (R + n ) Ar
X/(n + n) X/Cn + n) X/(n + n)
Vol. 36, No. 11
o<o 8 ■£) rO * o o = <2 £
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on Up