HEATING AND COOLING RECTANGULAR AND CYLINDRICAL SOLIDS This paper deals with the estimation of surface and central temperatures as functions of time in cases of heating or cooling solids of various shapes by convection. A pair of Schack-type charts is presented for the infinite cylinder, and Schack's charts are reproduced for the infinite slab. It is shown how the charts can be used singly or in pairs to estimate the surface and central temperatures of finite shapes such as rectangular bars, brick shapes (rectangular parallelepipeds),short cylinders, etc., when these shapes are exposed to convective heating or cooling. Mathematical proofs of these new methods are presented. The same methods can be applied to the Gurney-Lurie charts which have been reproduced in several reference
ALBERT B. NEWMAN Cooper Union, New York, N. Y.
T
HE purpose of this paper is to review the
published work on the solution of problems involving heat transfer by convection to and from solid slabs and cylinders, and to extend the utility of the methods which have been developed to other rectangular and cylindrical shapes. In 1923 Gurney and Lurie (3) presented a set of charts for the slab, infinite cylinder, sphere, and semi-infinite solid. By the use of the appropriate chart, it is possible to predict the temperature a t any time and a t any location within the solid, given the constant temperature of the fluid medium to or from which heat is being transferred by convection, the initial uniform temperature of the solid, the surface coefficient of heat transmission (film coefficient), and the density, specific heat, thermal conductivity, and significant dimension of the solid. The Gurney-Lurie charts have been widely accepted by American authors of reference books (4, 5 , 9, 12). I n 1930 Schack (IO) published a chart based upon the same theory as used by Gurney and Lurie. Instead of attempting to present on a single chart the information necessary to determine the temperature and any location in the solid, Schack used one chart for the surface of the solid and another for the center. Other Schack charts could be plotted for intermediate locations within the solid, but in most practical cases the temperature history a t the surface and a t the center is all that is needed. The Schack charts for the slab have also been published in reference works (1, 11) and are reproduced here as Figures 1 and 2. Gurney and Lurie did not publish the mathematical equations underlying their charts, but they were derived independently by the writer in connection with the analogous case of liquid water transfer in the drying of porous solids (6) and checked with the equations published by Grober and Erk ( 2 ) . In order to show how the use of the charts may be extended, it is necessary to present a brief mathematical analysis. The following nomenclature is a modification of that used by McAdams (6A):
works.
Rectangular Shapes The general partial differential equation for heat flow in rectangular coordinates is:
If heat is permitted to flow only in the x direction, there is no temperature gradient in the y or z directions, and the equation reduces to that of the slab:
For a slab of t'hickness 2a, the central plane being a t x = 0, the case of heating or cooling by convection at the surfaces x = T a is represented by the above partial differential equation and by the following conditions :
* k -6t 62
Nomenclature t = temp., varying with time and location in solid
= h(ts
- t) a t x
= =,= a
(2)
Each side of this equation represents the heat quantity passing across unit surface area in unit time.
initial uniform temp. of solid constant temp. of surroundings (fluid medium) 5,v,z, r = distance from center of solid in x, y, z, or T direction 0 = time k = thermal conductivity C = specific heat p = density LY = k/Cp = thermal diffusivity h = surface or film coefficient of heat transmission R = radius of cylinder a, b, e = half-thickness of rectangular solid in 5, y, and z directions, respectively ma, mb, mc, mR = respectively, k/ha, k/hb, k/hc, k/hR, dimensionless ratios Nu,, Nub, Nu,, NUB = modified Nusselt No. = l / m a , l/mb, etc. X,, xb, X , , X R = &/a2, Cyt?/b2, etc. to = la =
6t
k-
62
= Oatz = 0
(3)
representing the condition where there is no heat flow across the central plane of the slab on account of symmetry, t = t o when 8 = 0
(4)
indicating uniformity of temperature a t the beginning of heating or cooling. t = ts when 8 = 545
m
(5)
INDUSTRIAL AND ENGINEERING CHEMISTRY
546 10
VOL. 28, NO. 5
This is the case of a rectangular bar of which the length is very great as compared with the dimensions of a cross section. The thickness of the bar in the x direction is 2a and in the y direction 2b. It can be proved that an equation satisfying the conditions for this case is:
0.9
OB 07 06 Y
a5
t,
-t
- = YXY, t s - to
04
0.3
The proof is carried out by evaluating the partial derivatives:
02 01
'0
002
004
006
008
01
0.3
05
09 I 2
0.7
4
6
8
10
12
14
16
18
20
Nu
FIGURE 1. CHART FOR DETERMINING THE TEMPERATURE HISTORY OF FACES OF RECTAXGULAR SHAPES
indicating that eventually the whole slab will be heated or cooled to the temperature of the surroundings.
P O I N T S ON T H E
SUR-
Substituting these values in the partial differential equation for the bar gives :
An equation, in dimensionless form, satisfying the diff erential equation and the other imposed conditions is From the definitions of Y , and Y , in the cases of unidirectional heat flow, where
and Pn is defined by cot P = map, and values PI, P 2 , etc., being the first, second, etc., roots of this equation. Tabulated values of /3 were published by Newman and Green (8). Y , is set up as a convenient symbol for later use. Figure 1 gives Y , plotted as ordinate against hTu, = l/m, as abscissa for the case of x = =F a, and each curve represents a fixed value of X a = aB/a2. Figure 2 gives corresponding values for the case of x = 0. If it were desired to set up the condition that heat was to flow only in the y direction then the partial differential equation would be
Substituting these values in the equation immediately preceding, both sides become identical. By similar methods, all of the other conditions can be shown to be satisfied and the equation proved valid. For the rectangular parallelepiped (brick shape), the solution is t, -t - to
- Y,Y,Y,
t8
and the general form of differential Equation 1 is satisfied. Thus the single pair of charts for the case of a slab (Figures 1 and 2) can be used for rectangular bars heated or cooled on four sides and brick shapes on six sides. The procedure is
and the solution, complying with the same type of boundary conditions, would be
n=l
cos (0"
s)
=
Yy
It is obvious that Figures 1 and 2 would apply equally well to this case. If h e a t w e r e p e r m i t t e d t o flow in the x and y directions, but not in the z d i r e c t i o n , the differential equation would be:
I,=.($++) 6t
6st
N"
FOR DETERMINING THE TEMPERATURE HISTORY OF POIXTS AT THE CENTDRS FIGURE 2 . CHART OF
RECTANGULAR SHAPES
INDUSTRIAL .AND ENGINEERING CHEMISTRY
MAY, 1936
547
simply the multiplication of the numerical values of the proper ordinate8 read from the charts.
Numerical Example for Heating Rectangular Shapes
Afire brick of dimensions 9 X x 2.5 inches, initially a t 70"F., is suspended in a flue through which furnace gases a t 300" F. are traveling a t a rate which prod u c e s a s u r f a c e coefficient of h = 4.1 B . t . u . per square foot per hour per " F. Estimate the temperatures at t h e f o l l o w i n g points a t the end of one hour: the center of the brick, any corner of FOR DBTERMIKING THE TEMPERATURE HISTORY OF POINTS ON FIGURE3. CHART the brick, the center of the 9 X 4.5 CYLINDRICAL SCRFACES inch faces, the centers of the 9 X 2.5 inch faces, t h e c e n t e r s of Center of brick: the 4.5 x 2.5 inch faces, the middle of the long edges. The 300 - = (0.98)(0.75)(0.43) = 0.316 230 (z = 0 ; y = 0 ; z = 0) data are as follows: t 300 - 72.6 227.4" F. 300 t Corner of brick: ___ = (0.325) (0.29)(0.245) = 0.023 k = 0.3 B. t . u. ft./sq. ft. hr. O F. 230 (Z = a ; y = b ; z = C ) p = 103 lb./cu. ft. t = 300 - 5.3 = 294.7' F. 4.5
'
C = 0.25 B. t. u./lb." F. CY = k / C o = 0.01164 sa. - ft./hr. . t, = 300; to = 70; e = I a = 0.375 ft.; b = 0.1875 ft.; c = 0.104 ft.
Center of 9 X 4.5 face: 300 - t 12 = 0 ,: u- = 0 : z =
(0.98)(0.75)(0.245) = 0.18 t = 300 - 41.5 = 258.5" F.
230
Center of 9 X 2.5 face: 300 - t
From the above :
(z = 0'; y = b ; z = 0)
x, = 0.0828;
NU,
C)
X b = 0.3313; X , = 1.073 5.125; Nub = 2.562; N U , = 1.425
230
Center of 4 . 5 X 2.5 face: 300 (z = a ; y = 0 ; z = 0 )
230
t t
=
= =
(0.98)(0.29)(0.43) = 0.122 300 - 28 = 278" F.
(0.325)(0.75)(0.43) = 0.105 t = 300 - 24.2 = 275.8' F. =
From the charts : For
2,
y, z = 0
0.98 0.75 0.43
For x,y, z = 0.325 0.29 0.245
For all cases: t, - t ~
t, - t o
= _300 ~__
a, b , c
. -
t
300 - 16 = 284" F.
As would be expected, the temperatures are in the following order, from lowest to highest: center of brick, center of 9 x 4.5 face, center of 9 X 2.5 face, center of 4.5 X 2.5 face, middle of longest edge, corner.
t - 300 - t
300 - 70
E
~
230
Cylindrical Shapes The simplest case is a circular solid cylinder of infinite length or its equivalent (short cylinder with insulated ends). For this case, the partial differential equation in cylindrical coordinates is :
I .o
09
0.8 0.7 0.6
Y 0.5
For cylinder of radius r = R, the axis being a t r = 0, the case of heating or cooling by convection is represented by the differential equation and by
0.4
0.3 0.2
0.I
k
0
61 = h(t, 6r
t) at r = R
(2A)
NU
FIGURE4.
CHART FOR DETERMINING THE TEMPERATURE HIBTORY AT THE AXESOF CYLINDRICAL SHAPES
6t
k -6r = O a t r = 0
(3A)
INDUSTRIAL AND ENGINEERING CHEMISTRY
548 t = to when e = 0 = t, when 0 =
Center of gravity:
t
(x = 0;
An equation, in dimensionless form, satisfying these conditions is
n=l
where
A, =
+
(1
m, Pn‘m,’)
[JO(Pn)
I
@)
se
where x is the distance along the axis, and the point x = 0, r = 0 is the center of gravity. If the length is 2a and the diameter 2R, the solution for this case is ta - t = Y,Y, t a - to
and solutions of numerical problems may be effected by the use of Figures 1 to 4.
Numerical Example for Cooling a Cylindrical Shape A circular fire-clay cylinder 4.5 inches in diameter and 5 inches long, initially a t a uniform temperature of 300” F., is suspended in a cooling duct through which air at 70” F. is passing a t a rate which produces a surface coefficient of h = 4.1 B. t. u. per square foot per hour per O F. Estimate the temperatures a t the following locations a t the end of one hour: the center of gravity, the ends of the axis, the cylindrical surface midway between the ends, the edges a t the ends. The data are the same as were given for the previous example except : ta = 70; t o = 300 a = 0.208 ft.; R = 0.1875 ft. From the above: X, = 0.2682; Xr = 0.3313 NU, = 2.85; Nut = 2,562 From the charts : For x,P = 0 0.83 0.625
YI
-1
- to
70--1 - t - 7 0 70 - 300 230
I
U
-
For x,r a, R 0.36fi 0.215
For all cases: ta t,
= 0)
Ends of axis: (a: = a; T = 0 )
‘-230 -
70 - (0.83)(0.525)
t = 70 t------= - 70
230
t
=
+ 100 = 170” F.
(0.36) (0.525) 70 43.5 = 113.5’ F.
+
Cvlindrical surface midway between ends: t-- 70 - (0.83)(0.215) 230 t = 70 41 = 111”F. Edges a t ends: 1 70 -= (0.36) (0.215) (I:= a; T = R) ’ 230 t = 70 17.8 = 77.8” F.
+
and &, is defined by J@) = nar/?J1(/3);/?I, 62, etc., being the first, second, etc., roots of this equation. A table of values of p was published by the writer (6). The essential properties of the Bessel functions Jo( ) and J1( ) and the method of transformation from rectangular to cylindrical coordinates were also given by the writer (7). Taking as variables the three dimensionless quantities Y,, X,,and Nu,, Equation 6 was reduced to a set of tables which were used to plot the curves of Figures 3 and 4, analogous to Figures 1 and 2. The case of the right circular solid cylinder with convection a t the cylindrical surface and the plane ends is represented by 6t 6% 1 6t 6=t = a (6r2 + r 5. +
YZ
T
VOL. 28, NO. 5
+
Miscellaneous Notes In any of the cases given in this paper, except the infinite cylinder, if any of the plane faces not parallel to each other are completely heat-insulated, it is necessary only to double the dimension perpendicular to each insulated face and then to carry out the calculations in the usual way. If a pair of parallel faces is insulated, the corresponding dimension is eliminated. In the case of a laminated solid, in which the thermal conductivity may be different along the various axes, t h e value of thermal diffusivity may be correspondingly adjusted for the different directions, and the calculation carried out as usual. If the surface coefficient, h, is different a t the various faces, these different values of la may be used, the only limitation being that of maintaining symmetry; the value of h must be the same for any pair of parallel faces. The data for the solution of problems may be taken from the Gurney-Lurie plots; the Schack-type plots are preferred by the writer merely on account of easier reading and interpolation.
Acknowledgment The writer wishes to acknowledge the assistance rendered by Boris Gouguell, who computed the tables underlying Figures 3 and 4 and checked the derivations and calculations given in this paper.
Literature Cited Fishenden and Saunders, “Calculation of Heat Transmission,” London, H. M. Stationery Office, 1932. Grober and Erk, “Die Grundgesetae der Wiirmeubertragung,” Berlin, J. Springer, 1933. Gurney and Lurie, IND. ENQ.CHEM.,15, 1170 (1923). Haslam and Russell, “Fuels and Their Combustion,” New York, McGraw-Hill Book Co., 1926. McAdams, W. H., “Heat Transmission,” New York, McGrawHill Book Co.. 1933. (5A) Ibid., p. 839. (6) Newman, A. B., Trans. Am. Inst. Chem. Engrs., 27, 203 (1931). (7) Ibid., 27, 310 (1931). (8) Newman and Green, Trans. Electrochem. Soc., 56, 345 (1934). (9) Perry, Chemical Engineers’ Handbook, New York, McGrawHill Book Co., 1934. (10) Schack, A., Stahl u. Eisen, 50, 1290-2 (1930). (11) Schack, Goldschmidt, and Partridge, “Industrial Heat Transfer,’’ New York, John Wiley & Sons, 1933. (12) Trinks, W., “Industrial Furnaces,” 2nd ed., Vol. 1, New York, John Wiley & Sons, 1926. RECEIVED February 11, 1936.
It
The paper “Heat Transfer Coefficients on Inclined Tubes,” by D. F. Jurgensen, Jr., and G. H. Montillon, IND.ENG. CHEM.,27, 1466-1475 (Dee.. 19351, was also presented as part of this symposium. Other papers will appear in the June issue.
