Simplified Unsteady-State Conduction Calculations JAMES H. WIEGAND Southwest Research Institute, San Antonio 6 , Tex.
For the infinitely wide plate
20
A =
I 1 1 1 1 1 I I I I I
A,
\I
I
2 sin 4
(c
+ sin 9 cos @)
[COS(&)l
(4)
The value b can be calculated from the same basic variables by the relationships:
I t
For the infinitely long solid cylinder
15
bs2 -=-
t-c
CY
For the sphere a
10
002
x2
2.303
-
*2
2.303
0
004
For the infinitely wide plate Figure 1. Value of A for n = 0 as a function of resistivity for the three shapes
A
I n each of these cases, x, $, and $I are the values (normally written with subscript 1) for the first terms of the solution. The values of these three quantities were obtained from Jakob’s
LTHOUGH the mathematics of conduction of heat in the unsteady state has attracted sufficient interest so that complete equations have been derived for a number of models, the practical use of these equations has been largely limited t o the use of graphs showing a temperature function us. a time function. The possibility of simpler use of these relationships has been pointed out recently by Chung and Jackson ( 1 ) for the case of the solid cylinder and by Geckler ( 8 ) for the hollow cylinder. Both utilize a simple relationship of the temperature difference ratio, Y , t o the time, 6, assuming constant source temperature, to, and constant thermal resistivity, k / h s ,
which is accurate for times e above some minimum. The possible greater utility of this concept is believed worth further discussion and this paper shows several useful methods that can be applied. Under the conditions where only the first term in the solution of the unsteady-state equations applies, and hence Equation 1 is valid, the term A has the following values:
0 10 0
For the infinitely long solid cylinder
..
W.J
For the sphere
A =
n Figure 2.
817
Value of N as a function of the position ratio n for gweral values of resistivity for the infinitely wide plate,
818
INDUSTRIAL AND ENGINEERING CHEMISTRY
voi. 48, N ~ 4.
approaches this value of n, the t,rue temperature curve', which always st,arts a t Y = 1 for 0 = 0, approaches a straight line. (No mathematical proof of the exist,ence of a straight line of actual temperature value through Y = 1 has been attempted, but experimentally determined lines with t,his property have been observed.) A t the value of n giving A = 1, the true temperature curve on a graph of log Y us. 0 is t,hus a st,raight line through Y = 1 a t 0 = 0. iilthough this may appear a t first sight as of only academic interest', it has very useful consequences. The analysis of experimental data N 0.5 in determining thermal diffusivity as shown by Chung and Jackson ( I ) , for example, entails plotting log Y us. 0 and drawing the best straight line through the later data. If, however, all of t h e data lie on a straight line through Y = 1, the data can be analyzed simply by calculation of the least square line through Y = 1, using all of the observed temperature values. This has particular utility, because the largest temperature differences exist at the start, so that the early values of temperature difference are the most accurately determined. 0 The usual method of plotting this type of data 0 0.5 I is t o calculate Y from the temperature differn ences, ( t o - t ) and (lo - t i ) , and plot the reFigure 3. Value of N as a function of the position ratio for several sulting values of Y against the time, e, on values of resistivity for an infinitely long cylinder semilog graph paper. This involves a good deal of tabular calculation of data before plotting can be performed. The constant, b can be quickly obtained by a knowledge only of the temperature at Tables 13-9, 13-10, and 13-8, respectively ( S ) , and are given as the point in quest'ion as a function of time. This approach functions of the resistivity ratio, m, in Table I together with the was derived by the writer for use in t'he analysis of B.O.D. calculated values of bs2/a: for each shape. [In t h e tables of (biochemical oxygen demand) values (4). A similar analysis is Jakob ( S ) , the values of the function are given in terms of bs, useful in heat transfer work, when it can be assumed as for and in the terminologv used here, the resistivitv ratio m = l / b s . ] The calculation of 2- is facilitated by separately calculating the terms J d x n ) , sin(@)/( +n),and cos(+n) for values of the position ratio, n; these terms are denoted as N , . Since a t n = 0, KO = 1, the remaining term can be denoted as Ao, and multiplication of AO by X,, will give the values of A for t h a t value of n. The values of N , a t values of n of 0.2, 0.4, 0.6, 0.8, and 1.0 are tabulated in Table I, together with AO and the resulting values of A . The term AO is shown plotted against m in Figure 1, and the term N , is shown plotted against n in Figures 2, 3, and 4. The value of N on Figure 3 for the solid cylinder is a good average approximation for m = 1, 2, 10, and 20 for all three shapes. The relationships shown in Figures 2, 3, and 4 can also be considered as the relative values of temperature difference P a t different positions on the cross section relative to the value of Y a t the center. This is true, of course, only when Equation 1 is applicable. The value of bs2/a: is shown in Figure 5 for the three cases. T h e term A is shown as a function of n in Figures 6, 7, and 8. It can be seen from Equation 1 that a t e = 1 0, Y = A , and if A = 1, then Y = 1 Thus for A = 1, the straight line on a graph of log I 0 I' us. 0 goes through Y = 1, and Equation 1 applies from 8 = 0. It will be noted from Figures 6, 7 , and 8 that there is a value of n Figure 4. Value of N as a function of position ratio for several for which A = 1 a t each value of m. As one values of resistivity for the sphere
I
m*m
I l I l i l I i I I l I I l l l 1 ~ ~
INDUSTRIAL AND ENGINEERING CHEMISTRY
April 1956
Values of bs2/cr, N , and A for the Three Shapes
Table I. m 0 0.02 0.05 0.10 0.25 1.00 2.00
10.00 20 IO0 m
bsa/a
G 3.14 3.08 2.98 2.84 2.45 1.57 1.17 0.54 0.39 0.00
4.29 4.12 3.86 3.50 2.61 1.07 0.594 0.127 0.066 0.00
No.9
0.936 0.938 0.940 0.947 0.960 0.984 0,991 0.998 0.999 1.000
No.4 0,757 0.765 0.779 0.798 0.851 0.936 0.964 0.992 0.996 1.000
x 0 0.02 0.05
0.10 0.25 1 .oo 2.00 10.00 20.00 m
2.405 2.360 2,290 2.170 1.906 1.253 0.940 0.443 0.315 0.000
2.511 2.397 2.277 2.044 1.577 0.681 0.383 0.0852 0.0430 0.000
NQ.~ NI A0 Sphere 0.505 0.223 0.000 2.000 0.520 0.254 0.0199 1.996 0.547 0.290 0.0559 1.975 0.582 0.337 0.1046 1.926 0.677 0.472 0.260 1.717 0.859 0.757 0.637 1.274 0.920 0.860 0.787 1.145 0.983 0.969 0.952 1.029 0.991 0.984 0.975 1.015 1.000 1.000 1.000 1.000 Infinitely Long Solid Cylinder No.6
0.542 0.561 0.579 0.620 0.699 0.864 0.922 0.983 0.991 1,000
+ 0.00 0.02 0.05 0.10 0.25 1.00 2.00 10.00 m
1.57 1.54 1.50 1.43 1.26 0.86 0.65 0.31 0.00
1.07 1.03 0.98 0.89 0.69 0.321 0.183 0.042 0.00
0.0610 0.1271 0.2783 0.644 0.791 0.952 0.975 1.000 Infinitely Wide Plate
0.310 0.332 0.362 0.414 0.534 0.773 0.868 0.969 1,000
A0.4
1.872 1.872 1.857 1.823 1.648 1.253 1.135 1.028 1.014
1.514 1.527 1.538 1.536 1.460 1.192 1.104 1.022 1.011
1.273 1.273 1.270 1.262 1.223 1.119 1.069 1.016 1,000
1.000
1.511 1.510 1.508 1.492 1.416 1.190 1,103 1.023 1.011 1.000
1.252 1.265 1.275 1.284 1.263 1.134 1.075 1.017 1.008
1.212 1.213 1.213 1.211 1.189 1.102 1.060 1.013 1.000
1.031 1.039 1.083 1.061 1,075 1.053 1.033 1.008 1,000
1,000
AO.B
1.010 1.035 1.082 1.122 1.162 1.096 1.053 1.013 1,006 1.000
A0.s
A1
0.446
0,000 0.040 0.110 0.201 0.447 0,812 0.901 0.980 0,990 1.000
0,508
0,573 0.649 0.811 0.965 0.985 0,997 0.999 1.000
0.868
0.000 0.0460 0.0971 0,1989 0.409 0.778 0.881 0.975 0.987
0,898
0.922 0,970 1.027 1.045 1.027 1.007 1.003 1.000
0.749 0.767 0.790 0.825 0.893 0.973 0.989 0,999 1,000
1,000
0.394 0.423 0.460 0.522 0.655 0.864 0.928 0.985 1,000
0 000 0.0392 0,0899 0.1772 0.375 0,730 0.851 0.968 1,000 I
ture a t equal time intervals necessitates only a subtraction to allow direct plotting of the data on semilog graph paper during the actual test. An example of this method is shown in Table I1 for the data given in Table I of Chung and Jackson (1). Figure 9 shows t h a t the values of log (At/AO) us. 0 and the resulting values of b = 0.112 and 0,105 min.-l compare well with the values of 0.112 and 0.109 obtained by the usual graph of log Y us. 0. Only the temperature values for times greater than e = 5 can be used to determine the desired straight line. The advantage of using the ca8e of A = 1 is evident from this example, as only a portion
I
oc
Ao.2
1.000
0.000 0,0288
5
bs'
819
05
01
00s
m Figure 5. Values of the exponent bst/a! as a function of resistivity ratio for the three shapes
Equation 1 that the thermal resistivity ratio, k / h s , and the source temperature, to, are both constant. From Equation 1 we can write, t = to
- (to
- 1,) A
10-*0
I
(8)
Differentiating, we have dt/d0 = b ( t o - t i ) A (In 10) 1 0 - b @
(9) 05
A graph of log (At/Ae) us. 0 will then be a straight line of slope -b, and intercept log[Ab(ta - t,) In lo]. The value of b can then be used to calculate the quantity for the values of e and a graph of t us. will give, as can be seen from Equation 8, an intercept of t o and a slope of A(to - t,). One advantage of this method is t h a t recording of tempera-
. 0 0
n
I
Figure 6. Value A as a function of position ratio n at severaI values of resistivity for the infinitely wide plate
Figure 7. Value A as a function of position ratio at several values of resistivity for the infinitely long solid cylinder
INDUSTRIAL AND ENGINEERING CHEMISTRY
820
Vol. 48, No, 4
50
40 30
20 15
IO
A t
ne 5 4
3
2
I
5
7
9
II
13
15 1
7
f3,min
Figure 9. Graphical determination of the exponent from the data of Chung and Jackson Figure 8. Value A as a function of position ratio at several values of resistivity for the sphere
cooling of rocket grains follow this relationship w e n for the case of nonhomogeneous hollow cylinders.
of the temperature data are pertinent with A # 1. The values of b obtained from Figure 9 are then used to calculate and these values are plotted against t ; the values of to obtained are 50.0 and 48.7, and the observed slopes, A(ta are 323 and 304. These values of t o and A(to - t , ) and the value of L of 220 result in A = 1.90 and 1.79 for the two cases, as compared Kith the theoletical values of A = 1.60 for this case of m 0.
CONCLUSION
for unsteadgTCe utility of the relationship '1 = A X state conduction has been further illustrated and simplifiedplotting procedures for analysis of temperature-time data are presented. The use of a thermocouple position such that A = 1 is shoivn to increase the data available for determination of h and -4. NOMENCLATURE
Table 11.
Differentiated Time and Temperature Data of Chung and Jackson ( I ) for ti = 220" F.
Time, Min. 1
to
53.2
t 220
3
50.3
187.3
5
49.5
139.0
49.4
103.4
49.4
82.5
9
11
48.6
69.0
13
49.7
61.2
15
49.9
5G.8
17
49.9
54.3
At/A@,
F./rnin.
16.35 24.15 17.8 10.45 6.75 3.9 2.2 1.25
to
61
t 220.2
54.3
187.5
51.6
140.5
51.0
106.0
51.0
85.0
51.0
71.7
51.2
63.8
51.2
58.8
51.3
56.0
= constant in Equation 1
At/W
F./min.
S
=
SO, Sa.2, 16.35 23.5 17.25
Y b h k
= = =
=
7n 12
=
6.65
r
=
3.95
9
=
I
10.5
=
2.5
tt
= =
1.4
to
=
CY
=
$
=
x$
==
e
As has been shown by Geckler ( d ) , experimental determination of b allows calculation of the heat transfer coefficient if the thermal diffusivity is known, and vice versa, for those shapes for which the relation of b and m is known. It is worth particular note that a simple relationship of the type Y = A x is characteristic of the temperature-time relationship in a wide variety of shapes, as long as the constancy of the heat transfer coefficient and of the source temperature can be assumed. Thus as Geckler ( 2 ) shows, the heating and
=
values of A a t n = 0, 0.2, etc.
(x) = Ressel function of first kind and zero order of x J 1 ( x ) = Bessel function of first kind and first order of x J,
Trial 2
Trial 1
A
-40,A,.,, A,.,, Ao.cr Ao.8, A1
=
values of A/Ao
A-o.6, -\rl = values of A' a t n = 0, 0.2, etc. temperature difference ratio, defined by Equation 1 exponent in Equation 1 heat transfer coefficient thermal conductivity resistivity ratio = k / ( h s ) position ratio = r / s distance from center t o measured point, fwt distance from center to outer surface, feet temperature a t point r and time 8, O F. value of t a t e = 0, F. source temperature, F. thermal diffusivity, k / ( pC,), sq. feet/hour t;lme, hours first positive roots of the transcendental equations fol each shape
I
LITERATURE CITED (1) (2) (3) (4)
Chung, P. K , Jackson. AI. L., IND. ENG.CHEM.46, 2563 (1954) Geckler, R D., Jet Propulsion 25, 31-5 (1955). J a k o b , M a x , "Heat Transfer," vol. 1, Wiley, S e w York, 1949 V'iegand, J. H., Sewage & Ind. Wastes 26, 160-3 (1954).
RECEIVED for review M a y 6, 1955.
ACCEPTED November 7, 1955.
