Temperatures in Solids during Heating or Cooling Tables for the Numerical Solution of the Heating Equation F. C. W. OLSON
0. T. SCHULTZ
American Can Company, Maywood, Ill.
3925 Davis Place, N. W., Washington, D. C.
f i F T H E numerous attempts to
diffusion theory have the same form The tables Dresented here afford a arrive at a practical Solution rapid of obtaining accurate as the heating equation. Newman (3, 5, 6 ) published considerable work of the heating equation, the Of many problems in the charts of Gurney and Lurie (2) have on the drying of porous solids, using perhaps been the most generallyappliOf heat. The underlymethods identical with those for heat cable to problems arising in industry. ing theory is described briefly, and conduction. Recent work by Biot (1) examples are given to illustrate the on the settlement of soils indicates Schack (8) published a modified type of chart for the heating of the method of application. that the heating equation may play an important part in consoliinfinite slab. Later Newman (4) extended Schack’s work with a dation theory. It is believed, thereSchack type chart for an infinite cylinder and showed how the fore, that these tables will prove useful not only for problems two may be combined to yield solutions for various rectanguin the conduction of heat, but in other fields as well. lar and cylindrical solids. However, particularly in the canning industry, certain instances exist where greater accuracy Theory is required than can be obtained with any of these charts. This is especially true in the early stages of heating or cooling. Although an understanding of the theory underlying the The tables presented here have been prepared to give a rapid construction and application of the tables is not necessary for their use, a brief outline will be given. I n the mathematical solution of the heating equation with an accuracy suficient theory of the conduction of heat, it is convenient to refer for any industrial application. temperatures to a “theoretical” temperature scale. Let T The use of these tables is not limited to the solution of problems in the conduction of heat. The basic equations of be the temperature on a “practical” scale such as Fahrenheit
TABLEI. PHYSICAL EQUIVALENTS AND HEATCONDUCTION FORMULAS OF TWELVE SOLIDS Name Infinite cylinder Semiinfinite cylinder Finite cylinder Semiinfinite solid Quarter-infinite solid Eighth-infinite solid
Physical Equivalent Region of a long cylinder remote from both ends, or a finite cylinder with insulated ends Region of a long cylinder near one of the ends Cylinder whose length is of the same order of magnitude as its diameter Region near a plane face of a large solid Region near the edge or intersection of two perpendicular faces of a large solid Region near the corner or intersection of 3 mutually perpendicular faces of a large solid
Point in Object On axis On axis, distance d from end Geometric center
u = C ( k t / r z ) X P(d/2&)
Distance d from surface Distance dl from one surface, d2 from second surface Distance dl from one surface, dz from second surface, d8 from third iMidway between planes Midway between paralle1 planes, distance d from end Midway between paralle1 planes, distance dl from one edge, d2 from other edge On axis
u = P(d/S&t) u = P(d1/2&)
X P(d2/2fit)
u = P(d1/2&)
X P(d~/2&)
Infinite slab Semiinfinite slab
Region of a large slab remote from the edges Region of a large slab near one plane surface perpendicular to the faces of the slab
Quarter-infinite slab
Region of a large slab near the intersection of two perpendicular surfaces each of which is aeraendicular to the faces of the slab - -
Infinite rectangvlar rod
Region of a long rod of rectangular cross-section remote from both ends, or a rectangular rod with insulated ends Regionof along rod of rectangular cross-section On axis, distance d from near one of the ends end Parallelopiped or brick whose length, width, and Geometric center height are of the same order of magnitude
Semiinfinite rectangular rod Brick
a74
Temperature Given by: u = C(kt/rf)
u = C ( k t / r * ) x S(kt/aZ)
X
p(da/2&) u = S(kt/a*)
u = S(kt/az) X P ( d l 2 G t ) u = S(kt/~z)X P(&pdkt)
P(di/2&)
X
u = S(kt/a*) X S(kt/bz) = S(kt/az) P(dl26t) u = S(kt/a2) S(kt/c2)
u
x X
S(ktlb2)
x
S(]ct/b3
X
INDUSTRIAL AND ENGINEERING CHEMISTRY
July, 1942
or centigrade, TIbe the temperature of the heating or cooling medium, and TObe the initial temperature of the object. Then the theoretical temperature, u, is related to T by u = (TI
- T)/(Tl- To)
(1)
The convenience of this scale is a t once apparent. All temperatures, regardless of the particular heating or cooling conditions, have values between 0 and 1; the heating or cooling temperature u1 is always zero, and the initial temperature % is always unity. The dimensionless quantity u may also be thought of as the ratio of actual temperature rise to maximum possible temperature rise. The heating equation may be written,
kvw
= au/at
(2)
where Va is the Laplacian operator. I n rectangular Cartesian coordinates the heating equation becomes: k(ivU/w
+ awayz +
awaZ2)
= au/at
(3)
The boundary and initial conditions assumed here are: u = ut = 0 at boundaries wJ=l
This corresponds to the case where an object originally at the same temperature throughout its volume is maintained at a constant temperature at its surface. It is assumed that the surface resistivity of the object is zero. The bounding surfaces considered here are: infinite plane, two parallel infinite planes, and infinite circular cylinder. These surfaces form a semiinfinite solid, an infinite slab, and a n infinite cylinder, respectively. The surfaces may be combined in various ways, but they must intersect at right angles. Thus, a pair of infinite parallel planes intersecting an infinite cylinder a t right angles form a right circular finite cylinder, and three pairs of parallel planes intersecting a t right angles form a parallelopiped or brick. The twelve solids which may be formed in this way and their physical equivalents are given in Table I. A certain amount of discretion must be exercised in interpreting the expression “remote from both ends”. Ordinarily a distance four times the diameter of a cylinder or four times the shortest dimension of a rectangular rod is sufficient to be considered remote. Even if great accuracy in estimating the temperature is required, the distance need not be more than ten times the diameter of the cylinder. Let S(0) be the temperature at any point midway between two parallel planes, C(0) the temperature on the axis of an infinite cylinder, and P(0) the temperature at a distance d from the surface of a semiinfinite solid. It can be shown that the temperature at the center of a finite cylinder is C(0) X S(0). Although Williamson and Adams (9) indicated that C(0) and S(0) could be multiplied to give the temperature at the center of a finite cylinder, they merely mentioned the fact and gave no proof. This result was obtained independently by Newman (6) who not only gave a proof but later (4) extended the idea to the case of finite surface resistance. The generalization of this result is immediate and offers no unusual dificulties. For instance, the temperature at the center of a brick is X(0J X X(&) X S(&),where 0 is a function of the thermal diffusivity and dimensions of the brick as well as of the time. Numerical values of C(0) and S(0) may be computed from the series,
c(e)
-
+
+ Aae-R:e + ...
where A,, = 2/RaJt(Rn); 0 = kt/rP
(4)
TAB^ 11.
VALUES OF
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019 0* 020 0.021 0.022 0.023 0.024 0.025 0.026 0.027 0.028 0.029 0.030 0.031 0.032 0.033 0.034 0.035 0.036 0.037 0.038 0.039 0.040 0.041 0.042 0.043 0.044 0.045 0.046 0.047 0.048 0.049 0.050 0.051 0.052 0.053 0.054 0.065 0.056 0.057 0.058 0.059 0.060 0.061 0.062 0.063 0.064 0.065 0.066 0.067 0.068 0.069 0.070 0.071 0.072 0.073 0.074 0.075 0.076 0.077 0.078 0.079 0.080 0.081 0.082 0.083 0.084 0.086 .0.086 0.087 0.088 0.089 0.090 0.091 0.092 0.093 0.094 0.095 0.096 0.097 0.098 0.099 0.loo
875
s(e) AND
LOQs(e)
0
1.00000 1.00000 1.00000 1.00000 1.00000 0,99999 0.99996 0.99984 0.99961 0.99919 0.99850 0.99750 0.99614 0.99439 0.99222 0.98962 0.98661 0.98318 0.97936 0.97516 0.97061 0.96572 0.96052 0.95504 0.94930 0.94333 0.93715 0.93078 0.92424 0.91755 0.91072 0.90379 0.89675 0.88963 0.88244 0.87519 0.86788 0.86055 0.85318 0.84580 0.83840 0.83100 0.82360 0.81622 0.80884 0.80148 0.79414 0.78684 0.77956 0.77231 0.76510 0.75793 0.75080 0.74372 0.73668 0.72968 0.72274 0.71584 0.70900 0.70220 0.69546 0.68877 0.68214 0.67555 0.66903 0.66256 0.65614 0.64978 0.64347 0.63722 0.63103 0.62489 0.61881 0.61278 0,60680 0.60088 0.59602 0.58921 0.58346 0.57776 0.57211 0.56651 0.56097 0.55548 0.55004
0.54466 0.53932 0.53404 0.52881 0.52363 0.51850 0.51342 0.50838 0.50340 0.49846 0,49357 0.48873 0.48394 0.47919 0.47449
10.00000-10 10.00000 10.00000 10.00000 10.00000 10.00000 9.99998 9.99993 9.99983 9.99964 9.99935 9.99891 9.99832 9.99756 9.99660 9.99547 9.99415 9.99263 9.99094 9.98908 9.98704-10 9.98485 9.98251 9.98002 9.97740 9.97466 9.97181 9.96885 9.96578 9.96263 9.95939 9.95607 9.95267 9.94921 9.94568 9.94210 9.93846 9.93478 9.93104 9.92727 9.92345-10 9.91960 9.91572 9.91181 9.90786 9.90389 9.89990 9.89588 9.89185 9.88779 9.88372 9.87963 9.87552 9.87141 9.86728 9.86313 9.85898 9.85482 9.85064 9.84646 9.84227-10 9.83807 9.83387 9.82966 9.82544 9.82122 9,81700 9.81277 9.80853 9.80429 9.80005 9.79580 9.79155 9.78730 9.78305 9.77879 9.77453 9.77027 9.76601 9.76174 9,7574s-IO 9.75321 9.74894 9.74467 9.74040 9.73612 9.73185 9.72758 9 72330 9:$1902 9.71475 9.71047 9.70619 9.70191 9.69763 9.69335 9.68907 9.68479 9.q8051 9.67622
0.101 0.102 0.103 0.104 0.105 0.106 0.107 0.108 0.109 0.110 0.111 0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.120
0.46983 0.46522 0.46066 0.45614 0.45166 0.44723 0.44254 0.43849 0.43419 0.42992 0.42570 0.42152 0.41738 0.41329 0.40923 0.40521 0.40124 0.39729 0.39339 0.38953
0.121 0.122 0.123 0.124 0.125 0.126 0.127 0.12s 0.129 0.130 0.131 0.132 0.133 0.134 0.135 0.136 0.137 0.138 0.139 0.140 0.141 0.142 0.143 0.144 0.145 0.146 0.147 0.148 0.149 0.150 0.151 0.152 0.153 0.154 0.155 0.156 0.157 0.158 0.159 0.160 0,161 0.162 0.163 0.164 0.165 0.166 0.167 0.168 0.169 0.170 0.171 0.172 0.173 0.174 0.175 0.176 0.177 0.178 0.179 0.180 0.181 0.182 0.183 0.184 0.185 0.186 0.187 0.188 0.189 0.190 0.191 0.192 0.193 0.194 0.195 0.196 0.197 0.198 0.199 0.200
0.38571 0.38192 0.37817 0.37446 0.37078 0.36714 0.36353 0.35996 0.35643 0.35293 0.34946 0.34603 0.34263 0.33927 0.33593 0.33264 0.32937 0.32613 0.32293 0.31976 0.31662 0.31351 0.31043 0.30738 0.30436 0.30138 0.29841 029548 0.29258 0.28971 0.28686 0.28405 0.28126 0.27850 0.27576 0.27305 0.27037 0.26772 0.26509 0.26248 0.25990 0.25735 0.25482 0.25232 0.24984 0.24739 0.24496 0.24256 0.24017 0.23781 0.23548 0.23317 0.23088 0.22861 0.22636 0.22414 0.22194 0.21976 0.21760 0.21646 0.21335 0.21125 0.20918 0.20712 0.20509 0.20307 0.20108 0.19910 0.19715 0.19521 0.19330 0.19140 0.18952 0.18766 0.18581 0.18399 0.18218 0.18039 0.17862 0.17687
9.6719410 9.66766 9.66335 9.65909 9.65481 9.65053 9.64624 9.64196 9.63768 9.63339 9.62911 9.62482 9.62054 9.61625 9.61197 9.60768 9.60340 9.69911 9.59483 9.59054 9.68626-10 9.58197 9.57768 9.57340 9.56911 9.56483 9.56054 9.55626 9.55197 9.54768 9.54340 9.53911 9.53453 9.53054 9.52625 9.52197 9.51768 9.51340 9.50911 9.50482 9.50054-10 9.49625 9.49196 9.48768 9.48339 9.47911 9.47482 9.47053 9.46625 9.46196 9.45768 9.45339 9.44910 9.44482 9.44053 9.43624 9.43196 9.42767 9.42339 9.41910 9.41481-10 9.41053 9.40624 9.40196 9.39767 9.39338 9.38910 9.38481 9.38052 9.37624 9.37195 9.36766 9.36338 9.35909 9.35481 9.35052 9.34623 9.34195 9.33766 9.33337 9.32909-10 9.32480 9.32051 9.31623 9.31194 9.30765 9.30337 9.29908 9.29480 9.29051 9.28622 9.28194 9.27765 9.27337 9.26908 9.26479 9.26051 9.25622 9.25193 9 24765
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
876
TABLE 111. 6
VALUES
OF
c(e)ASD E
Vol. 34, No. 7
LOGc(e) Log C ( 0 )
6
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
C(0) 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
10.00000-10 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000
0.101 0.102 0.103 0.104 0.105 0.106 0.107 0,108 0.109 0.110
0.84466 0.84095 0.83722 0.83349 0.82975 0.82899 0.82223 0.81846 0.81469 0.81090
9.92668-10 9.92477 9.92284 9,92090 9.91895 9.91698 9.91499 9.91300 9.91099 9.90897
0.201 0.202 0.203 0.204 0.205 0.206 0.207 0.208 0.209 0.210
C(0) 0.49865 0.49583 0.49303 0.49024 0.48746 0.48470 0.48196 0,47922 0.47651 0.47380
9.69780-10 9,69534 9.69287 9.69041 9.68794 9.68548 9.68301 9.68054 9.67807 9.67560
0.301 0.302 0.303 0.304 0.305 0.306 0.307 0.308 0.309 0.310
0.011 0,012 0.013 0.014 0.015 0.016 0.017 0.018 0,019 0.020
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1,00000 0,99999
10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000 10.00000
0.111 0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.120
0.80712 0.80333 0.79953 0.79573 0.79193 0.78813 0.78433 0.78053 0.77673 0.77293
9.90694 9.90489 9.90284 9.90077 9.89869 9.89660 9.89450 9.89239 9.89027 9.88814
0.211 0.212 0.213 0.214 0.215 0.216 0.217 0.218 0.219 0.220
0.47111 0.46844 0.46578 0.46313 0.46050 0.45788 0.45528 0.45268 0.46011 0.44755
9.67313 9,67065 9.66818 9.66570 9.66323 9.66076 9.65827 9.65580 9,65332 9.66084
0.021 0.022 0.023 0.024 0.025 0.026 0.027 0,028 0.029 0.030
0.99999 0.99998 0.99996 0.99994 0,99991 0.99987 0.99981 0,99974 0.99965 0.99953
0.76913 0.76534 0.76155 0.75776 0.75397 0.75019 0.74642 0.74266 0.73888 0.73613
9.88600-10 9.88385 9.88170 9.87953 9.87736 9.87517 9.87298 9.87078 9.86858 9.86636
0.221 0.222 0.223 0.224 0.225 0.226 0.227 0.228 0.229 0.230
0.44800 0.44246 0.43994 0.43743 0.43494 0.43246 0.42999 0.42753 0.42509 0.42267
0.031 0.032 0.033 0.034 0.035 0.036 0.037 0.038 0.039 0.040
0,99939 0.99921 0,99900 0.99876 0.99847 0.99813 0.99775 0.99731 0.99682 0,99627
0.73137 0.72763 0.72389 0.72017 0.71645 0.71273 0.70903 0.70534 0.70165 0.69798
9.86414 9.86191 9.85968 9.85743 9.85518 9.85293 9.85067 9.84840 9.84612 9.84384
0.231 0.232 0.233 0.234 0.235 0.236 0.237 0.238 0.239 0.240
0.041 0.042 0.043 0.044 0.045 0.046 0.047 0.048 0.049 0.050
0.99666 0.99499 0.99425 0.99345 0,99257 0.99162 0.99060 0.98951 0.98834 0.98710
0.69432 0.69066 0.68702 0.68339 0.67977 0.67616 0.67256 0,66897 0.66540 0.66183
9.84156-10 9.83927 9.83697 9,83467 9.83236 9.83005 9.82773 9.82541 9.82308 9.82075
0.051 0.052 0.053 0.054 0.055 0.056 0.057 0.058 0.059 0.060
0.98578 0.98439 0.98292 0.98137 0.97975 0.97806 0.97629 0.97445 0.97253 0.97054
0.65828 0.66475 0.65122 0.64771 0.64421 0.64073 0.63725 0.63379 0.63035 0.62692
0.061 0.062 0.063 0.064 0.065 0.066 0.067 0.068 0.069 0.070
0.96848 0.96635 0.96416 0.96189 0.95956 0.95717 0.95471 0.95219 0.94961 0.94697
9.98609-10 9.98514 9.98415 9.98313 9.98207 9.98099 9.97987 9.97872 9.97754 9.97634
0.161 0.162 0.163 0.164 0.165 0.166 0.167 0.168 0,169 0.170
0.071 0.072 0.073 0.074 0.075 0.076 0.077 0.078 0.079 0.080
0.94427 0.94152 0.93872 0.93586 0.93295 0.93000 0.92699 0,92394 0.92085 0.91772
9.97510 9.97383 9.97284 9.97121 9.96986 9.96848 9.96708 9.96565 9.96419 9.96271
0.081 0,082 0.083 0.084 0,085 0.086 0.087 0,088 0,089 0,090
0.91454 0 . 9 1 132 0.90807 0.90478 0.90145 0.89810 0.89471 0.89129 0.88784 0.88436
0.091 0.092 0.093 0.094 0.095 0.096 0.097 0.098 0,099 0.100
0.88086 0.87733 0.87378 0.87020 0.86661 0.86299 0,85936 0.85571 0.85204 0.84836
C(6) 0.28086 0.27924 0.27764 0.27604 0.27445 0.27287 0.27130 0.26973
Log C(0)
0.26818 0.26664
9.44849-10 9.44598 9.44348 9.44097 9.43846 9.43595 9.43345 9.43094 9.42843 9.42592
0.311 0.312 0.313 0.314 0.315 0.316 0.317 0.318 0.319 0.320
0.26510 0.26358 0.26206 0.26055 0.25905 0.25756 0.25607 0.25460 0.25313 0.25167
9.42341 9.42090 9.41840 9.41589 9.41338 9.41087 9.40836 9.40585 9.40334 9.40083
9.64836-10 9.64588 9.64339 9,64091 9.63843 9,63594 9,63346 9.63097 9.62849 9.62600
0.321 0.322 0.323 0.324 0.325 0.326 0.327 0.328 0.329 0.330
0.25022 0.24878 0,24735 0.24592 0.24451 0.24310 0.24170 0.24030 0,23892 0.23754
9.39833-10 9.39582 9.39331 9.39080 9.38829 9.38578 9.38327 9.38076 9.37828 9.37574
0.42025 0.41785 0.41546 0.41309 0.41073 0.40838 0.40604 0.40372 0.40141 0.39912
9.62351 9.62102 9,61863 9.61604 9.61355 9.61106 9.60857 9.60608 9.60359 9.60110
0.331 0.332 0.333 0.334 0.335 0.336 0.337 0.338 0.339 0.340
0.23617 0,23481 0.23346 0.23212 0.23078 0.22945 0.22813 0.22681 0.22850 0.22420
9.37323 9.37072 9.36821 9.36570 9.36319 9.36068 9.35817 9.36566 9.35315 0.36064
0.241 0.242 0.243 0.244 0.245 0.246 0.247 0.248 0.249 0.250
0.39683 0.39456 0.39230 0.39005 0.38782 0.38560 0.38339 0.38119 0.37901 0.37684
9.59861-10 9.5961 1 9.59362 9.59112 9.58863 9.68614 9.58364 9.58114 9.57865 9.57615
0.341 0.342 0,343 0.344 0.345 0.346 0.347 0.348 0.349 0.360
0.22291 0.22163 0.22035 0.21908 0.21782 0.21666 0.21531 0.21407 0.21284 0.21161
9.34813-10 9,34562 9.34311 9.34060 9.33809 9.33568 9.33307 9.33066 9.32805 9.32554
9.81841 9.81607 9.81373 9.81138 9.80903 9.80667 9.80431 9,80195 9.79958 9.79721
0.251 0.252 0.253 0.254 0.255 0.256 0.267 0.258 0.259 0.260
0.37467 0.37253 0.37039 0.36827 0.36615 0.36405 0.36196 0.35989 0,35782 0.35577
9.57365 9.57116 9.56866 9.66616 9.56366 9.56116 9.55867 9.55617 9,55367 9.55117
0.351 0.352 0.353 0.354 0.355 0.356 0.357 0.358 0.359 0.360
0.21039 0.20918 0.20798 0.20678 0.20558
9.32303 9.32052 9.31801 9.31550 9.31299 9.31048 9.30797 9.30546 9.30295 9.30044
0.62350 0.62009 0.61670 0.61332 0.60996 0.60661 0,60328 0.59996 0.59668 0.59336
9.79483-10 9.79246 9,79008 9,78769 9.78530 9.78291 9.78052 9.77812 9,77572 9.77332
0.261 0.262 0.263 0.264 0.265 0.266 0.267 0,268 0.269 0,270
0.35373 0.35170 0,34968 0,34767 0.34567 0.34369 0.34171 0.33975 0.33780 0.33586
9.54867-10 9.54617 9.54367 9.54117 9.53866 9.53616 9.53366 9.53116 9.52866 9.62615
0.361 0.362 0.363 0,364 0.366 0.366 0.367 0.368 0.369 0.370
0.19858 0.19743 0,19629 0.19516 0.19404 0.19292 0.19181 0.19070 0.18960 0.18851
9.29793-10 9.29542 9.29291 9.29040 9.28788 9.28537 9.28286 9.28035 9.27784 9.27533
0.171 0.172 0.173 0.174 0.175 0.176 0.177 0.178 0.179 0.180
0.59009 0.58683 0.58358 0,58034 0.57713 0.57392 0.57074 0,66786 0.56440 0.56126
9.77092 9.76851 9.76610 9.76369 9.76127 9.75885 9.75643 9.75401 9.75159 9.74916
0.271 0,272 0,273 0,274 0.275 0.276 0.277 0,278 0.279 0.280
0.33393 0.33201 0.33010 0.32820 0.32632 0,32444 0.32258 0.32072 0.31888 0.31704
9.52365 9.52115 9.51865 9.51614 9.51364 9.51114 9.50863 9,50613 9.50362 9.50112
0.371 0.372 0.373 0.374 0.375 0.376 0.377 0.378 0.379 0.380
0.18742 0.18634 0.18527 0.18420 0,18314 0.18208 0.18103 0.17999 0.17895 0.17792
9.27282 9.27031 9.26780 9.26529 9.26278 9.26026 9,25775 9.25524 9.25273 9.25022
9.06120-10 9.95967 9.95812 9.95654 9.95494 9.96332 9.95168 9.95002 9.94833 9.94663
0,181 0,182 0.183 0.184 0,185 0.186 0.187 0.188 0.189 0.190
0.55813 0.55501 0.55191 0,54883 0.54576 0.54270 0.53966 0.53664 0.53363 0.53063
9.74673-10 9.74430 9.74187 9.73944 9.73700 9.73456 0.73212 9,72968 9,72724 9.72479
0.281 0,282 0,283 0.284 0.285 0.286 0,287 0,288 0.289 0,290
0.31522 0.31341 0.31160 0.30981 0.30803 0,30626 0,30450 0.30275 0.30100 0.29927
9.49861-10 9.49611 9.49360 9.491 10 9,48859 9.48609 9.48388 9.48108 9.47857 9.47607
0.381 0.382 0.383 0.384 0.385 0.386 0.387 0.388 0.389 0.390
0.17689 0.17887 0.17486 0.17385 0.17285 0.17185 0.17086 0.16988 0.16890 0.16792
9.24771-10 9.24520 9.24269 9.24018 9.23767 9.23515 9.23264 9.23013 9.22762 9.2251 1
9.94491 9.94316 9.94140 9.93962 9.93782 9.93601 9.93417 9.93232 9.93046 9.92858
0.191 0.192 0.193 0.194 0.195' 0.196 0.197 0.198 0.199 0.200
0.52765 0.52468 0.52173 0.51880 0,51587 0.51297 0,51008 0,50720 0 50434 0.50149
9.72235 9.71990 9.71745 9.71600 9.71254 9 71009 9.70763 9,70618 9.70272 9.70026
0.291 0,292 0.293 0.294 0,295 0.296 0.297 0.298 0.299 0.300
0.29755 0.29584 0.29414 0,29244 0.29076 0.28909 0.28742 0.28412 0.28577
9.47356 9,47106 9.46855 9.46604 9.46353 9.46103 9.45852 9,45601 9.45351 9.45100
0.391 0.392 0.393 0.394 0.395 0.396 0.397 0.398 0.399 0.400
0.16695 0.16599 0.1F504 0.16408 0.16314 0.16220 0.16126 0.16033 0.15941 0.16849
9.22260 9.22009 9.21758 9.21507 9.21258 9.21004 9.20753 9.20502 9.20251 9.20000
9.99811-10 9.99782 9.99750 9.99714 9.99676 9 . 99635 9.99590 9.99542 9.99491 9.99436
0.141 0.142 0.143 0.144 0.145 0 , i46 0.147 0.148 0.149 0.150
0,28249
0.20440 0.20322 0.20205 0.20089 0.19973
INDUSTRIAL AND ENGINEERING CHEMISTRY
July, 1942
-f and s(e) = ?4r where 0 = kt/L2; L = a, b, c
e-9s2a
+ 1 e-25a2a -
semiinfinite cylinder. What is the temperature a t a point on the axis, 8 cm. from the exposed end after heating 200 seconds? The equation used for the solution of this problem is
5
P(0) is the well known probability integral,
u
C(kt/r2)
x ~(d/24i)
u = C(0.112) X P(0.665)
The functions S(0) and C(0) found in Tables I1 and 111 have been computed for a range of values of 0 likely to cover any encountered in practice. Since tables of the probability function are readily available (7) they are not given here. Values of S(0) and log S(0) are given for values of 0 up to 0.200. For values of 0 greater than 0.200, S(0) and log X(0) may be calculated accurately to a t least five decimal places by s(e)= 1.27324 10 -1 (-4.2863ie) log s(e) = 0.10481 - 4 . 2 8 6 3 ~ Values of C(0) are given for 0 up to 0.400. For values of 0 greater than 0.400, C(0) and log C(0) may be calculated accurately to a t least five decimal places by
c(e) = 1.60197 log-1 (-2.5116ie) c(e) = 0.204654 - 2.5ii6ie
log
From Table I11 and a table of the probability integral, C(0.112) = 0.80333 and P(0.665) = 0.65301, whence
u = (100 - T)/(100 T = 47.6" C.
= 0.803 X 0.653 = 0.524
S ( k t / a z ) = S(O.009) = 0.99961 S(kt/b2) = S(0.085) = 0.55004 S ( k t / c 2 ) = S(O.110) = 0.42992
The following three examples in heat conduction will serve to illustrate the method of using the tables. EXAMPLE I. An iron cylinder of length 24 cm. and radius 18 cm. is initially at 0" C. throughout. The surface is raised to 100" C., and it is desired to know what the temperature a t the center will be after 10 minutes. For iron, k = 0.181 cm.2 sec.-L. Expressing the time in seconds, the data used in the solution are k = 0.181, a = 24, r = 18, TO= 0, TI = 100, t = 600. To indicate the accuracy that may be attained with the tables in this example, we will regard these numbers as exact. Table I gives the formula to be used in this problem: u = (Ti - T ) / ( T , - Tu) = C(kt/+) X S ( k t / a 2 ) 0.181 X 600/(18)z = 0.335185, and kt/a2 = 0.188542. Entering Table I11 with 0 of 0.335185, we find C(0.335185) = 0.23053; and corresponding to the value 0.188542, we find in Table I1 that X(0.188542) = 0.19804. Hence, u = (100 - T)/(100 - 0) = 0.23052 X 0.19804 = 0.045578 T = 95.4346' C. = 203.7823' F. =
To illustrate the use of logarithms of C(0) and S(0), the same problem will be solved using logarithms, and for further illustration the given quantities will be expressed in English instead of metric units. Converting the above numbers to these units, k = 1.683297 square inches per minute, a = 9.44880 inches, T = 7.08660 inches, TO= 32" F., TI= 212" F., t = 10 minutes. The arguments computed from these quantities agree with those already obtained. From Tables I1 and 111, ~ O E Cf0.335185) = 9.36273 - 10 -
10
=
- 0)
EXAMPLE 111. What is the minimum temperature attained in a 6-pound can of spiced ham after heating 2 hours if the initial temperature is 50" F. and the bath temperature is 170" F.? The dimensions of a 6-pound can are nominally: a = 12.625, b = 4.125, c = 3.625 inches. For spiced ham, k = 0.012 square inch per minute (unpublished experimental work at American Can Company laboratory). The formula required for this problem is the last one in Table I. From Table 11,
Application
?
=
as found in Table I. The required data are: k = 0.181, t = 200, r = 18, d = 8, 2'0 = 0, TI = 100. Substituting in the equation,
where e = d / 2 4 % x = integration variable
It is found that kt/r2
a71
-
203.7822' F.
EXAMPLE11. If the iron cylinder of example 1,were insulated a t one end, it would be the physical equivalent of a
therefore,
-
-
(170 T)/(170 u = (TI T)/(T1 - To) = 0.430 X 0.550 X 1.000 = 0.237 T = 141.6" F.
- 50)
Nomenclature The notation used here differs somewhat from the customary notation. We have used a, b, and c to indicate full length, thickness, and width instead of 2a, 26, and 2c as used by most writers on this subject. Thus we are able to express 0 = kt/L2 in the same form in both Tables I1 and 111. length of finite cylinder, thickness of infinite slab, width of rectangular prism, or length of rectangular parallelopiped (brick) b = thickness of rectangular prism, width of brick c = thickness of brick d = distance from bounding surface of semiinfinite solid into solid Jm(z) = Bessel function of z of order m = thermal diffusivity of object = radius of finite or infinite cylinder = nth root of Jo(x) = 0 = temperature on "practical" scale, such as Fahrenheit centigrade, etc. = temperature of heating or cooling medium = initial temperature of object = temperature on "theoretical" scale, related to T by u = (TI T)/(Ti - To) = time = argument of functions given in Tables I1 and I11 where e = kt/LZ and L is a, b, c, or r a
=
-
Literature Cited Biot, M. A,, J . Applied Physics, 12, 155-64 (1941). Gurney, H. P., and Lurie, J., IND. EXG.CAEM.,15, 1170-2 (1923). Newman, A. B., Chem. & Met. Eng., 38,710-13 (1931). Newman, A. B., IND.ENQ.CHEM.,28, 545-8 (1936). Newman, A. B., Trans. Am. Inst. Chem. Engrs., 27, 203 (1931). Ibid., 27, 310 (1931). Peirce, B. O., "A Short Table of Integrals", Boston, Ginn and co., 1910. Schack, A., Stah2 u. Eisen, 50, 1290-2 (1930). Williamson, E. D., and Adams, L. H., P h y s . Rev., 20, 601-6 (1922).
