1NDUSTRIAL A N D ENGINEERING CHEMISTRY
1170
Vol. 15, No. 11
Charts for Estimating Temperature Distributions in Heating. or Cooling Solid Shapes’ u
By H. P. Gurney and J. Lurie BOSTON BELTING Co., BOSTON, MASS.
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N many technical procIn Figs. 1 to 5, the abThis paper is abstracted from a more complete exposition entitled esses large solid shapes scissas 0 to 5.5 are in units “On the Heating and Cooling of Certain Solid Shapes.” a special are subjected to temobtained by multiplying paper by one of the writers deposited in the library of the Massaperature changes in order time, t , into the thermal difchusetts Institute of Technology through the courtesy of the Chem.to attain certain physical fusivity, h2,of the material, ical Engineering Department. I n the original it has not been posor chemical results. In so and then dividing this prodsible to go exhaustiuely into the effect of time-temperature relations doing the temperature-time uct by the square of some upon the physical properties of the materials mentioned. Some of relations, irrespective of linear dimension of the these relations are fairly well understood. A better Ayowledge of what point may be taken shape in question, such as time-temperature relations in such processes as the vulcanizing as the time origin, may difthe radius of the sphere or of rubber, the hardening of steel, or the annealing of glass in confer considerably and in concylinder, or by one-half the junction with graphical application of the formulas of heat diffusion sequence cause to exist difthickness of the slab, R2. may help to realize Fourier’s wish expressed in the introduction of erent degrees of physical The expression generally the memoir, “Theorie du mouuement de la chaleur dans Ies corps inhomogeneity. Technical employed in denoting this solides (1812),” that his mathematics be applied to indusfry. examples that may be cited relation is 7 = th2/Rz,where are found to abound in rubthe symbols employed are ber, steel, glass, and other industries. in the order mentioned. The temperature-time relations in the interior of solid The ordinates 1 to 0.002 refer to the ratios of unaccombodies which are either being heated or cooled may be plished temperature change to the total, limiting, or maximum empirically determined by inserted thermometers, thermo- possible temperature change that the thermal environment couples, or other temperature-measuring apparatus, or may can impress upon any part of the body-that is, the temperabe calculated from the assumed conditions, in conjunction ture a t the start is conceived to be unit and to approach or with the physical constants of the materials, the surrounding eventually become zero. At any time, t , in question the contours, shapes, and media. The fact that thermal physi- temperature is conceived to differ from the final temperature cal data are not so complete as they might be, or that tech- by a temperature ratio, A, which approaches zero as a limit. nical conditions can rarely be made to coincide with the the- The ordinates are functions of two other variables besides oretical prototypes, does not negative the value of such time. The first variable, p , is the ratio of the distance from estimations, but does require that the precision of the com- the center, axis, or midplane to the point in question to the puted results should not be overestimated. Where em- distance from the surface to the axis, center, or midplane. pirical observations and theoretical calculations may be This ratio, p, is therefore zero at the center, axis, or midplane, made concurrently, new physical constants may be deter- and unit a t the surface, with intermediate values a t points mined which will be found to be more reliable in predicting between. In Fig. 5, dealing with the semi-infinite solid, other time-temperature relations under different, though only the point in question is considered; hence, p is not involved. similar conditions. The other function, m, is the ratio of the thermal conThe curves in Figs. 1, 2, 3, 4, and 5 were obtained by converting some of the more common formulas for heat diffusion ductivity, k , of the material to the product of the thermal into expressions containing pure ratios or nondimensional surface conductivity, E , into the radius, R , or semithickness variables only, thereby enormously reducing the necessary of the shape. This is expressed symbolically by m = k / E R . basic calculations as well as extending the field of applica- If the surface emissivity is infinity, E = a. That is, if the surface of the body instantly assumes the temperature of bility. 1 Received
May 25, 1923.
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0.5
0.5
0.2 0.1 0.0 5
005
n 0.2
a
0.1 002 0.0 I
002 00I 0,005
0.005 0.002
0002 I
2 RELATIVE TIME
FIG.I-UNACCOMPLISRED
3
4
5
UNITS
TEMPERATURE-CHANGE RATIOSO F A S L A B I N fTfALLY AT UNIFORM TEMPERATURE A N D APPROACHING ANOTHER UNIFORld TEMPERATURE. p DEFINES RELATIVE DISTANCE FROM MIDPLANE; m DEFINES
RELATIVE SURFACE
RESISTlVITY
’
R E L A T ~ ETIME
PIG. 2-UNACCOMPLISWED
UNITS
’
TEMPERATURE-CHANGE RATIOS OF A CYLINDER INITIALLY AT UNIFORM TEMPERATURE AND APPROACHING ANOTHERU N I FORM TEMPERATURB. p DEFINESRELATIVE DISTANCE FROX AXIS; m DEFINES RELATIVESURFACE RESISTIVITY
INDUSTRIAL A N D ENGINEERING CHEMISTRY
November, 1923
the surrounding media, then m = zero. If the surface is perfectly insulated (E = 0), then m = 00. m may also be expressed as the ratio of surface tliermal resistivity to the radius multiplied into the resistivity of the solid material. This follows because of the fact that these are reciprocals, respectively, of E and k.
1171
The relation between temperature Fahrenheit, 8,and A may be graphically determined by connecting points A = 1, 8 = 80" F., and A = 0, 8 = 280" F., by a straight line or by relation
e
=
280
-
~(280 -
so>
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I 0.5 n 02 0.; 005
0.5 n 0.2 0.I
005 00 2 0.0 I I
002 00 I
0005
0.005
0.002
0~002 I
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I FIG. 3 -UNACCOMPLISHED TEMPERATURE-CHANGE RATIOS OF A SPHERE INITIALLY AT UNIFORM TEMPERATURE AND APPROACHING ANOTHER UNIFORM TEMPERATURE. p DEFINES RELATIVE DISTANCE FROM CENTER;m DEFINES RELATIVESURFACE RESISTIVITY
Figs. 1, 2 , and 3 exhibit values of A (between 1 and 0.001) from T = 0 to 7 = 5.5, for p = 0, 0.2, 0.4, 0.6, 0.8, and 1.0, and m = 0, 0.5, 1, and 2 . For constant values of A, is nearly linearly proportional to m; hence, interpolations up to 2 are quite safe, and since the proportionality is even closer the greater m is, extrapolation may be carried up to m = 10 without a greater error than usually resides in the assumed physical constants. Fig. 4 exhibits the values of A for p = 0, 0.2, 0.4, 0.6, 0.8, in a slab, cylinder, or sphere, where the surface is subjected to uniform change in temperature, the total temperature change at any time, t, being considered unity. Fig. 5 was prepared to take care of a special case of either Figs. 1, 2, or 3. This figure shows not only the relative temperittures A for m = 0, 0.5, 1, and 2 , but its differential, d A/dT, with respect to r for points relatively near the surface of any type of solid body-that is, where p lies between 0.9 and 1.0. For lower values of p there would be an increasing error. I n this figure r = th2/R2is replaced by T = th2/X2, where X is the linear distance from the point in question to the surface. To illustrate and apply these figures more definitely, three examples will be adduced, and the transformation from arithlog to arithmetic ordinates will be made. A slab of rubber 2.5 inches in thickness (hence 3.17 centimeters from surface to midplane) is a t a temperatureof 80" F. uniformly distributed, and is placed suddenly in surroundings (such as a heated press) maintained a t a constant temperature of 280" F., and the body is allowed to heat up. It is desired to ascertain the temperature-time relations a t the midplane (where p = 0 ) and a t a point 0.25 inch from the surface (where p = 0.8). The thermal conductivity, K , of the rubber is 0.0004 calorie per centimeter cube per second, while the surface emissivity, E, is 0.0005, and the thermal diffusivity, h,2, is 0.00087 sq. cm. per second. Under these conditions k 0.0004 =
ER
= 0.0005 X
tE
Ra
X 0.00087 3 . 1 7 X 3.17
= t
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2 3 4 RELATIVETIME UNITS
F I G . 4-UNACCOMPLISHED
TEMPERATURE-CHANGE RATIOSAT
!
5
1
POINTS I N
A SLABCYLINDER OR SPHERE INITIALLY AT UNIFORM TEMPERATURE, BUT WITH SURFACE TEMPERATURE UNIFORMLY VARYING FROM INITIAL VALUE
I n Fig. 6, upper part, Fig. 1 is duplicated for p = 0 and m = 0 and 0.5. New dotted lines for m = 0.25 are drawn between m = 0 and m = 0.5, interpolating proportionally between intersections of these lines and lines of constant A. At the right the ordinates are renumbered in terms of temperature Fahrenheit. In Fig. 6, lower part, these dotted lines are converted into solid lines on arithmetic ordinates. A large ingot of steel is suddenly cooled at its surface from 1100" to 100" F., hence undergoing a drop in temperatureof 1000" F. It is desired to ascertain the time-temperature rate of change relations at a distance of 1 em. from the surface, assuming the thermal diffusivity of steel to be 0.15 sq. em. per second. At 7 = 1, expression T = thZ/X2is satisfied if t = 6.35 seconds, and in place of dA/d.r, de/&, 1000" F. per 6.35 seconds or 157.5" F. per second. I n Fig. 7, upper part, the ordinates are correspondingly renumbered, while in Fig. 7, lower part, the relations are converted over and expressed on arithmetic ordinates. A slab of glass 1.5 inches thick (hence 1.9 centimeters from surface to midplane) is subjected to a uniform temperature rate of increase of l oC. per minute starting a t 20" C. and proceeding uniformly for 30 minutes so that the surface p = 0.8, and
I 0.5 d A
82 0.1 005 002 001
0005
0002
3.17 = 0'25
and a t T = 1, t must, equal 11,600 seconds, or 3.22 hours, in order that the following condition be satisfied: 1 =
I
RELATIVE TIME
UNITS
FIG.
5-UNACCOMPLISHED TEMPSRATURE-CHANGE R A T I O S (4 U P P E R C U R V E S ) VARIATION PER U N I T OF TIME C U R V E S ) WITB RATESOF (LOWER FOR A P O I N T IN A SEMI-INFINITE SOLIDINITIALLY AT UNIFORM TEMPERA-
TURE.
m DEFINESRELATIVE SURFACE RESISTIVITY
INDUSXRlAL A N D ENGINEERING CHEMISTRY
l d 72
RELPTIVE TIME UNlTS
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Vol. 15, No. 11
RELATIVE TIME U N I T S
T
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TEMPERATURE C U R V E S AT THE C E N T E R A N D INCH SURFACE OF A SLAB OF RUBBERBEINGHEATED FROM 80' TO 280' F.
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FIG.?-RAT1
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RELATIVE T I M E
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OB COOLING I N A S T E E L INGOT 1 CY. FROY T H E SURFACECOOLINQ FROM
F I G . 6-ESTIMATED
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'
llOOo TO 100" F.
then arrives at a temperature of 50" C. It is desired to ascertain the temperature-time relations a t 0, 0.15, 0.30, 0.45, 0.60, and 0.75 inch from the surface--that is, a t p = 1, 0.8, 0.6, 0.4, 0.2, and 0 from 0 to 30 minutes. T o satisfy equation T = th2/R2when T = 1, t must equal 600 seconds, or 10 minutes. Fig. 8, upper part, presents Fig. 4 renumbered a t T = 1, 2, and 3 to t = 10, 20, and 30, a t which times the surface temperatures are by definition 30", 40°, and 50" C., where A = 1 successively stands for loo, 20°, and 30" C. Fig. 8, lower part, pFesents the same relations on arithmetic ordinates. It will be noted here that after a short time the actual temperature differences between different layers equidistant from the midplane become uniform and the conditions approach a state of uniform flow. It must be perfectly apparent from these examples that where temperature ratios are employed it is immaterial what temperature scale is used. I n evaluation of T in the equation T = th2/RZ, care must be taken that h2 is expressed in the same units as t and R2. It shoiild also be noted that Figs. 1 to 5, and upper parts of 6, 7, and 8 are expressed in arithlog ordinates-that is, the abscissas progress according to equal arithmetic increments, while the ordinates progress in a logarithmic order, equal increments represent equal ratio or percentage increases (or decreases). Because of the fact that many of the curves are nearly straight lines when plotted on arithlog paper, its use is particularly recommendable, especially if extrapolation is to be done. Figs. 6, 7 , and 8, lower parts, are plotted on arithmetic ordinates. Both abscissas and ordinates progress by equal arithmetic incremeii ts
.
On the occasion of the dedication of the Jesse Laboratory of Chemistry at Brown University on October 10, the degree of doctor of science was conferred on James W. McBain of t h e University of Bristol, who made the principal address. The F ~ G %-TEMPERATURES . I N A GLASS S L A B THE S U R F A C E TEMPERATURE degree of doctor of laws was conferred upon James R. Angell, BEINGRAISED1' c. PER MINUTESTARTING AT 20' c. president of Yale. University.
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