From Figure 9 Tzl//TM = 0.820 and T Z ~ J= 86.1’ C. Thus with the measurements of a mean temperature, the ratio of test section length to cylinder length and the operating conditions as described by 7 , the temperature gradient across the test length as well as the midcylinder temperature can be determined. Acknowledgment T h e authors are indebted to Neal R. Amundson for help in overcoming several mathematical difficulties. They thank Robert Kegler and Dale Biehoffer for making the numerical calculations. This work was directed by one of the authors and was supported by the Atomic Energy Commission, whose financial assistance is gratefully acknowledged. Nomenclature -4 = surface area, sq. feet A,, A,’ = constants a = constant = - QoL (-1)n B(n) n k b = constant C,’, C1, CI’, CZ, Ca = constants = wire diameter, feet; or a conD stant d n [ U l m ] , ds[UlZ] = Jacobian elliptic functions E = constant = 2hoa/kR e = Napierian base F = constant = (2ho/kR - Qoa//I) = constant = 2ho/kR Fo G = constant = Qo/k h = heat transfer coefficient, B.t.u./(hr.)(sq. ft.) (” F.) = heat transfer coefficient a t ho T = 0, B.t.u./(hr.)(sq. ft.) (” F.1 = zeroth order Bessel function Jo[ I = first order Bessel function J1[ 1 k = thermal conductivity a t T , B. t.u./(hr.) (sq. ft.) (” F.)/ft. = thermal conductivitv a t T = ko 0, B.t.u./(hr.)(iq. ft.)(” F.)/ft. L = cyfinderlength, feet 1 = constant = QOu/k - , M I , Ms, m = constants M rn = complementary (or function) parameter of elliptic functions A‘, XI, n‘2, n = constants P = constant = i%ob/kR P = substitution variable, d T / d Z Q,Qo = volumetric heat generation a t T and TO,respectively, B.t.u.(hr.)(cu. ft.) = heat transfer rate, B.t.u./hr. = radius of cylinder, feet r = radial distance, feet s n [ U / k ] ,sd[Ulm] = Jacobian elliptic functions T = temperature difference, t -
--[
” F.
T,
= integrated mean temperature
TM
or measured mean temuerature difference of cylinder. O F. = temperahre difference a t midlength of finite cylinder, F.
848
Liferatore Cited
substituted vari-
able
= substituted variable
-a t T = To=O - 61 = substituted variaTu ._- dz bleat T = Tw elliptic function argument, Y’.M
~~
dGZTAVf T, - T dimensionless variable, ratio of test section length to cylinder length, 2Z”/L dimensionless variable, 2Z/L constants imaginary zeroth order Bessei function length along a finite cylinder, feet 2 - L/2, length along a finite cvlinder. feet one‘ half test section length, feet
c,,
to,
__” _ -$1/92
~
T = Fourier sine transform of T To, Tp, T y = roots of cubic equation in T
iu.
T - 42
‘1
1
temperature difference a t radius R of cylinder, ” F. temperature difference a t ends of test length, ” F. G / F = temperature difference a t midlength of infinitely long cylinder, ” F. ambient temperature, F. temperature a t any spot within or on surface of cylinder, ” F.
GREEKALPHABET CY
= temperature coefficient of re-
P
= temperature
7 , YO
sistivity, (” F.) coefficient of thermal conductivity,
=
(” F.)-l d F L dF0 L ~
,
~
, respectively,
constants
hl, XZ
= constants
7r 7
= substituted variable, = (1
+
T) = 3.14159 = dimensionless group = T m - T m
l m
@
= substituted variable =
B(n)/ W n )
T
+
, ~ j = ~ roots of converted quadratic
polynomial
(1) Biot, J. B., Biblio. Brit. 27, 310 (1804); Trait6 dephys. 4, 669 (1816). (2) Carslaw, H. S., Jaeger, J. C., “Conduction of Heat in Solids,” Oxford Clarendon Press, Oxford, 1947. (3) Clark, J. W., Neuber, R. E., J . Appl. Phys. 21, 1084 (1950). (4) Fischer, Johannes, Arch. Elektrotech. 49, 140-.71 (1951); 262-74 (1952). (5) Fourier, J. B. J., “Thtorie analytique de la chaleur,” Paris, Firmin Didot, 1822. (6) Jacob, M., “Heat Transfer,” vol. 1, Chap. 8, IO, 12, p. 245, Wiley, New York, 1949. (7) Jahnke, P. R. E., Emde, R., “Tables of Functions.” 3rd ed.. G. E. Stechert & Co., New York, 1938. (8) Kohlrausch, Fr., Ann. Physik 4, 200 (1872); 2. Instrumentenk. 18, 139 (1898). (9) McAdams, W. H., Addoms, J. N., Rinaldo, P. M., Day, R. S., Chem. Eng. Progr. 44, 640-51 (1948). (10) Milne-Thomson, L. M., “Jacobian Elliptic Function Tables,” Dover Publications, New York, 1950. (11) Prins, J. A,, Schenk, J., Dumore, J. M., J . Appl. Sci. Research A(3) 4, 272-8 (1952). (12) Rinaldo, P. M., M.S. thesis, Massachusetts Institute of Technology, 1947. (13) Shaw, F. S., “Introduction to Kelaxation Methods,” Dover Publications, New York, 1953. (14) Wilhelm, R. H., Johnson, W. C., Acton, F. S., IND.ENG. CHEM. 35. 562 11953). (15) Will&, Fr. A., “Practical Analysis,” by R. T. Beyer, p. 402, Dover Publications, New York, 1947. 1
j
RECEIVED for review July 6, 1956 ACCEPTED November 18, 1957
CORRESPBN
Thermodynamics of Solutions SIR: I n connection with our paper, “Thermodynamics of Solutions. Determination of Bubble Points at Various Pressures for Prediction of Vapor-Liquid Equilibria” [IND.ENG.CHEM.49, 176974 (1957)], we have received a letter from W. Swietoslawski, Institute of Physical Chemistry, Polish Academy of Sciences, Warszawa 22, Poland, commenting on the operation .of the ebulliometer. Professor Swietoslawski called our attention to the fact that we erroneously quoted his work (“Ebulliometric Measurements,” Reinhold, New York, 1945)
INDUSTRIAL AND ENGINEERING CHEMISTRY
concerning the matter of drop rate variation for optimum operation when using different liquids. H e states that his measurements have shown that the optimum drop rate varies inversely with the enthalpy of vaporization and the density of liquid and can be considerably higher than 7 to 25 drops per minute; in fact, for benzene the lowest and highest number of drops per minute vary from 40 to 50 u p to 150 to 160. Measurements in our laboratory are in accordance with his findings of many years ago. In a forthcoming paper, reporting additional work, we plan further discussion of this matter.
€1. WILLIAMPRENGLE, JR. University of Houston Houston 4, Tex.
