4042
NOTES
ties (perhaps this is not too surprising when the complex structure of this biopolymer is considered) and that the previous history of the sample can result in different mechanical properties at higher temperatures. Although differences in the high-temperature properties are found by different investigators, there seems to be a persistent loss peak at 180’ and most sources agree that the major melting occurs near 220’.
A Rotating-Disk Thermocell. I. Theory -.
by Benson R. Sundheim and Werner Sauerwein
=
9 0
u
8.8
Department of Chemistry, ;Vew York University, A-ew Y o T ~ , N e w York 10003 (Received April 29, 1466)
8 6
”r:’
8.4
8 0
~
,
‘ r - -
~, 1
~
7 8
-180
-140
-100
-60
-20
20
Iempernture,
~
60 ‘C
100
140
1:
180
220
Figure 2 . Real and loss components of t h e complex shear modulus for Gelatin-I. Note (see text) t h a t there is considerable uncertainty about t h e sample dimensions at elevated temperatures a n d the open circle G’ and G” data are apparent values. T h e solid point shows the true value of G‘ after correction for dimensional changes t h a t have occurred.
be disregarded, for the specimen was very brittle after the run and seemed to have frothed during the latter stages of the experiment. It is well to note that although G’ and G” are dependent on sample dimensions, which obviously are in doubt at the higher temperatures, the mechanical loss, ,“/GI, is independent of dimensional changes. Results similar to these were also obtained with Gelatin-I1 except that we found peaks a t 90 and 115’ with G’ starting its increase at 120’. There were also indications of a melting or R large loss peak in the neighborhood of 180-190’. To summarize, this torsion pendulum work indicates that gelatin has a mechanical loss peak, whose temperature position is unaffected by water, occurring at -85’. For the gelatin-water systems studied, devitrification occurred at the expected temperature of - 10’. When the high-temperature data we obtained are compared with the current literature, there are indications that gelatin from different sources can have different properThe Journal of Physical Chemistry
Studies of thermoelectric properties entail the determination of electrode potentials across accurately known temperature and concentration distributions. A major experimental problem is the maintenance of these distributions free of convective perturbations.’ A related problem, that of determining the concentration profile produced under certain circumstances by the passage of electricity through an electrochemical cell, has been satisfactorily treated by establishing a known forced convection pattern.? The rotating-disk electrode was used for this purpose since complete solutions to the hydrodynamic and diffusion problems are known in this case.2 Here we explore the properties of the rotating-disk electrode system when the electrode and solution are held at different temperatures. One-Component Systenz. For a one-component electrolyte, e.g., fused AgKO3, no compositjon gradients can occur. The thermopotential is readily expressed in terms of the temperature gradient3 so that a knowledge of the temperature distribution is sufficient. A multicomponent system in which the convectioii is sufficient to prevent the establishment of the Sor&t effect will behave in the same way. The meaning of the various symbols employed below is given in the Glossary. (1) H. J. V. Tyrell, “Diffusion and Heat Flow in Liquids,” Butterfield Scientific Publications, London, 1961. This work gives a critical review of experiment and theory as well as extensive references. (2) (a) V. G. Levich, Acta Physicochim. URSS, 17, 257 (1942); (b) V. G. Levich, ibid., 19, 117, 133 (1944); (c) Y. G. Levich, “Physicochemical Hydrodynamics” (in English), Prentice-Hall, Inc., New York, N. Y., 1962. (3) (a) H . Holtan, Thesis, University of Utrecht, 1953; (b) J N. Agar in “The Structure of Electrolytic Solutions,” W.Hamer, Ed., John ‘Niley and Sons, Inc., New York, N. Y., 1959, p. 200; (c) B. Sundheim, “Fused Salts,” JfcGraw-Hill Book Co., Inc., New York. N. Y., 1964, p. 201.
KOTES
4043
For a uniform liquid the heat, balance equation takes the form (1)
q = -kVT
Combining this with the equation of continuity for the heat flux in the absence of sources, sinks, or viscous dissipat,ion4
place within the region 6, which is to be compared with the hydrodynamic boundary layer, 60. 6 = x(To
- Tl)/J
EO . ~ ( X / V ) ~ / ~ ~ O
60 = 3 . 6 ~ / ~
For usual values of cm.2/sec.
x, 1.6
X
for H20, Y =
6 E 0.1560
we obtain
(3)
v,
For the convective velocity? we use the velocity found by solving the hydrodynamic equations for the rotating d i ~ k . ~ , ~ -The - ’ velocity is given as a tabulated function or in asymptotic approximations expressed analytically. For the steady state, in cylindrical coordinates (3) becomes bT br
vr-
bT
4- v+-
a+
bT = by
4- vu-
We seek a solution of the form T u = T,(Y). Then dT ’ ”dy
2) ( > l ) -
k d2T d2T = pc, dy2 ‘dy2
-
~
(5)
T ( 0 ) = To; T(m) = T I The solution to ( 5 ) meeting the appropriate boundary conditions is
T a1
+ al s,” exp[ x
= TO
=
s‘ A s‘ o
v,(z)dz]dl
(6)
(TI- T0)/Lm exp[ x o u,(z)da]dt
The required integrals have been computed* from the series expression for v. An approximate expression2 for the integral leads to an equation for T
T
-
ly
To = (TI - To)
y
=
e-”3du/km e - U 3 d ~(7)
y/61/3x1/Jv1/6/w1/Z
An estimate of the heat flux at the surface can be obtained from ( 7 ) q
=
0.62”’3~-1’6~1’2(T1 - To)
(8)
The major Portion of the change in temperature takes
An important feature of the results is that the thickness of the effective boundary layer (and the detailed distribution) is independent of r. I t is because of this property that the system has been characterized in diffusion as a “uniformly accessible surface.” A graphical representation of eq. 6 (ref. 2, p. 69) shows a nearly linear rise in (T - To)/TQ over approximately the first 80% of its range. The temperature change from surface to bulk naturally is unaffected by the rate of rotation of the disk although the distribution itself is proportional to w-1/2. Consider now a thermocell composed of two cheniically identical electrodes in an electrolytic solution. Let one be a rptating disk held at temperature T, and the second stationary at To (the temperature of the bulk of the solution). It is the conclusion of this study that the temperature distribution between these electrodes is that given in eq. 6 without interference from casual convective mixing. I n view of the generally observed’ linear dependency of E on AT over short temperature spans, it is expected that AEIAT observed by this method will be independent of the speed of rotation. I t is not to be expected that a Sor& effect will be detected at moderate rotation speeds. An experimental test of this method will be published shortly.
Glossary q Heat flux T Temperature E Internal (first-law) energy p Density (independent of T ) cy Heat capacity a t constant volume (independent of T) 7 Viscosity Y = 7 / p Kinematic viscosity k Thermal conductivity Z Convective velocity y, r , + Cylindrical coordinates (4) E.g., R. Bird, W. Stewart, and E. Lightfoot, “Transport Phenomena,” John Wiley and Sons, Inc., New York, N. Y., 1960,p. 315. (5) T. Karman, 2.angew. Math. Mech., 1, 244 (1921). ( 6 ) B. Cochran, Proc. Cambridge Phil. Soc., 30, 365 (1934). Sparrow !I. and J. L,Gregg, J. Heat Transfer, 81,249 (1959). (7) E.> (8) H. c. Riddiford and R. Gregory, J. Chem. SOC., 3756 (1956).
Volume 69, Number 11 November 1966
NOTES
4044
x = k/PC, t Time Y Defined in eq. 7 w Angular velocity 6 Boundary layer thickness for thermal diffusion Boundary layer thickness for ordinary diffusion Acknowledgment. It is a pleasure to acknowledge assistance to this work from the Office of Naval Research (Contract Nonr-285-37 with New York University).
The Vapor Pressure and Heat of Sublimation of Chromium by D. S. Dickson,' J. R. Myers, and R. K. Saxer Department of Mechanics, Air Force Institute of Technology, Wright-Patterson AFB, Ohio (Received June 11, 1966)
A number of references describing the vapor pressure of solid chromium appear in the literature. These data were determined by a variety of techniques and are in fairly good agreement with the results of this investigation; however, the temperature ranges covered by previous investigators did not exceed 1675°K. The vapor pressures determined during this study by means of the Knudsen effusion technique included the temperature range 1560-1800°K. Because hightemperature vapor pressure data are lacking, it was considered that the results obtained during this study were significant.
Experimental Section Equipment and Procedure. The Knudsen method developed by means of statistical thermodynamics and kinetic theory postulates that the rate of effusion of a gas through an orifice into a high vacuum is related to the vapor pressure above the metal. The rate of effusion is related to the vapor pressure of the metal by the expression -,
At
I-
- m2.258 X 1 0 - z dTn
(1)
In these expressions, P is the vapor pressure in atmospheres, W is weight loss in grams during the effusion time interval, A is orifice area in square centimeters, t is effusion time in seconds, R is the universal gas constant, T is absolute temperature in OK., M is the molecular weight of the metal, and m is the effusion rate in grams per second per square centimeter. The Journal of Physical Chemistry
It is assumed in the above expressions that the orifice is ideal ( i e . , infinite thinness). In actual practice, the orifice does have measurable thickness and approximates a short tube or channel. Speiser2 has given equations to correct for this condition. The need for correction factors was avoided in this work by reaming the orifice to a knife-edge of 30" included angle. Calculations based on Ba1son1s3 derivation showed the orifice to be nearly ideal. The vacuum chamber used was fabricated from a 7in. diameter brass cylinder which measured 13 in. high. Copper tubing was used to circulate coolant water around the chamber and through hollow electrode leads to the resistance furnace. A mechanical forepump and an oil diffusion pump were used to create an operating vacuum of at least loF6 mm. A liquid nitrogen trap prevented water and oil vapor in the diffusion pump from reaching the vacuum chamber. Operating temperatures were obtained with a wound, resistance-type, tungsten-wire furnace surrounded by three cylindrical tantalum heat shields. Power was supplied to the furnace by means of a constant-voltage transformer. The Knudsen cells which contained the samples were fabricated from seamless tantalum tubing of 1-in. outside diameter and 0.020-in. wall thickness; cell bases and covers were formed from 0.010-in. thick tantalum sheet and welded in an argon atmosphere to the cells. A Leeds and Northrup disappearing-filament type optical pyrometer, calibrated against a Xational Bureau of Standards pyrometer by means of a standard tungsten ribbon filament lamp, was used to determine temperatures. The calibration was performed with the glass viewport in place to avoid corrections for window transmissibility. All temperature measurements were made at the orifice of the cell, which closely approximated blackbody conditions and eliminated the need for emissivity corrections. Effective times at temperature were calculated to compensate for heating and cooling periods. The effective time was calculated from the equation t e f f = ZAt,[e- A H v / R T A + A H ~ / R T R
1
(2) where At1 is the time interval between any two temperatures, TA is the average temperature during that interval, TR is the temperature of the test, and AHv is the heat of vaporization of chromium in calories per (1) Submitted in partial fulfillment of the requirements for the degree of Master of Science. (2) R. Speiser, "Vapor Pressure of Metals," Engineering Experiment Station, The Ohio State University, Columbus, Ohio, Vol. 19, No. 5. 1947, p. 12. (3) E. W. Balson, J . Phys. Chem., 6 5 , 1151 (1961).
