Thermodynamic Consistency Test Methods Two consistency tests for excess molal properties, zE, of binary mixtures are derived. The functions zzE/xl' employed here to check the reliability of thermodynamic data are more sensitive to errors than the commonly used functions zlE and Z Z ~The . proposed methods supply a consistency criterion for individual points as well as for the experimental curve in the whole eoncentration range,
Z I ~ / X ~ and *
T h e problem of checking the reliability of experimental data of solutions is old but still of great interest and the literature dealing with such methods is very extensive (Barieau, 1970; Hala, et al., 1967; Herington, et al., 1969; van Ness and Mrazek, 1959). Consistency tests are not only useful to prove the mutual consistency of the data, but they also permit the evaluation of the partial molal property of one constituent of a binary mixture from that of the other. All of these methods are generally based on the Gibbs-Duhem equation. Theory
The excess molal property of a binary solution, z E , is related to the partial molal properties of its components, z,E, by the equation ZE
=
21Z1E
+
XZZ2E
(14
Consequently
Differentiation of eq l b with respect to x2 gives
i l n alternate consistency test is obtained upon adding eq 3a and 31)
Both methods permit one to check individual experimental points as well as the experimental curye in the whole concentration range. An advantage of the proposed procedure when compared with some other previously published tests is that the functions zlE/xn2 and z 2 E / x I 2 vs. mole fraction are more sensitive to experimental errors than the commonly used functions zlE and z ~ E . Equation 5 can be applied in a similar way as the "cornposition-resolution" test of van Kess and hIrazek (1959), which is based on the fact that although a given curve of an excess property of mixing, zE, us. concentration can be obtained from an infinite number of experimentally determined partial molal excess quantities z l E and ZZE, only one consistent set of z l E and ZZE values can result from a given curve of zE us. concentration. The procedure is the following. From smoothed values of z l E and zZE vs. composition a curve z E / x ~ x Z vs. composition may be drawn. The data are consistent if the curve obtained in the way is identical n5th that calculated from eq 5. When applied to the activity coefficients of a binary mixture then z E = g E and z L E = RT In y i and eq 2a and 2b assume the form
and similarly for component 1
I):(
Considering that the sums of the last two terms on the right-hand side of eq 2a and 2b are equal to zero (according to the Gibbs-Duhem equation) and integrating, we find
[a
d
Xi
=
RT
(- 9+ i)
(6b)
P,T
where E is taking into consideration the fact that the experimental data are obtained either a t constant pressure or a t constant temperature. Thus dP By multiplying eq 3a b y xz and eq 3b by x1 and subtracting the resulting equations, we obtain
RT
280
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
(74
and E =
As is obvious from this equation, the experimental data are consistent, when a t any mole fraction, 0 < xz < 1, the area ~ concentration multiplied by x2 under the curve Z ~ E / X ?us. is equal to the area under zZE 'x12 vs. concentration multiplied b y 21.
(for constant T data)
-(&)E
(for constant P data)
(7b)
For isothermal conditions e is usually negligibly small (Chao, 1959). Table I shows the application of the proposed consistency test to the activity coefficients of the systems pyridine-
Table I. Consistency Test for the System Pyridine-Tetrachloroethylene. First a n d Second Term in E q 4 a n d gE/x1x2 Calculated f r o m the Derived Activity Coefficients a n d f r o m E q 5 in Cal/Mole gE Calcd I.
from
XIXO
XPyr
0.1 0.3 0.5 0.7 0.9
First term
51 111 127 107 49
Eq, 4 Second term
Activity coefficients
Eq 5
565 528 510 512 534
57 0 532 510 510 535
51 112 128
I07 48
tetrachloroethylene measured by Fried, et al. (1967), a t 60°C. The table gires the first and second terms of eq 4 and compares the values of gE/xlx2 obtained from the activity coefficients and from eq 5. As is obvious from the table, both niet>hods prove that the data are thermodynamically consistent.
E R S S T LIERER3I;INS1 VOJTECH F R I E D * Brooklyn College of the City I'niversity of !Yew York Brooklyn, .T. 1'. 11210 I ~ I : C P : I Vfor ~ : Dreview June 9, 1971 XCCP:PTI.:L) February 24, 1972 Postdoctoral fellow f i o m the Uiiiver>it~of \-ieniia, \.ieiina, Austria.
On Mass Transfer from Bubbles in Non-Newtonian Fluids at l o w Reynolds Numbers. An Appraisal of the Thin-Boundary-Layer Approximation By means of CI coordinate transformation, the convective diffusion equation is converted formally to Fick's second law with a variable equivalent diffusivity. A comparison between the previously assumed and the actual diffusivity terms leads to the conclusion that the Sherwood number for bubbles in power-law nonNewtonian fluids given b y the thin-boundary-layer approximation is the minimum value.
M a s s transfer from moving bubbles in non-r\Tewtonian power-law fluids with small Reynolds numbers has been investigated theoretically by sereral workers. Hirose and Moo-Young (1969) previously derived
using their own approximate stream function (eq 8) and employing a thin-boundar y-layer approximation for the concentration profile as established by Levich (1962). Rellek and Huang (1970) have pointed out recently that eq 1 gives higher values of Sh by about 10% than their numerical results using the same stream function, eq 7 ; they ascribed this discrepancy to the thin-boundary-layer approximation on which eq 1 is based. The purpose of this note is to show that eq 1 gives the minimum values of Sh; istill lower values cannot exist.
The present authors reported an approximate form of $ for creeping flow of power-law fluids (Hirose and Moo-Young, 1969) as
+ f(Y)
=
3
'[I
f(y) sin2 e
(7 )
6 n ( n - 1) (Y2 - Y) - 2n + 1 (y l n y
+ 61 y-1 -
-
(8) Transforming the spherical coordinates into new coordinates
( 4 x)
Forced Convective Diffujiion Equation
Forced convective mass transfer is formulated in spherical coordinates (y, e) as follows.
=
where
t
=
'l' 4
sin3 Ode
1 (I 4
= -
- cos e)2(2 + COS e)
(9)
Equation 2 reduces to C = 1 (at y = 1)
(3)
C+O ( a s y + m) (4) where dimensional terms are used. The velocity profiles are expressed in terms of a stream function, +, as
where the prime on f(y) refers to the differentiation with respect to y, which in turn is a function of t and 2. Equation Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
281
