performed for mixtures of the same mean composition over a range of rotational speeds, resulting concentrations difference ratios can be plotted in the manner suggested by Equation 18 to yield the elements of [ P L ] from the slopes and intercepts. Experimental evaluation of these procedures will be the subject of a later paper.
] p
p
Acknowledgment
6
Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research.
w
= = = = = = = = = =
specific chemical potential of species i specific chemical potential of pure species i matrix with elements, (bpt/dx,) matrix with elements, (bPL/bx,) a n element of the matrix, [PI, evaluated at the logarithmic mean composition total mass density average density defined by Equation 12 potential function angular velocity
[a]
SUBSCRIPTS
1, . .2 = radial positions = components 1, 2 . . . n L = logarithmic mean z , ~
Nomenclature
[a]
(F)
= matrix defined following Equation 9 = column vector defined as the average sedimentation
potential vector by Equation 14 F, = an element of (17) [ I ] = the identity malrix M , = molecular weight of species t ( M ) = column vector of molecular weights [M,] = diagonal matrix of molecular weights P = pressure R = gas constant = average radial position AR = difference in radial positions r = radial coordinate T = absolute temperature p, = partial molar volume of species i (7)= column vector of partial molar volumes (0) = column vector of partial mass volumes x, = mole fractions of species i = column vector of mole fractions (x) [ x , ] = diagonal matrix of mole fractions [fL\] = diagonal matrix of logarithmic mean mole fractions Ax? = mole fraction dilyerence between two radial positions ( A x ) = column vector ~ i t elements h Ax, (X) = column matrix of driving forces defined by Equation 2 (Z) = column vector with elements, In x I z / ~ I 1 GREEKLETTERS = activity coefficient of species i y, [y] = matrix with elements, (a In yt/d In x , ) = chemical potential of species z p,
OPERATORS
v -1
= gradient = inverse
Literature Cited
Cullinan, H. T., Cusick, M. R., IND.ENG.CHEM.FUNDAMENTALS 6, 72 (1967). Fitts, D. D., “Non-equilibrium Thermodynamics,” McGrawHill. New York. 1962. Guggenheim, E. ’A., “Thermodynamics,” North Holland Publishing Co., Amsterdam, 1950. Hooyman, G. J., Holtan, H., Mazur, P., de Groot, S. R., Physica 19, 1095 (1953). Miller. D. G.. Am. J . Phvs. 24. 555 (19561. Peller,’L., J . Chem. Phys. 3 (the case n = 2 is trivial). I n a ternary system-for example, Equation 10 gives
Thus the only choice of weighting function which will allow the L i to be independent is Z ~ = R
5 X,
i =
I, 2 . . ., n
(6)
Li
L3 =
where X may depend upon concentration. X plays no role in the subsequent development. This choice reduces the restraints on Equation 1 from two to one since it reduces Equation 2 t o Equation 3. T h e peculiar reference velocity corresponding t o this choice of Z t R is (Toor, 1962)
=
Dl?D13/#
(Qa)
Lz =
Dl*D28/$
(12b)
L3 =
D13D23/$
(12c)
Ll
(7) C,2/L, i=l
If the peculiar reference velocity is not chosen, then Equation 5 relates all the L i and allows only one kinetically independent
with
i,j=l,2,..,n-l where aJR’,j = 1, 2 . , , , n, is volume fraction, mass fraction, or mole fraction in volume, mass, and molar reference frames, respectively; in the solvent reference frame, ajR‘is zero for j # n and one for j = n. Consistency with the O.R.R. is most easily checked in the solvent frame where Equation 9 becomes
LiJS =
CtC, L , = L j l S , i z j , i, j = 1, 2 . . ., n C,
Lias = (?)‘L,
+ Li,
i
=
1, 2 . . ., n
-1
-1
(loa)
(lob)
T h e O.R.R. require that L,,s = L J > (Kirkwood et al., 1960) and this is seen to hold. Thus, the necessary condition for the existence of both Equation 1 and a peculiar reference frame is satisfied. However, the sufficient condition requires that a set of Lt exist for 320
I&EC FUNDAMENTALS
LlZS
so that given an L i j matrix, one can always compute the Li and determine the peculiar reference frame. Hence, all ternary diffusion matrices can be made diagonal by this procedure. T h e ternary Stefan-Maxwell ideal gas mixture, for example, gives
n
diffusion coefficient. Clearly then, if Equation 1 is written for a multicomponent system, it must be written in a peculiar reference frame. But this implies restraints on the diffusion matrix in the common reference frames which may or may not hold. A necessary condition is that these restraints are consistent with the Onsager reciprocal relationships (O.R.R.). T o investigate these restraints, one converts from the pecu1ia.r reference frame to an R‘ frame using Equations 1 and 6 (Toor, 1962). T h e result, using the minimum ( n - 1)2setof diffusion coefficients is
c 3 2 ~
ClC2
$ =
For n
XlDD23
f
x2D13
+
x3D12
(12 4
> 3, Equation 10a requires that
and only if these relationships hold (and they are much stronger than the O.R.R.) will it be possible to find a peculiar reference frame which diagonalizes the diffusion matrix. They do not generally hold in Stefan-Maxwell ideal gas mixtures for n >3 and there are no data on liquid mixtures for n > 3. Indeed, the test of the validity of Equation 1 apparently requires data in a system of a t least four components. If Equation 13 does not hold in liquids, then the peculiar reference frame is merely a peculiarity of ternary systems, and any liquid theories based on Equation 1 should be viewed with caution. If concentration instead of chemical potential were used in Equation 1, the O.R.R. would not generally be satisfied, so no peculiar reference velocity could exist with this choice, and the coefficients could not be independent. I t is, then, a matter of definition to make them all equal in ordinary reference frames as was done by Toor and Arnold (1965). Nomenclature
c, Li Lij
= = = = =
n
=
vi
=
6ij
Pi
= =
X
=
Dtj
Ji
v = xi = z, =
concentration of i binary diffusivity flux of i with respect to a reference velocity diffusion coefficient defined by Equation 1 multicomponent diffusion coeffjcient number of components velocity of i with respect to a fixed coordinate reference velocity mole fraction of i weighting function on i Kronecker delta chemical potential of i function of concentration
SUBSCRIPTS i, j , k , I = component indices n
= nth component
Toor, H. L., A.I.Ch.E. J . 8, 561 (1962). Toor, H. L., Arnold, K. R., IND.ENC.CHEM.FUNDAMENTALS 4,
SUPERSCRIPTS R, R’ = arbitrary reference frames S
363 (1965).
= solvent reference frame
C. M. YON
H. L. TOOR
Literature Cited
Carnegie-Mellon University
Cullinan, H. T., Jr., Cusiclc, M. R., IND.ENG.CHEM.FUNDAMEN- Pittsburgh, pa. 75273 TALS
6 , 7 2 (1967).
Kirkwood, J. G., Baldwin, R. L., Dunlop, P. J., Gosting, L. J., Kegeles, G., J . Chem. Phyr. 33, 1505 (1960).
RECEIVED for review June 29, 1967 ACCEPTED December 11, 1967
CONDENSATION OF STEAM IN T H E PRESENCE OF AIR Experimental Mass Transfer Coeficients in a Direct-Contact System Experiments were run in a direct-contact condensation system in which steam, containing various amounts of air, was condensed onto a flowing stream of cool water. The rates of condensation were measured and, from the experimental data, mass transfer coefficients were calculated. These coefficients agreed with boundary layer theory as developed for mass transfer to a moving flat plate with suction.
I
hen steam is condensizd on a cool surface in the presence of
W a small amount of air, the air from the bulk of the system is carried over by the steam toward the vapor-condensate interface. There: the ste,am condenses leaving a higher concentration of air than in the bulk. Since the air concentration a t the interface is now higher than in the bulk of the steam, it is transferred away from the interface by diffusion. A steady state is soon reached when the rate of air carried over by the steam toward the interface is equal to the rate of air diffusing away from it. At a constant total system pressure, the saturation temperature of steam is lowered when the concentration of noncondensables is increased. At any point along the steam-condensate interface, therefore, where the air concentration may be high, the steam saturation temperature ( T i ) is lower than the bulk saturation temperature ( T - ) . T h e true driving force for the heat transfer during the condensation is thus reduced; consequently, a t identical bulk temperatures and pressures the condensation rates are lower than in the absence of air. As a result of this? heat transfer coefficients calculated based on the over-all driving force ( T , - T,) are lower than in the absence of noncondensables (Figure 1). Allowance must be made, therefore, for this mass transfer resistance t o evaluate the rates of condensation correctly (Ilohsenow and Choi, 1961). T h e local rate of steam condensation ( W , moles per unit time per unit area) may be expressed in terms of the heat transfer equation. This is:
I+’
= h(2’i
-
Tc)/(4H)
(1)
Here, h is the true local heat transfer coefficient across the condensate film, and ( A H ) is the latent heat of vaporization per mole of steam. I n this equation, sensible heat effects have been neglected. In terms of the mass transfer rate, if one defines a mass transfer coefficient based on the diffmion flux alone, one may say that (Bird et al., 1962)
It’
-
W(l
- Xi)= kp,(X, - X,)
(2)
T.
I
I
-
-
Figure 1 . Temperature and concentration profiles in the direct-contact condensation tray system
or that
I n Equation 2, k is the local mass transfer coefficient, p m is the vapor density in mo!es per unit volume, and Xiand X, are the mole fractions of air a t the interface and in the bulk of the vapor, respectively. T i in Equation 1 and X , in Equation 2 are related together by the steam saturation curve a t the total pressure of the system-Le.,
X, = X ( T i , P)
(3)
I n cases where rapid condensation rates are encountered, the coefficient k is dependent on the rate of mass transfer. This is a result of the concentration field distribution being dependent on the velocity field within the system. Boundary VOL. 7
NO. 2 M A Y 1 9 6 8
321
