Page 709 - Bird R.B. Transport phenomena

§22.4 Definition of Transfer Coefficients in Two Phases 689

Exceptions to this are: (i) extremely high mass-transfer rates, observed for gas phases at
high vacuum, where N approaches р /Л/2тгМ ЯТ, the equilibrium rate at which gas
A0 А0 А
molecules impinge on the interface; and (ii) interfaces contaminated with high concen-
trations of adsorbed particles or surfactant molecules. Situation (i) is quite rare, and situ-
ation (ii) normally acts indirectly by changing the flow behavior rather than causing
deviations from equilibrium. In extreme cases surface contamination can provide addi-
tional transport resistances.
To describe rates of interphase transport, one can either use Eqs. 22.4-2 and 3 to cal-
culate interface concentrations and then proceed to use the single-phase coefficients, or
else work with overall mass transfer coefficients

N = K° (y - y ) = ^ ( x - x ) (22.4-4)
A0 y>loc Ab A e o c A e A b
Here y Ac is the gas phase composition in equilibrium with a liquid at composition x ,
Ab
and x Ae is the liquid phase composition in equilibrium with a gas at composition y . The
Ab
quantity Щ is the overall mass transfer coefficient "based on the gas phase," and KP
loc x]oc
is the overall mass transfer coefficient "based on the liquid phase." Here again the molar
flux N is taken to be positive for transfer to the liquid phase.
A0
Equating the quantities in Eqs. 22.4-2 and 4 gives two relations
*чюс(*л, " x ) = k° (x - x ) (22.4-5)
A b Xtloc A0 A b
К^УАЪ ~ Уле) = ЩхЛУль - УАО) (22.4-6)
connecting the two-phase coefficients with the single-phase coefficients.
The quantities x Ae and y Ae introduced in the above three relations may be used to de-
fine quantities m and m y as follows:
x
_ У Ah ~ УАО, _ УАО ~ УАе . „ .
(ZZ.4-7, o)
Щ — ~z—z~^—; nt — — _ п о ft
-
Ae A0 -MO Ab
X X У u X
As we can see from Fig. 22.4-2, m is the slope of the line connecting points (x , y )
x A0 A0
and (x , y ) on the equilibrium curve, and m y is the slope of the line from (x Ab/ y ) to
Ae
Ae
Ab
From the above relations we can then eliminate the concentrations and get relations
among the single-phase and two-phase mass transfer coefficients:
*Vloc т у*у,\ос ^у,1ос *JC,1OC
The first of these was obtained from Eqs. 22.4-5, 2, and 7, and the second from Eqs. 22.4-
6, 2, and 8. If the equilibrium curve is nearly linear over the range of interest, then m —
x
m y = m, which is the local slope of the curve at the interfacial conditions. We see, then,
that the expressions in Eqs. 22.4-9, 10 both contain a ratio of single-phase coefficients
weighted with a quantity m. This quantity is of considerable importance:
(i) If к° /тЩ « 1, the mass-transport resistance of the gas phase has little ef-
хМс 1ос
fect, and it is said that the mass transfer is liquid-phase controlled. In practice, this
means that the system design should favor liquid-phase mass transfer.
(ii) If /c^ioc/m/Cy > > 1, then the mass transfer is gas-phase controlled. In a practical
loc
situation, this means that the system design should favor gas-phase mass
transfer.
(iii) If 0.1 < k Q /mk < 10, roughly, one must be careful to consider the interac-
x>loc yloc
tions of the two phases in calculating the two-phase transfer coefficients. Out-
side this range the interactions are usually unimportant. We return to this point
in the example below.