InterfaceMTCoeff

Chapter 13 Interface mass transfer (Material presented in this chapter are based on those in Chapters 8 and 13, �Diffusion mass transfer in fluid systems� second edition by EL Cussler) In chapter (7), methods to estimate mass transport coefficients for transport from a homogeneous solutions into wall were derived. There are several situations in which the transport of a solute species occurs from one fluid phase into another fluid phase, separated by a region called the interface. In the case where the bulk concentration in the two phases are constant, at the contact location there can be two possibilities.

The first possibility is that there might be a jump discontinuity at the contact point of the two phases. This is possible only if the diffusivity in both the phases are infinity. As the diffusivity of the solute is finite in the phase, a jump discontinuity is unlikely to exist. This leads to the second possibility where there is a concentration gradient he near the contact point in the phase into which the solute is transported. This region where a concentration gradient exists is termed as the interface. In this chapter, different theories that characterize the concentration profile in the interface will be considered.

Methods to estimate the overall mass transfer coefficients will be presented.

13.1

Mass transport driving force

13.1.1

Mass transport between two liquids

Consider the case of two well-mixed liquids that are immiscible. Suppose if the denser fluid, say 1 is placed in the container and lighter fluid, say 2 177

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above the heavier one (Fig. 13.1). As these two fluids are immiscible, an interface will be formed at the contact point. Assume the interface to be planar. Suppose that the solute species A concentration in the fluid 2 is concentration C2 and that in fluid 1 is C1 .

Figure 13.1: Mass transport of a solute from one phase and another phase.

If the solute concentrations in both fluids are equal at start, that is, C1 = C2 , then after a certain time, the solute species will equilibrate between the two phases and C2 > C1 if the solubility of the solute in fluids 2 is greater than that in fluid 1.

Consider a similar problem in heat transport. If the two fluids are at temperature T1 and T2 then the driving for heat transport is T2 − T1 and if the temperatures are equal, then there is no heat transport. This property of no heat transfer when both are same temperature analogy does not exist in mass transfer between two phases as even if the concentrations in the two phases are same, mass transport can still occur. This does not mean that mass transfer can occur at zero driving force. It simply means that the concentration gradient of the solute between the two phases is not the driving force. The question then arises as to what is the driving force.

The driving force for mass transfer of solute across an interface between

13.1. MASS TRANSPORT DRIVING FORCE

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two phases is essentially the difference between the actual concentration of the solute in one phase and the concentration if it were in equilibrium with the other phase, that is ∆C = C2 − C2e = C2 − HC1

(13.1)

where C2e is the concentration of the solute in fluid 2 at equilibrium that corresponds the concentration of the solute in fluid 1, C1 is the concentration of the solute in fluid 1 and H is the partition coefficient which is the ratio at equilibrium of the concentration of the solute in fluid 2 to that in fluid 1. Note that the gradient presented in Eq. (13.1) will result in a non-zero concentration gradient even if the solute concentration in both the phases are equal.

Based on the concentration gradient in Eq. (13.1), the mass transfer flux can be defined as N = K∆C (13.2)

13.1.2

Mass transport between a liquid and a gas

Consider the case where a gas (air) is present on top of a liquid containing the solute. Assume that the solute vaporized from the liquid phase and gets transported into the gas phase. Initially, the solute concentration in the liquid phase is higher than that in the gas phase. However, after certain amount of evaporation, the concentration in gas phase is higher than that in the liquid phase.

While quantity of the solute in the liquid phase is expressed in terms of concentration (moles/lit), in the gas phase, it is expressed in terms of the partial pressures. The question arises as to how the quantity of solute expressed in different forms can be equated when the transport occurs across the interface. (Although different forms are used to express the quantity of the solute in the two phases, the fundamental quantity that characterizes the amount of solute in a fluid is chemical potential which can be converted into concentration or pressures appropriately.) A modified partition coefficient or Henry’s law constant can be used to relate the partial pressure of the solute in gas phase with the concentration in the liquid phase. At equilibrium between the gas and the liquid phases, the partial pressure of the solute at the interface is given by P1i = HC1i (13.3)

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where P1i is the partial pressure of the solute at the interface in the gas phase side, C1i is the concentration of the solute at the interface at the liquid phase side, H is the Henry’s constant or partition coefficient, subscript 1 refers to the solute 1.

13.2

Overall mass transport coefficient

Consider the a liquid in contact with gas phase (Fig. 13.2). Assume that the solute is getting transported from the gas phase to the liquid phase.

Figure 13.2: Mass transfer of a solute from gas phase to liquid phase.

The flux of transport of the solute from the bulk to the interface on the gas phase is N1 = kP (P10 − P1i ) (13.4) where kP is the gas phase mass transport coefficient (in cmmol 2 satm ), P10 and P1i are the bulk and the interface partial pressures of the solute.

13.2. OVERALL MASS TRANSPORT COEFFICIENT

181

Assuming thin interfacial region, the flux of solute transport in the liquid phase from the interface to the bulk is given by N1 = kL (C1i − C10 )

(13.5)

where kL is the liquid phase mass transport coefficient, typically in cm/sec.

At equilibrium, P1i = HC1i and kP (P10 − P1i ) = kL (C1i − C10 ) ⇒ C1i =

P1i kp P10 + kL C10 = H kp H + kL

(13.6)

Substituting Eq. (13.6) into Eq. (13.4) leads to the following expression for the mass transfer flux: N1 =

1 kP

1 +

H kL

(P10 − HC10 )

(13.7)

the partial pressure at equilibrium is P1e = HC10 .

As the partial pressure gradient for mass transfer is P10 − HC10 , where HC10 is the hypothetical partial pressure that would be in equilibrium with the bulk gas, the overall gas phase side mass transport coefficient for transport of solute across the interface is given by 1 1 H = + KP kp kL

(13.8)

If the driving force is the concentration gradient PH10 − C10 , where PH10 is the hypothetical liquid concentration that would be in equilibrium with bulk, then, the overall liquid phase side mass transport coefficient is given by 1 1 1 = + KL kL kp H

(13.9)

If the bulk partial pressure in the gas phase and concentrations are known, then the overall mass transport coefficient can be estimated using Eqs (13.8) and (13.9).

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