Mass transfer is the transport of a substance (mass) in liquid and gaseous media. Depending on the conditions, the nature, and the forces responsible for mass transfer, four basic types are distinguished: (1) diffusion in a quiescent medium, (2) mass transfer in laminar flow, (3) mass transfer in the turbulent flow, and (4) mass exchange between phases.

The simplest case is mass transfer in a medium at rest in which the driving force is the difference of concentrations in adjacent regions of the medium and the mechanism is molecular diffusion. The substance flows, by virtue of the statistical character of molecule motion, from a high concentration region to a low concentration one tending to equalization of concentration throughout the entire volume. This mass transfer is described by an equation known as Fick's law which, when applied to a binary mixture, has the form

where
is the flow of substance A, kg/m^{2}s, in the reverse direction to the concentration gradient of this substance dC_{A}/dy (kmole/m^{3}m),
is the molecular weight of component A (kg/kmole) and D_{AB} (m^{2}/s) is the interdiffusion coefficient of substance A in substance B and is determined by the physical properties of these substances. D_{AB} has been determined experimentally for many gas pairs and can also be calculated using a molecular-kinetic theory. D_{AB} is known to depend on temperature and pressure. The diffusion coefficient in gases under normal conditions is of the order of 10^{−4} m^{2}/s, while in liquids it is about five orders lower (see Diffusion and Diffusion Coefficient for more details).

Actually, in addition to concentration gradient, temperature and pressure gradients which affect mass transfer via thermal and pressure diffusion may come into play. These effects are most significant in gas mixtures with a widely varying molecule size, e.g., He—Cs. Thermal diffusion underlies one of the methods of separation of uranium isotope U^{235}.

In laminar flow of gaseous and liquid mixtures, calculation of mass transfer does not present particular difficulties. For instance, for a plate in the stream of incompressible liquid, the set of Equations (2) describing the velocity and concentration fields has the form

where x and y are the longitudinal and transverse coordinates, u and v the longitudinal and transverse components of velocity, ν and D the coefficients of kinematic viscosity and molecular diffusion, respectively, and C the local concentration of a substance, C = f(x,y).

Under the boundary conditions
=
= 0 and C = C_{1}, at y = 0,
=
and C = C_{0} at x < 0 or y = ∞ the solution to equation system (2) yields

where β is the local mass transfer coefficient at the distance x from the leading edge of the plate (β = [
/
(C_{1} - C_{0})], Re_{x} = x
/ν is the Reynolds number, and Sc = ν/D the Schmidt number. Note that there is a departure from the linear dependence of
or (C_{1} - C_{0}) at high rates of mass transfer [see Sherwood et al., (1975)].

If the total length of the plate is L, then the length-averaged mass transfer coefficient is found from the equation

where Re_{L} = Lu_{0}/ν.

Mass transfer fundamentally changes in transition to a turbulent flow. Its vortex flow characteristics lead to a large-scale transport of fluid. This transport commonly has rates which are orders of magnitude higher than molecular ones and promotes a faster equalization of the concentration field and, given a substance source, rapid propagation of the substance over the flow cross section. Since a rigorous theory of turbulence is lacking, it is desirable to describe the flow itself and heat and mass transfer in it by a set of equations similar to (2) for the laminar flow using averaged velocity values and replacing v and D by their effective values conforming to the conditions in the turbulent flow, i.e., by the coefficients of "eddy viscosity" and "turbulent diffusion" (see Diffusion and Diffusion Coefficient). The molecular diffusion also occurs in the turbulent flow, e.g., between and inside eddies. Its role is enhanced (relative to turbulent transport) as the channel surface is approached and becomes predominant near it. It is commonly believed that the molecular D_{M} and turbulent D_{T} diffusion coefficients are additive, i.e., D = D_{M} + D_{T}.

Since in a developed turbulent flow the substance, energy, and momentum transport occurs via large-scale eddies, the transport rate is considered identical and D_{T}, a_{T}, and ν_{T} are about equal. (This is a triple analogy between the transport of the substance, energy, and momentum.) This makes it possible to use empirical dimensionless equations describing heat transfer for calculation of mass transfer.

Mass exchange between a gas and a droplet is commonly encountered in engineering. For a medium at rest the solution can be written thus

where β is the mass transfer coefficient, d_{d} the droplet diameter, and D the diffusion coefficient of the exchanged substance in the gas. When the drop moves with respect to a medium in the Re_{d} ≤ 200 range the Frössling-Marshall formula

(where Re_{d} = d_{p}u/ν with u and v being the velocity of the gas relative to the drop and the gas kinematic viscosity, respectively) is quite consistent with the experiment. This formula also holds in the case of evaporation of droplets in a gas stream provided the evaporation rate is small or moderate.

The rate of mass transfer from the surface of liquid film (e.g. evaporation) flowing on the inner surface of the tube toward the central gas flow can be calculated using an empirical formula

where d is the diameter of the tube (cf. Nu = 0.023 for heat transfer).

One more example of mass transfer is diffusion of some substance, such as A from one moving medium to another through interface (two-film theory). If it is assumed that there is no concentration jump on the boundary, i.e., C_{ABi} = C_{AEi} (Fig. 1a), then the flow of substance A can be represented in the form

where β_{AB}, β_{AE}, and K_{A} are the coefficients of mass transfer for substance A in media B and E and the overall mass transfer coefficient, respectively, and C_{A} the concentration of substance A at the points indicated in Figure 1.

**Figure 1. Mass transfer of component A between media B and E with no concentration jump at the interface.**

Hence,

i.e., the total resistance to mass transfer is a sum of resistances in each medium.

In many cases, the concentrations of the transferring substance are not identical at the interface in the two respective media. For instance, if B is a liquid phase and E is gas, and an equilibrium on the boundary obeys Henry's law C_{ABi} = H C_{ABi} (Figure 2), then

whence

Given the flow parameters for both films, β can be determined using, e.g., the above formulas.

**Figure 2. Mass transfer of component A between media B and E with a concentration jump at the interface.**

Mass transfer, mostly in combination with heat transfer, is widely employed in industry, in chemical process equipment, metallurgy, power engineering, and so on. The equipment includes fractionating towers, absorbers and extractors, driers and cooling towers, combustion chambers, heterogeneous catalysis apparatuses, and many others.

#### REFERENCES

Sherwood, T. K., Pigford, R. L., and Wilke, C. R. (1975) *Mass Transfer.* McGraw Hill, New York.

Spalding, D. B. (1963) *Convective Mass Transfer.* Arnold Publ. London.