# MAXWELL-STEFAN EQUATIONS

The Maxwell-Stefan (M-S) equations [Maxwell (1866), Stefan (1871)] describe the process of diffusion, where diffusive fluxes, Ji, of species through a plane, across which no net transfer of moles occurs, depend on all (n-1) independent driving forces in a mixture of n species. Their predictions, in particular that a species need not diffuse in the direction of its own driving force, have been confirmed by reliable experimental studies, e.g., Duncan & Toor (1962). A recent text describing their formulation, development and applications is Taylor and Krishna (1993) who quote them as,

(1)

In ideal mixtures at constant temperature and pressure, the left hand side simplifies to or for one space dimension. The ik diffusivities are then the widely available binary diffusivities, ik. Further simplification for a binary gives Fick's Law. Equation (1) has been used to describe many processes, notably Distillation and Condensation. For a film model of interfacial transport, Krishna & Standart (1977) show an exact solution. The M-S equations are often written in matrix form, explicit in diffusive flux. Here braces and square brackets denote (n-1) column and square matrices, respectively, and normal matrix products are implied,

(2)

The second term in Equation (2) is the generalized form of Fick's Law and the composition dependence of the diffusion coefficients [D] is clear through the stated relationship with the M-S equations. Equation (2) is the origin of the Linearized Theory approach of Toor (1964) and Stewart & Prober (1964), which can be applied generally to film, surface renewal and boundary layer models of mass transfer. It also shows the dependence on activity coefficient, γ, in nonideal mixtures. The ik then become composition dependent, but may in some cases be estimated from binary diffusivities, °, at infinite dilution, Taylor & Krishna (1993). The Maxwell-Stefan equations are the correct description of multicomponent mass transfer and their usage is steadily broadening as experimental evidence supporting their better applicability mounts.

#### REFERENCES

Duncan, J. B. and Toor, H. L. (1962) An Experimental Study of Three Component Gas Diffusion, AIChEJ, 8, 38-41.

Krishna, R. and Standart, G. L. (1976) A Multicomponent Film Model Incorporating an Exact Matrix Method of Solution to the M-S Equations, AIChEJ, 22, pp 383-389.

Maxwell, J. C. (1866) On the dynamic theory of gases, Phil. Trans. Roy. Soc, 157, 49-88.

Stefan, J. (1871) über das Gleichgewicht und die Bewegung, insbesondere die Diffusion Von Gasmengen, Sitzungsber. Akad. Wiss. Wien, 63, 63-124.

Stewart, W. E. and Prober, R. (1964) Matrix Calculation of Multi-component Mass Transfer in Isothermal Systems, Ind Eng Chem Fundam, 3, pp. 224-235.

Taylor, R. and Krishna, R. (1993) Multicomponent Mass Transfer. Wiley Interscience.

Toor, H. L. (1964) Solution of the Linearised Equations of Multi-component Mass Transfer, AIChEJ, 10, 448-465.