Show that in a binary system the mass flux with respect to the mass average velocity is J1=-J2 a

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Show that in a binary system the mass flux with respect to...

Key Concepts

Binary Mixture

A binary mixture is a system composed of two distinct species where the interactions, transport, and exchange properties between the two species are of interest. In the analysis of transport phenomena, the binary mixture model simplifies the mathematical treatment by ensuring that any excess of one component corresponds to a deficiency in the other, which, due to mass conservation, often leads to symmetry in flux relationships.

Mass Average Velocity

The mass average velocity is defined as the velocity field of the mixture weighted by the mass fractions of the individual components. It provides a frame of reference in which the overall mass transport is analyzed. In this frame, because the net mass in the system is conserved, the individual mass fluxes of the components are equal in magnitude but opposite in direction, leading to the relationship J1 = -J2.

Molar Average Velocity

The molar average velocity is similar to the mass average velocity but is computed by weighting the velocity field by the molar concentrations (or mole fractions) of the species. This concept is crucial when the transport analysis is concerned with the movement of molecules rather than mass. In a binary mixture, using the molar average velocity as the reference frame results in a cancellation of the net molar flux, thereby establishing that the molar fluxes of each species are also equal and opposite, i.e. J1* = -J2*.

Mass Flux

Mass flux refers to the amount of mass passing through a unit area per unit time. When measured with respect to the mass average velocity, the flux of each species accounts for both convective transport and diffusive contributions. In a binary system, the condition of mass conservation dictates that any gain in mass flux by one species is exactly compensated for by an equivalent loss by the other species, leading to J1 = -J2.

Molar Flux

Molar flux is the rate at which moles of a substance pass through a unit area per unit time. In mixtures, it is often more insightful to analyze transport in terms of moles rather than mass. When the molar flux is defined relative to the molar average velocity, conservation of moles in a binary system ensures that the molar contributions from each species cancel each other, resulting in the relationship J1* = -J2*.

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