Velocity based on volume-fraction weighting, If the volume fraction φa is used as the weighting factor, one can define an average velocity vV as vV = ∑φvvi Derive a form of Fick’s law based on this definition of average velocity for a binary mixture

Velocity based on volume-fraction weighting, If the volume fraction φa is used as the weighting factor, one can define an average velocity vV as vV = ∑φvvi Derive a form of Fick’s law based on this definition of average velocity for a binary mixture. Note that the volume fraction of solute i is related to the partial molar volume of solute, Vˆi, in the mixture by the following relation: φi = CiVˆi Also verify that the volume-fraction-averaged velocity is the same as the mole-fractionaveraged velocity when the total molar concentration (molar density) is constant, e.g., for a gas mixture. Verify that volume-fraction-averaged velocity is the same as the mass-fraction-averaged velocity when the total mass density is constant, e.g., for a liquid mixture. These observations led Cussler to conclude that the use of the volume-average velocity is more general, but most textbooks disguise the use of this averaging.