Dimensional Analysis of Flux Equation for cross filtration

[582153236]

Homework Statement

The following equation is presented in my textbook, with very little context and J is simply described as the "trans-membrane fluid flux, which can be modeled by using Darcy’s law."

Typically in this bioseparations class, when we discuss permeate flux, it is a unit volume per (unit area*unit time), so L*m^-2*s^-1.
However, in this case, the dimensions don't really seem to match up.

Homework Equations

The Attempt at a Solution

p is the pressure difference across the membrane, for which I used pascals
σ is a dimensionless rejection coefficient
π is the osmotic pressure, also in pascals
μ is the viscosity given in cP, but I converted to Pa*s
Rm and Rp are appropriate resistivity values given in units of m^-1

So a dimensional analysis shows that the units of J are m/s, which does not represent the permeate flux. My ultimate goal is to calculate J in order to solve for the volumetric flow rate out of the membrane so that I can do a material balance for the system and model the concentration as a function of time.

 

[Chestermiller]

If J has units of velocity, then it can be considered the superficial velocity of the permeate. It is the same as volumetric throughput rate per unit area of membrane. So it is the volumetric flux of permeate.

Chet