Equation Governing Volume Transfers in the EQ format
I derived the governing equation for the volume transfer from SepCon devices in the wet/wet (EQ) format. It is …
where the equilibrium volume for the cup can be calculated apriori using the cross sectional areas of the cup and the surrounding annulus …
with the total volume of interest given by
and the time scale is set by the parameter …
in this equation ‘epsilon’ is the hydraulic permeability we need to calculate, ‘rho’ is the water density, and ‘g’ is the centrifugal acceleration experienced by the sample. Note that this is not the traditional ‘acceleration due to gravity’.
We can solve for permeability with a knowledge of the equilibrium and geometry values and a measurement of the remaining cup volume at time t …
It may be simpler in practice to replace the cross-sectional area measurements in the denominator of this equation with the equilibrium volumes …
Lets do a dimensional check on our work. Hydraulic permeability is in units of Volume/[Area-time-Pressure] = L^3/[L^2 – time – Force/L^2] = L^3/[time-force]. Since rho*g is a force per unit volume, the dimensions of t*rho*g are [time-force]/L^3. And since this quantity is in the denominator, our dimensions do check out.
Tom wrote to say that a final equation based on volumes presents problems because our geometry isn’t really cylindrical like assumed in this theory.
The better equation to employ for our current set-up is …
where the h’s are heights of the cup volume measured from the membrane.
Tom, Jess – lets see what happens to our permeability values with this equation.