# Filter Area for given flow rate

Background

For a washing process after a crystallization we want to know how much filter area we would need and if said area will work for our fixed volume flow.

Calculation

$\dot{V_F}$ has a given value

Velocity $u$ should be the superficial velocity and is at a fixed value we found in our mechanical engineering script.

With $\frac{\dot{V_F}}{u} = A$ we can calculate our needed filter area. Question is, will that area work for our volume flow.

With $\dot{V_F} = \frac{A \cdot \Delta p}{\eta_F(\alpha_w \cdot L )+ f_M}$

we can then calculate $L$.

$\Delta p = \Delta p_{filter medium} + \Delta p_{filter cake}$

$\Delta p_{filter cake} = \frac{(1-\epsilon)^2}{\epsilon^3 \cdot x^2_{32}} \cdot 150 \cdot L \cdot \eta_F \cdot u$

Now we insert $L$ in the equation for $\Delta p_{filter cake}$ and we get the exact $\Delta p_{filter cake}$ that we originally assumed when calculating $L$. Actually we expect to get a different value in order to find all variables via iteration.

Somewhere we made a mistake in our assumptions I think, but I can't see where right now.

I know that $\dot{V_F}$ is a function of $L$ however we can't change $\dot{V_F}$. And we insert it anyway in our equation for $L$ so I think that assumption is valid.

English is not my mother tongue, so please excuse any spelling / grammar mistakes. Furthermore this is not a given tasks, but a problem we stumbled upon during our group project (designing a plant).

Your math seems to be sound. It looks like you calculated everything correctly up to where you calculated $L$. At that point you should have all of the variables solved for and there should be no need to refine your numbers.