Dead End Filtration Model

[AntoniaBerger]

The next unit should be a dead end filtration model. Here is the discussion how we would like to model this unit.

Key properties of dead end filtration

• liquid is pressed (with pressure) through a filter medium.
• All particles that are larger than the pores of the filter are separated on the filter surface.
• During the filtration process, a particle cake collects on the filter surface -> Cake increases the flow resistance

Model Approach in CADET

Objective: The model should simulate the concentration of the permeate after the filter and the pressure development of the process.

Challenge: CADET is currently limited in the sense that it can only process volume flow and concentration as input variables, so @schmoelder and @daklauss have developed a design based on the CSTR system.

• F: Feed
• C: Cake
• P: Permeate
• PT: Permeate Tank

With the following differential equations:

• $$\frac{d(c_i^{PT}V^{PT})}{dt} = Q^P c_i^{in} - Q_{out}c_i^{PT}$$

$$ V^{c}(t) = Q^{c}(t_0) + \int_{t_0}^{T}Q^{c}(t)dt $$

• $$V^{PT}(t) = Q^{PT}(t_0) - Q^{O}(t_0) + \int_{t_0}^{T}Q^{PT}(t)- Q^{O}(t)dt$$

With $Q^p = (1-r_i)Q_{in}$ and $Q^c = r_iQ_{in}$, where $r_i$ is the ratio of how much of $c_i$ is rejected (how do we get to $r_i$? With R(t)?).
We use Darcy's law to calculate the other objectives:

$$ \Delta p = \frac{\mu R(t) Q^p(t)}{A} $$

• $$R(t) = R_m + \alpha \frac{V^p}{A}$$

[AntoniaBerger]

Other Ideas:

An other but simular approach would be to model the filtration inside the CSTR:
We start again with the CSRT equations:
$$\frac{d(c_i^{P}V^{P})}{dt} = Q^P c_i^{in} - Q_{out}c_i^{P}$$
But instead of using $r_i$ and $Q_{out}$ as parameters, we calculate $Q_{out}$ with Darcys law and a given pressure $\Delta p$:
$$Q_{out} = \frac{dV_p(t)}{dt} = A \cdot \frac{\Delta p}{\mu \cdot (R_{m}+R_{c}(t))}$$ and with
$$R_{c}(t) = \alpha \cdot h_{cake}(t) = \alpha \cdot \gamma \cdot \frac{V_p(t)}{A} $$

[AntoniaBerger]

I had some thought about flow splitting: @daklauss @schmoelder

• Filtration model flow splitting

  • In CADET we can give the following initial values: $Q_{in}$ (total volume flow into the system)and $c_i^F$ (initial concentration of each component)
    • This allows us to calculate the initial value of the particle flux for each component: $\dot{n_i}^F = c_i^F Q^{in}$
  • After constructing the model, we now define a parameter $r_i$ that determines how much of the particle stream (for each component) is filtered or not filtered: $\dot{n_i}^c = r_i \dot{n_i}^F$ (Cake) and $\dot{n_i}^p = (1-r_i)\dot{n_i}^F$ (Permeate)
  • The relationship $\rho_i = \frac{m_i}{V}$ results in two volume flows $Q^c_i = r_i Q_i^{in}$ (Cake) and $Q^p_i = (1-r_i) Q_i^{in}$ (Permeate)
  • This means that the following must apply in general:

$$ Q_i^c = r_i Q_i^{in} = \frac{M_i \dot{n_i^c}}{\rho_i} $$

With that we can avoid the problem of "Volume out of nothing"- problem, if we give $Q_i^{in}$ for every componets and calculate $Q^{in}$ in a pre-process step to use the CADET interface.