Add model equations for simple Dead-End Filtration Model

[daklauss]

In GitLab by @schmoelder on Jul 2, 2024, 15:22

In previous works, Tarek had written down some equations for a simple 0D-filtration model. Some of these equations were in a jupyter notebook, others already implemented in Python. Because I find it hard to keep track, I would like to ask you @d.klauss and @AntoniaBerger to consolidate them and write them down here in this thread (including all the assumptions and extensions). Then, we will use this to implement the model in the CADET-Python-Simulator.

[daklauss]

In GitLab by @AntoniaBerger on Jul 5, 2024, 09:00

Simple model for Dead-End

A simple model for Dead-end filtration yielding the filtrate/permeate volume using Darcy's law can be:

$$\frac{dV_p(t)}{dt} = JA = A \cdot\frac{ \Delta p}{\mu \cdot R(t)} = A \cdot \frac{\Delta p}{\mu \cdot (R_{m}+R_{c}(t))}$$
With

• $V$ volume of the filtrated collection (permeate) -> Answer to Question
• $J$ Filtration flux (Flow rate per unit area)
• $A$ Filtration area
• $R_m$ resistance of the medium
• $R_c$ resistance of the cake
• $\Delta p$ pressure difference
• $\mu$ dynamic Viscosity

Where the cake's resistance is:
$$R_{c}(t) = \alpha \cdot h_{cake}(t) = \alpha \cdot \gamma \cdot \frac{V_p(t)}{A}$$
With

• $h_{cake}$ Height of the cake
• $\gamma$ concentration of the suspended particles in the feed ( Should be revised ; [c] = mol / $m^3$; [ $\gamma_c$] = -)
• $\alpha$ Specific cake resistance

$\color{red}{\text{Assumptions}}$:

1. $\gamma$, $\Delta p$ constant
2. $\alpha$ is constant $\Rightarrow$ incompressible cake.
3. Constant $R_{m}$ $\Rightarrow$ no membrane fouling.
4. Membrane has $100%$ efficiency $\Rightarrow$ all particles above certain size excluded.
5. Single component
6. No cake volume $V_{c} = 0$ $\Rightarrow$ $V_{f} = V_{p}$
7. Perfect separation $V_c = V_s$; $V_p = V_l$
8. $\gamma$ $\approx$ $c_{f}$. where $c_{f}$ is the feed's concentration.

[daklauss]

In GitLab by @AntoniaBerger on Jul 5, 2024, 09:04

Model Extension with new Assumptions:

1. Cake has non volume zero and is time dependent:

• $V_f = V_c + V_l$
• $V_c(t) = h_{c}(t) \cdot A$
• Has a individual volume flow
  $$\dot{V_c} = J A $$
• With Carman-Kozeny we can add porosity of the cake:
  $$\dot{V_c} = A \cdot \beta \cdot \frac{\Delta P \cdot D^2}{\mu \cdot h_c \cdot K}$$
  With
• Porosity factor $\beta = \frac{\varepsilon^3}{(1-\varepsilon)^2}$
• Height of the cake $h_c = \frac{V_c}{A}$
• $D$ the particle size

2. $\Delta p$ is time dependent

3. Multi component

4. Non perfect membrane

[daklauss]

Wrote our simple model with constant flow $\dot{V}$ and every other constant = 1 as residualf and calculatet with ida. Because $R_c$ growth with $V$ linear, $\Delta p$ does too:
$$0 = \dot{V} -1$$
and
$$0 = -\Delta p \cdot A + \mu \left( R_m + \alpha\cdot \gamma \frac{V}{A} \right)\dot{V} $$

[schmoelder]

Hey @daklauss,

how did you calculate this? Did you already integrate this into CADET-Python-Simulator or was this calculated "outside"?

[daklauss]

Calculated outside, wanted to test sundials ida a bit to get confortable with residuals, and the solver itself. The simple model confuses me a bit. Because how it is written down above it doesn't say anything about the Seperated Volume $V_s$. Assumption 6. and 7. is a bit confusing because 6. states $V_c = 0$ and 7. states $V_s = V_c$. So $V_s = 0$. Does it mean this simple model is there to calculate preasure drop?

Hey @daklauss,

how did you calculate this? Did you already integrate this into CADET-Python-Simulator or was this calculated "outside"?

[daklauss]

Simple Model for Dead End V2

Darcy's law provides a simple model for calculating the preasure Drop at a membrane:

$$\dot{V}_P = \Delta p \cdot \frac{A}{\mu} \cdot \frac{1}{R}$$

• $\dot{V}_P$ flow of the Permeate through the membrane
• $\Delta p$ pressure drow of the unit operation, so combined drop of cake and membrane?
• $A$ membrane area
• $\mu$ viscosity of our liquid
• $R$ hydraulic Resistance of our Unit Operation

With $R$ as the combined Resistance of the Cake and Membrane:

$$R = R_m + R_c $$

• $R_m$ Membrane resistance
• $R_c$ Cake resistance

With $h_c = V_c / A$ we can describe the Cake Resistance as:

$$R_c (t) = \alpha \cdot h_{cake} (t) = \alpha \cdot \frac{V_c (t)}{A}$$

• $h_c$ Cake height
• $\alpha$ Specific cake resistance
• $V_c$ Cake volume

We can split the Volume of the feed into two seperate parts:

$$ V_f = V_c + V_p \Rightarrow \dot{V}_f = \dot{V}_c +\dot{V}_p $$

• $V_f$ Volume of the feed

Assuming our filtration efficency is $100$% $= 1$, our Cakevolume is:

$$ \dot{V}_c = \gamma_s \cdot \dot{V}_f $$

• $\gamma$ Proportion of filtered material

Proportion of the filtered material is:

$\gamma_s = c^n _s \cdot \frac{M}{\rho ^m} = \frac{c^n _s}{\rho^n}$

or if Volume is zero:

$\gamma_s = V^n \cdot c^n _s $

• $M$ Molar Mass of suspended material
• $\rho^n$ Molar density of suspended material
• $V^n$ Molar Volume of suspended material
• $C^n _s$ molar concentration of suspended material
• $\rho^m$ mass density of suspended material

Assumptions

1. our filtration efficency is constant equal to $1$
2. Our Filter does filter as soon as the stream enters the cake
3. Single Component
4. Constant $R_m$

Open Questions

1. What about the viscosity? it should change inbetween $V_f$ and $V_P$ with Filtered particles

[schmoelder]

1. What about the viscosity? it should change inbetween $V_f$ and $V_P$ with Filtered particles

viscosity is a property of the fluid. As long as we do not discretize the cake, let's (for now) assume that vicosity is simply given by the feed stream.

$K$ weird konstant?

Where does the $K$ come from?

[daklauss]

As far as if looked for reference $K$ is described as "an empirical constant which depends on bed tortuosity."