Add tracking of c/n in cake

[daklauss]

This pull request adds c_cake and n_cake for the DeadEndFiltration model. It also reworks the entire state structure of the unit operation as the following graphics and equations show.

So now a short summary about the residual equations:
Inlet (note that $\tilde{c}_{in}$ is the connection value between uo.):

$$0=c^{in}_i-\tilde{c}^{in}_i$$ $$0 = \dot{n}^{in}_i -Q^{in}\cdot c^{in}_i$$

Retentate (rej is based on the rejection model):

$$0=\dot{n}^R_i - rej_i \cdot \dot{n}^{in}_i$$

mw are molecular weights, $\rho$ is the density

$$0=\dot{V}_R - \sum_i \dot{n}^R_i * mw_i / \rho_i$$

if \dot{V}_R > 1e-16:

$$0=c^R_i -\dot{n}_i^R / \dot{V}^R$$

else:

$$0=c^R_i$$

Permeate:

$$0=\dot{n}^P_i - rej_i \cdot \dot{n}^{in}_i$$ $$0=\dot{V}_P - \sum_i \dot{n}^P_i \cdot mw_i / \rho_i$$

if \dot{V}_P > 1e-16:

$$0=c^P_i -\dot{n}_i^P / \dot{V}^P$$

else:

$$0=c^P_i$$

Cake Equations:

$$0 = V^C - \sum_i n^C \cdot mw_i/ \rho_i$$ $$0=\dot{n}^C_i - \dot{n}^R_i$$

if V^C> 1e-16:

$$0=\dot{c}^C_i -(\dot{n}_i^C - c^C \cdot \dot{V}^C_i ) / V^C$$

else:

$$0=\dot{c}^C_i$$

Permeate tank equations:

$$0 = \dot{c}^PT_i - (\dot{n}^P_i - Q^{out}\cdot c^{PT}_i - \dot{V}^{PT} \cdot c^{PT}_i) / V^{PT}$$ $$0 = \dot{V}^PT - \dot{V}^P + Q^{out}$$ $$0 = n^{PT}_i - c^{PT}_i \cdot vol^{PT}$$

Pressue equation:
CV is the volume of a specific component

$$CV_i = n^C_i mw_i / \rho_i$$

$R^C$ is the cake resistance, $\alpha$ is the specific cake resistance:

$$R^C= \sum_i \alpha_i \cdot \rho CV_i$$

so the pressure equations is:
if not np.sum(\dot{n}^P) < 1e-16:

$$0 = \Delta P - \mu \cdot \dot{V}^P \cdot (R^M + R^C) / A$$

else:

$$0 = \Delta P$$

[daklauss]

Tests have been updated and additionally n_cake is part of the Unit Operation. For now c_cake is neither tracked nor somehow calculated. The user has to manually calculate c_cake by using n_cake and volume. This can't be used withing the residual because of numerical issues. (Using c= n/V leads to IDA not converging while it is actually not necessary). In a further issue it is maybe necessary to implement an additional functionality to track states/values that aren't used for the residual Calculation.

[schmoelder]

Let's move the formatting commit to its own PR.

[schmoelder]

Thanks for taking care of this @daklauss

For now c_cake is neither tracked nor somehow calculated.

Is there a technical reason for this? I believe, the equations should be the same as for c_permeate_tank.

[daklauss]

They are the same but even the permeate tank is crashing with Volume = 0? But in comparsion it is not reasonable to have an initial cake? I've tried today to vary them in some regards but it leads to IDA not beeing able to converge. I don't know.

In hindsight im a little bit confused why n_in_cake and n_cake are both tracked even when they are (at least for now?) the same.

[schmoelder]

In hindsight im a little bit confused why n_in_cake and n_cake are both tracked even when they are (at least for now?) the same.

Hmm, they should not be the same. n_in_cake is a stream. so maybe we should consider calling it n_dot_cake_in, whereas n_cake is the cumulative molar amount of the components.

[daklauss]

In hindsight im a little bit confused why n_in_cake and n_cake are both tracked even when they are (at least for now?) the same.

Hmm, they should not be the same. n_in_cake is a stream. so maybe we should consider calling it n_dot_cake_in, whereas n_cake is the cumulative molar amount of the components.

No particles are leaving the cake, so every particle streaming into the cake is part of the cake? It would be more of an issue to use different n if idk some particles react or we would use a more complicated model where cake particles could flow through the membrane statisticly.

[schmoelder]

I don't think so. If n_in = 0, that does not mean that n_cake will probably become 0. It just won't change.

[daklauss]

$$\dot{n}_{in} = \dot{n}_{feed} \cdot rejection$$

So $n_{in}$ tracks all incoming Particles. $n_{cake}$ is calculated by:

$$\dot{n}_{cake} = \dot{n}_{in}$$

So i guess in our case they are the same. I mean the only difference would be if you would to set the filtercake sectionwise to 0, then $n_{cake}$ would track the current particlecount and $n_{cake}$ the whole.

[daklauss]

Needed a bit to long for this png ... one question is still that $c^{out}_i$ and $c^{PT}_i$ are the same. So do we track them separately or not?

[schmoelder]

one question is still that $c^{out}_i$ and $c^{PT}_i$ are the same.

Yes, they are the same since it's assumed an ideal stirred tank.

So do we track them separately or not?

I's suggest to track them separately. While they are numerically the same, we want to differentiate the unit operation outlet (which is a general interface which should be consistent across all unit operations) and the internal state.

How is this done in the CSTR?

[daklauss]

In the CPS CSTR we have inlet and Bulk and $c^{out}_i$ is the $c_i$ of the bulk. But tbh. i'm not sure how functionable this cstr is. Haven't touched anything beside the DEF in a while. I dont't think this matters to much numerically tbh....

[schmoelder]

then for now, treat it the same.

[daklauss]

So i kind of reworked state, residual equations and Initialization but it currently fails. Aiming to write the residuals equations later here. IDA is currently not converging.

[schmoelder]

So i kind of reworked state, residual equations and Initialization but it currently fails. Aiming to write the residuals equations later here. IDA is currently not converging.

You need help with that?

[daklauss]

So i kind of reworked state, residual equations and Initialization but it currently fails. Aiming to write the residuals equations later here. IDA is currently not converging.

You need help with that?
Maybe later.

[daklauss]

So now a short summary about the residual equations:
Inlet (note that $\tilde{c}_{in}$ is the connection value between uo.):

$$0=c^{in}_i-\tilde{c}^{in}_i$$ $$0 = \dot{n}^{in}_i -Q^{in}\cdot c^{in}_i$$

Retentate (rej is based on the rejection model):

$$0=\dot{n}^R_i - rej_i \cdot \dot{n}^{in}_i$$

mw are molecular weights, $\rho$ is the density

$$0=Q_R - \sum_i \dot{n}^R_i * mw_i / \rho_i$$

if Q_R > 1e-16:

$$0=c^R_i -\dot{n}_i^R / Q^R$$

else:

$$0=c^R_i$$

Permeate:

$$0=\dot{n}^P_i - rej_i \cdot \dot{n}^{in}_i$$ $$0=Q_P - \sum_i \dot{n}^P_i \cdot mw_i / \rho_i$$

if Q_P > 1e-16:

$$0=c^P_i -\dot{n}_i^P / Q^P$$

else:

$$0=c^P_i$$

Cake Equations:

$$0 = V^C - \sum_i n^C \cdot mw_i/ \rho_i$$ $$0=\dot{n}^C_i - \dot{n}^R_i$$

if V^C> 1e-16:

$$0=\dot{c}^C_i -(\dot{n}_i^C - c^C \cdot \dot{V}^C_i ) / V^C$$

else:

$$0=\dot{c}^C_i$$

Permeate tank equations:

$$0 = \dot{c}^PT_i - (\dot{n}^P_i - Q^{out}\cdot c^{PT}_i - \dot{V}^{PT} \cdot c^{PT}_i) / V^{PT}$$ $$0 = \dot{V}^PT - Q^P + Q^{out}$$ $$0 = n^{PT}_i - c^{PT}_i \cdot vol^{PT}$$

Pressue equation:
CV is the volume of a specific component

$$CV_i = n^C_i mw_i / \rho_i$$

$R^C$ is the cake resistance, $\alpha$ is the specific cake resistance:

$$R^C= \sum_i \alpha_i \cdot \rho CV_i$$

so the pressure equations is:
if not np.sum(\dot{n}^P) < 1e-16:

$$0 = \Delta P - \mu \cdot Q^P \cdot (R^M + R^C) / A$$

else:

$$0 = \Delta P$$

This is still work in Progress. I highly doubt, $Q_P$ and $Q_R$ are viable here to use and we have to rely on $V_P$ and $V_R$ and their derivatives, because otherwise the differential index is not 1 ?

[daklauss]

Yeah so to sum it up, it seems to converge but wether the solution does make sense or not i can't say yet.

[schmoelder]

Could you please move the figure and the corresponding equations to your first post? That would make it easier to find the relevant information, should we ever need to come back to this PR.

[daklauss]

Could you please move the figure and the corresponding equations to your first post? That would make it easier to find the relevant information, should we ever need to come back to this PR.

Done