Modelling of two phase flow in packed bed (continued)

In summary: I don't know actually, but I think you will be right about the CO2 depositing temporarily on the bed. What I thought would happen (assuming a bed colder than the freezing point of CO2) was that the ambient CO2 enriched stream would enter the cold bed and immediately the CO2 at the 'front' of the stream would freeze. The pure air would carry on through the bed. Then the newly entering stream - which is at ambient temperature - would vaporise the frozen CO2, and the vaporised CO2 plus the CO2 'behind' it in the stream would now be frozen/deposited slightly further downstream. This process repeats until you
  • #421
Updating on an initial attempt to reformulate the sharp front model on a mole basis:

Converting their capture step sharp front equations to a mole basis -

Component mass balance for defrost front:

and are the molar flow rates after and before the defrost front, respectively (in moles per second), and are the molar fractions of component 'i' after and before the defrost front, respectively. is the cross-sectional area of the bed through which the gas flows (in square meters). is the number of moles of component 'i' deposited per unit of bed volume (in moles per cubic meter). is the velocity of the defrost front (in meters per second), calculated as the distance the defrost front travels divided by the time taken.

Component mole balance for frost front:


So we can see here that the molar flow out of the defrost front is equal to the molar flow into the frost front.

The frost and defrost front velocities dont change for our model. I am guessing that we will use the value to track the position of the front:



Energy balances on a mole basis:



where the heat capacities are in

Overall mole balance for each front:




We also have the molar desublimation rate equation which might be useful to calculate $m_i$:


Just one question - have I missed a relation here? It looks like we have 7 equations and 8 unknowns. I'm defining the unknowns as: , the molar fluxes out of the fronts and , the mole fractions out of the fronts and , the front velocities and , and the amount of solid buildup in mol/m3.

If this is not the intended reformulation just let me know and I can reform these equations
 
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  • #422
Initial attempt to generate the sharp front heat balances from our equations:

The heat balance for the gas phase is given by:

And the heat balance for the packed bed:

Assuming an infinite heat transfer coefficient means that Tg = Tb. Also assuming axial dispersion = 0 we can discount the dispersion term.

If we make these assumptions, and add the two heat balances we get:

We know the relationship between and so we can sub out using:

Subbing in and cancelling:


Assuming an infinite heat transfer coefficient means that all co2 in a spatial element is either sublimated or desublimated depending on the temperature (this happens instantly). So is actually replaced with , where A is the CSA and v is the velocity (of the front):

At this point I'm not sure how to finish this out. If we let subscript 0 denote before the defrost front, subscript 1 denote between the defrost and frost fronts, and subscript 2 denote after the frost front, we can say that = for defrost and = for frost.

I can recreate the RHS of their heat balance for the defrost front for example knowing that:

That leaves the LHS as below, which I am not sure how to simplify to what Tuinier has:

They seem to have factored out but we only have one velocity term on the LHS. Are we going to integrate the time partial derivatives here or do we know the value of this term?

For reference, their LHS is = , and their RHS is the same as ours
 
  • #423

I need to think about this some more. If the deposition term were not there, we Ould have
 
  • #424
Chestermiller said:
I need to think about this some more. If the deposition term were not there, we Ould have
Interesting. I assume this form is useful because it is a form of the advection equation, where everything before the spatial partial derivative is the velocity of the front. This equation seems to apply when we're outside the defrost/frost zone (as is zero)

If we did have the deposition term (which is the case in between the frost and defrost fronts):

We get a first-order linear non-homogeneous PDE, which is the same as you had with an additional source term:

If we're going the road of using a step function then I suppose the general solution is below but it doesn't seem useful:

Can we equate the velocity in the deposition term and the velocity term in the advection equation defined by you above?
 
  • #425
Hi Chet, I just wanted to clear up the wording on the constant dispersion length assumption - are we saying the first or second statement below?
1) the dispersion length is assumed to be equal to
2) the spatial increment is assumed to be equal to , where is the dispersion length

I realise that these are very similar. In our discussion it seems that you're saying its "let = ", but when we get to this point in our discretisation scheme:


We sub in = and we dont sub in for :




Is it true then to say that we are actually saying "assume = /2", and not "assume = 2*l?

Apologies for the verbose question
 
  • #426
casualguitar said:
Interesting. I assume this form is useful because it is a form of the advection equation, where everything before the spatial partial derivative is the velocity of the front. This equation seems to apply when we're outside the defrost/frost zone (as is zero)

If we did have the deposition term (which is the case in between the frost and defrost fronts):

We get a first-order linear non-homogeneous PDE, which is the same as you had with an additional source term:

If we're going the road of using a step function then I suppose the general solution is below but it doesn't seem useful:

Can we equate the velocity in the deposition term and the velocity term in the advection equation defined by you above?
I'm not so sure this is exactly right. I'm struggling with this, and am having trouble getting the mental concentration to do this right. I'm usually much better at formulation than this.
 
  • #427
Chestermiller said:
I'm not so sure this is exactly right. I'm struggling with this, and am having trouble getting the mental concentration to do this right. I'm usually much better at formulation than this.
As a side point, I'm not exactly sure if the time required here is worth what it would give us? It strengthens the case for saying the model is 'validated' yes, however we have already validated this model at least somewhat with the Tuinier experimental data
 
  • #428
casualguitar said:
As a side point, I'm not exactly sure if the time required here is worth what it would give us? It strengthens the case for saying the model is 'validated' yes, however we have already validated this model at least somewhat with the Tuinier experimental data
I have a different perspective. It looks like the simple sharp front model tells 90 % of the story, except for a small amount of dispersion at the fronts. It certainly would be much simpler to do the real modeling and design with the sharp front model, and the results would be easier to interpret physically.
 

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