Modelling of two phase flow in packed bed (continued)

[casualguitar]

The molar flow rate seems reasonable, but not the holdup

If the molar flow is ok, and the molar holdup is dependent on the molar flow, surely this means the bug is in the molar holdup equation itself (rather than in any question that feeds the molar holdup equation)?

 

[Chestermiller]

I've calculated the molar holdup by saying that:

$$m_j = \frac{P}{RT_g}A_C*\delta z*\epsilon$$

If these times were increased, is it correct to say that we would see the temperature variation 'spread out' across the bed rather than be concentrated at the inlet?

The value I calculated was for the entire column, not 1 tank. Our values should differ by a factor of 30. Your value was 0.0012 for a single tank, and mine would be 0.00012 moles for a single tank, a factor of 10 lower. I've checked my number, and it seems correct. please check yours.

 

[Chestermiller]

If the molar flow is ok, and the molar holdup is dependent on the molar flow, surely this means the bug is in the molar holdup equation itself (rather than in any question that feeds the molar holdup equation)?

Your molar holdup equation looks correct.

 

[casualguitar]

The value I calculated was for the entire column, not 1 tank. Our values should differ by a factor of 30. Your value was 0.0012 for a single tank, and mine would be 0.00012 moles for a single tank, a factor of 10 lower. I've checked my number, and it seems correct. please check yours.

Heres the exact molar holdup calculation:

And here's the output:

Other values:
Bed length = 3m
Bed diameter = 0.035m
n = 30
dz = bed length/n

Possibly the most likely different value is the bed diameter. Are you also using 0.035m?

 

[Chestermiller]

Heres the exact molar holdup calculation:
View attachment 304982
And here's the output:
View attachment 304983

Other values:
Bed length = 3m
Bed diameter = 0.035m
n = 30
dz = bed length/n

Possibly the most likely different value is the bed diameter. Are you also using 0.035m?

Their bed length is 0.3 m, not 3m.

 

[casualguitar]

Their bed length is 0.3 m, not 3m.

Whoops my mistake. I changed that and now yes I'm getting 0.00012 moles also for a single tank

Here are the plots with the correct bed length:

What I noticed was that this 'max levelling off temperature' of around -85C is not constant. If I zoom in:

The temperature at the inlet seems to actually slightly decrease over time

These plots are definitely quite a bit closer to the Tuinier plots though. The main things that are not the same are the temperature profiles near the inlet (not sure why the temperature doesn't increase further than -86C, and also possible the max mass deposition in our simulation being about double what is seen in Tuinier et al

 

[Chestermiller]

This looks much better. Please cut the calculated results off at 200 sec so we don't get distracted from their results.

Please run a calculation with the mass transfer totally turned off (k=0). This should tell us a lot about the temperature profile issues.

 

[casualguitar]

This looks much better. Please cut the calculated results off at 200 sec so we don't get distracted from their results.

Please run a calculation with the mass transfer totally turned off (k=0). This should tell us a lot about the temperature profile issues.

Temperature plot for k=0, much closer to Tuinier et al again:

The temperatures near the inlet are now very close to the Tuinier values. We just don't have the constant temperature phase change section yet. Would we expect to have this though for k=0?

 

[Chestermiller]

Temperature plot for k=0, much closer to Tuinier et al again:
View attachment 304988

The temperatures near the inlet are now very close to the Tuinier values. We just don't have the constant temperature phase change section yet. Would we expect to have this though for k=0?

Yes. This is about what I expected to see. The speed at which the profiles are advancing along the bed is on the order that I expected. How do the bed temperature profiles compare?

 

[casualguitar]

Yes. This is about what I expected to see. The speed at which the profiles are advancing along the bed is on the order that I expected. How do the bed temperature profiles compare?

Those plots above are with $U_g$ being calculated (non-constant), and the values are between 20 and 50 W/m2.K. $U_b$ has a constant value of 20 W/m2.K. Here are the temperature profiles for gas and bed:

To get closer to the Tuinier simulation I'll change the U_b value to a very high number. The output temperature profiles with very high $U_b$ are below. The temperature increases faster (as expected) and the temperature profiles are more similar than with the smaller heat transfer coefficient:

Note: all of these plots are with ki = 0!

 

[Chestermiller]

Those plots above are with $U_g$ being calculated (non-constant), and the values are between 20 and 50 W/m2.K. $U_b$ has a constant value of 20 W/m2.K. Here are the temperature profiles for gas and bed:
View attachment 305015View attachment 305016

To get closer to the Tuinier simulation I'll change the U_b value to a very high number. The output temperature profiles with very high $U_b$ are below. The temperature increases faster (as expected) and the temperature profiles are more similar than with the smaller heat transfer coefficient:
View attachment 305017
View attachment 305018

Note: all of these plots are with ki = 0!

Why is the abscissa 0 to 29? Why isn't it 1 to 30? Also, let's plot the points at the coordinates of the center of the tank in cm: 0.5, 1.5, ...,29.5.

It seems like there is something wrong with the mass transfer description in the model. We seem to be getting much high heat releases from the desublimation than Tunier. Try scaling back the k's in the model until we get a better match with Tunier. Keep track of the scale down factor.

 

[Chestermiller]

what is your logic for shutting off the water vaporization and CO2 desublimation when the amount of deposited liquid and CO2 goes to zero?

 

[casualguitar]

Why is the abscissa 0 to 29? Why isn't it 1 to 30? Also, let's plot the points at the coordinates of the center of the tank in cm: 0.5, 1.5, ...,29.5.

Thats just because I'm plotting from n=0 rather than n=1, its still 30 tanks but just shifted by one index. I should be able to change it but the output will be the same

To get these centre points I guess I should average adjacent points i.e.
$$T = \frac{T_j + T_{j-1}}{2}$$

Try scaling back the k's in the model until we get a better match with Tunier. Keep track of the scale down factor.

So are you saying to start with the k values I was using previously and keep dividing by say 10 for example until we see similar solid mass buildup profiles?

what is your logic for shutting off the water vaporization and CO2 desublimation when the amount of deposited liquid and CO2 goes to zero?

I'm passing a function to the integrator in which the ODEs are set up. Inside this function I've also got an if statement that just says if the solid/liquid deposited goes under 0, then set it equal to zero. So I don't think I've shut it off per se but just stopped the mass from taking a negative value

 

[Chestermiller]

Thats just because I'm plotting from n=0 rather than n=1, its still 30 tanks but just shifted by one index. I should be able to change it but the output will be the same

To get these centre points I guess I should average adjacent points i.e.
$$T = \frac{T_j + T_{j-1}}{2}$$

No. you plot T1 and x=0.5 cm, T2 at x=1.5 cm, ..., T30 at x = 29.5 cm

So are you saying to start with the k values I was using previously and keep dividing by say 10 for example until we see similar solid mass buildup profiles?

yes

I'm passing a function to the integrator in which the ODEs are set up. Inside this function I've also got an if statement that just says if the solid/liquid deposited goes under 0, then set it equal to zero. So I don't think I've shut it off per se but just stopped the mass from taking a negative value

What is set to zero, the deposition rate or the amount deposited? You need to shut the deposition rate off.

 

[casualguitar]

What is set to zero, the deposition rate or the amount deposited? You need to shut the deposition rate off.

I turned the deposition rate off also. So now if the solid mass deposited goes below zero, both deposition rate and the amount deposited are set to zero. This did not change the plots. I think the reason the plots are unchanged is that the amount and rate are set to zero before $\dot{m}_j$ is actually calculated

yes

Doing this currently (the scale down of $k_i$)

 

[casualguitar]

I turned the deposition rate off also. So now if the solid mass deposited goes below zero, both deposition rate and the amount deposited are set to zero. This did not change the plots. I think the reason the plots are unchanged is that the amount and rate are set to zero before $\dot{m}_j$ is actually calculated

Doing this currently (the scale down of $k_i$)

k_i/100000:

ki/10000:

ki/1000:

ki/100:

Actually at ki/100 the gas temperature was still visually basically the same, however the bed temperature profile did start to show some hint of a constant temperature section:

 

[casualguitar]

Just splitting these up as there is a memory limit:
ki/10:

Unusual behaviour here for the bed temperature.

So it seems like the real scaledown factor is somewhere between 10 and 100? I'm basing this on the max amount of solid deposited on the MCO2 plots. It does seem to be fairly close to 100 though. I'll post all of the ki/100 plots (Tg, Tb, MCO2)

 

[casualguitar]

ki/100:

MCO2:

Zoomed in on the relevant section:

Tb:

Tg:

Note: the legend doesn't seem to show up on some plots. It is the same legend for all plots

 

[Chestermiller]

ki/100:

MCO2:
View attachment 305094
Zoomed in on the relevant section: View attachment 305095

Tb: View attachment 305096

Tg:
View attachment 305097

Note: the legend doesn't seem to show up on some plots. It is the same legend for all plots

Try k/30

 

[casualguitar]

Try k/30

k/30:

Here is a plot of the rate of desublimation of CO2 for the same selection of times:

The MCO2 plot seems to suggest that there are two plugs of CO2 moving through the bed rather than one. However the dM_CO2_dt plot doesn't suggest this

 

[Chestermiller]

k/30:
View attachment 305128
View attachment 305130

View attachment 305129
Here is a plot of the rate of desublimation of CO2 for the same selection of times:
View attachment 305150

The MCO2 plot seems to suggest that there are two plugs of CO2 moving through the bed rather than one. However the dM_CO2_dt plot doesn't suggest this

So what is your interpretation of what is happening here?

Please provide a documentation of the finite difference equations for heat and mass transfer that are currently being used in the model.

 

[casualguitar]

So what is your interpretation of what is happening here?

I had turned both the MCO2 value and the dM_CO2_dt values to zero in the case where the solid buildup goes below zero, but really its just the MCO2 value that needs to be zero I think

Changing this (so now in the case of negative solid buildup the value is set to zero), here is the output for k/30:

Just checking some more ki/x values now to try the solid deposition to peak at a similar value to Tuinier et al

 

[casualguitar]

Just checking some more ki/x values now to try the solid deposition to peak at a similar value to Tuinier et al

After turning off the if statement that forced the rate of sublimation to be zero the plots look fairly close to Tuinier et al with some differences (k/100):

Adjusting the heat transfer coefficients:
If Ug and Ub are both = 100000 we get the plot below for Tb (I think very high Ub and Ug approximates the Tg=Tb assumption made in the Tuinier model?):

And here's Ub = 100000 and Ug = between 20 and 50 (calculated for each tank):

For the MCO2 plot, is their 'fudge factor' as you previously called it coming into play here? The max solid deposited seems to be in the ballpark of the Tuinier model, however our rate of sublimation of this solid seems to be a lot lower (their peaks are sharper).

Lastly, when Ug is not a very large number the constant temperature zone seems to appear for Tb, but not for Tg just yet

 

[Chestermiller]

After turning off the if statement that forced the rate of sublimation to be zero the plots look fairly close to Tuinier et al with some differences (k/100):
View attachment 305166
View attachment 305167
View attachment 305168

Adjusting the heat transfer coefficients:
If Ug and Ub are both = 100000 we get the plot below for Tb (I think very high Ub and Ug approximates the Tg=Tb assumption made in the Tuinier model?):
View attachment 305169

And here's Ub = 100000 and Ug = between 20 and 50 (calculated for each tank):

View attachment 305171
View attachment 305173

For the MCO2 plot, is their 'fudge factor' as you previously called it coming into play here? The max solid deposited seems to be in the ballpark of the Tuinier model, however our rate of sublimation of this solid seems to be a lot lower (their peaks are sharper).

Lastly, when Ug is not a very large number the constant temperature zone seems to appear for Tb, but not for Tg just yet

Refresh my memory. What do you use for Ub in the standard situation, and show its calculation.

 

[casualguitar]

Refresh my memory. What do you use for Ub in the standard situation, and show its calculation.

Apologies for the delay (public holiday here) -

$U_b$ is calculated by:
$$U_b = \frac{k_p}{\frac{d_p}{\beta}}$$
where $k_p$ = 0.8W/m., $d_p$ is 0.00404m and $\beta$ = 10

Using the above values, $U_b$ = 1980 w/m2.K

I checked the $k_p$ value (ranged it from 0.1 to 50) and changing the value of $k_p$ does not seem to affect the plots visually

 

[Chestermiller]

Apologies for the delay (public holiday here) -

$U_b$ is calculated by:
$$U_b = \frac{k_p}{\frac{d_p}{\beta}}$$
where $k_p$ = 0.8W/m., $d_p$ is 0.00404m and $\beta$ = 10

Using the above values, $U_b$ = 1980 w/m2.K

I checked the $k_p$ value (ranged it from 0.1 to 50) and changing the value of $k_p$ does not seem to affect the plots visually

Looks good.

 

[casualguitar]

Looks good.

Here is the output for $U_b$ calculated as above, and $U_g$ calculated at each interval

I think generally the output above is similar to the Tuinier output. The main two differences seem to be:
1) the rate of sublimation of the solid buildup (solid to gas) is lower in our model (gradually tails off rather than immediately sublimates
2) there is no constant temperature gas section showing up in our model

Checking the dTg_dt plot shows that there is no section where the value of dTg_dt reaches zero (a constant temperature section):

In the Tuinier model, what is the driving force that actually keeps the gas temperature almost constant while the phase is changing?

 

[Chestermiller]

Here is the output for $U_b$ calculated as above, and $U_g$ calculated at each interval
View attachment 305254View attachment 305255View attachment 305256View attachment 305257

I think generally the output above is similar to the Tuinier output. The main two differences seem to be:
1) the rate of sublimation of the solid buildup (solid to gas) is lower in our model (gradually tails off rather than immediately sublimates
2) there is no constant temperature gas section showing up in our model

Checking the dTg_dt plot shows that there is no section where the value of dTg_dt reaches zero (a constant temperature section):
View attachment 305259
In the Tuinier model, what is the driving force that actually keeps the gas temperature almost constant while the phase is changing?

Is it correct to say that these results were obtained using the corrected mass transfer description together with all other parameters standard?

I think the difference is that we have heat transfer resistance between the gas and bed and a different mass transfer parameterization (and different dispersion parameterization). We can make our results more like theirs by increasing Ug (I think). You can see that the bed temperature is closer to constant than the gas temperature at -90.

We can also decrease the dispersion by adding more tanks, holding the length of the column constant.

 

[casualguitar]

Is it correct to say that these results were obtained using the corrected mass transfer description together with all other parameters standard?

Yes these results were obtained using the Tuinier dimensions, flow and initial conditions as much as possible, with the corrected mass transfer coefficients of $k_{CO_2}$/100 and $k_{H_2O}$/100. The $U_b$ value is calculated according to the equation in post 305, and the$U_g$ value is calculated using the correlations from BSL

We can make our results more like theirs by increasing Ug (I think). You can see that the bed temperature is closer to constant than the gas temperature at -90.

Ok I see what you're saying. If we increase $U_g$ then the gas temperature will more closely follow the bed temperature. Checking a value of $U_g$= 100000 gives this $T_g$ profile which is visually identical to the $T_b$ profile:

As a note, if I calculate the $U_g$ values at each interval (which gives a range of about 20-50W/m2.K), then $T_b$ starts to show the almost constant temperature section, which doesn't happen at high $U_g$ values. The $T_g$ profile is unchanged from the one above:

Running the opposite (calculated $U_g$ and a high $U_b$ value gives identical output to the $T_b$ and $T_g$ profiles above)

We can also decrease the dispersion by adding more tanks, holding the length of the column constant.

Checking n=100 now for calculated $U_g$ and calculated $U_b$ values (no very high values). This will take a fair while to run so I'll post these results once it finishes

Is this considered 'model tuned'? I guess this is a judgement call rather than anything but it seems pretty close. If this is 'model tuned' I was thinking about taking what was learned during debugging here and applying it to the original liquefaction model (I'm guessing I can find some bugs in that now, and the model is simpler in comparison)

But besides this, I was thinking about some other things. The Tuinier paper (in my view) isn't that useful for a new reader because it doesn't actually say how good the system is (or can be), but rather just shows output for the system they used. What I mean is that they don't define performance parameters that would allow someone to compare this system with others or get a feel for how good this system actually is

To do something like this would involve (I think) answering questions like "what performance parameters are actually useful to calculate here?", "what simulations best show how these performance parameters vary with varied input" and lastly "does the model need to be developed further to be 'different enough' from the Tuinier model". Or something like these anyway

Regarding the performance parameters, I guess there are standard ones. However, possibly one meaningful parameter is the separation efficiency (how good can the system get at separating CO2 from both N2 and H2O in theory).

And lastly one other thing Tuinier didn't do is vary the ICs/BCs to arrive at an optimal solution (optimising the performance parameters). This sounds interesting to do and would also be novel as far as I can see (just trying to think of ways to further separate this model from Tuinier and make it of use to a future reader)

 

[Chestermiller]

Yes these results were obtained using the Tuinier dimensions, flow and initial conditions as much as possible, with the corrected mass transfer coefficients of $k_{CO_2}$/100 and $k_{H_2O}$/100. The $U_b$ value is calculated according to the equation in post 305, and the$U_g$ value is calculated using the correlations from BSL

So you used our k divided by 100. Please double check the finite difference equation used in our model for the mass flux from gas to solid to make sure we are not missing something (like say a factor of $\Delta z$).

Ok I see what you're saying. If we increase $U_g$ then the gas temperature will more closely follow the bed temperature. Checking a value of $U_g$= 100000 gives this $T_g$ profile which is visually identical to the $T_b$ profile:View attachment 305285

As a note, if I calculate the $U_g$ values at each interval (which gives a range of about 20-50W/m2.K), then $T_b$ starts to show the almost constant temperature section, which doesn't happen at high $U_g$ values. The $T_g$ profile is unchanged from the one above:
View attachment 305288

Running the opposite (calculated $U_g$ and a high $U_b$ value gives identical output to the $T_b$ and $T_g$ profiles above)Checking n=100 now for calculated $U_g$ and calculated $U_b$ values (no very high values). This will take a fair while to run so I'll post these results once it finishes

Is this considered 'model tuned'? I guess this is a judgement call rather than anything but it seems pretty close. If this is 'model tuned' I was thinking about taking what was learned during debugging here and applying it to the original liquefaction model (I'm guessing I can find some bugs in that now, and the model is simpler in comparison)

Model tuning is when you tune it to your own data. What we are doing here is BENCHMARKING the model against the Tunier model to make sure we are consistent. It seems to me that, at present, we are using much lower values of k than the equivalent they are using.

I suggest calculating the total amount of CO2 solid deposited on the bed so that we can compare it with the amount of CO2 that entered the bed up to any time. See what you get with the standard k and with k/100.

But besides this, I was thinking about some other things. The Tuinier paper (in my view) isn't that useful for a new reader because it doesn't actually say how good the system is (or can be), but rather just shows output for the system they used. What I mean is that they don't define performance parameters that would allow someone to compare this system with others or get a feel for how good this system actually is

To do something like this would involve (I think) answering questions like "what performance parameters are actually useful to calculate here?", "what simulations best show how these performance parameters vary with varied input" and lastly "does the model need to be developed further to be 'different enough' from the Tuinier model". Or something like these anyway

Regarding the performance parameters, I guess there are standard ones. However, possibly one meaningful parameter is the separation efficiency (how good can the system get at separating CO2 from both N2 and H2O in theory).

And lastly one other thing Tuinier didn't do is vary the ICs/BCs to arrive at an optimal solution (optimising the performance parameters). This sounds interesting to do and would also be novel as far as I can see (just trying to think of ways to further separate this model from Tuinier and make it of use to a future reader)

These are all good ideas once you are comfortable that you have a working model that matches your experimental data.

 

[casualguitar]

So you used our k divided by 100. Please double check the finite difference equation used in our model for the mass flux from gas to solid to make sure we are not missing something (like say a factor of Δz).

Yes k/100

When you say finite difference equation for the gas-solid mass flux do you mean this:
$$\dot{M}_i^"=k_i\left(\frac{Py_i-p_i(T_I)}{RT_I}\right)$$
Is this considered a 'finite difference equation'? If this is the equation then I'll check for missing terms. I could well have missed something in the $Sh_{loc,i}$ equation or something

I suggest calculating the total amount of CO2 solid deposited on the bed so that we can compare it with the amount of CO2 that entered the bed up to any time. See what you get with the standard k and with k/100.

Can do

These are all good ideas once you are comfortable that you have a working model that matches your experimental data.

Sure, I'll save this and return to it after this k/100 issue is sorted

 

[Chestermiller]

Yes k/100

When you say finite difference equation for the gas-solid mass flux do you mean this:
$$\dot{M}_i^"=k_i\left(\frac{Py_i-p_i(T_I)}{RT_I}\right)$$
Is this considered a 'finite difference equation'? If this is the equation then I'll check for missing terms. I could well have missed something in the $Sh_{loc,i}$ equation or something

No, I don't think this is the equation I was concerned with. I was concerned more with $$\frac{dM_i}{dt}=\dot{M}_i^"A_s$$where $$A_s=\frac{6}{d_p}(1-\epsilon)A_c\Delta z$$But I think we already checked this out and agreed that it is correct. Anyway, please check again.

I think we should also check the overall mass balance on the CO2 at the selected times. Please calculate the total moles of CO2 that have flowed into the bed up to time t, the total number of moles of CO2 that has exited the bed up to time t, the total molar holdup of CO2 in the gas phase in the column at time t, and the total molar holdup of CO2 on the bed at time t. In addition to these giving us a check on the CO2 mass balance, they will also tell the fraction of the CO2 that is captured by the bed up to a given time.

 

[casualguitar]

Please calculate the total moles of CO2 that have flowed into the bed up to time t

$$M_{IN,TOTAL} = \dot{m} * t\tag{1}$$

the total number of moles of CO2 that has exited the bed up to time t

Hmm. I think we could do that using the total molar holdup and the inlet flow, something like this:
$$M_{OUT,TOTAL} = \sum_{t=1}^{t}{\dot{m}_{IN} - m_{HOLDUP}}\tag{3}$$
Which I think is the sum of the outlet flows at all times in the bed

the total molar holdup of CO2 in the gas phase in the column at time t

Hmm so we have to consider each tank individually here given the temperature profile.
The molar holdup of a single tank is $(P/RT)(A_cdz\epsilon)$
So the total molar holdup would be:
$$M_{GAS} = \sum_{j=0}^{n}(P/RT_j)(A_cdz\epsilon)\tag{4}$$

and the total molar holdup of CO2 on the bed at time t

I guess this is solid phase only, so:
$$M_{SOLID} = \sum_{j=0}^{n}(M_j)\tag{5}$$

they will also tell the fraction of the CO2 that is captured by the bed up to a given time.

When you say 'fraction of CO2 captured' what does this mean? Do we define 'captured' as meaning solid phase? So the fraction would be the solid mass of CO2 at a point divided by the total mass of CO2 at that point?

I'll work away on the above calculations and if they're incorrect I can adjust

 

[casualguitar]

$$M_{IN,TOTAL} = \dot{m} * t\tag{1}$$

Hmm. I think we could do that using the total molar holdup and the inlet flow, something like this:
$$M_{OUT,TOTAL} = \sum_{t=1}^{t}{\dot{m}_{IN} - m_{HOLDUP}}\tag{3}$$
Which I think is the sum of the outlet flows at all times in the bed

Hmm so we have to consider each tank individually here given the temperature profile.
The molar holdup of a single tank is $(P/RT)(A_cdz\epsilon)$
So the total molar holdup would be:
$$M_{GAS} = \sum_{j=0}^{n}(P/RT_j)(A_cdz\epsilon)\tag{4}$$

I guess this is solid phase only, so:
$$M_{SOLID} = \sum_{j=0}^{n}(M_j)\tag{5}$$

When you say 'fraction of CO2 captured' what does this mean? Do we define 'captured' as meaning solid phase? So the fraction would be the solid mass of CO2 at a point divided by the total mass of CO2 at that point?

I'll work away on the above calculations and if they're incorrect I can adjust

I'll separate these plots out, however just for an overview these are the values mentioned above plotted versus time (note I've assumed CO2 captured is the mass of solid CO2 in the column divided by the total mass of CO2 in the column):

 

[Chestermiller]

$$M_{IN,TOTAL} = \dot{m} * t\tag{1}$$

This needs to be multiplied by $y_{in}$ for CO2

Hmm. I think we could do that using the total molar holdup and the inlet flow, something like this:
$$M_{OUT,TOTAL} = \sum_{t=1}^{t}{\dot{m}_{IN} - m_{HOLDUP}}\tag{3}$$
Which I think is the sum of the outlet flows at all times in the bed

This assumes that the mass balance is truly satisfied. The objective of this exercise is to check to see if that is the case. If $\dot{m}_i(t)$ is the total molar flow rate out of tank i at time t, then the total molar amount of CO2 leaving the column up to time t is$$\int_0^t{\dot{m}_n(t')y_{CO2}(t')dt'}$$

Hmm so we have to consider each tank individually here given the temperature profile.
The molar holdup of a single tank is $(P/RT)(A_cdz\epsilon)$
So the total molar holdup would be:
$$M_{GAS} = \sum_{j=0}^{n}(P/RT_j)(A_cdz\epsilon)\tag{4}$$

The total molar holdup of CO2 in the gas would be $$M_{CO2} = \sum_{j=1}^{n}(P/RT_j)y_{j,CO2}(A_cdz*\epsilon)\tag{4}$$
Note that the summation is from tank 1 to tank n, the mole fraction of CO2 multiplies the total number of moles, and that you multiply by the void fraction rather than dividing.

I guess this is solid phase only, so:
$$M_{SOLID} = \sum_{j=0}^{n}(M_j)\tag{5}$$

Again, the sum is from 1 to n.

When you say 'fraction of CO2 captured' what does this mean? Do we define 'captured' as meaning solid phase?

Yes

So the fraction would be the solid mass of CO2 at a point divided by the total mass of CO2 at that point?

I don't know what this means, but it doesn't sound correct.

I'll work away on the above calculations and if they're incorrect I can adjust