Modelling of two phase flow in packed bed (continued)

[Chestermiller]

Well here is the gas temperature vs position (for a range of times):
View attachment 304702
The gas temperature profile seems to stop changing in any significant way after about 2000s. The final range of temperatures across the bed (210K to about 180K) does seem to be significant because as you said its this range that sublimation/desublimation occurs, suggesting that there is something not working with the sublimation/desublimation coding logistics

For reference, here's the time vs gas temperature plot for the 10 tank setup:
View attachment 304703

I'm now wondering why the evaporation bug would occur across a range of temperatures (210 to 180K), rather than at one specific temperature (as the pressure is constant). It seems slightly more intuitive that the temperature profile of the bed would level out at a single temperature (say 210K), rather than trend towards the profile it has trended towards

The results don't look anything like the results in the paper. Try running a calculation for the same operating data as theirs and see how close you can come to matching them: column design, packing, initial temperature, inlet flow rate and mole fractions.

 

[Chestermiller]

The leveling off at 180 K could be consistent with CO2 desublimation. On their graphs, this leveling is at -95 C. Of course, the times are very different.

 

[casualguitar]

The leveling off at 180 K could be consistent with CO2 desublimation. On their graphs, this leveling is at -95 C. Of course, the times are very different.

The times are different yes I suppose because the flow, dimensions, initial conditions etc are different, but yes the temperature could be consistent with CO2 desublimation.

Why a higher $k_i$ value would cause this levelling off, and not the lower value is not obvious to me. there's a point at around $k_i$ = 10^-6 somewhere where the temperature no longer gets to inlet temperature but instead levels off at a temperature around the desublimation temperature. I can try find a more accurate value for this $k_i$ value

 

[Chestermiller]

The times are different yes I suppose because the flow, dimensions, initial conditions etc are different, but yes the temperature could be consistent with CO2 desublimation.

Why a higher $k_i$ value would cause this levelling off, and not the lower value is not obvious to me. there's a point at around $k_i$ = 10^-6 somewhere where the temperature no longer gets to inlet temperature but instead levels off at a temperature around the desublimation temperature. I can try find a more accurate value for this $k_i$ value

Please don’t bother. Just use their input operating conditions please. This will tell us a lot.

 

[casualguitar]

Please don’t bother. Just use their input operating conditions please. This will tell us a lot.

So using their operating conditions as much as is possible (I can lay out the exact values used if necessary), this is the position vs gas temperature output:

And the Tuinier et al equivalent:

So they are not that similar in that our one stops at -90C (around the desublimation temperature), and that there is no clear constant temperature section.

One thing I noticed - here's the plot of CO2 solid buildup versus position:

This is in moles. Notice how the solid builds up at each position and then decreases, suggesting that sublimation is also occurring. But how can this happen if the temperature is below the sublimation temperature? I don't know the exact temperature of sublimation but even if the gas temperature is slightly above that it would still likely be slow desublimation, not like the above

The mechanics for the sublimation/liquefaction pressure are as follows:

and for water:

The above is just to show what I do outside the temperature bounds. I either set the sublimation/liquefaction pressure equal to 0 or a very large number

It seems odd that the solid buildup would take on a normal trend, while the gas temperature stays below (or on) the sublimation temperature?

EDIT: I mentioned above that there is no clear constant temperature section in the plot, when actually this is probably not true as the constant temperature section would occur at the maximum temperature on this plot. I guess if the temperature went higher (above sublimation temperature) we would see that constant section

 

[casualguitar]

In addition, one thing I have found is that the $Q_{IB}$ and $Q_{GI}$ values at the left boundary start at a few hundred and trend gradually to zero (as expected), however the internal node values of $Q_{IB}$ and $Q_{GI}$ start at a decimal value and trend to very small numbers (10^-10), suggesting that heat transfer in the inner nodes is much less. Looking into this

 

[Chestermiller]

So using their operating conditions as much as is possible (I can lay out the exact values used if necessary), this is the position vs gas temperature output:

View attachment 304767
And the Tuinier et al equivalent:

View attachment 304768
So they are not that similar in that our one stops at -90C (around the desublimation temperature), and that there is no clear constant temperature section.

One thing I noticed - here's the plot of CO2 solid buildup versus position:View attachment 304769
This is in moles. Notice how the solid builds up at each position and then decreases, suggesting that sublimation is also occurring. But how can this happen if the temperature is below the sublimation temperature? I don't know the exact temperature of sublimation but even if the gas temperature is slightly above that it would still likely be slow desublimation, not like the above

The mechanics for the sublimation/liquefaction pressure are as follows:
View attachment 304770
and for water:
View attachment 304771
The above is just to show what I do outside the temperature bounds. I either set the sublimation/liquefaction pressure equal to 0 or a very large number

It seems odd that the solid buildup would take on a normal trend, while the gas temperature stays below (or on) the sublimation temperature?

EDIT: I mentioned above that there is no clear constant temperature section in the plot, when actually this is probably not true as the constant temperature section would occur at the maximum temperature on this plot. I guess if the temperature went higher (above sublimation temperature) we would see that constant section

You need to compare these on a more common basis. Remember that $\Delta x=L/n$, where n is the total number of tanks. The tank centers are therefore at $\Delta x/2, 3\Delta x/2, 5\Delta x/2,\ etc$. Plot the data at these distances, not tank numbers. There is also a data point at x = 0, of course, corresponding to the inlet conditions.

I'm guessing that you are using 10 tanks for the whole bed. They seem to be using a finer grid, with approximately 3x your resolution. So,, for comparison, please use 30 tanks.

You should also be comparing the mass buildup per unit area of bed, not the molar solid buildup within each tank. Please, for comparison, show the mass buildups per unit bed area within each tank.

I'm definitely not going to go through your coding. If you want to show a logic diagram for how the deposition is calculated, I'll consider it.

What is this all about: "I either set the sublimation/liquefaction pressure equal to 0 or a very large number". I thought we are assuming the total pressure is 1 bar.

 

[casualguitar]

I'm guessing that you are using 10 tanks for the whole bed. They seem to be using a finer grid, with approximately 3x your resolution. So,, for comparison, please use 30 tanks.

I'm currently using 10 tanks yes. Ah ok I did attempt to make the grids comparable, using the below statement from the paper (and knowing that the bed is 300m in length):

I thought that 300mm length divided into 3cm increments would be about 10 'tanks'. Is this incorrect? If so I can switch to 30 but I don't yet see how you got 30

You should also be comparing the mass buildup per unit area of bed, not the molar solid buildup within each tank. Please, for comparison, show the mass buildups per unit bed area within each tank.

I noticed this also yes that they use $kg.m^3$ not just $kg$. I guess multiplying my molar solid buildup value by a tank volume ($dz * A_c$) works here?

I'm definitely not going to go through your coding. If you want to show a logic diagram for how the deposition is calculated, I'll consider it.

Yes that was my intention to show the logic above. I'll make a logic diagram for how this currently works

What is this all about: "I either set the sublimation/liquefaction pressure equal to 0 or a very large number". I thought we are assuming the total pressure is 1 bar.

Yes the total pressure is 1 bar however the sublimation rate is dependent on the sublimation pressure at a given temperature. Above the critical temperature I set the sublimation/liquefaction pressure equal to a very large number to stop any desublimation happening

 

[Chestermiller]

I'm currently using 10 tanks yes. Ah ok I did attempt to make the grids comparable, using the below statement from the paper (and knowing that the bed is 300m in length):View attachment 304777
I thought that 300mm length divided into 3cm increments would be about 10 'tanks'. Is this incorrect? If so I can switch to 30 but I don't yet see how you got 30

OK. I was confused. I missed the point that their total length of bed was 30 cm. However, the locations of their thermocouples do not necessarily correspond to the grid spacing in their model. I would suggest using more tanks in your calculations. Their graphs seem to suggest that the used a finer resolution than $\Delta x = 3\ cm.$. I suggest 30 tanks, so that $\Delta x = 1 \ cm$ and so that the center of the first tank is at x = 0.5 cm and the center of the last tank is at 29.5 cm.

The locations where the temperatures are changing substantially in their model do not seem to correspond to where they are changing substantially in your model. All the variation seems to be happening closer to the inlet in your model. Is there a scaling problem on time?

I noticed this also yes that they use $kg.m^3$ not just $kg$. I guess multiplying my molar solid buildup value by a tank volume ($dz * A_c$) works here?

I'll let you work this geometric conversion out. But please provide the rationale and equations for the conversion that you develop. It is not simply Adz.

 

[Chestermiller]

On second thought, surface area to volume ratio of a particle is $$\frac{4\pi r_p^2}{\frac{4}{3}\pi r_p^3}=\frac{3}{r_p}=\frac{6}{d_p}$$The particle volume per bed column volume is $1-\epsilon$. The column volume per tank is $A_c\Delta z$. So, the available deposition area per tank is $$\frac{6}{d_p}(1-\epsilon)A_c\Delta z$$

 

[casualguitar]

I suggest 30 tanks, so that Δx=1 cm and so that the center of the first tank is at x = 0.5 cm and the center of the last tank is at 29.5 cm

Change made

The locations where the temperatures are changing substantially in their model do not seem to correspond to where they are changing substantially in your model. All the variation seems to be happening closer to the inlet in your model. Is there a scaling problem on time?

The variation happening close to the inlet is something I noticed also and I did spot something yesterday on that (my post #246 from yesterday above, I'll quote it here):

In addition, one thing I have found is that the $Q_{IB}$ and $Q_{GI}$ values at the left boundary start at a few hundred and trend gradually to zero (as expected), however the internal node values of $Q_{IB}$ and $Q_{GI}$ start at a much smaller decimal value and trend to very small numbers (10^-10), suggesting that heat transfer in the inner nodes is much less.

i.e. the variation is happening closer to the inlet. Possibly suggesting I've set up the boundary conditions incorrectly

Also I don't think there's a time scaling problem. I checked the simulation length time and it equals the length of the time array for the solution, meaning that there is a 1:1 matching

I'll let you work this geometric conversion out. But please provide the rationale and equations for the conversion that you develop. It is not simply Adz.

Will do

 

[casualguitar]

On second thought, surface area to volume ratio of a particle is $$\frac{4\pi r_p^2}{\frac{4}{3}\pi r_p^3}=\frac{3}{r_p}=\frac{6}{d_p}$$The particle volume per bed column volume is $1-\epsilon$. The column volume per tank is $A_c\Delta z$. So, the available deposition area per tank is $$\frac{6}{d_p}(1-\epsilon)A_c\Delta z$$

The units here seem to be $mol.m^2$ whereas the units in Tuinier et al are $kg.m^3$. Is $m^2$ right here?

Also, if the above is correct then I've potentially made a mistake elsewhere -

The solid phase mass balance is:

And I have defined $a_s$ (the specific surface area), as $\frac{6}{d_p}(1-\epsilon)A_c\Delta z$. Is this incorrect? Everywhere we have $a_s$ in the model equations I have used the above

Looking into the 'all the variation happening at the inlet' issue now

 

[Chestermiller]

The units here seem to be $mol.m^2$ whereas the units in Tuinier et al are $kg.m^3$. Is $m^2$ right here?

Remember, we decided to work in terms of moles rather than mass. This is signified by using M rather than m. Also, the subscript "I" signifies that it is averaged over tank i.

Also, if the above is correct then I've potentially made a mistake elsewhere -

The solid phase mass balance is:
View attachment 304803

In our development, $a_s$ is the surface area available for deposition in tank i.

And I have defined $a_s$ (the specific surface area), as $\frac{6}{d_p}(1-\epsilon)A_c\Delta z$. Is this incorrect? Everywhere we have $a_s$ in the model equations I have used the above

In our development, this is correct to use. Of course, you have to be able to convert from our notation to theirs to compare the results. Are you not able to do this?

 

[casualguitar]

Remember, we decided to work in terms of moles rather than mass. This is signified by using M rather than m. Also, the subscript "I" signifies that it is averaged over tank i.

Yes apologies for the confusion I'm ok with the kg and mol difference however their plot here seems to have m3 in the units whereas ours has $m^2$ instead

In our development, this is correct to use. Of course, you have to be able to convert from our notation to theirs to compare the results. Are you not able to do this?

So I'm not fully clear on how to convert between yet. $m^2$ makes sense to me as we're looking for the amount of solid available for deposition. Why would they use $m^3$ here?

 

[casualguitar]

In addition, the difference in $Q_{GI}$ values stems from the $\Delta T$ between gas and bed at the inlet being higher than the delta T for the internal nodes:

I'm not sure why this happens yet. I would have expected the blue curve to be similar to the orange and green curves. Looking into it

 

[casualguitar]

Final update (apologies). So the reason that there is more variation at the inlet is this -

The gas phase heat balance is this:

When setting up this equation, I have one equation for the boundary tank (which I think is tank zero), and one equation for the other tanks.

$\dot{m}_{j-1}$ is the inlet molar flow rate for tank zero (boundary tank) and $T_{j-1}$ is the inlet flow temperature.

For the rest of the system I just take the outlet flow and temperature from the previous tank and use this for the $j-1$ values.

So plotting $T_{j-1} - T_j$ for all tanks gives:

which shows that there is a much larger temperature change occurring at the boundary always. Is this expected?

 

[Chestermiller]

Yes apologies for the confusion I'm ok with the kg and mol difference however their plot here seems to have m3 in the units whereas ours has $m^2$ instead

View attachment 304805

So I'm not fully clear on how to convert between yet. $m^2$ makes sense to me as we're looking for the amount of solid available for deposition. Why would they use $m^3$ here?

Yes. My bad. Their $m_i$ is supposed to be mass per unit of bed volume (actually column volume). So their $a_s$ is equal to $\frac{6}{d_p}(1-\epsilon)$, and is the deposition surface area per unit volume of column. Our $M_i$ is supposed to be total moles of species deposited on solid surface in tank and our $M_i^"$ is the moles of species deposited per unit time per unit of deposition surface in tank. So, $$M_i=\frac{1000m_iA_c\Delta z}{W_i}$$and$$M_i^"=\frac{1000m_i^"A_c\Delta z}{W_i}$$where $W_i$ is the molecular weight of the species.

 

[Chestermiller]

Final update (apologies). So the reason that there is more variation at the inlet is this -

The gas phase heat balance is this:
View attachment 304808
When setting up this equation, I have one equation for the boundary tank (which I think is tank zero), and one equation for the other tanks.

$\dot{m}_{j-1}$ is the inlet molar flow rate for tank zero (boundary tank) and $T_{j-1}$ is the inlet flow temperature.

For the rest of the system I just take the outlet flow and temperature from the previous tank and use this for the $j-1$ values.

So plotting $T_{j-1} - T_j$ for all tanks gives:
View attachment 304809
which shows that there is a much larger temperature change occurring at the boundary always. Is this expected?

What is your exact equation for tank 1?

 

[casualguitar]

What is your exact equation for tank 1?

By tank 1 (if we're talking about the boundary tank), then my exact equation gas phase heat balance is:
$$\frac{\partial T_g}{\partial t} = \frac{\dot{m}_{in}C_{p}(T_{in} - T_1) - q_{g,I}A_s}{m_1C_{p}}$$
where $\dot{m}_{in}$ is the inlet molar flow and $T_{in}$ is the temperature of the inlet flow, and $m_1$ is the molar holdup in tank 1

Is this reasonable?

and here is the general equation:
$$\frac{\partial T_g}{\partial t} = \frac{\dot{m}_{j-1}C_{p}(T_{j-1} - T_j) - q_{g,I}A_s}{m_1C_{p}}$$

 

[Chestermiller]

By tank 1 (if we're talking about the boundary tank), then my exact equation gas phase heat balance is:
$$\frac{\partial T_g}{\partial t} = \frac{\dot{m}_{in}C_{p}(T_{in} - T_1) - q_{g,I}A_s}{m_1C_{p}}$$
where $\dot{m}_{in}$ is the inlet molar flow and $T_{in}$ is the temperature of the inlet flow, and $m_1$ is the molar holdup in tank 1

Is this reasonable?

and here is the general equation:
$$\frac{\partial T_g}{\partial t} = \frac{\dot{m}_{j-1}C_{p}(T_{j-1} - T_j) - q_{g,I}A_s}{m_1C_{p}}$$

Looks OK

 

[casualguitar]

Looks OK

So the boundary condition equation is set up ok. This graph doesn't seem intuitive though ($T_{j-1}$ - $T_j$ for all tanks). This delta T discrepancy is driving the higher rate of variation at the inlet:

Maybe one of the heat transfer terms driving heat transfer axially through the bed is low? The $Q_GI$ term at the inlet is much higher than $Q_GI$ for the other tanks also. I don't think this is to be expected.

I've checked the molar desublimation rates across the bed and they seem ok everywhere.

I messed around with the heat of vaporisations/desublimations just on the off chance they were incorrect. Currently the heat of desublimation and vaporisation are 26000 and 40650 J/mol respectively. If I just divide them both by 1000 (no particular reason other than maybe the current units are wrong), then the gas temperature levels off at a higher temperature. Still too low, but higher. Suggesting maybe that this is a value error rather than a desublimation mechanics error?

Plot of gas temperature with the heat of vaporisation/desublimation values divided by 1000:

Original values:

 

[casualguitar]

Slightly more interestingly, if I divide those heat of vaporisation/desublimation values by 1000 again then we get a relatively normal looking profile (note these temperatures are degrees C):

This might suggest that I haven't divided by 1000 somewhere where I should have. 1000 would come up when converting from kg to mol so maybe I haven't done that somewhere? Complete guess but will check

 

[Chestermiller]

Have you checked what the literature gives for these heats?

i’n having trouble understanding what seems wrong.

 

[Chestermiller]

If it wouldn't be too much trouble, could you please plot up the 30 tank results of temperature vs distance and CO2 deposited vs distance (with time as a secondary parameter) using the units in the Turnier paper so that we can compare directly with their paper. Thanks.

 

[Chestermiller]

By tank 1 (if we're talking about the boundary tank), then my exact equation gas phase heat balance is:
$$\frac{\partial T_g}{\partial t} = \frac{\dot{m}_{in}C_{p}(T_{in} - T_1) - q_{g,I}A_s}{m_1C_{p}}$$
where $\dot{m}_{in}$ is the inlet molar flow and $T_{in}$ is the temperature of the inlet flow, and $m_1$ is the molar holdup in tank 1

Is this reasonable?

and here is the general equation:
$$\frac{\partial T_g}{\partial t} = \frac{\dot{m}_{j-1}C_{p}(T_{j-1} - T_j) - q_{g,I}A_s}{m_1C_{p}}$$

In these equations, we use $m_j$ to represent the total number of moles of all species in tank j, and $\dot{m}_{j}$ to represent the total number of moles per unit time exiting tank j and entering tank j-1.

 

[Chestermiller]

Yes. My bad. Their $m_i$ is supposed to be mass per unit of bed volume (actually column volume). So their $a_s$ is equal to $\frac{6}{d_p}(1-\epsilon)$, and is the deposition surface area per unit volume of column. Our $M_i$ is supposed to be total moles of species deposited on solid surface in tank and our $M_i^"$ is the moles of species deposited per unit time per unit of deposition surface in tank. So, $$M_i=\frac{1000m_iA_c\Delta z}{W_i}$$and$$M_i^"=\frac{1000m_i^"A_c\Delta z}{W_i}$$where $W_i$ is the molecular weight of the species.

On second thought, I think we should let $A_s$ represent the deposition area available within each tank: $$A_s=\frac{6}{d_p}(1-\epsilon)A\Delta z$$and we should let $M_i^"$ represent the molar deposition rate per unit deposition area available in tank "I":$$M_i^"=\frac{1000m_i^"}{W_i}$$So $$\frac{dM_i}{dt}=M_i^"A_s$$

 

[casualguitar]

Have you checked what the literature gives for these heats?

Yes and the values I had originally are the literature values

i’n having trouble understanding what seems wrong.

So the last two plots in post #261 show the temperature levelling off at a temperature that is below the inlet temperature (100C). I was pointing out in post #262 that if the heat of vaporisation/sublimation is multiplied by a factor of about 1000 then we get the correct 'temperature level off'. Obviously we can't just multiply by 1000 though so I thought maybe a factor of 1000 was missing somewhere else (possibly I hadn't converted between kg and mol properly somewhere). Just a thought though. But more importantly it shows that the desublimation/vaporisation mechanics aren't fully broken in that the gas temperature can go beyond the desublimation/vaporisation temperature

If it wouldn't be too much trouble, could you please plot up the 30 tank results of temperature vs distance and CO2 deposited vs distance (with time as a secondary parameter) using the units in the Turnier paper so that we can compare directly with their paper. Thanks.

Can do

In these equations, we use mj to represent the total number of moles of all species in tank j, and m˙j to represent the total number of moles per unit time exiting tank j and entering tank j-1.

Should this read 'entering tank j+1'?

On second thought, I think we should let $A_s$ represent the deposition area available within each tank: $$A_s=\frac{6}{d_p}(1-\epsilon)A\Delta z$$and we should let $M_i^"$ represent the molar deposition rate per unit deposition area available in tank "I":$$M_i^"=\frac{1000m_i^"}{W_i}$$So $$\frac{dM_i}{dt}=M_i^"A_s$$

Understood. Actually this was the representation I was using. So to convert between their CO2 solid buildup plots all that is needed is to multiply the mole values by $\frac{W_i}{1000*A_C*dz}$?

Doing those plots now

 

[casualguitar]

Here are the plots of position vs Mass of solid CO2 build up and position versus gas temperature (for a selection of times):

Some notes:
- The CO2 mass buildup is in the general ballpark of the Tuinier et al paper. The max buildup I see (about 120 kg/m3) is about double what the Tuinier et al paper shows

- Secondly, as you said earlier the activity in the above plots seems to be concentrated at the inlet much more so than the Tuinier et al plots (unless I'm reading it incorrectly)

- Lastly the gas temperature seems to max out at about -90C which is not the case in the Tuinier paper. We initially thought this was to do with sublimation mechanics but now that we can fix this by 'dividing the heat of vaporisation/sublimation by 1000', maybe this isn't the case

I'll take a look for forgotten 'divisions by 1000'. If there's any other useful plots I could do just let me know

 

[Chestermiller]

Yes and the values I had originally are the literature values

So the last two plots in post #261 show the temperature levelling off at a temperature that is below the inlet temperature (100C). I was pointing out in post #262 that if the heat of vaporisation/sublimation is multiplied by a factor of about 1000 then we get the correct 'temperature level off'. Obviously we can't just multiply by 1000 though so I thought maybe a factor of 1000 was missing somewhere else (possibly I hadn't converted between kg and mol properly somewhere). Just a thought though. But more importantly it shows that the desublimation/vaporisation mechanics aren't fully broken in that the gas temperature can go beyond the desublimation/vaporisation temperatureCan do

Should this read 'entering tank j+1'?

Yes. My mistake.

Understood. Actually this was the representation I was using. So to convert between their CO2 solid buildup plots all that is needed is to multiply the mole values by $\frac{W_i}{1000*A_C*dz}$?

Doing those plots now

Yes. Another mistake of mine. I misread the units as kg/m^2

 

[casualguitar]

Yes. My mistake.

Yes. Another mistake of mine. I misread the units as kg/m^2

Those changes had been incorporated anyway. The plots above (post #268) are the plots you mentioned earlier and use as much of the Tuinier data as possible. In places I had to make assumptions (like picking very high $U_g$ and $U_b$ values to approximate their infinite gas-solid heat transfer coefficient). Unless I've read the plots incorrectly they're not very similar yet. I'm looking into the possibility that I've forgotten to convert between kg and mol somewhere along the way

Maybe I've misunderstood again but this plot (Fig 7 in Tuinier) doesn't seem to make sense):

The x-axis is time in seconds and the temperature is the outlet temperature of the packed bed. Does this not say that their temperature reaches a maximum at roughly -90C also? But surely this is at odds with Fig 6 and 5 which show the temperature going above this. Or have I misunderstood? Because fig 7 looks a lot like our time vs Tg plot (besides the time scale)