Modelling of two phase flow in packed bed (continued)

[Chestermiller]

One other note - the simulation running properly is dependent on the mass transfer coefficient (which is currently constant) being below a certain threshold. The current simulation uses a constant 8 x 10^-8 which is small enough, however the mass transfer coefficient calculator function does not return values in this range but rather in the 10^1 range. Is there anything wrong with this that you can see:
View attachment 304339

Where the following approximate ranges/values apply:
Reynolds number: 10<Re<10000
Schmidt Number: 0.5 < Sc < 0.7
Particle Diameter (dp) = 0.005
D CO2-N2 = 0.14 x 10^-4
D H2O-N2 = 0.259 x 10^-4

These values give a number in the 10^1 range, which causes the simulation to break, however the 10^-8 values produce expected results. Have I missed something?

How do the mass transfer coefficients compare with the values in the paper? Any value of the mass transfer coefficient should not cause the calculation to crash. What values of the heat transfer coefficient does the correlation give?

 

[casualguitar]

Release the mass transfer coefficient correlation next.

Will do

It's just what they gave in the other papers.

## U_b = \frac{k_p}{d_p/\beta}##?

Did you mean "heat capacity correlation" or did you mean "heat transfer correlation?"

I meant heat capacity correlation but yes actually ##U_g## is constant in the current model also, so I'll need to add the heat transfer correlation in also. This seems more straightforward though so maybe I can leave that until the mass transfer correlation works

How do the mass transfer coefficients compare with the values in the paper?

The ones produced by my code are much larger than both the random constant value we have been using and the Tuinier et al values. Both of those are around 10^-7, whereas the calculation I'm doing produces a value around 10^1

Any value of the mass transfer coefficient should not cause the calculation to crash.

Yes it technically doesn't crash (it does run to completion), however the gas/bed temperatures level out at 180C rather than the inlet temperature of 300C. This does not happen when I use 'normal' values. It could be a tolerance issue for the larger mass transfer coefficients so I'll try that now. But either way, the Tuinier paper has mass transfer coefficients that are 10^7 times smaller so I'm likely doing something wrong in the mass transfer coefficient calculation

EDIT: I've checked all the terms in the mass transfer coefficient correlation and the only possible term that could be wrong is the diffusion coefficient (for both CO2 and H2O in N2). It is my guess that my units/magnitude is off for this. I need a term in the 10^-8 to 10^-9 range. The term I'm currently using is 10^-4
The terms seem to vary quite a bit across gas and liquid species. Am I right to say I'm looking for gaseous diffusion terms for both CO2 and H2O in air?

 

[Chestermiller]

Will do

## U_b = \frac{k_p}{d_p/\beta}##?

Correct.

I meant heat capacity correlation but yes actually ##U_g## is constant in the current model also, so I'll need to add the heat transfer correlation in also. This seems more straightforward though so maybe I can leave that until the mass transfer correlation works

The ones produced by my code are much larger than both the random constant value we have been using and the Tuinier et al values. Both of those are around 10^-7, whereas the calculation I'm doing produces a value around 10^1

Yes it technically doesn't crash (it does run to completion), however the gas/bed temperatures level out at 180C rather than the inlet temperature of 300C.

This indicates that there is something wrong. Maybe something like evaporation or sublimation is occurring without shutting off when liquid/solid is depleted. I don't know. But it can't be 180 C at long times.

This does not happen when I use 'normal' values. It could be a tolerance issue for the larger mass transfer coefficients so I'll try that now. But either way, the Tuinier paper has mass transfer coefficients that are 10^7 times smaller so I'm likely doing something wrong in the mass transfer coefficient calculation

Let's see the detailed hand calculation for a typical case. Also, the range of Reynolds numbers that you show for your bed seems very large to me. Please plot the Re vs tank number at a selection of times for a typical case. The only things that should affect the Re are the mass flow rate and the viscosity.

Also, please plot the heat transfer coefficient vs tank number at a selection of times. Also the mass transfer coefficient vs tank number at a selection of times.

EDIT: I've checked all the terms in the mass transfer coefficient correlation and the only possible term that could be wrong is the diffusion coefficient (for both CO2 and H2O in N2). It is my guess that my units/magnitude is off for this. I need a term in the 10^-8 to 10^-9 range. The term I'm currently using is 10^-4

The values you gave are in the right ballpark for m^2/s units.

The terms seem to vary quite a bit across gas and liquid species. Am I right to say I'm looking for gaseous diffusion terms for both CO2 and H2O in air?

In nitrogen.

 

[casualguitar]

The values you gave are in the right ballpark for m^2/s units.

Just one question before I do the above - is the right ballpark the 10^-4 one or the 10^-7 one?

 

[Chestermiller]

Just one question before I do the above - is the right ballpark the 10^-4 one or the 10^-7 one?

-4

 

[casualguitar]

-4

I attempted to find the point that the simulation deviates from what should happen (when I change from 10^-8 to 10^-4) however I couldn't.

I'll start on the plots above first thing tomorrow

 

[casualguitar]

I attempted to find the point that the simulation deviates from what should happen (when I change from 10^-8 to 10^-4) however I couldn't.

I'll start on the plots above first thing tomorrow

Will add these in as I do them.

The ##U_b## calculation:
##U_b## = ##\frac{k_p}{d_p/\beta}##
##k_p## = 15 W/m.K
##d_p## = 0.01 m
##\beta## = 10 for spheres

##U_b## = 15,000 W/m2.K

Seems very high?

Doing the plots now

 

[casualguitar]

As for the Reynolds Number vs Tank number plot (for inlet flow = 0.5 mol/s), the trend seems to be that initially there is a high Re (t=0), which levels off almost immediately:

This does correspond to the molar flow which jumps up to about 1 mol/s initially and then gradually levels off to about 0.56 mol/s. This flow should surely level off to 0.5 not 0.56 though?

Notes:
As for the mass transfer coefficient and heat transfer coefficient plots - the heat/mass transfer coefficients have been constant in all simulations so far, so all I'm doing is generating the mass and heat transfer coefficients from the Reynolds number etc, which aren't actually used in the simulation as constant values are used

Also one further note, after I changed the ##U_b## value to the new calculated value of 15,000 I also had to change ##U_g## to be a high number (in the thousands) to get the simulation to converge

Doing the mass/heat transfer coefficient plots now

 

[Chestermiller]

As for the Reynolds Number vs Tank number plot (for inlet flow = 0.5 mol/s), the trend seems to be that initially there is a high Re (t=0), which levels off almost immediately:
View attachment 304476
This does correspond to the molar flow which jumps up to about 1 mol/s initially and then gradually levels off to about 0.56 mol/s. This flow should surely level off to 0.5 not 0.56 though?

Yes.

Notes:
As for the mass transfer coefficient and heat transfer coefficient plots - the heat/mass transfer coefficients have been constant in all simulations so far, so all I'm doing is generating the mass and heat transfer coefficients from the Reynolds number etc, which aren't actually used in the simulation as constant values are used

That's OK. We're just checking to see what they would come out to be and to compare them with the constant values that you have used. So please provide these plots. Also, please specify the units.

 

[casualguitar]

Yes.

Will provide details on the mass flow calculation

That's OK. We're just checking to see what they would come out to be and to compare them with the constant values that you have used. So please provide these plots. Also, please specify the units.

Doing this currently

 

[casualguitar]

Will provide details on the mass flow calculation

Doing this currently

One further question. Just looking at my ##U_g## calculation. Will we still have the contribution from the bed in the ##U_g## calculation i.e. 1/##U_b## as we had in the previous model where we lumped them, or will ##U_g## just be a function of re, pr, mu, etc? i.e. no bed properties involved

 

[Chestermiller]

One further question. Just looking at my ##U_g## calculation. Will we still have the contribution from the bed in the ##U_g## calculation i.e. 1/##U_b## as we had in the previous model where we lumped them, or will ##U_g## just be a function of re, pr, mu, etc? i.e. no bed properties involved

Ug does not involve any bed properties. Ub does. The overall U combines both. This is the same for both models.

 

[casualguitar]

Ug does not involve any bed properties. Ub does. The overall U combines both. This is the same for both models.

Plots of the CO2 and H2O mass transfer coefficients versus position:

And ##U_g## versus position:

The ##U_g## plot seems to return values in a reasonable range. The mass transfer coefficient seems to be too high by a factor of about 1000. I'll confirm if all of the nested values in the ki function are in the right range

EDIT: for some reason the times didnt show up on the ki graphs. The legend is the same as for the ##U_g## plot

 

[casualguitar]

These are the values, ranges and equation used by the simulation to recalculate the mass transfer coefficient values (for both CO2 and H2O):
Reynolds Number: 3500 - 7000
Schmidt Number: Approx 0.7 average
##D_{CO2,N2}## = 0.144 * 10^-4
##D_{H2O,N2}## = 0.259 * 10^-4

The equation I'm using:
##k_i## ##=## ##(2.19Re^{1/3} + 0.78Re^{0.619})Sc^{1/3}D_i(1-\epsilon)/d_p##

and it produces values (as in the graph above) of approx 10^-1

Anything obviously wrong with this?

The Reynolds numbers seem reasonable. Also the Schmidt number does seem to line up with BSL. And you say the D values are reasonable. Possibly suggesting that my ##k_i## equation is being used incorrectly? Or that my recalculation of ##k_i## is wrong somewhere (checking this now)

 

[Chestermiller]

Plots of the CO2 and H2O mass transfer coefficients versus position:
View attachment 304485View attachment 304486
And ##U_g## versus position:
View attachment 304487

The ##U_g## plot seems to return values in a reasonable range. The mass transfer coefficient seems to be too high by a factor of about 1000. I'll confirm if all of the nested values in the ki function are in the right range

EDIT: for some reason the times didnt show up on the ki graphs. The legend is the same as for the ##U_g## plot

Please show a hand calculation of k for a Reynolds number of 4000.

 

[casualguitar]

Please show a hand calculation of k for a Reynolds number of 4000.

Is the above (just posted) suitable?

 

[Chestermiller]

Is the above (just posted) suitable?

The k values calculated for these graphs with the equation you used seem correct to me. Also, the Ug's seem reasonable to me.

In using the k values, you multiply by the molar density of the gas by the difference in mole fractions between gas bulk and interphase equilibrium at the gas-bed interface temperature to get the molar flux at the surface. Correct?

 

[Chestermiller]

Please send me a link to that Tuinier et al paper again. I have lost track of it in my files.

 

[casualguitar]

In using the k values, you multiply by the molar density of the gas by the difference in mole fractions between gas bulk and interphase equilibrium at the gas-bed interface temperature to get the molar flux at the surface. Correct?

Not fully following the above, but I think so yes. To use the k values I'm using the molar deposition rate equation which is:
$$\dot{M}_i^"=k_i\left(\frac{Py_i-p_i(T_I)}{RT_I}\right)$$

Where P is pressure, ##y_i## is the gas phase mole fraction of species i, ##p_i(T_I)## is the saturation pressure of species i evaluated at the interface temperature, R is the gas constant and T is the gas temperature. Is this an equivalent to what you said above (if I factor out the molar density from the equation above)?

 

[casualguitar]

Please send me a link to that Tuinier et al paper again. I have lost track of it in my files.

https://www.sciencedirect.com/scien...jNb8NEK3mdGAivU7_KCet9gdzeQ2PUKUvtRz0izC8Wyrg

 

[Chestermiller]

https://www.sciencedirect.com/scien...jNb8NEK3mdGAivU7_KCet9gdzeQ2PUKUvtRz0izC8Wyrg

What is the value of g that they use?

 

[casualguitar]

What is the value of g that they use?

It seems they use these three values (using g as a tuning parameter I think). 1x10^-5 and 1x10-6 fit the data better than 1x10^-7 so I suppose 1x1-^-5 and 1x10^-6 are the g values here

 

[Chestermiller]

It seems they use these three values (using g as a tuning parameter I think). 1x10^-5 and 1x10-6 fit the data better than 1x10^-7 so I suppose 1x1-^-5 and 1x10^-6 are the g values here
View attachment 304579

What value did they use in their main calculations? I’ve derived a relationship between their g and our k:
$$k=\frac{1000RT}{M}g$$where T is the temperature and M is the molecular weight of the diffusing species. R is in J/mole-K, k is in m/s, and g is in s/m. What values of k do you calculate at 250 K using M=18 for water and M = 44 for CO2? How do they compare with the values from our correlation?

 

[casualguitar]

What value did they use in their main calculations? I’ve derived a relationship between their g and our k:
$$k=\frac{1000RT}{M}g$$where T is the temperature and M is the molecular weight of the diffusing species. R is in J/mole-K, k is in m/s, and g is in s/m. What values of k do you calculate at 250 K using M=18 for water and M = 44 for CO2? How do they compare with the values from our correlation?

It looks like they don't really have a 'main calculation' but they do use a range of g values and then conclude that g = 1x10^-6 is the value that best fits the experimental data.

This value of g gives us k = 2.0785/M

which gives ##k_{H2O}## = 0.115 and ##k_{CO2}## = 0.047

Our correlation returns values of ##k_{H2O}## = 0.26 and ##k_{CO2}## = 0.14

They're not the same, but they are in an approximate ballpark range. Is this reasonable closeness? If so that would mean that the mass transfer coefficients are ok and it is something else that is causing the temperature to level out at an unusual temperature

 

[Chestermiller]

This is certainly reasonable closeness. I didn’t expect it to be this close.

 

[casualguitar]

This is certainly reasonable closeness. I didn’t expect it to be this close.

Interesting so the mass transfer coefficients check out, and the ##U_g## (and ##U_b##) values are also ok. The gas temperatures level out at values lower than the inlet gas temperature though so there is still some bug present. Right now I have just 'recalculated' the ##k_i## and ##U_g## values. I could implement them in the actual simulation and see how the output looks? Or do you think it is worth checking for the bug elsewhere before doing this?

 

[Chestermiller]

Interesting so the mass transfer coefficients check out, and the ##U_g## (and ##U_b##) values are also ok. The gas temperatures level out at values lower than the inlet gas temperature though so there is still some bug present. Right now I have just 'recalculated' the ##k_i## and ##U_g## values. I could implement them in the actual simulation and see how the output looks? Or do you think it is worth checking for the bug elsewhere before doing this?

You need to determine why the temperature doesn’t get to the inlet temperature first. It must have something to do with the evaporation coding logistics.

 

[casualguitar]

You need to determine why the temperature doesn’t get to the inlet temperature first. It must have something to do with the evaporation coding logistics.

The two 'clues':
- The simulation works as expected (gas temperature approaching the inlet gas temperature), when the ##k_i## value is sufficiently small
- The temperature at each point in the bed does not level off to the same value (shown below). This seems like a big hint as for where the simulation breaks. Will take a look

 

[Chestermiller]

The two 'clues':
- The simulation works as expected (gas temperature approaching the inlet gas temperature), when the ##k_i## value is sufficiently small
- The temperature at each point in the bed does not level off to the same value (shown below). This seems like a big hint as for where the simulation breaks. Will take a look
View attachment 304682

How does this compare with the results in the paper? What does it look like as a function of position at constant times?

 

[casualguitar]

How does this compare with the results in the paper? What does it look like as a function of position at constant times?

Well here is the gas temperature vs position (for a range of times):

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:

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

 

[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. Theres 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. Theres 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