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Equilibrium expression as weak constraint?
Posted 07.11.2011, 00:36 GMT-5 Version 4.2 4 Replies
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Hi,
I am trying to model the dehumidification of a process air stream through a channel coated with a desiccant material, with adsorption taking place inside the solid domain and diffusive transport of the two species through the desiccant. The model involves the coupled calculation of velocity, temperature and concentration fields which is done using the laminar flow, heat transfer and transfer of diluted species modules.
The relative coverages c1/c1,sat and c2/c2,max are linked together with an equilibrium equation where c1,sat and c2,max are constants:
c1/c1,sat = R c2 / (c2,max –(1-R) c2) <=> (c2,max) c1 – (c1,sat) c2 – (1-R) c1 c2 = 0
My problem is now that I don’t know how to implement the equilibrium exression in COMSOL. I tried to use a weak constraint formulation on the desiccant domain and put test operators into the expression but got an error message because the expression is not linear in the test functions.
The transport in porous media module gives me the option to define an adsorption equilibrium equation, but not the opportunity for the diffusive transport of absorbed species.
I am new to COMSOL and would really appreciate any help. It seems to be a minor thing but I just can’t seem to get it right.
Many thanks,
Thorsten
I am trying to model the dehumidification of a process air stream through a channel coated with a desiccant material, with adsorption taking place inside the solid domain and diffusive transport of the two species through the desiccant. The model involves the coupled calculation of velocity, temperature and concentration fields which is done using the laminar flow, heat transfer and transfer of diluted species modules.
The relative coverages c1/c1,sat and c2/c2,max are linked together with an equilibrium equation where c1,sat and c2,max are constants:
c1/c1,sat = R c2 / (c2,max –(1-R) c2) <=> (c2,max) c1 – (c1,sat) c2 – (1-R) c1 c2 = 0
My problem is now that I don’t know how to implement the equilibrium exression in COMSOL. I tried to use a weak constraint formulation on the desiccant domain and put test operators into the expression but got an error message because the expression is not linear in the test functions.
The transport in porous media module gives me the option to define an adsorption equilibrium equation, but not the opportunity for the diffusive transport of absorbed species.
I am new to COMSOL and would really appreciate any help. It seems to be a minor thing but I just can’t seem to get it right.
Many thanks,
Thorsten
4 Replies Last Post 22.11.2011, 20:28 GMT-5
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Posted:
1 decade ago
Hi
this is also very close to a subject I'm scratching my head on, any news/models would be of interest.
I expect that the people at Eidhoven i.e. Mr. van Schijndel might have interesting models, but so far I haven't found the way, to set one up completely via their articles.
But there might be others out here ...
--
Having fun Comsoling
Ivar
this is also very close to a subject I'm scratching my head on, any news/models would be of interest.
I expect that the people at Eidhoven i.e. Mr. van Schijndel might have interesting models, but so far I haven't found the way, to set one up completely via their articles.
But there might be others out here ...
--
Having fun Comsoling
Ivar
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Posted:
1 decade ago
Hi Ivar and others,
I have tried a different approach now using a general form PDE module, which allows me to formulate the adsorption equilibrium directly in the equation system of the domain. For the mass transfer I basically have one PDE system with two dependent variables (moisture content in air and desiccant).
The governing equations are:
for air
rho_a (Y_t)+rho_a*ux (Y_x)=0
(W_t)=0
for desiccant
epsilon*rho_a(Y_t) + rho_d (W_t)=rho_a*D_A*(Y_xx+Y_rr)+rho_d*D_S*(W_xx+W_rr)
1/W_max*(W)=phi/ (R+(1-R)*phi)
The model converges now for simulation times less than 10s but I have the problem that I see transfer of adsorbed species into the air.
I think I will have to calculate the mass transport in air and desiccant material in two different modules, but am not sure how to link them, i.e.
air module
rho_a (Ya_t)+rho_a*ux (Ya_x)=0
desiccant module
epsilon*rho_a(Yd_t) + rho_d (W_t)=rho_a*D_A*(Yd_xx+Yd_rr)+rho_d*D_S*(W_xx+W_rr)
1/W_max*(W)=phi/ (R+(1-R)*phi)
interface boundary
Ya=Yd
I have attached a short exemplary model with only the laminar flow and mass transfer calculations using an average temperature and a pdf with a few more details on the governing equations.
Any comments would help me a lot!
Many thanks,
Thorsten
I have tried a different approach now using a general form PDE module, which allows me to formulate the adsorption equilibrium directly in the equation system of the domain. For the mass transfer I basically have one PDE system with two dependent variables (moisture content in air and desiccant).
The governing equations are:
for air
rho_a (Y_t)+rho_a*ux (Y_x)=0
(W_t)=0
for desiccant
epsilon*rho_a(Y_t) + rho_d (W_t)=rho_a*D_A*(Y_xx+Y_rr)+rho_d*D_S*(W_xx+W_rr)
1/W_max*(W)=phi/ (R+(1-R)*phi)
The model converges now for simulation times less than 10s but I have the problem that I see transfer of adsorbed species into the air.
I think I will have to calculate the mass transport in air and desiccant material in two different modules, but am not sure how to link them, i.e.
air module
rho_a (Ya_t)+rho_a*ux (Ya_x)=0
desiccant module
epsilon*rho_a(Yd_t) + rho_d (W_t)=rho_a*D_A*(Yd_xx+Yd_rr)+rho_d*D_S*(W_xx+W_rr)
1/W_max*(W)=phi/ (R+(1-R)*phi)
interface boundary
Ya=Yd
I have attached a short exemplary model with only the laminar flow and mass transfer calculations using an average temperature and a pdf with a few more details on the governing equations.
Any comments would help me a lot!
Many thanks,
Thorsten
Attachments:
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Posted:
1 decade ago
Hi
have you progressed ?
Having stolen a few minutes to look at your file, I notice you ahve 2 set of solvers, one for the fluid, one for the transport, that is OK but you do not say to the "transport" to use the fluid flow results as input conditions. That I beleive is missing
2 ways around:
1) you leave as is but you select in the 2nd solver Dependent Variables Initial VAlues, Solution Solver 1
2) you have one study with two consecutive steps, one per physics as you have selected now. The fact taht you define them in the same study will make COMSOL to link them such that the second step uses automatically the results from the first step
--
Good luck
Ivar
have you progressed ?
Having stolen a few minutes to look at your file, I notice you ahve 2 set of solvers, one for the fluid, one for the transport, that is OK but you do not say to the "transport" to use the fluid flow results as input conditions. That I beleive is missing
2 ways around:
1) you leave as is but you select in the 2nd solver Dependent Variables Initial VAlues, Solution Solver 1
2) you have one study with two consecutive steps, one per physics as you have selected now. The fact taht you define them in the same study will make COMSOL to link them such that the second step uses automatically the results from the first step
--
Good luck
Ivar
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Posted:
1 decade ago
Hi,
haven't had any real success yet, unfortunately. I tried to calculate the transport mechanisms in the two domains using two separate "transfer of diluted species"-modules and tried linking them together using equivalent boundary conditions on the interface for each domain:
a) two equivalent Dirichlet conditions, imposing Ya=Yd
b) two equivalent boundary flux terms, imposing D_va*(Ya_r)=D_A*(Yd_r), with D_va being the diffusion coefficient in the air domain.
Comparing these two modelling results to those using only one "transfer of diluted species" module defined for both domains, I get three different results:
The adsorbate transport seems to be the same in all three cases, while...
... in option a) there is no gradient of Ya-field (air domain), leading to a different distribution of Yd in the desiccant
... in b) the Ya is actually negative although have a similar shape to model 1, while Yd-field (solid domain) looks completely different to the one of the original model (one "diluted species" module) and does not show any resemblance to the profile of the Wd distribution.
(See the attached pdf for the plots)
Have I defined the boundary conditions on the interface correctly?
If I exchange the flux terms to
N1=-N2= D_A*(Yd_r)-D_va*(Ya_r)
I get the same profile only with slightly different numbers.
I have attached another pdf with the comparison of the different results I got and the exemplary model.
Would really appreciate any help!
Cheers,
Thorsten
PS: Ivar, to your comment above -
I have set up the calculations of velocity and concentration fields using two separate studies because the flow calculation is independent of the transfer calculations and I did not want to calculate it again every time I changed something. I have set them up, so that the second time-dependent study of the mass transfer uses the solution of the laminar flow module, so I selected in the 2nd solver under "dependent variables not solved for", "Solution" and "Solution Solver 1".
haven't had any real success yet, unfortunately. I tried to calculate the transport mechanisms in the two domains using two separate "transfer of diluted species"-modules and tried linking them together using equivalent boundary conditions on the interface for each domain:
a) two equivalent Dirichlet conditions, imposing Ya=Yd
b) two equivalent boundary flux terms, imposing D_va*(Ya_r)=D_A*(Yd_r), with D_va being the diffusion coefficient in the air domain.
Comparing these two modelling results to those using only one "transfer of diluted species" module defined for both domains, I get three different results:
The adsorbate transport seems to be the same in all three cases, while...
... in option a) there is no gradient of Ya-field (air domain), leading to a different distribution of Yd in the desiccant
... in b) the Ya is actually negative although have a similar shape to model 1, while Yd-field (solid domain) looks completely different to the one of the original model (one "diluted species" module) and does not show any resemblance to the profile of the Wd distribution.
(See the attached pdf for the plots)
Have I defined the boundary conditions on the interface correctly?
If I exchange the flux terms to
N1=-N2= D_A*(Yd_r)-D_va*(Ya_r)
I get the same profile only with slightly different numbers.
I have attached another pdf with the comparison of the different results I got and the exemplary model.
Would really appreciate any help!
Cheers,
Thorsten
PS: Ivar, to your comment above -
I have set up the calculations of velocity and concentration fields using two separate studies because the flow calculation is independent of the transfer calculations and I did not want to calculate it again every time I changed something. I have set them up, so that the second time-dependent study of the mass transfer uses the solution of the laminar flow module, so I selected in the 2nd solver under "dependent variables not solved for", "Solution" and "Solution Solver 1".
Attachments: