ABSTRACT (original article): Many chemical engineering programs teach gas permeation because of its important applications in the oil and gas industry. In fact, it is considered a good replacement for classical separation techniques such as cryogenic technologies, PSA, physical and chemical absorption etc. Gas permeation module modeling can play an important role in both simulation and design of permeation units. In fact, the simulation allows the prediction of the separation performance of gas permeation modules as well as the selection of optimal operating parameters, stage cut and pressure ratio . On the other hand, the design provides useful data such as the module area. Here, we present the theoretical background of this unit operation, and the modeling of a single-stage gas permeation module used to separate multicomponent gas mixtures for the following flow patterns: (1) perfect mixing, (2) one-side mixing, (3) cross flow, (4) co-current flow, (4) counter-current flow. Our numerical simulation methodology is based on Chebyshev orthogonal collocation and arc-length continuation techniques. We apply the proposed methodology to two industrially important systems, which are described in reference (Shindo et al. in Sep Sci Technol 20:445–459, 1985. https:// doi. org/ 10. 1080/ 01496 39850 80606 92): (1) separation of NH3, H2, and N2 gaseous mixture by means of a polyethylene membrane. Such a gas mixture arises in the Haber process. (2) Separation of H2, CH4, CO, N2, and CO2 mixture through a microporous glass membrane. This mixture is important in the partial oxidation of fuels with air and in the steam gasification of biomass. Whenever possible, our results are benchmarked against data obtained from the open literature or predicted using the software Aspen-Plus®. Furthermore, we performed sensitivity analyses where we vary either the stage cut or the pressure ratio in order to investigate the effect of these parameters on both the permeate and retentate compositions as well as on the surface area of the permeation module. Finally, we provide useful tips and computer coding to readers who wish to investigate similar gas permeation problems using our new methodology. CITATION (original article): H. Binous, U. Zahid and A. Bellagi, Modeling for Multi-component Gas Permeation with Orthogonal Collocation and Arc-Length Continuation, Chemistry Africa (2024). DOI: https://doi.org/10.1007/s42250-024-01167-y
We consider the separation of a mixture of He, CO2, N2, and CH4.
Choi et al. (see Ref. 1) performed experimental separations using a multilayer thin-film composite hollow fiber membranes for helium extraction from its mixtures. Choi’s experimental set-up is based on the co-current flow pattern.
Medi et al. (see Ref. 2) proposed a theoretical investigation for the separation of such mixtures and for co-current flow pattern.
Here, we compute the mole fraction of Helium in the permeate stream versus the stage cut again for the co-current flow pattern. Here, our calculation is based on a methodology first proposed by Binous et al. (see Ref. 3). Such results are superimposed to their counterparts given in Fig. 4 of Ref. 2. Close agreement between all data provides another validation of the new methodology proposed by Binous et al. (Ref. 3).
Data given in Tables below are used for the simulation and are summarized in Medi et al. (2022):
Choi et al. (see Ref. 1) performed experimental separations using a multilayer thin-film composite hollow fiber membranes for helium extraction from its mixtures. Choi’s experimental set-up is based on the co-current flow pattern.
Medi et al. (see Ref. 2) proposed a theoretical investigation for the separation of such mixtures and for co-current flow pattern.
Here, we compute the mole fraction of Helium in the permeate stream versus the stage cut again for the co-current flow pattern. Here, our calculation is based on a methodology first proposed by Binous et al. (see Ref. 3). Such results are superimposed to their counterparts given in Fig. 4 of Ref. 2. Close agreement between all data provides another validation of the new methodology proposed by Binous et al. (Ref. 3).
Data given in Tables below are used for the simulation and are summarized in Medi et al. (2022):
A, B, C, D stand for Helium, Methane, Carbon Dioxyde and Nitrogen, respectively
Permeate side composition at the gas separation module’s entrance
eq1i=/()qA(-/)/(qB(-/));eq2i=/()qA(-/)/(qC(-/));eq3i=/()qA(-/)/(qD(-/));eq4i=+++1;
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Feed composition
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Pressure ratio
g=1.036/21.7;
Permeance ratios
qA=1;qB=6.3610^-2/2.61;qC=1.7810^-2/2.61;qD=3.6810^-2/2.61;
Feed flow rate
q
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Permeate side pressure
p
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Derivative matrix
Np=20;
d[0,0]=(2Np^2+1)/6//N;
d[Np,Np]=-(2Np^2+1)/6//N;
yc[i_]:=Cos[iPi/Np]//N;
d[i_,i_]:=-yc[i]/(2(1-yc[i]^2))//N;
d[j_,k_]:=If[j≠k&&k≠0&&k≠Np&&j≠0&&j≠Np,(-1)^(j+k)/(yc[j]-yc[k])]//N;
d[Np,0]=-1/2(-1)^Np;
d[0,Np]=1/2(-1)^Np;
d[j_,0]:=If[j≠Np&&j≠0,-1/2(-1)^j/(1-yc[j])];
d[j_,Np]:=If[j≠Np&&j≠0,1/2(-1)^(j+Np)/(1+yc[j])];
d[0,j_]:=If[j≠Np&&j≠0,2(-1)^j/(1-yc[j])];
d[Np,j_]:=If[j≠Np&&j≠0,-2(-1)^(j+Np)/(1+yc[j])];
Add=Table[d[i,j]//N,{i,0,Np},{j,0,Np}];
Discretization of derivatives of the permeate and retentate mole fractions - Af is total area of the separation module
xAp[i_]:=2/stAdd[[i,j]]xA[j];
Np+1
∑
j=1
yAp[i_]:=2/stAdd[[i,j]]yA[j]
Np+1
∑
j=1
xBp[i_]:=2/stAdd[[i,j]]xB[j];
Np+1
∑
j=1
xCp[i_]:=2/stAdd[[i,j]]xC[j];
Np+1
∑
j=1
yBp[i_]:=2/stAdd[[i,j]]yB[j];
Np+1
∑
j=1
yCp[i_]:=2/stAdd[[i,j]]yC[j];
Np+1
∑
j=1
yDp[i_]:=2/stAdd[[i,j]]yD[j];
Np+1
∑
j=1
qp[i_]:=2/stAdd[[i,j]]q[j];
Np+1
∑
j=1
xDp[i_]:=2/stAdd[[i,j]]xD[j];
Np+1
∑
j=1
Governing equations
eq1[i_]:=-xAp[i]q[i]qA(xA[i]-/yA[i])-xA[i](qA(xA[i]-/yA[i])+qB(xB[i]-/yB[i])+qC(xC[i]-/yC[i])+qD(xD[i]-/yD[i]))
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eq2[i_]:=-xBp[i]q[i]qB(xB[i]-/yB[i])-xB[i](qA(xA[i]-/yA[i])+qB(xB[i]-/yB[i])+qC(xC[i]-/yC[i])+qD(xD[i]-/yD[i]))
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eq3[i_]:=-xCp[i]q[i]qC(xC[i]-/yC[i])-xC[i](qA(xA[i]-/yA[i])+qB(xB[i]-/yB[i])+qC(xC[i]-/yC[i])+qD(xD[i]-/yD[i]))
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eq5[i_]:=qp[i]-(qA(xA[i]-/yA[i])+qB(xB[i]-/yB[i])+qC(xC[i]-/yC[i])+qD(xD[i]-/yD[i]))
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eq6[i_]:=(1-q[i])yA[i]-q[i]xA[i]
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eq7[i_]:=(1-q[i])yB[i]-q[i]xB[i]
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eq8[i_]:=(1-q[i])yC[i]-q[i]xC[i]
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Summation rules
yD[i_]:=1-yA[i]-yB[i]-yC[i]xD[i_]:=1-xA[i]-xB[i]-xC[i]
Boundary conditions
xA[Np+1]=;xB[Np+1]=;xC[Np+1]=;yA[Np+1]=;yB[Np+1]=;yC[Np+1]=;q[Np+1]=;q[1]=(1-theta);
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Unknowns
unknown=Join[Flatten[Join[Table[{{xA[i],0.001},{xB[i],0.6},{xC[i],0.04}},{i,1,Np}],Table[{{yA[i],0.02},{yB[i],0.01},{yC[i],0.2}},{i,1,Np}],Table[{{q[i],0.7}},{i,2,Np}]],1],{{theta,0.3},{,0.06},{,0.2},{,0.2},{,0.06}}];
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All equations
sys=Flatten@Join[Table[{eq1[i],eq2[i],eq3[i],eq5[i],eq6[i],eq7[i],eq8[i]},{i,1,Np}],{eq1i,eq2i,eq3i,eq4i}];
Solution for a particular value of the separation module’s total area or for st=1 using FindRoot
sol=Quiet@FindRoot[sys/.st0.1,unknown,MaxIterations100,AccuracyGoal20]
Out[]=
{xA[1]0.00126493,xB[1]0.673197,xC[1]0.0440586,xA[2]0.00126514,xB[2]0.673198,xC[2]0.0440583,xA[3]0.00126577,xB[3]0.673202,xC[3]0.0440572,xA[4]0.0012668,xB[4]0.673208,xC[4]0.0440554,xA[5]0.00126822,xB[5]0.673216,xC[5]0.044053,xA[6]0.00126997,xB[6]0.673227,xC[6]0.04405,xA[7]0.00127204,xB[7]0.673239,xC[7]0.0440465,xA[8]0.00127436,xB[8]0.673252,xC[8]0.0440426,xA[9]0.00127688,xB[9]0.673267,xC[9]0.0440384,xA[10]0.00127954,xB[10]0.673283,xC[10]0.0440339,xA[11]0.00128228,xB[11]0.673299,xC[11]0.0440293,xA[12]0.00128503,xB[12]0.673315,xC[12]0.0440247,xA[13]0.00128772,xB[13]0.67333,xC[13]0.0440202,xA[14]0.00129028,xB[14]0.673345,xC[14]0.044016,xA[15]0.00129265,xB[15]0.673358,xC[15]0.0440121,xA[16]0.00129477,xB[16]0.67337,xC[16]0.0440086,xA[17]0.00129659,xB[17]0.673381,xC[17]0.0440056,xA[18]0.00129805,xB[18]0.673389,xC[18]0.0440032,xA[19]0.00129912,xB[19]0.673395,xC[19]0.0440014,xA[20]0.00129978,xB[20]0.673399,xC[20]0.0440004,yA[1]0.0187863,yB[1]0.774679,yC[1]0.0147621,yA[2]0.0187887,yB[2]0.774678,yC[2]0.014762,yA[3]0.0187959,yB[3]0.774674,yC[3]0.0147617,yA[4]0.0188076,yB[4]0.774667,yC[4]0.0147612,yA[5]0.0188237,yB[5]0.774658,yC[5]0.0147605,yA[6]0.0188437,yB[6]0.774646,yC[6]0.0147596,yA[7]0.0188671,yB[7]0.774633,yC[7]0.0147587,yA[8]0.0188934,yB[8]0.774617,yC[8]0.0147575,yA[9]0.018922,yB[9]0.774601,yC[9]0.0147563,yA[10]0.0189522,yB[10]0.774583,yC[10]0.0147551,yA[11]0.0189832,yB[11]0.774565,yC[11]0.0147537,yA[12]0.0190143,yB[12]0.774547,yC[12]0.0147524,yA[13]0.0190447,yB[13]0.77453,yC[13]0.0147512,yA[14]0.0190737,yB[14]0.774513,yC[14]0.0147499,yA[15]0.0191005,yB[15]0.774497,yC[15]0.0147488,yA[16]0.0191244,yB[16]0.774483,yC[16]0.0147478,yA[17]0.0191449,yB[17]0.774471,yC[17]0.014747,yA[18]0.0191614,yB[18]0.774462,yC[18]0.0147463,yA[19]0.0191735,yB[19]0.774455,yC[19]0.0147458,yA[20]0.0191809,yB[20]0.77445,yC[20]0.0147455,q[2]0.998011,q[3]0.998047,q[4]0.998107,q[5]0.998189,q[6]0.998291,q[7]0.998411,q[8]0.998545,q[9]0.99869,q[10]0.998842,q[11]0.998999,q[12]0.999156,q[13]0.999308,q[14]0.999453,q[15]0.999587,q[16]0.999707,q[17]0.999809,q[18]0.999891,q[19]0.999951,q[20]0.999988,theta0.00200157,0.0274224,19.819,-30.5776,11.7311}
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Solution based on arc-length continuation
In[]:=
xA[Np+1]=.;xB[Np+1]=.;xC[Np+1]=.;q[Np+1]=.;q[1]=.;
In[]:=
yA[Np+1]=.;yB[Np+1]=.;yC[Np+1]=.;
In[]:=
xA[Np+1][s_]:=;xB[Np+1][s_]:=;yA[Np+1][s_]:=[s];yB[Np+1][s_]:=[s];xC[Np+1][s_]:=;yC[Np+1][s_]:=[s];q[Np+1][s_]:=;q[1][s_]:=(1-theta[s]);
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Initial conditions
init=Join[Flatten[Table[{xA[i][0](xA[i]/.sol),yA[i][0](yA[i]/.sol),xC[i][0](xC[i]/.sol),yC[i][0](yC[i]/.sol),xB[i][0](xB[i]/.sol),yB[i][0](yB[i]/.sol)},{i,1,Np}],1],Flatten[Table[q[i][0](q[i]/.sol),{i,2,Np}],1],{theta[0](theta/.sol),[0](/.sol),st[0]0.1,yA[Np+1][0](/.sol),yB[Np+1][0](/.sol),yC[Np+1][0](/.sol)}]
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Out[]=
{xA[1][0]0.00126493,yA[1][0]0.0187863,xC[1][0]0.0440586,yC[1][0]0.0147621,xB[1][0]0.673197,yB[1][0]0.774679,xA[2][0]0.00126514,yA[2][0]0.0187887,xC[2][0]0.0440583,yC[2][0]0.014762,xB[2][0]0.673198,yB[2][0]0.774678,xA[3][0]0.00126577,yA[3][0]0.0187959,xC[3][0]0.0440572,yC[3][0]0.0147617,xB[3][0]0.673202,yB[3][0]0.774674,xA[4][0]0.0012668,yA[4][0]0.0188076,xC[4][0]0.0440554,yC[4][0]0.0147612,xB[4][0]0.673208,yB[4][0]0.774667,xA[5][0]0.00126822,yA[5][0]0.0188237,xC[5][0]0.044053,yC[5][0]0.0147605,xB[5][0]0.673216,yB[5][0]0.774658,xA[6][0]0.00126997,yA[6][0]0.0188437,xC[6][0]0.04405,yC[6][0]0.0147596,xB[6][0]0.673227,yB[6][0]0.774646,xA[7][0]0.00127204,yA[7][0]0.0188671,xC[7][0]0.0440465,yC[7][0]0.0147587,xB[7][0]0.673239,yB[7][0]0.774633,xA[8][0]0.00127436,yA[8][0]0.0188934,xC[8][0]0.0440426,yC[8][0]0.0147575,xB[8][0]0.673252,yB[8][0]0.774617,xA[9][0]0.00127688,yA[9][0]0.018922,xC[9][0]0.0440384,yC[9][0]0.0147563,xB[9][0]0.673267,yB[9][0]0.774601,xA[10][0]0.00127954,yA[10][0]0.0189522,xC[10][0]0.0440339,yC[10][0]0.0147551,xB[10][0]0.673283,yB[10][0]0.774583,xA[11][0]0.00128228,yA[11][0]0.0189832,xC[11][0]0.0440293,yC[11][0]0.0147537,xB[11][0]0.673299,yB[11][0]0.774565,xA[12][0]0.00128503,yA[12][0]0.0190143,xC[12][0]0.0440247,yC[12][0]0.0147524,xB[12][0]0.673315,yB[12][0]0.774547,xA[13][0]0.00128772,yA[13][0]0.0190447,xC[13][0]0.0440202,yC[13][0]0.0147512,xB[13][0]0.67333,yB[13][0]0.77453,xA[14][0]0.00129028,yA[14][0]0.0190737,xC[14][0]0.044016,yC[14][0]0.0147499,xB[14][0]0.673345,yB[14][0]0.774513,xA[15][0]0.00129265,yA[15][0]0.0191005,xC[15][0]0.0440121,yC[15][0]0.0147488,xB[15][0]0.673358,yB[15][0]0.774497,xA[16][0]0.00129477,yA[16][0]0.0191244,xC[16][0]0.0440086,yC[16][0]0.0147478,xB[16][0]0.67337,yB[16][0]0.774483,xA[17][0]0.00129659,yA[17][0]0.0191449,xC[17][0]0.0440056,yC[17][0]0.014747,xB[17][0]0.673381,yB[17][0]0.774471,xA[18][0]0.00129805,yA[18][0]0.0191614,xC[18][0]0.0440032,yC[18][0]0.0147463,xB[18][0]0.673389,yB[18][0]0.774462,xA[19][0]0.00129912,yA[19][0]0.0191735,xC[19][0]0.0440014,yC[19][0]0.0147458,xB[19][0]0.673395,yB[19][0]0.774455,xA[20][0]0.00129978,yA[20][0]0.0191809,xC[20][0]0.0440004,yC[20][0]0.0147455,xB[20][0]0.673399,yB[20][0]0.77445,q[2][0]0.998011,q[3][0]0.998047,q[4][0]0.998107,q[5][0]0.998189,q[6][0]0.998291,q[7][0]0.998411,q[8][0]0.998545,q[9][0]0.99869,q[10][0]0.998842,q[11][0]0.998999,q[12][0]0.999156,q[13][0]0.999308,q[14][0]0.999453,q[15][0]0.999587,q[16][0]0.999707,q[17][0]0.999809,q[18][0]0.999891,q[19][0]0.999951,q[20][0]0.999988,theta[0]0.00200157,[0]11.7311,st[0]0.1,[0]0.0274224,[0]19.819,[0]-30.5776}
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Initial conditions for the derivatives
Auxiliary equation
Unknowns
All unknown variables are functions of s or the arc-length
Overall system of equations
Solution of the DAE system using NDSolve
Helium in the permeate
Plot of the mole fraction of Helium in the permeate stream versus stage cut
Comparison of our result (dotted red line) with the experimental data of Choi et al. 2019 (blue diamonds) and the theoretical predictions of Medi et al. 2022 (dashed blue curve). This comparison indicates, as expected, close agreement between our results and Medi’s theoretical predictions.
References
References
◼
[1] S-H Choi, M. M. Ben Sultan, A; A. Alsuwailem and S. M. Zuabi, Preparation and characterization of multilayer thin-film composite hollow fiber membranes for helium extraction from its mixtures, Separation and Purification Technology, Volume 222, 2019, Pages 152-161, https://doi.org/10.1016/j.seppur.2019.04.036.
◼
[2] B. Medi, M. Vesali-Naseh and M. Haddad-Hamedani, A generalized numerical framework for solving cocurrent and counter-current membrane models for gas separation, Heliyon, Volume 8, Issue 3, 2022. https://doi.org/10.1016/j.heliyon.2022.e09053
◼
[3] H. Binous, U. Zahid and A. Bellagi, Modeling for Multi-component Gas Permeation with Orthogonal Collocation and Arc-Length Continuation, Chemistry Africa (2024). https://doi.org/10.1007/s42250-024-01167-y
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Co-current membrane model for Helium enrichment from natural gas
by Housam Binous and Ahmed Bellagi
Wolfram Community, STAFF PICKS, December 25, 2024
https://community.wolfram.com/groups/-/m/t/3344617
by Housam Binous and Ahmed Bellagi
Wolfram Community, STAFF PICKS, December 25, 2024
https://community.wolfram.com/groups/-/m/t/3344617