We consider the enrichment of nitrogen from air using gas permeation. In particular, we compute nitrogen recovery versus nitrogen purity for different polymer membrane. We successfully reproduce Fig. 7a of Ref. 1 using the methodology described in Ref. 2. Membrane properties are reported in Table 7 of Ref. 1. Feed contains 79 mole % nitrogen. The pressure ratio is set equal to Φ = 11.We assume cross-flow pattern.As expected, it is found that nitrogen recoveries decrease with increasing nitrogen purity.
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param={xA[1]xA[1][s],yA[1]yA[1][s],xA[2]xA[2][s],yA[2]yA[2][s],q[2]q[2][s],xA[3]xA[3][s],yA[3]yA[3][s],q[3]q[3][s],xA[4]xA[4][s],yA[4]yA[4][s],q[4]q[4][s],xA[5]xA[5][s],yA[5]yA[5][s],q[5]q[5][s],xA[6]xA[6][s],yA[6]yA[6][s],q[6]q[6][s],xA[7]xA[7][s],yA[7]yA[7][s],q[7]q[7][s],xA[8]xA[8][s],yA[8]yA[8][s],q[8]q[8][s],xA[9]xA[9][s],yA[9]yA[9][s],q[9]q[9][s],xA[10]xA[10][s],yA[10]yA[10][s],q[10]q[10][s],xA[11]xA[11][s],yA[11]yA[11][s],q[11]q[11][s],xA[12]xA[12][s],yA[12]yA[12][s],q[12]q[12][s],xA[13]xA[13][s],yA[13]yA[13][s],q[13]q[13][s],xA[14]xA[14][s],yA[14]yA[14][s],q[14]q[14][s],xA[15]xA[15][s],yA[15]yA[15][s],q[15]q[15][s],xA[16]xA[16][s],yA[16]yA[16][s],q[16]q[16][s],xA[17]xA[17][s],yA[17]yA[17][s],q[17]q[17][s],xA[18]xA[18][s],yA[18]yA[18][s],q[18]q[18][s],xA[19]xA[19][s],yA[19]yA[19][s],q[19]q[19][s],xA[20]xA[20][s],yA[20]yA[20][s],q[20]q[20][s],thetatheta[s],stst[s],[s],[s]};(*thiswilltransformallthevariableintofunctionsofsorthearclength*)
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Feed flow rate
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Pressure ratio
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Chebyshev orthogonal collocation first we build matrix Add
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}];
Derivatives (calculated at the collocation nodes
xAp[i_]:=2/stAdd[[i,j]]xA[j];
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∑
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yAp[i_]:=2/stAdd[[i,j]]yA[j]
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∑
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qp[i_]:=2/stAdd[[i,j]]q[j];
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Pressure ratio
g=1/11.;
Governing equations and boundary conditions
eq1[i_]:=-yA[i]qp[i]qA(xA[i]-/yA[i]);
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eq2[i_]:=-(1-yA[i])qp[i]qB((1-xA[i])-/(1-yA[i]));
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eq3[i_]:=yA[i]qp[i]q[i]xAp[i]+qp[i]xA[i];
yB[i_]:=1-yA[i]
xB[i_]:=1-xA[i]
xA[Np+1]=xAin;yA[Np+1]=;q[Np+1]=;q[1]=(1-theta);
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Unknowns
unknown=Join[Flatten[Join[Table[{{xA[i],0.8}},{i,1,Np}],Table[{{yA[i],0.2}},{i,1,Np}],Table[{{q[i],0.99}},{i,2,Np}]],1],{{theta,0.05},{,0.95},{,0.1}}];
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System of equations
sys=Flatten@Join[Table[{eq1[i],eq2[i],eq3[i]},{i,2,Np}],{eq1i,eq2i,eq1[1],eq2[1],eq3[Np+1]}];
PI Membrane
PI Membrane
Permeate mole fractions at the entrance of the module
eq1i=/()qA(xA[Np+1]-/)/(qB(xB[Np+1]-/));
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eq2i=+1;
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Feed composition
xAin=0.79;
Ratio of permeabilities
qA=1;qB=6.7;
Boundary conditions
xA[Np+1]=xAin;yA[Np+1]=;q[Np+1]=;q[1]=(1-theta);
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The solution we get using FindRoot when st=0.01
sol=FindRoot[sys/.st0.01,unknown,MaxIterations100];
Boundary conditions
xA[Np+1]=.;yA[Np+1]=.;q[Np+1]=.;q[1]=.;xA[Np+1][s_]:=xAin;yA[Np+1][s_]:=[s];q[Np+1][s_]:=;q[1][s_]:=(1-theta[s]);
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Initial conditions obtained for s=0.01
init=Join[Flatten[Table[{xA[i][0](xA[i]/.sol),yA[i][0](yA[i]/.sol)},{i,1,Np}],1],Flatten[Table[q[i][0](q[i]/.sol),{i,2,Np}],1],{[0](/.sol),[0](/.sol),theta[0](theta/.sol),st[0]0.01}];
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Initial conditions for derivatives
initdev=Join[Flatten[Table[{xA[i]'[0]-0.1,yA[i]'[0]-0.1},{i,1,Np}],1],Flatten[Table[{q[i]'[0]-0.1},{i,2,Np}],1],{theta'[0]-0.1,st'[0]-1,'[0]-0.1,'[0]-0.1}];
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The auxiliary equation
aux=Sum[xA[i]'[s]^2+yA[i]'[s]^2,{i,1,Np}]+Sum[q[i]'[s]^2,{i,2,Np}]+theta'[s]^2+st'[s]^2+'[s]^2+'[s]^21;
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The dependent variables we aim to find
unknown1=Join[Flatten[Table[{xA[i],yA[i]},{i,1,Np}],1],Flatten[Table[q[i],{i,2,Np}],1],{theta,st,,}];
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The system augmented with the auxiliary equation and the initial conditions all variable should now be a function of s, which is the arc-length
SYS=Join[sys/.param,{aux},init,initdev];
Solution of the DAE system using NDSolve
SOL=Quiet@NDSolve[SYS,unknown1,{s,-10,10},AccuracyGoal5];
bounds=SOL[[1]][[2,2]][[1,1]];
Plotting nitrogen recovery versus nitrogen purity
p1=ParametricPlot[Evaluate[{xA[1][s],xA[1][s](1-theta[s])/xAin}/.SOL[[1]]],{s,bounds[[1]]+0.01,bounds[[2]]-0.01},PlotRange{{0.9,1},{0,1}},FrameTrue,GridLinesAutomatic,AspectRatio1,PlotStyle{Thick,Dashed,Darker@Green},FrameTrue,FrameLabel{Style["Nitrogen Purity",16],Style["Nitrogen Recovery",16]}];
PPO Membrane
PPO Membrane
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eq1i=/()qA(xAin-/)/(qB((1-xAin)-/));eq2i=+1;xAin=0.79;qA=1;qB=4.4;(*ratioofpermeabilities*)xA[Np+1]=xAin;yA[Np+1]=;q[Np+1]=;q[1]=(1-theta);sol=FindRoot[sys/.st0.01,unknown,MaxIterations100];xA[Np+1]=.;yA[Np+1]=.;q[Np+1]=.;q[1]=.;xA[Np+1][s_]:=xAin;yA[Np+1][s_]:=[s];q[Np+1][s_]:=;q[1][s_]:=(1-theta[s]);init=Join[Flatten[Table[{xA[i][0](xA[i]/.sol),yA[i][0](yA[i]/.sol)},{i,1,Np}],1],Flatten[Table[q[i][0](q[i]/.sol),{i,2,Np}],1],{[0](/.sol),[0](/.sol),theta[0](theta/.sol),st[0]0.01}];initdev=Join[Flatten[Table[{xA[i]'[0]-0.1,yA[i]'[0]-0.1},{i,1,Np}],1],Flatten[Table[{q[i]'[0]-0.1},{i,2,Np}],1],{theta'[0]-0.1,st'[0]-1,'[0]-0.1,'[0]-0.1}];aux=Sum[xA[i]'[s]^2+yA[i]'[s]^2,{i,1,Np}]+Sum[q[i]'[s]^2,{i,2,Np}]+theta'[s]^2+st'[s]^2+'[s]^2+'[s]^21;unknown1=Join[Flatten[Table[{xA[i],yA[i]},{i,1,Np}],1],Flatten[Table[q[i],{i,2,Np}],1],{theta,st,,}];SYS=Join[sys/.param,{aux},init,initdev];SOL=Quiet@NDSolve[SYS,unknown1,{s,-10,10},AccuracyGoal5];bounds=SOL[[1]][[2,2]][[1,1]];
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p2=ParametricPlot[Evaluate[{xA[1][s],xA[1][s](1-theta[s])/xAin}/.SOL[[1]]],{s,bounds[[1]]+0.01,bounds[[2]]-0.01},PlotRange{{0.9,1},{0,1}},FrameTrue,GridLinesAutomatic,AspectRatio1,PlotStyle{Thick,Dashed,Darker@Red},FrameTrue,FrameLabel{Style["Nitrogen Purity",16],Style["Nitrogen Recovery",16]}];
PDMS Membrane
PDMS Membrane
We check our results against Fig . 7 a of Ref . 1
References
References
[1] T. Harlacher and M. Wessling, Gas–Gas Separation by Membranes, Chapter Thirteen, Progress in Filtration and Separation, Editor: Steve Tarleton, Elsevier, 2015, pp. 557-584.
[2] 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
[2] 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
Nitrogen recoveries with different polymer membranes
by Housam Binous and Ahmed Bellagi
Wolfram Community, STAFF PICKS, December 26, 2024
https://community.wolfram.com/groups/-/m/t/3345034
by Housam Binous and Ahmed Bellagi
Wolfram Community, STAFF PICKS, December 26, 2024
https://community.wolfram.com/groups/-/m/t/3345034