An Equilibrium Theory for the Parametric Pump
Author's Data: Housam Binous
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Reference :

R. L. Pigford, B. Baker III and D. E. Blum, An Equilibrium Theory for the Parametric Pump, I & EC Fundamentals, Vol 8, No 1, pp. 144- 149, 1969.

Finding and plotting the composition of the top and bottom products obtained by parametric pump. We obtain the same figure as Figure 4 in the above reference.

ZTransform[{yB[n + 1] ==  yB[n] (((1 - b)/(1 + b)))}, n, 
      z]

Solve[%, {ZTransform[yB[n], n, z]}]

yB[0] =  y0;

sol = 
      InverseZTransform[{(((1 + b) z yB[0])/(-1 + b + z + b z))}, z, n] // Simplify

RSolve[{yT[n] -  
              yT[n - 1] (((1 + b)/(1 - b))) (((1 - b)/(1 + b))) - (((2  b)/(1 + b))) (((((1 - b)/(1 + b))))^(n - 3)) y0 == 0, yT[2] ==  y0 (1 + ((2 b)/(1 + b)))}, yT[n], n] // 
      FullSimplify

<< Graphics`Graphics`

b = 0.5;

plt1 = 
      LinearLogListPlot[Table[{n, (((((1 - b)/(1 + b))))^n)}, {n, 0, 7}], 
        PlotStyle ->  RGBColor[1, 0, 0], 
        DisplayFunction ->  Identity, 
        PlotJoined ->  True]

y0 = 1;

plt2 = 
      LinearLogListPlot[
        Join[{{1, y0}}, 
          Table[{n, ((2 (1 + 2 b) y0)/(1 + b)) - (((-1 + (2/(1 + b))))^(-2 + n)) y0}, {n, 2, 7}]], 
        PlotStyle ->  RGBColor[1, 0, 0], 
        DisplayFunction ->  Identity, 
        PlotJoined ->  True]

b = 0.1;

plt3 = 
      LinearLogListPlot[Table[{n, (((((1 - b)/(1 + b))))^n)}, {n, 0, 7}], 
        PlotStyle ->  RGBColor[0, 0, 1], 
        DisplayFunction ->  Identity, 
        PlotJoined ->  True]

plt4 = 
      LinearLogListPlot[
        Join[ {{1, y0}}, 
          Table[{n, ((2 (1 + 2 b) y0)/(1 + b)) - (((-1 + (2/(1 + b))))^(-2 + n)) y0}, {n, 2, 7}]], 
        PlotStyle ->  RGBColor[0, 0, 1], 
        DisplayFunction ->  Identity, 
        PlotJoined ->  True]

Show[plt1, plt2, plt3, plt4, 
      DisplayFunction ->  $DisplayFunction, PlotRange ->  All, 
      AspectRatio ->  1.5, Frame ->  True]

Finding and plotting the separation factor. We obtain the same figure as Figure 6 in the above reference.

b =.

(Subscript[α, n])_ := (((2 (1 + 2 b) y0)/(1 + b)) - (((-1 + (2/(1 + b))))^(-2 + n)) y0)/((((((1 - b)/(1 + b))))^n) y0) // FullSimplify

(Subscript[α, n])

b = 0.125;

plt5 = 
      LinearLogListPlot[Table[{n, (Subscript[α, n])}, {n, 0, 70}], 
        PlotStyle ->  RGBColor[0, 0, 1], 
        DisplayFunction ->  Identity, 
        PlotJoined ->  True]

b = 0.083;

plt6 = 
      LinearLogListPlot[Table[{n, (Subscript[α, n])}, {n, 0, 70}], 
        PlotStyle ->  RGBColor[0, 0, 1], 
        DisplayFunction ->  Identity, 
        PlotJoined ->  True]

Show[plt5, plt6, DisplayFunction ->  $DisplayFunction, 
      PlotRange ->  All, AspectRatio ->  1.5, 
      Frame ->  True]