Expressions of Isothermal Effectiveness Factor 
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

References:
L. D. Schmidt, The Engineering of Chemical Reactions, Oxford University Press, 1998.
H. S. Fogler, Elements of Chemical Reaction Engineering, 3rd Edition, Prentice Hall, 1999.
O. Levenspiel, Chemical Reaction Engineering, 3rd Edition, Wiley 1999.

Porous Catalyst Slab

Solving differential equation resulting from shell balance and using boundary conditions: CA'[0]=0 and CA[l]=CAs with (λ^2)=\(TraditionalForm\`\(k\^'''\)\/D\_e\)

sol1 = 
      DSolve[D[CA[x], {x, 2}] ==  λ^2 CA[x], CA[x], 
        x]

CA[x_] := (E^(-x λ)) C1 + (E^(x λ)) C2

sol2 = 
      Solve[{CA'[0] ==  0, CA[l] ==  CAs}, {C1, 
          C2}]

C1 = sol2[[1, 1, 2]]

C2 = sol2[[1, 2, 2]]

CA[
              x] /. (E^(2 l λ)) ->  (Cosh[
                    2  l λ] + 
                  Sinh[2 l λ]) /. (E^(l λ - x λ)) ->  (Cosh[l λ - x λ] + 
                Sinh[l λ - 
                    x λ]) /. (E^(l λ + x λ)) ->  (Cosh[l λ + x λ] + 
              Sinh[l λ + x λ]) // Simplify

Computing the effectiveness factor

η = 
      Integrate[
          CAs Cosh[x λ] Sech[l λ], {x, 0, 
            l}]/(l CAs)

η = 
      Tanh[l λ]/(l λ) /. λ -> ϕ/
            l

Ploting concentration profile for different values of λ when CAs=1 and l=1.

When (Subscript[D, e])is large (λ is small), CA is close to CAs over most of the catalyst particle. There is no resistance to pore diffusion limitation and expect η to be close to unity. 

When (Subscript[D, e])is small (λ is large), CA is close to zero over most of the catalyst particle, there are strong pore diffusion effects and expect η to be small compared to one.

l = 1; CAs = 1;

CA[x] // Simplify

Plot[Evaluate[Table[CA[x], {λ, 0, 20, 2}]], {x, 0, 1}, 
      PlotStyle ->  Table[Hue[λ/30], {λ, 0, 30, 5}], 
      PlotRange -> {{0, 1}, {0, 1}}]

l =. ;  CAs =. ;

Porous Catalyst Spheres

Solving differential equation resulting from shell balance and using boundary conditions: CA finite at r=0 and CA[R0]=CAs with (λ^2)=\(TraditionalForm\`\(k\^'''\)\/D\_e\)

CA[x_] =.

C1 =. ; C2 =. ;

sol3 = 
      DSolve[1/r^2 D[
              r^2 D[CA[r], {r, 1}], {r, 1}] ==  λ^2 CA[r], 
        CA[r], r]

CA[r_] := (E^(-r (Sqrt[(λ^2)]))) ((C1/(2 r (Sqrt[(λ^2)]))) + (((-1 + (E^(2 r (Sqrt[(λ^2)])))) C2)/(2 r (Sqrt[(λ^2)])))) /. (E^(-r (Sqrt[(λ^2)]))) -> (Cosh[-r (Sqrt[(λ^2)])] + 
                Sinh[-r (Sqrt[(λ^2)])])  /. (E^(2 r (Sqrt[(λ^2)]))) -> (Cosh[2 r (Sqrt[(λ^2)])] + 
              Sinh[2 r (Sqrt[(λ^2)])])

CA[r] // Simplify

C1=0 because CA must be finite at r=0

Cosh[r λ]/(2 r λ) /. r -> 0

C1 = 0;

sol4 = Solve[CA[R0] ==  CAs, C2] // Simplify

C2 = sol4[[1, 1, 2]]

CA[r] // Simplify

Computing the effectiveness factor

η = 
      Integrate[
            4 Pi r^2 (( CAs R0 Csch[R0 (Sqrt[(λ^2)])] Sinh[r (Sqrt[(λ^2)])])/r), {r, 0, R0}]/(4  Pi/3 R0^3 CAs) // 
        Simplify

η = ((3 (-1 + R0 (Sqrt[(λ^2)]) Coth[R0 (Sqrt[(λ^2)])]))/((R0^2) (λ^2))) /. (Sqrt[(λ^2)]) -> ϕ/R0

η = ((3 (-1 + ϕ Coth[ϕ]))/((R0^2) (λ^2))) /. R0 -> ϕ/λ // Simplify

Ploting concentration profile for different values of λ when CAs=1 and R0=1. 

When (Subscript[D, e])is large (λ is small), CA is close to CAs over most of the catalyst particle. There is no resistance to pore diffusion limitation and expect η to be close to unity. 

When (Subscript[D, e])is small (λ is large), CA is close to zero over most of the catalyst particle, there are strong pore diffusion effects and expect η to be small compared to one.

R0 = 1; CAs = 1;

CA[r] // Simplify

Plot[Evaluate[Table[CA[r], {λ, 0.1, 15, 2}]], {r, 0, 1}, 
      PlotStyle ->  Table[Hue[λ/30], {λ, 0.1, 30, 5}], 
      PlotRange -> {{0, 1}, {0, 1}}]

R0 =. ;  CAs =. ;

Porous Catalyst Cylinders

Solving differential equation resulting from shell balance and using boundary conditions: CA finite at r=0 and CA[R0]=CAs with (λ^2)=\(TraditionalForm\`\(k\^'''\)\/D\_e\)

C1 =. ; C2 =. ;

CA[r_] =.

sol5 = 
      DSolve[1/r D[r D[CA[r], {r, 1}], {r, 1}] ==  λ^2 CA[
              r], CA[r], r]

CA[r_] := (E^(-r (Sqrt[(λ^2)]))) ((((E^(r (Sqrt[(λ^2)]))) BesselK[0, r (Sqrt[(λ^2)])] C1)/(Sqrt[π])) + (E^(r (Sqrt[(λ^2)]))) BesselI[0, 
                  r (Sqrt[(λ^2)])] C2) // Simplify

CA[r] // Simplify

BesselK[0, r (Sqrt[(λ^2)])] /. r -> 0

C1=0 because CA must be finite at r=0

C1 = 0;

sol6 = Solve[CA[R0] ==  CAs, C2]

C2 = sol6[[1, 1, 2]]

CA[r] // Simplify

Computing the effectiveness factor

η = 
      Integrate[
          2 Pi r (( CAs BesselI[0, r (Sqrt[(λ^2)])])/BesselI)[0, R0 (Sqrt[(λ^2)])], {r, 0, R0}]/(Pi R0^2 CAs)

η = ((2 BesselI[1, R0 λ])/(R0 λ BesselI[0, R0 (Sqrt[(λ^2)])])) /. (Sqrt[(λ^2)]) -> ϕ/R0 /. 
          R0 -> ϕ/λ // Simplify

Ploting concentration profile for different values of λ when CAs=1 and R0=1.

When (Subscript[D, e])is large (λ is small), CA is close to CAs over most of the catalyst particle. There is no resistance to pore diffusion limitation and expect η to be close to unity. 

When (Subscript[D, e])is small (λ is large), CA is close to zero over most of the catalyst particle, there are strong pore diffusion effects and expect η to be small compared to one.

R0 = 1; CAs = 1;

CA[r] // Simplify

Plot[Evaluate[Table[CA[r], {λ, 0.1, 15, 2}]], {r, 0, 1}, 
      PlotStyle ->  Table[Hue[λ/30], {λ, 0.1, 30, 5}], 
      PlotRange -> {{0, 1}, {0, 1}}]

R0 =. ;  CAs =. ;

Effectiveness factor versus Thiele modulus for porous particles of various shapes.

\(<< "C:\Program Files\Wolfram Research\Mathematica\4.0\AddOns\StandardPackages\Graphics\Graphics.m"\)

LogLogPlot[{Tanh[x]/x, 1/x (1/Tanh[3 x] - 1/(3 x)), 
        1/x BesselI[1, 2  x]/BesselI[0, 2  x]}, {x, 0.01, 100}, 
      PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0], 
          RGBColor[0, 0, 1]}, GridLines ->  True, 
      PlotRange -> {{0.01, 110}, {0.01, 5}}]