Computing of NonIsothermal 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:
M. B. Cutlip and M. Shacham, Problem Solving in Chemical Engineering with Numerical Methods, Prentice Hall, 1999.

Simultaneous diffusion and nonitothermal first order reaction in spherical catalyst pellets.

Integration of Heat Transfer Equation

T[r] = 
      Ts + Δ(Subscript[H, R]) De/
              ke (CA[r] - CAs) /. (Δ(Subscript[H, R]) De/
                ke -> -β Ts/CAs)

Rate constant

ka = 
      ks a Exp[-ϵ (Ts/T[r] - 1)] /. 
        a ks ->  ϕ^2 De/Rs^2

Numerical values of the parameters of the problem

Rs = 0.01; ϕ = 1; ϵ = 30; CAs = 0.01; 
    Ts = 400; Δ(Subscript[H, R]) = -8 10^7; β = 0.2; 
    De = 10^-6;

Split Boundary Problem solved using the Shooting Method. Must solve simultaneously for nonisothermal effectiveness factor.

myODEsoln[Ω_] := 
      NDSolve[{-De D[
                r^2 D[CA[r], {r, 1}], {r, 1}] ==  -ka CA[
                r] r^2, 
          D[η[r], {r, 1}] ==  
            3 Exp[-ϵ (Ts/T[r] - 1)] CA[
                r] r^2/(CAs Rs^3), η[10^-5] ==  
            0, CA '[10^-5] ==  0, 
          CA[10^-5] ==  Ω  }, {CA[r], η[
            r]}, {r, 10^-5, Rs}, MaxSteps -> 1000]

yend[Ω_?NumericQ] := 
      Flatten[(CA[r] /. myODEsoln[Ω]) /. 
          r -> Rs]
bc = FindRoot[
            yend[Ω] == CAs, {Ω, 0, 
              0.000001}, MaxIterations ->  1000][[1, 
            2]];

bc

Plot of concentration profile

Plot[Evaluate[CA[r] /. myODEsoln[bc]], {r, 0.00001, Rs}]

Plot of temperature profile

Plot[Evaluate[T[r] /. myODEsoln[bc]], {r, 0.00001, Rs}]

Checking Boundary conditions

T[r] /. myODEsoln[bc] /. r -> Rs

CA[r] /. myODEsoln[bc] /. r -> Rs

Temperature inside catalyst pellet is 479.813 K. This temperature is higher than Ts=400 K (temperature at surface of the catalyst pellet). This is due to the fact that reaction is exothermic (β =0.2>0). Thus, we expect η, the nonisothermal effectiveness factor, to be higher than 1. For ϕ = 1 and ϵ = 30, we find η = 5.81 which agrees with figure 7-10 of book by M.B.Cutlip and M.Shacham,Problem Solving in Chemical Engineering with Numerical Methods,Prentice Hall,1999.

T[r] /. myODEsoln[bc] /. r -> 10^-5

η[r] /. myODEsoln[bc] /. r -> Rs