Example 3.2.11  Axial Conduction and Diffusion in a Tabular Reactor 

Axial diffusion and conduction in an adiabatic tabular reactor can be described by:[6] 

 

Typesetting:-mrow(Typesetting:-mo( 

 

 

 

Typesetting:-mrow(Typesetting:-mo( 

 

Typesetting:-mrow(Typesetting:-mi( 

 

Typesetting:-mrow(Typesetting:-mo( 

 

Typesetting:-mrow(Typesetting:-mo(                                                          (3.2.31) 

 

Typesetting:-mrow(Typesetting:-mo((0) = θ(0)Typesetting:-mo(1                                               (3.2.32) 

 

and 

 

Typesetting:-mrow(Typesetting:-mo((1)=0                                                            (3.2.33) 

 

 

This coupled boundary value problem is solved in Maple below for the following value of parameters Pe = 10, B0 = 10, E = 18 

 

>
 

> with(plots):
 

> Eq[1]:=1/Pe*diff(y(x),x$2)-diff(y(x),x)=4*y(x)*exp(E*(1-1/theta(x)));
 

`+`(`/`(`*`(diff(diff(y(x), x), x)), `*`(Pe)), `-`(diff(y(x), x))) = `+`(`*`(4, `*`(y(x), `*`(exp(`*`(E, `*`(`+`(1, `-`(`/`(1, `*`(theta(x)))))))))))) (1)
 

> Eq[2]:=1/Bo*diff(theta(x),x$2)-diff(theta(x),x)=-4*beta*y(x)*exp(E*(1-1/theta(x)));
 

`+`(`/`(`*`(diff(diff(theta(x), x), x)), `*`(Bo)), `-`(diff(theta(x), x))) = `+`(`-`(`*`(4, `*`(beta, `*`(y(x), `*`(exp(`*`(E, `*`(`+`(1, `-`(`/`(1, `*`(theta(x)))))))))))))) (2)
 

> BCs:=[1/Pe*D(y)(0)-y(0)+1,D(y)(1),1/Bo*D(theta)(0)-theta(0)+1,D(theta)(1)];
 

[`+`(`/`(`*`((D(y))(0)), `*`(Pe)), `-`(y(0)), 1), (D(y))(1), `+`(`/`(`*`((D(theta))(0)), `*`(Bo)), `-`(theta(0)), 1), (D(theta))(1)] (3)
 

> pars:={Pe=10,Bo=10,E=18,beta=0.05};
 

{Bo = 10, E = 18, beta = 0.5e-1, Pe = 10} (4)
 

> for i to 2 do eq[i]:=subs(pars,Eq[i]);od;
 

 

`+`(`*`(`/`(1, 10), `*`(diff(diff(y(x), x), x))), `-`(diff(y(x), x))) = `+`(`*`(4, `*`(y(x), `*`(exp(`+`(18, `-`(`/`(`*`(18), `*`(theta(x))))))))))
`+`(`*`(`/`(1, 10), `*`(diff(diff(theta(x), x), x))), `-`(diff(theta(x), x))) = `+`(`-`(`*`(.20, `*`(y(x), `*`(exp(`+`(18, `-`(`/`(`*`(18), `*`(theta(x))))))))))) (5)
 

> eqs:=(eq[1],eq[2]);
 

`+`(`*`(`/`(1, 10), `*`(diff(diff(y(x), x), x))), `-`(diff(y(x), x))) = `+`(`*`(4, `*`(y(x), `*`(exp(`+`(18, `-`(`/`(`*`(18), `*`(theta(x)))))))))), `+`(`*`(`/`(1, 10), `*`(diff(diff(theta(x), x), x))... (6)
 

> vars:=(y(x),theta(x));
 

y(x), theta(x) (7)
 

> bcs:=op(subs(pars,BCs));
 

`+`(`*`(`/`(1, 10), `*`((D(y))(0))), `-`(y(0)), 1), (D(y))(1), `+`(`*`(`/`(1, 10), `*`((D(theta))(0))), `-`(theta(0)), 1), (D(theta))(1) (8)
 

> sol:=dsolve({eqs,bcs},{vars},type=numeric);
 

proc (x_bvp) local res, data, solnproc, ndsol, outpoint, i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; `:=`(_EnvDSNumericSaveDigits, Digits); `:=`(Digits, 14); if _EnvInF... (9)
 

> sol(0);
 

[x = 0., theta(x) = 1.01463595348572144, diff(theta(x), x) = .146359534857216172, y(x) = .707280930285567600, diff(y(x), x) = -2.92719069714432400]
[x = 0., theta(x) = 1.01463595348572144, diff(theta(x), x) = .146359534857216172, y(x) = .707280930285567600, diff(y(x), x) = -2.92719069714432400]
(10)
 

> sol(1);
 

[x = 1., theta(x) = 1.04981039182617586, diff(theta(x), x) = 0., y(x) = 0.379216347648523786e-2, diff(y(x), x) = 0.] (11)
 

> odeplot(sol,[x,y(x)],0..1,thickness=4,title="Figure Exp. 3.2.21.",axes=boxed);
 

Plot_2d
 

> odeplot(sol,[x,theta(x)],0..1,thickness=4,color=green,title="Figure Exp. 3.2.22.",axes=boxed);
 

 

Plot_2d
 

>