Example 5.5 

Consider the diffusion of a gas (A) through a stagnant liquid (B) in a container.[2]  A reacts with B according to the irreversible reaction.   

 

Both dimensionless and dimensional plots are obtained. 

> restart;
 

> with(linalg):with(plots):
 

> ge:=diff(u(x,t),t)=diff(u(x,t),x$2)+Phi^2*u(x,t);
 

diff(u(x, t), t) = `+`(diff(diff(u(x, t), x), x), `*`(`^`(Phi, 2), `*`(u(x, t)))) (1)
 

> k:=2e-7;Dab:=2e-9;ca0:=0.01;Lc:=10e-2;
 

 

 

 

0.2e-6
0.2e-8
0.1e-1
.10 (2)
 

> Phi:=sqrt(k*Lc^2/Dab);
 

1.000000000 (3)
 

> bc1:=u(x,t)-1;
 

`+`(u(x, t), `-`(1)) (4)
 

> bc2:=diff(u(x,t),x);
 

diff(u(x, t), x) (5)
 

> IC:=u(x,0)=0;
 

u(x, 0) = 0 (6)
 

> N:=10;
 

10 (7)
 

> L:=1;
 

1 (8)
 

> dydxf:=1/2*(-u[2](t)-3*u[0](t)+4*u[1](t))/h:
 

> dydxb:=1/2*(u[N-1](t)+3*u[N+1](t)-4*u[N](t))/h:
 

> dydx:=1/2/h*(u[m+1](t)-u[m-1](t)):
 

> d2ydx2:=1/h^2*(u[m-1](t)-2*u[m](t)+u[m+1](t)):
 

> bc1:=subs(diff(u(x,t),x)=dydxf,u(x,t)=u[0](t),x=0,bc1):
 

> bc2:=subs(diff(u(x,t),x)=dydxb,u(x,t)=u[N+1](t),x=1,bc2):
 

> eq[0]:=bc1;
 

`+`(u[0](t), `-`(1)) (9)
 

> eq[N+1]:=bc2;
 

`+`(`/`(`*`(`/`(1, 2), `*`(`+`(u[9](t), `*`(3, `*`(u[11](t))), `-`(`*`(4, `*`(u[10](t))))))), `*`(h))) (10)
 

> for i from 1 to N do eq[i]:=diff(u[i](t),t)= subs(diff(u(x,t),x$2) = subs(m=i,d2ydx2),diff(u(x,t),x) = subs(m=i,dydx),u(x,t)=u[i](t),x=i*h,rhs(ge));od:
 

> u[0](t):=(solve(eq[0],u[0](t)));
 

1 (11)
 

> u[N+1](t):=solve(eq[N+1],u[N+1](t));
 

`+`(`-`(`*`(`/`(1, 3), `*`(u[9](t)))), `*`(`/`(4, 3), `*`(u[10](t)))) (12)
 

> for i from 1 to N do eq[i]:=eval(eq[i]);od:
 

> eqs:=[seq(rhs(eq[j]),j=1..N)]:
 

> Y:=[seq(u[i](t),i=1..N)];
 

[u[1](t), u[2](t), u[3](t), u[4](t), u[5](t), u[6](t), u[7](t), u[8](t), u[9](t), u[10](t)] (13)
 

> A:=genmatrix(eqs,Y,'b1'):
 

> b:=matrix(N,1):for i to N do b[i,1]:=-eval(b1[i]);od:evalm(b);
 

Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mfrac(Typesetting:-mverbatim( (14)
 

> h:=eval(L/(N+1));
 

`/`(1, 11) (15)
 

> A:=map(eval,A):
 

> if N > 4 then A:=map(evalf,A);end:
 

> evalm(A);
 

Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn(
(16)
 

> mat:=exponential(A,t):
 

> mat:=map(evalf,mat):
 

> Y0:=matrix(N,1):for i from 1 to N do Y0[i,1]:=evalf(subs(x=i*h,rhs(IC)));od:evalm(Y0);
 

Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mtable(Typesetting:-mtr(Typesetting:-mtd(Typesetting:-mn( (17)
 

> s1:=evalm(Y0+inverse(A)&*b):
 

> Y:=evalm(mat&*s1-inverse(A)&*b):
 

> Y:=map(simplify,Y):
 

> Digits:=5;
 

5 (18)
 

> for i from 1 to N do u[i](t):=evalf((Y[i,1]));od:
 

> for i from 0 to N+1 do u[i](t):=eval(u[i](t));od;
 

 

 

 

 

 

 

 

 

 

 

 

1
`+`(`-`(`*`(0.45530e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `-`(`*`(0.17752e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `-`(`*`(0.38258e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `-`(`*`...
`+`(`-`(`*`(0.45530e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `-`(`*`(0.17752e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `-`(`*`(0.38258e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `-`(`*`...
`+`(`*`(0.86819e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `*`(0.29046e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `*`(0.46462e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `*`(0.43627e-1, `*`(ex...
`+`(`*`(0.86819e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `*`(0.29046e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `*`(0.46462e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `*`(0.43627e-1, `*`(ex...
`+`(`*`(0.86819e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `*`(0.29046e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `*`(0.46462e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `*`(0.43627e-1, `*`(ex...
`+`(`-`(`*`(0.12002e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `-`(`*`(0.29776e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `-`(`*`(0.18166e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `*`(0.3...
`+`(`-`(`*`(0.12002e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `-`(`*`(0.29776e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `-`(`*`(0.18166e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `*`(0.3...
`+`(`-`(`*`(0.12002e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `-`(`*`(0.29776e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `-`(`*`(0.18166e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `*`(0.3...
`+`(`*`(0.14203e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `*`(0.19676e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `-`(`*`(0.24399e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `-`(`*`(0.66975e-...
`+`(`*`(0.14203e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `*`(0.19676e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `-`(`*`(0.24399e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `-`(`*`(0.66975e-...
`+`(`-`(`*`(0.15081e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `-`(`*`(0.24177e-2, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `*`(0.47799e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `*`(0.11420e...
`+`(`-`(`*`(0.15081e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `-`(`*`(0.24177e-2, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `*`(0.47799e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `*`(0.11420e...
`+`(`*`(0.14554e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `-`(`*`(0.15718e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `-`(`*`(0.33647e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `*`(0.59188e...
`+`(`*`(0.14554e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `-`(`*`(0.15718e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `-`(`*`(0.33647e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `*`(0.59188e...
`+`(`-`(`*`(0.12671e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `*`(0.28139e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `-`(`*`(0.69347e-2, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `-`(`*`(0.51...
`+`(`-`(`*`(0.12671e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `*`(0.28139e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `-`(`*`(0.69347e-2, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `-`(`*`(0.51...
`+`(`-`(`*`(0.12671e-1, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `*`(0.28139e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `-`(`*`(0.69347e-2, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `-`(`*`(0.51...
`+`(`*`(0.96066e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `-`(`*`(0.30323e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `*`(0.42071e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `-`(`*`(0.23888e-...
`+`(`*`(0.96066e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `-`(`*`(0.30323e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `*`(0.42071e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `-`(`*`(0.23888e-...
`+`(`-`(`*`(0.56473e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `*`(0.21479e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `-`(`*`(0.44157e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `*`(0.68062e...
`+`(`-`(`*`(0.56473e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `*`(0.21479e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `-`(`*`(0.44157e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `*`(0.68062e...
`+`(`-`(`*`(0.56473e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t)))))))), `*`(0.21479e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `-`(`*`(0.44157e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t)))))))), `*`(0.68062e...
`+`(`*`(0.11620e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `-`(`*`(0.48222e-2, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `*`(0.11554e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `-`(`*`(0.22517e-...
`+`(`*`(0.11620e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `-`(`*`(0.48222e-2, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `*`(0.11554e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `-`(`*`(0.22517e-...
`+`(`*`(0.11620e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `-`(`*`(0.48222e-2, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `*`(0.11554e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `-`(`*`(0.22517e-...
`+`(`*`(0.34317e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `-`(`*`(0.13589e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `*`(0.30124e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `-`(`*`(0.52710e-...
`+`(`*`(0.34317e-2, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `-`(`*`(0.13589e-1, `*`(exp(`+`(`-`(`*`(438.99, `*`(t)))))))), `*`(0.30124e-1, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `-`(`*`(0.52710e-...
(19)
 

Semianalytical solutions are obtained in dimensionless form.  The dimensionless concentration profiles are plotted as: 

> setcolors(["Red", "Blue", "LimeGreen", "Goldenrod", "maroon", "DarkTurquoise", "coral", "aquamarine", "magenta", "khaki",  "sienna", "orange", "yellow", "gray"]);
 

[ (20)
 

> pp:=plot([seq(u[i](t),i=0..N+1)],t=0..0.4,thickness=4);
 

PLOT(CURVES([[0., 1.], [0.871886166666666594e-2, 1.], [0.163050965833333346e-1, 1.], [0.248366111666666682e-1, 1.], [0.334246781666666660e-1, 1.], [0.419719275833333322e-1, 1.], [0.498963219166666666e... (21)
 

> arw:=arrow(<0.1,1.01>,<0.1,-0.8>,width=[1/600,relative=false],head_width=[1/200,relative=false],head_length=[1/30,relative=false]):
pt:=textplot([[0.28,0.15,typeset("Follow the arrow: ",u[0],"(t), ..., ",u[N+1],"(t).")]]):
 

> display([pp,pt,arw],title="Figure Exp. 5.13.",axes=boxed,labels=[t,"u"]);
 

Plot_2d
 

> tf:=0.5;
 

.5 (22)
 

> M:=30;
 

30 (23)
 

> T1:=[seq(tf*i/M,i=0..M)];
 

[0., 0.16667e-1, 0.33333e-1, 0.50000e-1, 0.66667e-1, 0.83333e-1, .10000, .11667, .13333, .15000, .16667, .18333, .20000, .21667, .23333, .25000, .26667, .28333, .30000, .31667, .33333, .35000, .36667,...
[0., 0.16667e-1, 0.33333e-1, 0.50000e-1, 0.66667e-1, 0.83333e-1, .10000, .11667, .13333, .15000, .16667, .18333, .20000, .21667, .23333, .25000, .26667, .28333, .30000, .31667, .33333, .35000, .36667,...
[0., 0.16667e-1, 0.33333e-1, 0.50000e-1, 0.66667e-1, 0.83333e-1, .10000, .11667, .13333, .15000, .16667, .18333, .20000, .21667, .23333, .25000, .26667, .28333, .30000, .31667, .33333, .35000, .36667,...
(24)
 

> PP:=matrix(N+2,M+1);
 

array( 1 .. 12, 1 .. 31, [ ] ) (25)
 

> for i from 1 to N+2 do PP[i,1]:=evalf(subs(x=(i-1)*h,rhs(IC)));od:
 

> for i from 1 to N+2 do for j from 2 to M+1 do PP[i,j]:=evalf(subs(t=T1[j],u[i-1](t)));od;od:
 

> plotdata := [seq([ seq([(i-1)*h,T1[j],PP[i,j]], i=1..N+2)], j=1..M+1)]:
 

> surfdata( plotdata, axes=boxed, title="Figure Exp. 5.14.",labels=[x,t,u],orientation=[-75,75]);
 

Plot
 

The mass transafer flux at x = 0 in dimensionless form is given by: 

> flux:=-dydxf;
 

`+`(`*`(.14792, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `*`(.55030, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `*`(1.0972, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `*`(1.6478, `*`(exp(`+`(`-`(`*`(32...
`+`(`*`(.14792, `*`(exp(`+`(`-`(`*`(471.73, `*`(t))))))), `*`(.55030, `*`(exp(`+`(`-`(`*`(438.99, `*`(t))))))), `*`(1.0972, `*`(exp(`+`(`-`(`*`(387.95, `*`(t))))))), `*`(1.6478, `*`(exp(`+`(`-`(`*`(32...
(26)
 

> plot(flux,t=0..0.5,thickness=4,title="Figure 5.15.",axes=boxed,labels=[t,"flux"]);
 

Plot_2d
 

Next, the concentraiton profiles are converted to dimensionless form as: 

> for i from 0 to N+1 do ca[i](t):=u[i](t)*ca0;od:
 

> for i from 0 to N+1 do ca[i](t1):=subs(t=t1*Dab/Lc^2,ca[i](t));od:
 

> for i from 0 to N+1 do p[i]:=plot(ca[i](t1),t1=0..2e6,thickness=3);od:
 

> pp:=plot([seq(ca[i](t1),i=0..N+1)],t1=0..2e6,thickness=4);
 

PLOT(CURVES([[0., 0.100000000000000002e-1], [43594.3083333333270, 0.100000000000000002e-1], [81525.4829166666604, 0.100000000000000002e-1], [124183.055833333332, 0.100000000000000002e-1], [167123.3908... (27)
 

> arw:=arrow(<0.1e6,0.0105>,<1.0e6,-0.0085>,width=[1/1.0e5,relative=false],head_width=[1/5.0e3,relative=false],head_length=[1/30,relative=true]):
pt:=textplot([[1.4e6,0.0015,typeset("Follow the arrow: ",ca[0],"(t), ..., ",ca[N+1],"(t).")]]):
 

> display([pp,pt,arw],title="Figure Exp. 5.16.",axes=boxed,labels=[t1,"ca"]);
 

Plot_2d
 

> tf:=2e6;
 

0.2e7 (28)
 

> M:=30;
 

30 (29)
 

> T1:=[seq(tf*i/M,i=0..M)];
 

[0., 66667., 0.13333e6, 0.20000e6, 0.26667e6, 0.33333e6, 0.40000e6, 0.46667e6, 0.53333e6, 0.60000e6, 0.66667e6, 0.73333e6, 0.80000e6, 0.86667e6, 0.93333e6, 0.10000e7, 0.10667e7, 0.11333e7, 0.12000e7, ...
[0., 66667., 0.13333e6, 0.20000e6, 0.26667e6, 0.33333e6, 0.40000e6, 0.46667e6, 0.53333e6, 0.60000e6, 0.66667e6, 0.73333e6, 0.80000e6, 0.86667e6, 0.93333e6, 0.10000e7, 0.10667e7, 0.11333e7, 0.12000e7, ...
[0., 66667., 0.13333e6, 0.20000e6, 0.26667e6, 0.33333e6, 0.40000e6, 0.46667e6, 0.53333e6, 0.60000e6, 0.66667e6, 0.73333e6, 0.80000e6, 0.86667e6, 0.93333e6, 0.10000e7, 0.10667e7, 0.11333e7, 0.12000e7, ...
[0., 66667., 0.13333e6, 0.20000e6, 0.26667e6, 0.33333e6, 0.40000e6, 0.46667e6, 0.53333e6, 0.60000e6, 0.66667e6, 0.73333e6, 0.80000e6, 0.86667e6, 0.93333e6, 0.10000e7, 0.10667e7, 0.11333e7, 0.12000e7, ...
(30)
 

> PP:=matrix(N+2,M+1);
 

array( 1 .. 12, 1 .. 31, [ ] ) (31)
 

> for i from 1 to N+2 do PP[i,1]:=evalf(subs(x=(i-1)*h*Lc,0));od:
 

> for i from 1 to N+2 do for j from 2 to M+1 do PP[i,j]:=evalf(subs(t1=T1[j],ca[i-1](t1)));od;od:
 

> plotdata := [seq([ seq([(i-1)*h*Lc,T1[j],PP[i,j]], i=1..N+2)], j=1..M+1)]:
 

> surfdata(plotdata,title="Figure Exp. 5.17.",axes=boxed, labels=[z,t1,"ca"],orientation=[-75,75]);
 

Plot
 

Next, the mass transfer flux is converted to dimensionless form as: 

> Flux:=Dab*ca0/Lc*flux:
 

> Flux:=subs(t=t1*Dab/Lc^2,Flux);
 

`+`(`*`(0.29584e-10, `*`(exp(`+`(`-`(`*`(0.94346e-4, `*`(t1))))))), `*`(0.11006e-9, `*`(exp(`+`(`-`(`*`(0.87798e-4, `*`(t1))))))), `*`(0.21944e-9, `*`(exp(`+`(`-`(`*`(0.77590e-4, `*`(t1))))))), `*`(0....
`+`(`*`(0.29584e-10, `*`(exp(`+`(`-`(`*`(0.94346e-4, `*`(t1))))))), `*`(0.11006e-9, `*`(exp(`+`(`-`(`*`(0.87798e-4, `*`(t1))))))), `*`(0.21944e-9, `*`(exp(`+`(`-`(`*`(0.77590e-4, `*`(t1))))))), `*`(0....
`+`(`*`(0.29584e-10, `*`(exp(`+`(`-`(`*`(0.94346e-4, `*`(t1))))))), `*`(0.11006e-9, `*`(exp(`+`(`-`(`*`(0.87798e-4, `*`(t1))))))), `*`(0.21944e-9, `*`(exp(`+`(`-`(`*`(0.77590e-4, `*`(t1))))))), `*`(0....
`+`(`*`(0.29584e-10, `*`(exp(`+`(`-`(`*`(0.94346e-4, `*`(t1))))))), `*`(0.11006e-9, `*`(exp(`+`(`-`(`*`(0.87798e-4, `*`(t1))))))), `*`(0.21944e-9, `*`(exp(`+`(`-`(`*`(0.77590e-4, `*`(t1))))))), `*`(0....
(32)
 

> plot(Flux,t1=0..2e6,thickness=4,title="Figure Exp. 5.18.",axes=boxed,labels=[t,"Flux"]);
 

Plot_2d
 

>