Example4.11 rev 1.mw

Example 4.11 Current Distribution in an Electrochemical Cell

Primary and secondary current distributions in electrochemical cells are governed by the Laplace equation.[8] Consider a rectangular geometry governed by the following equation[9]

u(x,0) =1 (4.1.20)

(L,y) = u(x,∞) =0

Note that this geometry is of finite dimension in x (L) and semi-infinite in y. The following transformation is used to combine the variable:

η = (4.1.21)

The variable u in the new coordinate η is represented by U. The boundary conditions for U are:

U(0)=1

(4.1.22)

U(∞)=0

Example 4.11 is solved in Maple below. The program used for example 4.10 can be modified to solve this example. Only the variable transformation (equation (4.1.21)) has to be modified. The following results are obtained:

> | restart:

> | with(student):

> | with(plots):

> | eq:=diff(u(x,y),x$2)+diff(u(x,y),y$2);

| (1)

> | eq:=changevar(u(x,y)=U(eta(x,y)),eq):

> | eq1:=(simplify(subs(eta(x,y)=y/(L-x),eq))):

> | eq1:=subs(y=eta*(L-x),eq1):

> | eq1:=simplify(eq1*(L-x)^2):

> | eq2:=convert(eq1,diff);

| (2)

> | bc1:=U(0)=1;

| (3)

> | bc2:=U(infinity)=0;

| (4)

> | U:=rhs(dsolve({eq2,bc1,bc2},U(eta)));

| (5)

> | U:=expand(U);

| (6)

> | plot(U,eta=0..10,thickness=4,title="Figure Exp. 4.1.23.",axes=boxed);

> | u:=expand(subs(eta=y/(L-x),U));

| (7)

> | plot3d(subs(L=1,u),x=0..1,y=0..2,axes=boxed,title="Figure Exp. 4.1.24.",orientation=[30,60],labels=[x,y,"u"]);

The current distribution at the electrode (y=0) is given by:

> | curr:=subs(y=0,-diff(u,y));

| (8)

> |