Page 7

In the transient case, the fourth order runge-kutta method was used to ensure error
minimization. None of the equations contain a time variable, so the runge-kutta method
uses the function vector from the steady-state solution to output the transient
concentration profile.
Results and Discussion
Solutions to the Steady-State Case
For the steady-state problems, 40 nodes were used to determine the solutions with dx =
1/(40-1) = 1/39 because this step size produces and error of
= 6 x 10
4
which is less than 0.5% of the average value of y
A
and y
B
for each of the parts. A larger
number of nodes will take more time for computation, and a smaller number of nodes
will produce less accurate graphs. The tolerance for convergence of Newton’s method
was set to 10
-5
because MATLAB displays values to 10
-4
, so that no additional error of
notable size will incur from not allowing Newton’s method to converse fully to a solution.
Page 8

Case A:
Graph 1 below displays the solution for the steady-state case A obtained via Newtonian
method iteration, as plot of concentration of A vs position.
Graph 1: Concentrations at Steady-State (Case A)

Case B:
Graph 2 below displays the solution for steady-state case B obtained via Newtonian
method iteration, as a plot of concentration of B vs position.
Page 9

Graph 2: Concentrations at Steady-State (Case B)
Now, y
A
doesn’t decline as quickly as the previous case since gamma is smaller in this
case, but y
B
declines quicker than in the previous case because it is being turned into
species U, but there is no conversion of U to B since zeta is equal to zero.
Case C:
Graph 3 below displays the solution for steady-state case C obtained via Newtonian
method iteration, as a plot of concentration of U vs position.
Page 10

Graph 3: Concentrations at Steady-State (Case C)
The difference between the solution in Graph 3 and that in Graph 2 is that species U
can indeed react to form species B in Case C. This can be seen in the curve for y
U
that
has a final vale at position=1, which is slightly smaller that its final value in case B, and
y
B
’s final value, which is slightly higher than in the graph of case C than it is in case B.
Solutions to the Transient Case
The runge-kutta method was used to integrate the system of ordinary differential
equations in time, and the number of nodes used here was also 40. The smallest time
step that is possible here is determined as:
with
and set to 2. So,
the step size is 6.57 x 10
-4
and
= 1.3 x 10
-3
; a step size smaller than this
would increase computational time without a substantial increase in accuracy. To show
when the transient solution reached steady-state, one has to track when the transient
solution was within 99% of the known steady-state solution. A while loop was used to
Page 11