CBEE $212$

Studio $10$

Oregon State University

Three component system

In this exercise we will apply Euler's Method to a three$-$component system. Consider the following reaction in a batch process

$ABC$

Where the conversion of $A$ to be is described by $k_1$ and the conversion of $B$ is described by $k_2.$

The system is described by a set of differential equations.

$)=(0)=(0$

$)=(0$

For each species we can approximate the value at the next time$-$step as:

$C_{i+1}=C_i+f(C_i)t$

Where $f(C)$ is the value of the derivative at a given concentration of $C_A$ or $C_B$ respectively.

We can also track the time using:

$t_{i+1}=t_i+t$

The total time is the time step $(t)$ times the number of steps.

For the system described above, implement Euler's method for $t=1$ minutes, $k_1=0\mathrm{.}35mi{n}^{-1},k_2=0\mathrm{.}25mi{n}^{-1},$ and $C_{A0}=5\mathrm{.}0\frac{g}{L}.$ Initially there is no $B$ or $C$ in the system. Complete $10$ steps for a total model time of $10$ minutes. Plot the solution.

Now reduce the time step to $0\mathrm{.}5$min and take $20$ steps for a total model time of $10$min. What is the percent change in the final point of the solution? What happens if you decrease the time step to $0\mathrm{.}25$min $?$