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\noindent 3450:436/536 Mathematical Models, Kreider, Spring 2014
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\noindent Homework Set 1
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\noindent Due date: Monday 27 January
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Type your responses to the extent possible. If necessary, leave blank space in the document to write
equations by hand.
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\begin{enumerate}
\item (25 pts)
Consider a 2D rectangular domain that contains a concentration $C$ of some chemical. The concentration
outside the domain is $C_{bulk}$. The {\it transport boundary condition}, given by
\beqn
-D {\partial C \over \partial n} = K(C-C_{bulk})
\neqn
describes the transport of the chemical into (if $C_{bulk}>C$) or out of (if $C_{bulk}C_{bulk}$, so that the chemical is leaking out of the domain. Write an argument that justifies
the minus sign on the left. Look at the sign of each quantity to show that the left and right sides of the
equation have the same sign. Pick one side of the domain to analyze. This is a writing assignment, so focus
on making the argument clear and concise.
\item (15 pts)
In class, we derived the nondimensional chemical reactor equation $\beta \dot{\tilde{C}}= \beta (1-\tilde{C})
- \tilde{C}$, where we scaled concentration
$C$ by $C_{in}$ and time by $V/q$. The initial condition is $\tilde{C}(0)=\gamma = C_0/C_{in}$.
Consider the case where $\beta$ is large.
Provide a physical explanation of what this means, solve the ODE and write the solution in dimensional form.
Draw a qualitative sketch of the solution (by hand is fine) for the cases $C_0>C_{in}$ and $C_0