Optimization of Reactor Design
Much of reactor design uses optimization techniques to determine the
best theoretical design before engineering considerations are then
tackled. The optimization of a parallelpiped for the best shape is a
very simple example of this, but is instructive.
To prove that a cube is the best parallelpiped for minimal critical
volume, we start knowing the buckling and the volume equations for the
parallelpiped.
\[V=abc\]
\[B^{2}=\left(\frac{\pi}{a}\right)^{2}+\left(\frac{\pi}{b}\right)^{2}+\left(\frac{\pi}{c}\right)^{2}\]
Now, we want to pose the problem with a Lagrange multiplier,
\(\lambda\).
\[V\left(a,b,c,\lambda\right)=abc+\lambda\left[B^{2}-\left(\frac{\pi}{a}\right)^{2}-\left(\frac{\pi}{b}\right)^{2}-\left(\frac{\pi}{c}\right)^{2}\right]\]
This gives us an equation in four dimensions, where the minimum of that
hyperplane is the location where the volume is minimized and the
buckling criteria is satisfied. So, we take the derivatives and set them
equal to zero
\[\begin{split}\begin{align*}
\frac{\partial V}{\partial a}=0 & =bc-\frac{2\pi^{2}\lambda}{a^{3}}\\
\frac{\partial V}{\partial b}=0 & =ac-\frac{2\pi^{2}\lambda}{b^{3}}\\
\frac{\partial V}{\partial c}=0 & =ab-\frac{2\pi^{2}\lambda}{c^{3}}
\end{align*}\end{split}\]
Manipulation of these three equations leads us to another set of
equations
\[\begin{split}\begin{align*}
abc & =2\lambda\frac{\pi^{2}}{a^{2}}\\
abc & =2\lambda\frac{\pi^{2}}{b^{2}}\\
abc & =2\lambda\frac{\pi^{2}}{c^{2}}
\end{align*}\end{split}\]
for which the only solution (this can be solved with substitution if it
is not apparent) is
\[a=b=c\]
Which shows that the minimal volume for a parallelpiped critical reactor
is a cube, and volume is
\[V=a^{3}\]
and buckling is
\[B^{2}=3\left(\frac{\pi}{a}\right)^{2}\]
