Exercise $10\mathrm{.}3$: Changing particle coordinates

Population balance models can appear quite different in different coordinate systems.

Consider changing the characteristic size coordinate from particle length, $L,$ to particle

volume, $V.$ The coordinates are related by

$V=v_p{L}^3$

in which $v_p$ is again the particle shape factor for volume. The particle size distributions

in these two coordinate systems are related by

$f(L,t)=f(L(V),t)\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{L}}=$tilde$(f)(V,t)\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{L}}$

The corresponding population balances for the well$-$stirred batch reactors in these two

coordinate systems are

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{t}}=-\frac{\text{d}\text{e}\text{l}(fG)}{\text{d}\text{e}\text{l}\text{L}},\frac{\text{d}\text{e}\text{l}(tilde(f))}{\text{d}\text{e}\text{l}\text{t}}=-\frac{\text{d}\text{e}\text{l}((tilde(f))G_V)}{\text{d}\text{e}\text{l}\text{V}}$

Say we wish to express the rate of change of the total particle volume, $V_p^{}$

$V_p^{}$:$=\frac{d}{dt}{\displaystyle \underset{0}{\overset{}{}}}f(L,t)(v_p{L}^3)dL$

$V_p^{}$:$=\frac{d}{dt}{\displaystyle \underset{0}{\overset{}{}}}$tilde$(f)(V,t)VdV$

$($a$)$ Show that the results for the two coordinate systems are

$V_p^{}=3v_p{\displaystyle \underset{0}{\overset{}{}}}G{L}^{2}f(L,t)dL,V_p^{}={\displaystyle \underset{0}{\overset{}{}}}{G}_V$tilde$(f)(V,t)dV$

Hint: differentiate the defining integrals for $V_p^{}$ and substitute the corresponding

population balances.

Note that we used the first form to write the solute balance in Example $10\mathrm{.}2.$

$($b$)$ Although these two expressions look quite different, show that they are in fact

equivalent.

Hint: First differentiate the coordinate relationship to show that the two growth

rates are related by $G_V=3v_p{L}^{2}G$