Q$3$: For a batch crystalliser, the population balance equation is given as follows:

$\frac{deln(t,d_p)}{\text{d}\text{e}\text{l}\text{t}}+G\frac{deln(t,d_p)}{del{d}_p}=N+A-B$

Here, $d_p$ refers to a crystal diameter, and $n(t,d_p)$ is the number of crystal for each diameter, $d_p.$ The nucleation, $N,$ and growth, $G,$ terms are constant and the agglomeration, $A,$ and breakage, $B,$ terms are only function of crystal diameter, which are defined as follows:

$G(t,d_p)=0\mathrm{.}001,$

$N(t,d_p)=10,$

$A(t,d_p)=6{10}^5{d}_p,$

$B(t,d_p)=1\mathrm{.}18{10}^{10}{d}_p^2.$

The initial number density of crystal, $)=(0,d_p,$ is given by

$)=(0,d_p$

$($a$)$ Write the given equation in a discrete form using the explicit Euler method for time derivative term, $\frac{deln(t,d_p)}{\text{d}\text{e}\text{l}\text{t}},$ and upwind scheme for particle diameter derivative term, $\frac{deln(t,d_p)}{del{d}_p}.[15$ marks$]$

$($b$)$ Develop a MATLAB code to solve the discretized equations using $201$ nodes for particle diameter space from $0m$ to $100m$ and with timestep of $5{10}^{-4}s.$ Find the size distribution at $t=1s.$ Plot initial and final time particle size distribution with MATLAB.

$[25$ marks$]$