Series of reactions are carried out in a batch reactor at a constant temperature and a pressure,

and that the reactor initially contains only the component $A$ at a concentration of $1mo\frac{l}{L}.$ Given

an example set of rate constants and applying mass balance on three of the components and the

constraint that the concentration of four species must sum to one, results in the following

system of linear equations.

$C_A+C_B+C_C+C_D=1$

$-0\mathrm{.}3C_A+0\mathrm{.}02C_B+0\mathrm{.}05C_D=0$

$0\mathrm{.}1C_A-0\mathrm{.}82C_B+0\mathrm{.}1C_C+0\mathrm{.}1C_D=0$

$0\mathrm{.}5C_B=0\mathrm{.}11C_C+0\mathrm{.}1C_D=0$

where $C_A,C_B,C_C$ and $C_D$ are concentrations of species $A,B,C$ and $D$ respectively.

a$)$ Write the linear equations in matrix form.

b$)$ Use Gauss$-$Seidel iteration method to calculate the concentrations. Please do your

calculations by hand. Relative error can be taken as $10\%.$ First check the diagonally dominant

coefficient matrix, and make the necessary rearrangement of equations. Use $5$ significant

numbers in your hand calculations.

c$)$ Solve part $($b$)$ by MATLAB using Gauss$-$Siedel method. Relative error can be taken as $1\%\mathrm{.}1.$ Series of reactions are carried out in a batch reactor at a constant temperature and a pressure,

and that the reactor initially contains only the component A at a concentration of $1$ mol$/$L$.$ Given

an example set of rate constants and applying mass balance on three of the components and the

constraint that the concentration of four species must sum to one, results in the following

system of linear equations.

CA $+$ CB $+$ CC $+$ CD $=1$

$-0\mathrm{.}3$CA $+0\mathrm{.}02$CB $+0\mathrm{.}05$CD $=0$

$0\mathrm{.}1$CA $-0\mathrm{.}82$CB $+0\mathrm{.}1$CC $+0\mathrm{.}1$CD $=0$

$0\mathrm{.}5$CB $-0\mathrm{.}11$CC $+0\mathrm{.}1$CD $=0$

where CA$,$ CB$,$ CC$,$ and CD are concentrations of species A$,$ B$,$ C and D respectively.

a$)$ Write the linear equations in matrix form.

b$)$ Use Gauss$-$Seidel iteration method to calculate the concentrations. Please do your

calculations by hand. Relative error can be taken as $10\%.$ First check the diagonally dominant

coefficient matrix, and make the necessary rearrangement of equations. Use $5$ significant

numbers in your hand calculations.

c$)$ Solve part $($b$)$ by MATLAB using Gauss$-$Siedel method. Relative error can be taken as $1\%.$