L17

15/09/26 Tue 10:37am

Dynamics of Chemical Systems

Isothermal CSTR (Continuous Stirred Tank Reactor)

Mass balance equation

$\text{Rate of Change of Mass accumulation}=\text{In-Out+Generation-Consumption}$

L17-2026-09-15-10.45.16.excalidraw

$$C_A=\text{Concentration of A in reactor}$$ $$C_B=\text{Concentration of B in reactor}$$ $$C_{Ai}=\text{Inlet concentration of A}$$ $$F=\text{Volumetric flow rate}\left(\frac{m^3}{s}\right)$$ $$V=\text{reactor volume}$$ $$K=\text{reaction rate constant}$$ $$r=K{C}_A=\text{Reaction rate per unit volume}\frac{mol}{m^{3}s}$$ $$\boxed{V\frac{\delta C_A}{\delta t}=F.C_{Ai}-F.C_A+0-K.C_A.V}$$ $$\boxed{V.\frac{\delta C_B}{\delta t}=0-F.C_B+K.C_A.V-0}$$ $$\frac{\delta C_A}{\delta t}=\frac{F.C_{Ai}}{V}-K.C_A$$ $$\frac{\delta C_A}{\delta t}=\frac{F.C_{Ai}}{V}-\frac{F{C}_A}{V}-K.C_A$$ $$\frac{\delta C_B}{\delta t}=K.C_A-\frac{F.C_B}{V}$$

$⟹$

$$\frac{\delta C_A}{\delta t}=\frac{F}{V}(C_{Ai}-C_A)-K{C}_A$$ $$\frac{\delta C_B}{\delta t}=K.C_A-\frac{F.C_B}{V}$$

Now to model it in state space

$$\left[\begin{array}{c}\dot{C_A}\\ \dot{C_B}\end{array}\right]=\left[\begin{array}{cc}-K-\frac{F}{V}& 0\\ K& -\frac{F}{V}\end{array}\right]\left[\begin{array}{c}{C}_A\\ C_B\end{array}\right]+\left[\begin{array}{c}\frac{F}{V}\\ 0\end{array}\right]C_{Ai}$$

and the output will be

$$Y=C_B$$