Consider a system of reactors whose purpose is to form a reaction product from a glucose$-$rich solution. The reaction occurs in two steps and each step is performed in a separate reactor. We assume that both the reactors are perfectly mixed. The feed solution to the first reactor has a flow rate $Q(\frac{1}{d})$ and contains only glucose, at a concentration $g_{in}(\frac{\text{m}\text{m}\text{o}\text{l}}{l}).$ In the first reactor $($with constant volume $1$L$),$ glucose is converted to acetate. The reaction rate is proportional to the glucose concentration in $\frac{\text{m}\text{m}\text{o}\text{l}}{l}$ in the reactor with proportionality constant in Ka $($in $d^{-1}).$ In this first reactor, $1$ mole of glucose is converted to $2$ moles of acetate.

Liquid from the first reactor is directly fed to the second reactor. In addition, an acetate solution, with acetate concentration $a_{in}$ is fed separately to the second reactor at a rate $Q_a(\frac{1}{d}).$ In this reactor, acetate is converted to the end product in the presence of a biomass catalyst. $1$ mole of acetate produces $1$ mole of the end product. The rate of this reaction $(v)$ follows Haldane kinetics and is given by

$v=\frac{k_{p}a}{a+K_s+\frac{a^2}{K_i}}$

Where in mmo$\frac{l}{l}$ denotes the concentration of acetate in the second reactor; $k_p(\frac{\text{m}\text{m}\text{o}\text{l}}{1*d}),K_s(\frac{\text{m}\text{m}\text{o}\text{l}}{l})$ and $K_i(\frac{\text{m}\text{m}\text{o}\text{l}}{l})$ are constants.

Liquid is withdrawn at a rate $Q_3(\frac{1}{d})$ from the second reactor. This flow rate is controlled by a pump and is therefore variable. Hence the volume of liquid in the second reaction is also variable.

Draw a neat figure describing the reactor system wherein you mark the flow rates and concentrations at various points in the system. Write the differential equations describing the evolution of the variables of the system.