Question 10.1.1: Differential Balances on a Chemical Reactor A continuous sti......
Differential Balances on a Chemical Reactor
A continuous stirred-tank reactor is used to produce a compound R in the liquid-phase reaction A → R. Feed enters the reactor at a rate of \dot{V}_{0}(L/s); the concentration of the reactant in the feed is C_{A0}(mol A/L); and the volume of the tank contents is V(L). The vessel may be considered perfectly mixed, so that the concentration of A in the product stream equals that in the tank. For this process the rate of consumption of A equals kC_{A}[mol/(s · L of reaction volume)]. All fluids (the feed, the tank contents, and the product) may be taken to have the same density, ρ(g/L).
Write differential balances on total mass and on moles of A, expressing the balances in terms of the variables shown on the following diagram:
Basis: Given Quantities
Total Mass Balance (generation = 0, consumption = 0)
accumulation = input – output
Mass in reactor: M(g) = V(L)ρ(g/L)
\left. \Large{\Downarrow} \right. \\ accumulation (g/s) = \frac{dM}{dt}= \frac{d(V\rho )}{dt} = \rho \frac{dV}{dt} (since \rho is constant) \\ input(g/s) = \dot{V}_{0}(L/s)\rho (g/L)\\ output(g/s) = \dot{V}(L/s)\rho (g/L)\\ \left. \Large{\Downarrow} \right.\\ \rho \frac{dV}{dt}= \dot{V}_{0}\rho – \dot{V} \rho \\ \left. \Large{\Downarrow} \right. cancel \rho \\ \boxed{\begin{matrix}dV/dt= \dot{V}_{0} – \dot{V} \\ t = 0, V=V_{0}\end{matrix}}
where V_{0} is the initial volume of the tank contents.
Question: If = \dot{V}_{0} = \dot{V} what does the mass balance tell you?
Balance on A
accumulation = input – output – consumption
Moles of A in the reactor = V(L)C_{A}(mol/L)
\left. \Large{\Downarrow} \right. \\ accumulation (mol A/s) = \frac{d(VC_{A})}{dt} \\ input (mol A/s) = \dot{V}_{0}(L/s)C_{A0}(mol A/L) \\ output (mol A/s) = \dot{V}(L/s)C_{A}(mol A/L) \\ consumption (mol A/s) = kC_{A}[mol A/(s\cdot L)] V(L) \\ \left. \Large{\Downarrow} \right.\\ \boxed{\begin{matrix}\frac{d(VC_{A})}{dt} =\dot{V}_{0}C_{A0} – \dot{V}C_{A} – kC_{A}V \\ t=0, C_{A}= C_{A}(0)\end{matrix}}
where C_{A}(0) is the concentration of A in the initial tank contents. How you would solve this equation for the output concentration C_{A}(t) depends on how the quantities \dot{V}_{0}, \dot{V} , and C_{A0} vary with time.