CSTR with a Second-Order Reaction — Model and Transfer Function: 23-Chem-A6 Process Dynamics and Control, May 2014, Q7

Problem 7: CSTR with a Second-Order Reaction — Model and Transfer Function (20%)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Isothermal constant-volume CSTR; constant $\rho$ (so the volumetric flow $q=F/\rho$ is constant); second-order kinetics $r_A=k_1 C_A^2$.

Find. (a) the dynamic model and steady-state $C_{As}$; (b) $\delta C_A/\delta C_{A_o}$.

Problem 7: well-mixed isothermal CSTR of constant volume $V$; feed $q,\,C_{A_o}$, effluent $q,\,C_A$, with second-order consumption $r_A=k_1 C_A^2$ throughout the vessel.

Approach. Write an unsteady component-$A$ balance over the well-mixed volume, set the derivative to zero for the steady state, then linearise the non-linear reaction term about $C_{As}$ and Laplace-transform to obtain a first-order transfer function.

(a) Dynamic model. Component-$A$ balance on the constant volume $V$ (in $=$ out $+$ consumed $+$ accumulated), with volumetric flow $q=F/\rho$: $$\boxed{V\dfrac{dC_A}{dt}=q\,(C_{A_o}-C_A)-V k_1 C_A^{2}}.$$ The out-flow term uses the well-mixed effluent concentration $C_A$; the reaction removes $k_1C_A^2$ per unit volume.
Steady state. Set $dC_A/dt=0$: $q(C_{A_o,s}-C_{As})-Vk_1C_{As}^2=0$, i.e. $Vk_1C_{As}^2+qC_{As}-qC_{A_o,s}=0$. Solving the quadratic (positive root): $$\boxed{C_{As}=\dfrac{-q+\sqrt{q^{2}+4Vk_1 q\,C_{A_o,s}}}{2Vk_1}}.$$
Linearise the reaction term. Expanding $k_1C_A^2$ about $C_{As}$: $k_1C_A^2\approx k_1C_{As}^2+2k_1C_{As}\,(C_A-C_{As})$. In deviation variables $C_A'=C_A-C_{As}$, $C_{A_o}'=C_{A_o}-C_{A_o,s}$ the balance becomes $$V\dfrac{dC_A'}{dt}=q\,(C_{A_o}'-C_A')-2Vk_1C_{As}\,C_A'.$$
Collect into first-order form. $V\dot C_A'+(q+2Vk_1C_{As})C_A'=q\,C_{A_o}'$. Dividing by $(q+2Vk_1C_{As})$ gives $\tau\dot C_A'+C_A'=K C_{A_o}'$ with $$\tau=\dfrac{V}{q+2Vk_1C_{As}},\qquad K=\dfrac{q}{q+2Vk_1C_{As}}.$$
(b) Transfer function. Laplace-transforming, $$\boxed{\dfrac{\delta C_A(s)}{\delta C_{A_o}(s)}=\dfrac{K}{\tau s+1}=\dfrac{q}{V s+\big(q+2Vk_1C_{As}\big)}}.$$ It is a first-order lag; the gain $K=q/(q+2Vk_1C_{As})<1$ because the reaction consumes part of any inlet-concentration change, and the reaction shortens $\tau$ below the pure residence time $V/q$.

Result	Expression
Dynamic model	$V\,dC_A/dt=q(C_{A_o}-C_A)-Vk_1C_A^2$
Steady state	$C_{As}=\dfrac{-q+\sqrt{q^2+4Vk_1qC_{A_o,s}}}{2Vk_1}$
Time constant	$\tau=V/(q+2Vk_1C_{As})$
Gain	$K=q/(q+2Vk_1C_{As})<1$
Transfer function	$\delta C_A/\delta C_{A_o}=K/(\tau s+1)$

Check — flow convention

“Mass flowrate $F$” with constant density is converted to a constant volumetric flow $q=F/\rho$ for the concentration (mol per volume) balance. If one prefers to keep $F$ throughout, replace every $q$ above by $F/\rho$; the structure and conclusions are unchanged.

Concept & Theory

Topic. Dynamic modelling and linearisation of a non-linear (second-order-kinetics) CSTR.

Key relations. Component balance $V\,dC_A/dt=q(C_{A_o}-C_A)-Vr_A$; steady state sets the derivative to zero; linearise $r_A(C_A)\approx r_A(C_{As})+r_A'(C_{As})C_A'$ with $r_A'=2k_1C_{As}$; first-order lag $K/(\tau s+1)$.

Why this works. Around a steady state small deviations obey the tangent (linear) approximation, so the non-linear reactor behaves as a first-order system whose gain and time constant depend on the operating point through $C_{As}$. Reaction adds an effective “sink resistance” $2Vk_1C_{As}$ in parallel with the flow term $q$, which both lowers the gain and speeds the response.

Common pitfall. Linearising $C_A^2$ as $C_A^2$ instead of $2C_{As}C_A'$, or forgetting to convert the mass flowrate to a volumetric flow for a molar-concentration balance.

Source. Fogler, Elements of Chemical Reaction Engineering, Ch. 8 (CSTR dynamics); Seborg et al., Ch. 2–4 (modelling and linearisation); Coughanowr & LeBlanc, Ch. 5.

Question 7 of 8: CSTR with a Second-Order Reaction — Model and Transfer Function

Problem 7: CSTR with a Second-Order Reaction — Model and Transfer Function (20%)