Linearization of a Nonlinear (Second-Order-Reaction) CSTR: 20-Bio-A2 Process Dynamics and Control, May 2018, Q8

Problem 8: Linearization of a Nonlinear (Second-Order-Reaction) CSTR (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. Constant-volume, constant-density, isothermal CSTR; second-order reaction $r_A=kC_A^2$; volumetric flow $F$; inlet concentration $C_{A_o}$ is the input, $C_A$ the output.

Find. (a) the dynamic model for $C_A(t)$ and its steady-state value; (b) $\delta C_A(s)/\delta C_{A_o}(s)$ in standard gain/time-constant form.

Approach. Write the unsteady-state mole balance on species A, solve the resulting steady-state quadratic for the physically meaningful (positive) root, linearize the quadratic rate term by a first-order Taylor expansion about that steady state, and Laplace-transform.

(a) Dynamic model. Mole balance (constant $V$, constant $\rho$): accumulation $=$ in $-$ out $-$ reaction, $$\boxed{V\frac{dC_A}{dt}=F(C_{A_o}-C_A)-VkC_A^2.}$$
(a) Steady state. Setting $dC_A/dt=0$: $VkC_{As}^2+FC_{As}-FC_{A_os}=0$, a quadratic in $C_{As}$ whose physically meaningful (positive) root is $$\boxed{C_{As}=\frac{-F+\sqrt{F^2+4VkFC_{A_os}}}{2Vk}.}$$
(b) Linearize. $C_A^2\approx C_{As}^2+2C_{As}\,\delta C_A$. Substituting into the deviation-variable balance and subtracting the steady-state equation, $$V\frac{d(\delta C_A)}{dt}+(F+2VkC_{As})\,\delta C_A=F\,\delta C_{A_o}.$$
(b) Transfer function. Laplace-transforming from rest, $$\boxed{\frac{\delta C_A(s)}{\delta C_{A_o}(s)}=\frac{F}{Vs+(F+2VkC_{As})}=\frac{K}{\tau s+1},}$$ $$K=\frac{F}{F+2VkC_{As}}\ (0\lt K\lt1),\qquad \tau=\frac{V}{F+2VkC_{As}}\ \Big(\lt\frac{V}{F}\Big).$$

Result	Value
Dynamic model	$V\,dC_A/dt=F(C_{A_o}-C_A)-VkC_A^2$
Steady state	$C_{As}=\big[-F+\sqrt{F^2+4VkFC_{A_os}}\big]/(2Vk)$
Gain	$K=F/(F+2VkC_{As})$
Time constant	$\tau=V/(F+2VkC_{As})$

Concept & Theory

Topic. Linearization of a nonlinear (second-order reaction) CSTR material balance.

Key relations. Unsteady mole balance $V\,dC_A/dt=F(C_{A_o}-C_A)-VkC_A^2$; Taylor linearization $C_A^2\approx C_{As}^2+2C_{As}\delta C_A$; first-order lag standard form.

Why this works. The reaction term adds an extra, concentration-dependent "removal" pathway on top of the flow term, so the linearized time constant $\tau=V/(F+2VkC_{As})$ is always smaller than the pure residence time $V/F$: the reactor is more self-regulating (faster-responding) than a non-reacting tank, because higher $C_A$ accelerates consumption and pulls the system back toward steady state faster.

Common pitfall. Selecting the negative root of the steady-state quadratic (mathematically valid but physically meaningless, since $C_{As}\ge0$), or forgetting the factor of 2 from differentiating $C_A^2$.

Source. Seborg et al., Process Dynamics and Control, Ch. 2–4 (CSTR modeling, linearization of nonlinear reaction-rate terms).

Question 8 of 8: Linearization of a Nonlinear (Second-Order-Reaction) CSTR

Problem 8: Linearization of a Nonlinear (Second-Order-Reaction) CSTR (20%)