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23-Chem-A6 Process Dynamics and Control · December 2014

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

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams / EGBC — December 2014 — 04-Chem-A6 Process Dynamics & Control. Three-hour open-book examination; any non-communicating calculator is permitted. Eight problems are printed and any five constitute a complete paper (each worth 20%); all eight are solved below for completeness. The parts are quantitative throughout — two-capacitance thermal modelling, state-space transfer functions, IMC design with a right-half-plane zero, Nyquist stability of an open-loop-unstable plant, ultimate gain with sensor dead time, second-order damping regimes, a non-linear CSTR, and Bode gain-margin design — and every requested plot (step response, IMC servo response, Nyquist locus, damping family, Bode diagram) is drawn as a real figure.

Reference texts: D. E. Seborg, T. F. Edgar, D. A. Mellichamp & F. J. Doyle III, Process Dynamics and Control (4th ed., Wiley) — Laplace-domain modelling, transfer functions, feedback stability, frequency response and IMC design; G. Stephanopoulos, Chemical Process Control: An Introduction to Theory and Practice (Prentice Hall) — Nyquist and Bode stability, dead-time systems and controller design; D. R. Coughanowr & S. E. LeBlanc, Process Systems Analysis and Control (3rd ed., McGraw-Hill) — first- and second-order dynamics, linearisation of non-linear balances. Standard control conventions (deviation variables; unity valve/sensor gains unless stated) are used throughout.

Problem 7: Second-Order-Reaction CSTR — 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, constant-density CSTR; second-order rate $r_A=k_1C_A^2$; mass flow $F$ (volumetric $q=F/\rho$); inlet $C_{A_o}$.

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

CSTR V, C_A r_A=k₁C_A², isothermal q, C_Aₒ q, C_A
Problem 7: isothermal CSTR with an inlet disturbance $C_{A_o}$; the second-order reaction $r_A=k_1C_A^2$ provides an extra concentration-dependent removal path in addition to washout $q$.

Approach. Write a component mole balance, set the derivative to zero for $C_{As}$ (positive root of a quadratic), then linearise the non-linear rate about $C_{As}$ and transform to standard first-order form.

  1. (a) Component balance and steady state. $$V\,\frac{dC_A}{dt}=q\,(C_{A_o}-C_A)-Vk_1C_A^{2}.$$ Setting $\dot C_A=0$: $Vk_1C_{As}^2+qC_{As}-qC_{A_o}=0$, whose physically meaningful positive root is $$\boxed{C_{As}=\dfrac{-q+\sqrt{q^{2}+4Vk_1q\,C_{A_o}}}{2Vk_1}.}$$
  2. (b) Linearise the rate. The only non-linearity is $r_A=k_1C_A^2$; about $C_{As}$, $r_A'\approx2k_1C_{As}\,C_A'$. In deviation variables $$V\,\frac{dC_A'}{dt}=q\,C_{A_o}'-(q+2Vk_1C_{As})C_A'.$$
  3. (b) Transfer function. Transforming and rearranging into standard first-order form, $$\boxed{\dfrac{\delta C_A}{\delta C_{A_o}}=\dfrac{K}{\tau s+1},\quad K=\dfrac{q}{q+2Vk_1C_{As}},\quad \tau=\dfrac{V}{q+2Vk_1C_{As}}.}$$ Both the gain ($K<1$) and the time constant ($\tau<V/q$) are pulled below their no-reaction values by the term $2Vk_1C_{As}$, so the reactor is self-regulating and faster than the pure residence time would suggest.
QuantityExpression
(a) Model$V\dot C_A=q(C_{A_o}-C_A)-Vk_1C_A^2$
(a) Steady state$C_{As}=\dfrac{-q+\sqrt{q^2+4Vk_1qC_{A_o}}}{2Vk_1}$
(b) $K$$q/(q+2Vk_1C_{As})<1$
(b) $\tau$$V/(q+2Vk_1C_{As})<V/q$