Assignment 3 - 2014

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Due date(s): 03 February 2014, in class
Nuvola mimetypes pdf.png (PDF) Assignment questions
Nuvola mimetypes pdf.png (PDF) Assignment solutions

Assignment objectives: understand how to work with nonlinear systems; how to model time delays; further experience with transfer functions and block diagrams. Getting some practice for the midterm.

Question 1 [10]

Consider the system from the previous assignment. It is a CSTR where the reaction \(\text{A} \longrightarrow \text{B}\) is occurring. You do not have to derive the component material balance, i.e., the differential equation in the figure.

../figures/process-control/Marlin-slides/ode-system.png

Using the above nonlinear model, create a linearized model and find the Laplace transform representation of it. You will need to create deviation variables \(C_A'\) for outlet concentration, and \(C_{A0}'\) for inlet concentration. The transfer function will relate the outlet concentration \(C_A'\) to the incoming (input) concentration, \(C_{A0}'\).

Question 2 [8]

From a previous midterm

A thermometer having first-order dynamics, with a time constant of 0.2 minutes, is placed in a temperature bath, and after the thermometer comes to equilibrium, with the bath, the temperature of the bath is increased linearly with time, at a rate of 0.5 degrees per minute.

  1. Write down what the Laplace transfer expression would be for the thermometer temperature reading, \(T\). Use deviation variables.
  2. Find the analytical response of the thermometer temperature, \(T'(t)\).

Question 3 [12]

From a previous midterm

A second order reaction occurs in a well mixed tank (liquid phase reaction). The system is currently operating at steady state, however the inlet feed flow and concentration are both known to vary (i.e. there are two inputs to the system)

The reaction is given by \(-r_\text{A} = kC_\text{A}^2\), and you may assume the tank volume is constant, and the reactor is isothermal. The reaction rate constant has units of \(\text{m}^3.\text{s}^{-1}.\text{mol}^{-1}\).

  1. Derive a dynamic model that relates the outlet concentration of species A, denoted by \(C_\text{A}\), to the inlet flow rate, \(F\), and inlet concentration, \(C_\text{A,in}\).
  2. Use this model to derive the Laplace transform representation, linearizing where required.

Question 4 [10]

From a previous midterm

Apply a step input of magnitude 3 units to the following system:

\[G(s) = \dfrac{e^{-2s}}{\left(s^2+2s+5\right)\left(s+2\right)}\]

then, without explicitly inverting to the time domain, determine:

  1. whether the response is stable
  2. the steady value of the response
  3. whether the response will have oscillatory characteristics (justify your answer).
  4. Give a rough sketch of the system's output, starting from when the step response is applied.