Chemical Engineering

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Task 2.2 (2% of the marks)

Show that by combining Eqns (2.2) and (2.3) that the following 2nd order differential equation can be derived.

[pic 5] Eqn (2.4)

Identify the parameters τ and ζ.

end of task 2.2

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System data and further information

Initially the system is at a steady-state with the following parameters.

- V = 10 m3

- F1 = 0.08 m3 s-1

- F2 = 0.02 m3 s-1

- F = 0.1 m3 s-1

- CA01 = 5 kmol m-3

- CA02 = 1 kmol m-3

- k = 0.01 s-1

- CAs = 2.1 kmol m-3

- τp = 50 s

- Kp1 = 0.4

- Kp2 = 0.1

The probe (sensor) has the following parameters

- Km = 4 mA (kmol m-3)-1

- τm = 15 s

As part of the CSTR / probe system there is a feedback control system which is able to adjust the value of CA02 by manipulating a valve, depending on what the value of the set point CA,set is (note: without changing the value of F2). The feedback loop contains:

- An electronic controller with proportional (P-only) action, which has a gain, KC (mA mA-1). Initially KC = 0.5 mA mA-1.

- An electronic-pneumatic transducer (gain, Kep = 0.25 p.s.i mA-1) to convert the controller signal into an equivalent pneumatic signal for the control valve.

- A pneumatic control valve (which manipulates CA02) with first order dynamics. Model parameters KV = 4 (kmol m-3) (p.s.i)-1 and τV = 25 s.

(see Appendix A for further explanation of this)

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Further Tasks (You are advised to use the worksheet provided)

Derivation of a deviation variable mathematical model for the uncontrolled CSTR + probe system

Task 4.1 (a) (2% of the marks)

Using the following deviation variables

[pic 6]

Show that

[pic 7] Eqn (4.1)

Task 4.1 (b) (23% of the marks)

Solve Eqn (4.1) analytically to obtain the response of C*Am with time, t for:

- a step change in the concentration of A in stream 2,

C*A02 = 3 kmol m-3.

- a sinusoidal change in the concentration of A in stream 1,

C*A01 = [pic 8]

In case (i) C*A01 = 0 and in case (ii) C*A02 = 0. In both cases all flowrates are maintained at their steady state values.

end of task 4.1

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Testing of the Mathematical Model

Task 4.2 (20% of the marks)

Construct the following open loop models of the CSTR and sensor (probe). The values for the CSTR TF (transfer function) and Sensor TF are given in §3. The values for the input depend on Task 4.1 b (i) and (ii).

Model (i) – parameters in §3

[pic 9]

- Compare the numeric solution for C*Am generated by Simulink to the analytic solution you derived in Task 4.1 b (i). Run the simulation till an end ('stop') time of 250 seconds. Comment on your results.

- Comment on the differences between the values output to the workspace for CA_star (C*A) and CAM_star (C*Am).

Model (ii) – parameters in §3

[pic 10]

- Compare the numeric solution of C*Am generated by simulink to the analytic solution you derived in Task 4.1 b (ii). Run the simulation till an end time of 500 seconds. Comment on your results.

- Comment on the differences between the values output to the workspace for CA_star (C*A) and CAm_star (C*Am).

- You may wish to force Simulink to use more time points for its numeric solution (so you get ‘smooth’ curves). To do this select then . Under ‘Solver Options’ change the max step size from ‘auto’ to (say) ‘1’.

end of task 4.2

Further observations of the simulated open loop behaviour

Task 4.3 (a) (12% of the marks) – Investigating Model (i) from Task 4.2

For all of your simulations in this task use an end time of 1000 s, you may wish to add additional scopes or 'To worksheet' blocks to your simulations. Also from the Simulation menu choose ‘Configuration Parameters’ and change the ‘Max Step Size’ in the ‘Solver Options’ section from ‘auto’ to 1.

Investigate the following step changes, and sensor time constants τm.

Step changes: C*A02 = 0.1, 3, 10 (kmol m-3)

Time constant: τm = 15, 60 (s)

(This represents 6 simulations in total).

Observe and record

- The time taken to reach the final, settled value of C*Am and C*A.

- Compare the shapes of plots of C*Am and C*A against time.

- Comment on the effect of τm.

Task 4.3 (b) (13% of the marks) – Investigating Model