Answer Happy

please solve the tasks from 2 to 12 .
Wastewater system ACID Parameters V 9B (t) 9₁ (t) qBmax qamax qB qÀ q* qBmax MOTOR qA CA Control Valve pHc V, c(t) c(t), q(t) Figure 1. Neutralization tank with pH control Description Tank volume Unit Meters³ Base volumetric flow rate Meters³/Second Meters³/Second Meters³/Second Acid volumetric flow rate Maximum base volumetric flow rate Maximum acid volumetric flow rate Steady State base volumetric flow rate Meters³/Second Meters³/Second Meters³/Second Steady state acid volumetric flow rate Meters/Second Steady state output volumetric flow rate Maximum Steady State base volumetric flow rate Meters³/Second qB CB BASE Value 10 5 x 10-³ 5 x 10-3 5 x 10-3 25 x 10-4 5 x 10-3 5 x 10-³ 10-² 5 x 10-³

qAmax Meters³/Second 25 x 10-4 Maximum Steady State acid volumetric flow rate Steady state base feed concentration CB Moles/Liter -10 CA Steady state acid feed Moles/Liter 10 concentration C(t) Excess acid concentration Moles/Liter Cmax Moles/Liter 10-6 Maximum excess acid concentration Derivation of model According to the Arrhenius theory of acids and bases, when an acid is added to water, it contributes an H+ ion to water to form the molar concentration of hydronium ion H30+ (often represented by H+). The higher the concentration of H3O+ (or H*) in a solution, the more acidic the solution is. An Arrhenius base is a substance that generates hydroxide ions, OH, in water. The higher the concentration of OH in a solution, the more basic the solution is, i.e. H₂O = H+ + OH- pH is defined as the negative of the base-ten logarithm of the molar concentration of hydronium ions present in the solution. The unit for the concentration of hydrogen ions is Moles/Liter. pH can be determined as follows: pH = -log(H+) Consider the wastewater system outlined in Figure 1 that contains one single tank with volume V. Let C₁ (t) (Moles/Liter) and Con(t) (Moles/Liter) denote the concentration of H+ and OH ions, respectively. q(t) denotes flow rate. Let further subscript A denote acid, subscript B denote base and no subscript denote the outlet stream. Material balances for H+ and OH-yields V{CH(t)} =qA(t)CH,A + 9B (t)CH,B − q (t)CH +rV dt d V{COH (t)} = A (t)COH,A + 9B (t) COH,B - q(t)COH +rV dt

where r(moles/second.m³) is the rate for the reaction H₂0 H+ + OH¯ which for completely dissociated ("strong") acids and bases is the only reaction in which H* and OH- participate. We may eliminater from the equations by taking the difference to get a differential equation in terms of the excess of acid C(t) = CH (t)- СOH (t) Page 5 of 12 Hence -{C(t)} = A (t)C₁ + 9B (t) CB -q(t)C(t) (1) where CA = CH.A - COH,A and CB = CH.B - COH,B This is the material balance for mixing tank without reaction. The overall model is bilinear due to the product of flow rate and concentration q(t)C(t). Note that C(t) will take on negative values when pH is above 7. The acid and base feed concentrations C₁ and CB for both H+ and OH-are assumed to be constants. Linearising equation (1) around a steady-state nominal point (denoted with an asterisk) v {c(Đ)}+q°C(t) = qa(t)(C; – C") +qB(t)(CB – C*) dt * is used to denote steady-state values, and q* = C₁ + CB C* = 10-pH – 10-14+pH CA = CH.A - COH,A and CB=CH,BCOH,B

Scaled variables for the input are introduced for the input, output and the disturbance as follows 9₁ (t) 9B (t) y(t) = C (t) Cmax d(t): u(t)= qAmax qBmax Tasks 1) Explain how the performance of pH level control system can be compliant with environmental regulations and the treatment of wastewater. Provide two examples to explain the importance of a stable pH and its role in minimizing pollution in our ecosystem. [5 Marks] 2) For a neutral pH = 7, using Laplace transforms and assuming zero initial conditions, show that [C₁-C* 9Amax Св - Савмах 9(5)=571 G-C y(s) d(s) + ū(s)] + q* Cmax q* Cmax where the time constant is T = V/q*. [7 Marks]

3) Construct a block diagram depicting an open loop arrangement for the signals and transfer functions defined in 2). [3 Marks] 4) Assume zero initial conditions and a step input with magnitudes a and ß for each of d(s) and ū(s) respectively. Find the concentration output y(t). [10 marks] 5) Using MATLAB, produce a unit step response for the output y(t) and verify the result by comparing it with the analytical result derived in 4). Select the time scales so that both the transients and the steady state output are visible. [10 marks] 6) Assuming d(s) = 0, specify the parameter values that needs to be changed for the speed of the response to increase. Explain and justify your reasoning using appropriate mathematical functions and step response plots? [5 marks] 7) Assuming a unity negative feedback loop, derive the following transfer functions a. Gry(s) b. Gdy (S) c. Gre (s) d. Gde (s) [8 marks] 8) Verify that the closed-loop system is stable by graphically computing the poles and zeros. [4 marks] 9) Analytically calculate the steady state error due to the disturbance and the reference signal. What can you infer from the values obtained? [7 marks] 10) Prove that the output y(t) will only track steady-state targets if there is an integrator in either a feedforward controller C(s) or the plant G(s). What other condition is required? In addition, using a mathematical derivation, specify the requirement for disturbance signals to be totally rejected, that is to have no steady- state impact on the output? [12 marks]

11) Use MATLAB to investigate how offset and performance varies as you change the scalar controller gain Kp. Give some generic conclusions based upon what you observe. [7 marks] 12) Design two different feedforward controllers using MATLAB/SIMULINK to Page 7 of 12 Reduce the steady state error as much as possible. Raise damping to an optimum. Minimize the response time to reach the steady state value. You'll find that a compromise between these multiple goals will be necessary, and in some cases, you may not be able to meet all goals. Document your design choices and explain how you arrived at your final design. The controllers may consist of any combination of P, PI, PD and PID. Design is not merely for the specified overshoot, but also ensures that the steady state error is as small as possible. Start with the proportional controller first. You can combine several compensators but remember that the number of compensator zeros must never exceed the number of compensator poles (proper system). Document the final design in your report, i.e. List both compensator transfer functions and produce a SIMULINK block diagram of the overall system and graphically depict the location of all compensator and plant poles and zeros. • Document how you chose the controller, starting from the open-loop plant without the controller. After completing the design, show the step response of the improved loop transfer function including the controller.