Let's solve the problem step by step as requested: --- # **(a) Derive the Process Transfer Function** ## **System Description** - Two non-interacting tanks in series. - Each tank: \( V_1 = V_2 = 4\, m^3 \) - Flow rate: \( Q = 2\, m^3/\text{min} \) - Inlet temperature: \( T_{in} = 60^\circ C \) - Initial temperature: \( T_1 = T_2 = 60^\circ C \) - Heat capacity and density are constant. - Goal: Control outlet temp (\(T_2\)) by adjusting the inlet temp (\(T_{in}\)), using a control valve and controller. ## **Step 1: Write the dynamic equations** For **Tank 1** (non-interacting, well-mixed): \[ V_1 \frac{dT_1}{dt} = Q(T_{in} - T_1) \] \[ \Rightarrow \frac{dT_1}{dt} + \frac{Q}{V_1}T_1 = \frac{Q}{V_1}T_{in} \] For **Tank 2**: \[ V_2 \frac{dT_2}{dt} = Q(T_1 - T_2) \] \[ \Rightarrow \frac{dT_2}{dt} + \frac{Q}{V_2}T_2 = \frac{Q}{V_2}T_1 \] ## **Step 2: Laplace Transform** Let’s take Laplace transforms (assuming zero initial conditions): \[ V_1 [sT_1(s)] + Q T_1(s) = Q T_{in}(s) \] \[ \Rightarrow (V_1 s + Q)T_1(s) = Q T_{in}(s) \] \[ \Rightarrow T_1(s) = \frac{Q}{V_1 s + Q} T_{in}(s) \] Similarly, for Tank 2: \[ (V_2 s + Q) T_2(s) = Q T_1(s) \] \[ \Rightarrow T_2(s) = \frac{Q}{V_2 s + Q} T_1(s) \] ## **Step 3: Overall Transfer Function** Combine the above: \[ T_2(s) = \frac{Q}{V_2 s + Q} \cdot \frac{Q}{V_1 s + Q} T_{in}(s) \] Given \( V_1 = V_2 = 4 \), \( Q = 2 \): \[ T_2(s) = \left( \frac{2}{4s + 2} \right)^2 T_{in}(s) = \left( \frac{1}{2s + 1} \right)^2 T_{in}(s) \] So the **process transfer function** is: \[ \boxed{ G_p(s) = \frac{T_2(s)}{T_{in}(s)} = \frac{1}{(2s+1)^2} } \] --- # **(b) Write the Closed-Loop Equation** Let \( G_c(s) \) be the controller transfer function The block diagram is: - Controller: \( G_c(s) \) - Process: \( G_p(s) \) - Unity feedback. The closed-loop transfer function is: \[ T_{cl}(s) = \frac{G_c(s) G_p(s)}{1 + G_c(s) G_p(s)} \] Where: - \( G_p(s) = \frac{1}{(2s+1)^2} \) - \( G_c(s) \) depends on controller type (P, PI, PID). --- # **(c) Draw the Block Diagram** **Block Diagram Structure:** ``` +-----------+ +-----------+ +--------+ Setpoint --->| Controller |--->| Process |--->| Output | +-----------+ (G_p(s)) +--------+ | ^ +---------------+ ``` - Setpoint minus Output feeds the Controller. - Controller output feeds the Process (plant). - Process output is the system output (T2). --- # **(d) Response for Different Controllers** ## **1. Proportional Controller (P)** \[ G_c(s) = K_c \] Closed-loop TF: \[ T_{cl, P}(s) = \frac{K_c G_p(s)}{1 + K_c G_p(s)} = \frac{K_c}{(2s+1)^2 + K_c} \] ## **2. Proportional-Integral Controller (PI)** \[ G_c(s) = K_c \left( 1 + \frac{1}{\tau_I s} \right) \] Closed-loop TF: \[ T_{cl, PI}(s) = \frac{K_c \left(1 + \frac{1}{\tau_I s}\right) G_p(s)}{1 + K_c \left(1 + \frac{1}{\tau_I s}\right) G_p(s)} \] ## **3. Proportional-Integral-Derivative Controller (PID)** \[ G_c(s) = K_c \left( 1 + \frac{1}{\tau_I s} + \tau_D s \right) \] Closed-loop TF: \[ T_{cl, PID}(s) = \frac{K_c \left(1 + \frac{1}{\tau_I s} + \tau_D s\right) G_p(s)}{1 + K_c \left(1 + \frac{1}{\tau_I s} + \tau_D s\right) G_p(s)} \] --- # **Response to Step Input** For a step change of magnitude \( M \): - Input: \( R(s) = \frac{M}{s} \) - Output: \( T_2(s) = T_{cl}(s) \cdot R(s) \) **You can analyze the time response by inverse Laplace or using MATLAB/Simulink.** --- # **Parameter Values** Try **at least three different sets** for \( K_c, \tau_I, \tau_D \) to see the effect. --- # **Summary of Answers** ### **(a) Process Transfer Function** \[ \boxed{ G_p(s) = \frac{1}{(2s+1)^2} } \] ### **(b) Closed-Loop Equation** \[ \boxed{ T_{cl}(s) = \frac{G_c(s) G_p(s)}{1 + G_c(s) G_p(s)} } \] ### **(c) Block Diagram** - **Setpoint → Controller → Process → Output** - **Output is fed back (negative feedback) to compare with setpoint** ### **(d) Response with Different Controllers** - Use \( T_{cl}(s) \) for each controller type. - For a step input: \( T_2(s) = T_{cl}(s) \cdot \frac{M}{s} \) - Plot for at least three sets of controller parameters. --- ## **Tip** For simulation and plotting, use Simulink/MATLAB as instructed, and change \( K_c, \tau_I, \tau_D \) values to analyze the system's behavior under P, PI, and PID control. --- **If you need help with the Simulink implementation or specific parameter analysis, let me know!**