First Order System: K G(S) TS + 1 Unity-Gain Second Order System: wa G(S) 52 + 25 + wa Unit step response of unity-gain underdamped second order system: c(t) = 1 e-twnt cos(WnV1 - 32+ - ) where 0 = tan-'/1-32) V1- Wd = Wn1-72 Od = - 9 W XT Second Order Underdamped Response Damping Ratio TT Normalized Rise Time 1.104 To Wn11-12 0.1 0.2 %OS e-(//1-32) X 100 1.203 1.321 0.3 0.4 4 1.463 Ts 0.5 1.638 0.6 1.854 TE shown on right side 0.7 0.8 2.126 2.467 2.883 0.9 Governing equations of fluid tank systems: dh Aq = Adt Ah 9= R (h: water head. A: tank cross section area, R: valve resistance. q: flow rate) Governing equations of thermal (boiler) systems: de Ah = Mc ho = Gce ch: heat rate. M: mass capacity. G: mass flow rate, c: specific heat, 8: temperature change) dt Page 6 of 6 Q2 (3 marks)- Consider a water boiler with a mass capacity of L [kg] and a constant mass flow rate of F [kg/s]. Specific heat of water is c = 1 kcal/kg C. The heater is switched on at t=0 s. generating heat at a rate of h [kcal/s]. If the change in the temperature of the exiting water compared to the inflow is represented by @a, it can be shown that the transfer function of the system is: 6.S) 1 h(s) Les + Fc a) If the input heat rate is An [kcal/s], determine the required conditions for the values of L and F so that the steady state value of e, is 30. Provide a clear statement of conditions for both L and F. b) Now, assume the input heat rate is Bn [kcal/s). The time response of the system. i.e. 80. has been recorded as shown below. Determine the values of L and F (as a function of Bn) without finding the complete expression of the time response. Step Response 40 35 30 25 Amplitude 20 15 10 5 0 0 100 200 300 600 700 800 900 400 500 Time (seconds) First Order System: K G(S) TS + 1 Unity-Gain Second Order System: wa G(S) 52 + 25 + wa Unit step response of unity-gain underdamped second order system: c(t) = 1 e-twnt cos(WnV1 - 32+ - ) where 0 = tan-'/1-32) V1- Wd = Wn1-72 Od = - 9 W XT Second Order Underdamped Response Damping Ratio TT Normalized Rise Time 1.104 To Wn11-12 0.1 0.2 %OS e-(//1-32) X 100 1.203 1.321 0.3 0.4 4 1.463 Ts 0.5 1.638 0.6 1.854 TE shown on right side 0.7 0.8 2.126 2.467 2.883 0.9 Governing equations of fluid tank systems: dh Aq = Adt Ah 9= R (h: water head. A: tank cross section area, R: valve resistance. q: flow rate) Governing equations of thermal (boiler) systems: de Ah = Mc ho = Gce ch: heat rate. M: mass capacity. G: mass flow rate, c: specific heat, 8: temperature change) dt Page 6 of 6 Q2 (3 marks)- Consider a water boiler with a mass capacity of L [kg] and a constant mass flow rate of F [kg/s]. Specific heat of water is c = 1 kcal/kg C. The heater is switched on at t=0 s. generating heat at a rate of h [kcal/s]. If the change in the temperature of the exiting water compared to the inflow is represented by @a, it can be shown that the transfer function of the system is: 6.S) 1 h(s) Les + Fc a) If the input heat rate is An [kcal/s], determine the required conditions for the values of L and F so that the steady state value of e, is 30. Provide a clear statement of conditions for both L and F. b) Now, assume the input heat rate is Bn [kcal/s). The time response of the system. i.e. 80. has been recorded as shown below. Determine the values of L and F (as a function of Bn) without finding the complete expression of the time response. Step Response 40 35 30 25 Amplitude 20 15 10 5 0 0 100 200 300 600 700 800 900 400 500 Time (seconds)
