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5.1. Derive the transfer function H(sVQ(s) for the liquid-level system of Fig. P5-1 when (a) The tank level operates about the steady-state value of h, = 1 ft (b) The tank level operates about the steady-state value of h, = 3 ft 4, ft'/min

5.1. Derive the transfer function H(sVQ(s) for the liquid-level system of Fig. P5-1 when (a) The tank level operates about the steady-state value of h, = 1 ft (b) The tank level operates about the steady-state value of h, = 3 ft 4, ft'/min

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
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5:16 . g Vo) l 38% LTE1 kB/s LeBlanc.pdf PROBLEMS 5.1. Derive the transfer function H(s/Q(s) for the liquid-level system of Fig. P5-1 when (a) The tank level operates about the steady-state value of h, = 1 ft (b) The tank level operates about the steady-state value of h, = 3 ft q, ft'/min The pump removes water at a constant rate of 10 cfm (cubic feet per minute); this rate is inde- pendent of head. The cross-sectional area of the tank is 1.0 ft, and the resistance R is 0.5 ft/cfm. 5.2. A liquid-level system, such as the one shown in Fig. 5-1, has a cross-sectional area of 3.0 ft. The 2 ft valve characteristics are FIGURE P5-1 9 = 8A 4 Outlet flow where q = flow rate, cfm, andh= level above the valve, ft. Calculate the time constant for this system if the average operating level above the valve is 24 (a) 3 ft (b) 9 ft 1.0 5.3. A tank having a cross-sectional area of 2 ft² is operating at steady state with an inlet flow rate of 2.0 cfm. The flow-head characteristics are shown 0.3 1.0 h(ft) in Fig. P5-3. (a) Find the transfer function H(s)Q(s). (b) If the flow to the tank increases from 2.0 to 2.2 cfm according to a step change, calculate the level h two minutes after the change occurs. FIGURE PS-3 5.4. Develop a formula for finding the time constant of the liquid-level system shown in Fig. P54 when the average operating level is họ. The resistance R is linear. The tank has three vertical walls and one that slopes at an angle a from the vertical as shown. The distance separating the parallel walls is 1. ---B- -- FIGURE PS4 116 PART 2 LINEAR OPEN-LOOP SYSTEMS 5.5. Consider the stirrd-tank reactor shown in Fig. PS-5. C,F The reaction occurring is A - B Volume V -C,F and it proceeds at a rate FIGURE PS-5 = kC. where r= (moles A reacting)/(volume)(time) k = reaction rate constant C, (1) = concentration of A in reactor at any timet (mol Alvolume) V = volume of mixture in reactor Further, let F = constant feed rate, volume/time C,(1) = concentration of A in feed stream, moles/volume Assuming constant density and constant volume V, derive the transfer function relating the concentration in the reactor to the feed-stream concentration. Prepare a block diagram for the reactor. Sketch the response of the reactor to a unit-step change in C. 5.6. A thermocouple junction of area A, mass m, heat capacity C, and emissivity e is located in a furnace that normally is at T,"C. At these temperatures convective and conductive heat transfer to the junction is negligible compared with radiative heat transfer. Determine the linearized transfer function between the furnace temperature T, and the junction temperature To. For the case m = 0.1 g C = 0.12 cal/(g • C) e = 0.7 A = 0.1 cm? T = 1100°C plot the response of the thermocouple to a 10°C step change in furnace temperature. Cơ pare this with the true response obtained by integration of the differential equation. 5.7. A liquid-level system has the following properties: Tank dimensions: 10 ft high by 5-ft diameter Steady-state operating characteristics: Inflow, galh Steady-state level, ft 5,000 0.7 II 10,000 1.1 15,000 2.3 20,000 3.9 25,000 6.3
Transcribed Image Text:5:16 . g Vo) l 38% LTE1 kB/s LeBlanc.pdf PROBLEMS 5.1. Derive the transfer function H(s/Q(s) for the liquid-level system of Fig. P5-1 when (a) The tank level operates about the steady-state value of h, = 1 ft (b) The tank level operates about the steady-state value of h, = 3 ft q, ft'/min The pump removes water at a constant rate of 10 cfm (cubic feet per minute); this rate is inde- pendent of head. The cross-sectional area of the tank is 1.0 ft, and the resistance R is 0.5 ft/cfm. 5.2. A liquid-level system, such as the one shown in Fig. 5-1, has a cross-sectional area of 3.0 ft. The 2 ft valve characteristics are FIGURE P5-1 9 = 8A 4 Outlet flow where q = flow rate, cfm, andh= level above the valve, ft. Calculate the time constant for this system if the average operating level above the valve is 24 (a) 3 ft (b) 9 ft 1.0 5.3. A tank having a cross-sectional area of 2 ft² is operating at steady state with an inlet flow rate of 2.0 cfm. The flow-head characteristics are shown 0.3 1.0 h(ft) in Fig. P5-3. (a) Find the transfer function H(s)Q(s). (b) If the flow to the tank increases from 2.0 to 2.2 cfm according to a step change, calculate the level h two minutes after the change occurs. FIGURE PS-3 5.4. Develop a formula for finding the time constant of the liquid-level system shown in Fig. P54 when the average operating level is họ. The resistance R is linear. The tank has three vertical walls and one that slopes at an angle a from the vertical as shown. The distance separating the parallel walls is 1. ---B- -- FIGURE PS4 116 PART 2 LINEAR OPEN-LOOP SYSTEMS 5.5. Consider the stirrd-tank reactor shown in Fig. PS-5. C,F The reaction occurring is A - B Volume V -C,F and it proceeds at a rate FIGURE PS-5 = kC. where r= (moles A reacting)/(volume)(time) k = reaction rate constant C, (1) = concentration of A in reactor at any timet (mol Alvolume) V = volume of mixture in reactor Further, let F = constant feed rate, volume/time C,(1) = concentration of A in feed stream, moles/volume Assuming constant density and constant volume V, derive the transfer function relating the concentration in the reactor to the feed-stream concentration. Prepare a block diagram for the reactor. Sketch the response of the reactor to a unit-step change in C. 5.6. A thermocouple junction of area A, mass m, heat capacity C, and emissivity e is located in a furnace that normally is at T,"C. At these temperatures convective and conductive heat transfer to the junction is negligible compared with radiative heat transfer. Determine the linearized transfer function between the furnace temperature T, and the junction temperature To. For the case m = 0.1 g C = 0.12 cal/(g • C) e = 0.7 A = 0.1 cm? T = 1100°C plot the response of the thermocouple to a 10°C step change in furnace temperature. Cơ pare this with the true response obtained by integration of the differential equation. 5.7. A liquid-level system has the following properties: Tank dimensions: 10 ft high by 5-ft diameter Steady-state operating characteristics: Inflow, galh Steady-state level, ft 5,000 0.7 II 10,000 1.1 15,000 2.3 20,000 3.9 25,000 6.3
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