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Mass Equation (Recapitulation) PROBLEMS AuthorMessage

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Join date : 2009-10-23  Subject: Mass Equation (Recapitulation) PROBLEMS Fri Oct 30, 2009 7:37 pm --------------------------------------------------------------------------------Section-1: Mass Equation (Recapitulation) --------------------------------------------------------------------------------2-1-1 [m] Mass enters an open system with one inlet and one exit at a constant rate of 50 kg/min. At the exit, the mass flow rate is 60 kg/min. If the system initially contains 1000 kg of working fluid, determine the time when the system mass becomes 500 kg. [Manual Solution] Answers: 50 min Anim. 2-1-1 (click) --------------------------------------------------------------------------------2-1-2 [m] Mass leaves an open system with a mass flow rate of c*m, where c is a constant and m is the system mass. If the mass of the system at t = 0 is m0, derive an expression for the mass of the system at time t. [Manual Solution] Answers: m=m0exp(-ct) Figure 2-1-2 --------------------------------------------------------------------------------2-1-3 [m] Water enters a vertical cylindrical tank of cross-sectional area 0.01 m2 at a constant mass flow rate of 5 kg/s. It leaves the tank through an exit near the base with a mass flow rate given by the formula 0.2h kg/s, where h is the instantaneous height in m. If the tank is empty initially, develop an expression for the liquid height h as a function of time t. Assume density of water to remain constant at 1000 kg/m3. [Manual Solution] Answers: h=25(1-exp(-0.02t)) Anim. 2-1-3 (click) --------------------------------------------------------------------------------2-1-4 [m] A conical tank of base diameter D and height H is suspended in an inverted position to hold water. A leak at the apex of the cone causes water to leave with a mass flow rate of c*sqrt(h), where c is a constant and h is the height of the water level from the leak at the bottom. (a) Determine the rate of change of height h. (b) Express h as a function of time t and other known constants, rho (constant density of water), D, H, and c if the tank was completely full at t=0. [Manual Solution] Answers: h=[H^2.5-(10CH^2/(pi*rho*D^2))t]^0.4 --------------------------------------------------------------------------------2-1-5 [m] Steam enters a mixing chamber at 100 kPa, 20 m/s, with a specific volume of 0.4 m3/kg. Liquid water at 100 kPa and 25oC enters the chamber through a separate duct with a flow rate of 50 kg/s and a velocity of 5 m/s. If liquid water leaves the chamber at 100 kPa and 43oC with a volumetric flow rate of 3.357 m3/min and a velocity of 5.58 m/s, determine the port areas at the inlets and exit. Assume liquid water density to be 1000 kg/m3 and steady state operation. [Manual Solution] Answers: (a) 1190 cm2 (b) 100 cm2 (c) 100 cm2 Anim. 2-1-5 (click) --------------------------------------------------------------------------------2-1-6 [m] Air is pumped into and withdrawn from a 10 m3 rigid tank as shown in the accompanying figure. The inlet and exit conditions are as follows. Inlet: v1= 2 m3/kg, V1= 10 m/s, A1= 0.01 m2; Exit: v2= 5 m3/kg, V2= 5m/s, A2= 0.015 m2. Assuming the tank to be uniform at all time with the specific volume and pressure related through p*v=9.0 (kPa.m3), determine the rate of change of pressure in the tank. [ Manual Solution] Answers: 0.0315 kPa/s Anim. 2-1-6 (click) --------------------------------------------------------------------------------2-1-7 [m] A gas flows steadily through a circular duct of varying cross-section area with a mass flow rate of 10 kg/s. The inlet and exit conditions are as follows. Inlet: V1= 400 m/s, A1= 179.36 cm2; Exit: V2= 584 m/s, v2= 1.1827 m/kg. (a) Determine the exit area. (b) Do you find the increase in velocity of the gas accompanied by an increase in flow area counter intuitive? Why? [ Manual Solution] Answers: (a) 202.517 cm2 Anim. 2-1-7 (click) --------------------------------------------------------------------------------2-1-8 [TEST] Steam enters a turbine through a duct of diameter 0.25 m at 10 MPa, 600oC, 100 m/s. It exits the turbine through a duct of 1 m diameter at 400 kPa, 200oC. For steady state operation determine (a) the exit velocity, and (b) the mass flow rate of steam through the turbine. Use the flow state PC daemon to obtain the density of steam at the inlet and exit ports. (c) What-if Scenario How would the exit velocity change if the exit area was equal to the inlet area? [Manual Solution] [TEST Solution] Answers: (a) 86.9 m/s (b) 127.7 kg/s (c) 1392.27 m/s -------------------------------------------------------------------------------- Anim. 2-1-8 (click) 2-1-9 [TEST] Steam enters a turbine with a mass flow rate of 10 kg/s at 10 MPa, 600oC 30 m/s, it exits the turbine at 45 kPa, 30 m/s with a quality of 0.9. Assuming steady-state operation, determine (a) the inlet area, and (b) the exit area. Use the flow state PC daemon. [Manual Solution] [TEST Solution] Answers: (a) 0.01279 m2 (b) 1.075 m2 --------------------------------------------------------------------------------2-1-10 [TEST] Refrigerant R-134 enters a device as saturated liquid at 500 kPa with a velocity of 10 m/s and a mass flow rate of 2 kg/s. At the exit the pressure is 150 kPa and the quality is 0.2. If the exit velocity is 65 m/s, determine the (a) inlet and (b) exit areas. Use the flow state PC daemon. [Manual Solution] [TEST Solution] Answers: (a) 1.61 cm2 (b) 8.27 cm2 --------------------------------------------------------------------------------2-1-11 [TEST] Air enters a 0.5m diameter fan at 25oC, 100 kPa and is discharged at 28oC, 105 kPa with a volume flow rate of 0.8 m3/s. Determine for steady-state operation (a) the mass flow rate of air in kg/min, (b) the inlet and exit velocities. [Manual Solution] [TEST Solution] Answers: (a) 58.3 kg/min (b) 4.23 m/s (c) 4.07 m/s --------------------------------------------------------------------------------2-1-12 [TEST] Air enters a nozzle, which has an inlet area of 0.1 m2, at 200 kPa, 500oC, and 10 m/s. At the exit the conditions are 100 kPa and 443oC. If the exit area is 35 cm2, determine the steady-state exit velocity. [Manual Solution] [TEST Solution] Answers: 529.3 m/s Anim. 2-1-12 (click) --------------------------------------------------------------------------------Section-2: Energy Equation (Recapitulation) --------------------------------------------------------------------------------2-2-1 [mE] A 20kg block of solid cools down by transferring heat at a rate of 1 kW to the surroundings. Determine the rate of change of stored energy and (b) internal energy of the block. [Manual Solution] Answers: (a) -1 kW (b) -1 kW Anim. 2-2-1 (click) --------------------------------------------------------------------------------2-2-2 [mE] Suppose the specific internal energy in kJ/kg of the solid in the above problem is related to its temperature through u=0.5T, where T is the temperature of the solid in Kelvin, determine the rate of change of temperature of the solid. Assume the density of the solid to be 2700 kg/m3. [Manual Solution] Answers: -0.1 K/s --------------------------------------------------------------------------------2-2-3 [mE] A cup of coffee is heated in a microwave oven. If the mass of coffee (modeled as liquid water) is 0.2 kg and the rate of heat transfer is 0.1 kW, (a) determine the rate of change of internal energy. (b) Assuming the density of coffee to be 1000 kg/m3 and the specific internal energy in kJ/kg to be related to temperature through u=4.2T, where T is in Kelvin, determine how long does it take for the temperature of the coffee to increase by 20 oC. [Manual Solution] Answers: (a) 0.1 kW (b) 168 s Anim. 2-2-3 (click) --------------------------------------------------------------------------------2-2-4 [mE] An insulated tank contains 50 kg of water, which is stirred by a paddle wheel at 300 rpm while transmitting a torque of 0.1 kNm. At the same time, an electric resistance heater inside the tank operates at 110V, drawing a current of 2 A. Determine the rate of heat transfer after the system achieves steady state. [Manual Solution] Answers: -3.3616 kW Anim. 2-2-4 (click) --------------------------------------------------------------------------------2-2-5 [mE] A drill rotates at 4000 rpm while transmitting a torque of 0.012 kNm. Determine the rate of change of stored energy of the block initially. [Manual Solution] Answers: (a) 5 kW --------------------------------------------------------------------------------2-2-6 [mE] A 20 kg slab of aluminum is raised by a rope and pulley arrangement vertically at a constant speed of 10 m/min. At the same time the block absorbs solar radiation at a rate of 0.2 kW. Determine the rate of change of (a) potential energy (b) internal energy and (c) stored energy. [Manual Solution] Answers: (a) 0.2 kW (b) 0.327 kW (c) 0.233 kW Anim. 2-2-6 (click) --------------------------------------------------------------------------------2-2-7 [mE] An external force F is applied to a rigid body of mass m. If its internal and potential energy remain unchanged, show that an energy balance on the body reproduces Newton's law of motion. [Manual Solution] Answers: F =ma --------------------------------------------------------------------------------2-2-8 [mE] An insulated block with a mass of 100 kg is acted upon by a horizontal force of 0.02 kN. Balanced by frictional forces, the body moves at a constant velocity of 2 m/s. Determine (a) the rate of change of stored energy in the system and (b) power transferred by the external force. (c) How do you account for the work performed by the external force? [Manual Solution] Answers: (a) 0 (b) 40 W Anim. 2-2-8 (click) --------------------------------------------------------------------------------2-2-9 [mE] (a) Do an energy analysis of a pendulum bob to show that the sum of its kinetic and potential energies remain constant. Assume internal energy to remain constant, and neglect viscous friction and heat transfer. (b) What-if Scenario Discuss how the energy equation would be affected if viscous friction is not negligible. [Manual Solution] Anim. 2-2-9 (click) --------------------------------------------------------------------------------2-2-10 [mE] A rigid insulated tank contains 2 kg of a gas at 300 K and 100 kPa. A 1 kW internal heater is turned on. (a) Determine the rate of change of total stored energy. (b) If the internal energy of the gas is related to the temperature by u=1.1T (kJ/kg), where T is in Kelvin, determine the rate of temperature increase. [Manual Solution] Answers: (a) 1 kW (b) 0.455 K/s Anim. 2-2-10 (click) --------------------------------------------------------------------------------2-2-11 [TEST] A 10 m3 rigid tank contains air at 200 kPa, 150 oC. A 1 kW internal heater is turned on. Determine the rate of change of (a) stored energy (b) temperature and (c) pressure of air in the tank. Use the IG system state daemon. (Hint: Evaluate state-2 with stored energy incremented by the amount added in a small time interval, say, 0.1 s) [Manual Solution] Answers: (a) 1 kW (b) 0.0824 oC/s (c) 38.9 Pa/s -------------------------------------------------------------------------------- 2-2-12 [TEST] A 10 m3 rigid tank contains steam with a quality of 0.5 at 200 kPa. A 1 kW internal heater is turned on. Determine the rate of change of (a) stored energy (b) temperature and (c) pressure of steam in the tank. Use the PC system state daemon. (Hint: Evaluate state-2 with stored energy incremented by the amount added in a small time interval, say, 0.1 s) [Manual Solution] [TEST Solution] Answers: (a) 1 kW (b) -0.2026 oC/s (c) -2.683 kPa/s --------------------------------------------------------------------------------2-2-13 [mE] A piston-cylinder device, containing air at 200 kPa, looses heat at a rate of 0.5 kW to the surrounding atmosphere. At a given instant, the piston which has a cross-sectional area of 0.01 m2 moves down with a velocity of 1 cm/s. (a) Determine the rate of change of stored energy in the gas. [Manual Solution] Answers: -0.48 kW Anim. 2-2-13 (click) --------------------------------------------------------------------------------2-2-14 [mE] A piston-cylinder device contains a gas, which is heated at a rate of 0.5 kW from an external source. At a given instant the piston moves up with a velocity of 1 cm/s. Determine the rate of change of stored energy in the gas. Assume atmospheric pressure to be 101 kPa and the piston to be weightless. Also, neglect friction. [Manual Solution] Answers: (a) 0.49 kW (b) 0.5 kW Anim. 2-2-14 (click) --------------------------------------------------------------------------------2-2-15 [mE] A piston-cylinder device is used to compress a gas by pushing the piston with an external force. During the compression process, heat is transferred out of the gas in such a manner that the stored energy in the gas remains unchanged. Also, the pressure is found to be inversely proportional to the volume of the gas. Determine an expression for the heat transfer rate in terms of the instantaneous volume and the rate of change of pressure of the gas. [Manual Solution] Answers: Qdot=Vol*dp/dt Anim. 2-2-15 (click) --------------------------------------------------------------------------------2-2-16 [mE] A fluid flows steadily through a long insulated pipeline. Perform a mass and energy analysis to show that the flow energy j remains unchanged between the inlet and exit. What-if scenario How would this conclusion be modified if kinetic and potential energy changes were negligible? [Manual Solution] --------------------------------------------------------------------------------2-2-17 Water enters a constant-diameter, horizontal, insulated pipe at 500 kPa. Due to the presence of viscous friction the pressure drops to 400 kPa at the exit. At steady state, determine the changes in (a) specific kinetic energy and (b) specific internal energy between the inlet and the exit. Assume water density to be 1000 kg/m3. [Manual Solution] Answers: (a) 0 (b) 0.1 kJ/kg Anim. 2-2-17 (click) --------------------------------------------------------------------------------2-2-18 Water flows steadily through a variable diameter insulated pipe. At the inlet the velocity is 20 m/s and at the exit the flow area is half as that at the inlet. If the internal energy of water remains constant, determine the change in pressure between the inlet and exit. Assume water density to be 1000 kg/m3. [Manual Solution] Answers: 600kPa -------------------------------------------------------------------------------- 2-2-19 [mE] Oil enters a long insulated pipe at 200 kPa and 20 m/s. It exits at 175 kPa. Assuming steady flow, determine the changes in the following properties between the inlet and exit (a) j (b) ke and (d) h. Assume oil density to be constant. [Manual Solution] Answers: (a) 0 (b) 0 (c) 0 --------------------------------------------------------------------------------2-2-20 [TEST] Nitrogen gas flows steadily through a pipe of diameter 10 cm. The inlet conditions are as follows: pressure 400 kPa, temperature 300 K, and velocity 20 m/s. At the exit the pressure is 350 kPa (due to frictional losses). If the flow rate of mass and flow energy remain constant, determine (a) the exit temperature and (b) exit velocity. Use the IG (ideal gas) flow state daemon. (c) What-if-Scenario: How would the answer in (a) change if kinetic energy was neglected? [TEST Solution] Answers: (a) 299.94 K (b) 22.85 m/s (c) 300 K Figure 2-2-20 --------------------------------------------------------------------------------2-2-21 [TEST] A 5 cm diameter pipe discharges water into the open atmosphere at a rate of 20 kg/s at an elevation of 20 m. The temperature of water is 25oC and the atmospheric pressure is 100 kPa. Determine (a) Jdot (b) Edot (c) KEdot (d) Hdot (e) WdotF (f) How important is the flow work transfer compared to kinetic and potential energy carried by the mass? Use the SL flow state daemon. [Manual Solution] Answers: (a) 4.94 kW (b) 2.94 kW (c) 1.04 kW (d) -0.0266 kW (e) 2.0 kW -------------------------------------------------------------------------------- Figure 2-2-21 2-2-22 [TEST] Water enters a pipe at 90 kPa, 25oC with a velocity of 10 m/s. At the exit the pressure is 500 kPa and velocity is 12 m/s while the temperature remains unchanged. If the volume flow rate is 10 m3/min both at the inlet and exit. Determine (a) the flow rate of energy Jdot at the inlet, and (b) exit. (c) What-if-Scenario: How would the answer in part (b) change if the exit velocity was 15 m/s instead? [TEST Solution] Answers: (a) 6.42 kW (b) 78.4 kW (c) 85.14 kW --------------------------------------------------------------------------------2-2-23 [mE] Water at 1000 kPa, 25oC enters a 1-m-diameter horizontal pipe with a steady velocity of 10 m/s. At the exit the pressure drops to 950 kPa due to viscous resistance. Assuming steady-state flow, determine the rate of heat transfer necessary to maintain a constant specific internal energy. [Manual Solution] Answers: -392.7 kW Figure 2-2-23 --------------------------------------------------------------------------------2-2-24 [mE] An incompressible fluid (constant density) flows steadily downward along a constant-diameter, insulated vertical pipe. Assuming internal energy to remain constant, show that the pressure variation is hydrostatic. [Manual Solution] Answers: pi = pe + (rho*g*h)/1000 --------------------------------------------------------------------------------2-2-25 [mE] An incompressible fluid (constant density) flows steadily through a converging nozzle. (a) Show that the specific flow energy remains constant if the nozzle is adiabatic. (b) Assuming internal energy to remain constant and neglecting the inlet kinetic energy, obtain an expression for the exit velocity in terms of pressures at the inlet and exit and the fluid density. [Manual Solution] Answers: (b) ve = sqrt(2000(pi /rho-pe/rho)) Anim. 2-2-25 (click) --------------------------------------------------------------------------------2-2-26 [mE] Water flows steadily through an insulated nozzle. The following data are supplied. Inlet: p= 200 kPa, V= 10 m/s, z= 2; Exit: p=100 kPa, z=0. (a) Determine the exit velocity. Assume density of water to be 1000 kg/m3. Also assume the internal energy to remain constant. What-if scenario How would the answer change if (b) change in potential energy or (c) inlet kinetic energy was neglected in the analysis? [Manual Solution] Answers: (a) 18.42 m/s (b) 17.32 m/s (c) 15.47 m/s Anim. 2-2-26 (click) --------------------------------------------------------------------------------2-2-27 [mE] Water flowing steadily through a 2cm diameter pipe at 30 m/s goes through an expansion joint to flow through a 4cm diameter pipe. Assuming the internal energy to remain constant, determine (a) the change in pressure as the water goes through the transition. (b) Also determine the displacement of the mercury column in mm. Assume water density to be 997 kg/m3. [Manual Solution] Answers: (a) 421.9 kPa (b) 3164.3 mmHg --------------------------------------------------------------------------------2-2-28 [mE] An adiabatic work producing device works at steady state with the working fluid entering through a single inlet and leaving through a single exit. Derive an expression for the work output in terms of the flow properties at the inlet and exit. What-if scenario How would the expression for work simplify if changes in ke and pe were neglected? [Manual Solution] --------------------------------------------------------------------------------2-2-29 [mE] A pump is a device that raises the pressure of a liquid at the expense of external work. (a) Determine the pumping power necessary to raise the pressure of liquid water from 10 kPa to 2000 kPa at a flow rate of 1000 L/min. Assume density of water to be 1000 kg/m3 and neglect changes in specific internal, kinetic, and potential energies. What-if scenario (b) How would the answer change if pe was not negligible and z1=-10, z2=0? [Manual Solution] Answers: (a) -33.17 kW (b) -34.8 kW -------------------------------------------------------------------------------- Anim. 2-2-29 (click) 2-2-30 [mE] An adiabatic pump, working at steady state, raises the pressure of water from 100 kPa to 1 MPa while the specific internal energy remains constant. (a) If the exit is 10 m above the inlet and the flow rate of water is 100 kg/s, determine the pumping power. Neglect any change in kinetic energy. Assume density of water to be 997 kg/m3. (b) What-if scenario How would the answer change if change in potential energy was also neglected? [Manual Solution] Answers: (a) -100.1 kW (b) -90.3 kW --------------------------------------------------------------------------------2-2-31 [mE] Steam flows steadily through a single-flow device with a flow rate of 10 kg/s. It enters with an enthalpy of 3698 kJ/kg and a velocity of 30 m/s. At the exit, the corresponding values are 3368 kJ/kg and 20 m/s respectively. If the rate of heat loss from the device is measured at 100 kW, (a) determine the rate of work transfer. Neglect any change in potential energy. (b) What-if scenario How would the answer change if the change in kinetic energy was also neglected? [Manual Solution] Answers: (a) 3202.5 kW (b) 3200 kW Anim. 2-2-31 (click) --------------------------------------------------------------------------------2-2-32 [TEST] Steam enters an adiabatic turbine with a mass flow rate of 5 kg/s at 3 MPa, 600oC, 80 m/s. It exits the turbine at 40oC, 30 m/s with a quality of 0.9. Assuming steady-state operation, determine the shaft power produced by the turbine. Use the PC flow state daemon to evaluate enthalpies at the inlet and exit. [TEST Solution] Answers: --------------------------------------------------------------------------------2-2-33 [mE] A gas enters an adiabatic work consuming device at 300 K, 20 m/s, and leaves at 500 K, 40 m/s. (a) If the mass flow rate is 5 kg/s, determine the rate of work transfer. Neglect change in potential energy and assume the specific enthalpy of the gas to be related to its temperature in K through h=1.005T. (b) What-if scenario By what percent would the answer change if the change in kinetic energy was also neglected? [Manual Solution] Answers: (a) -1008 kW (b) -1005 kW (c) 0.3% Anim. 2-2-33 (click) --------------------------------------------------------------------------------2-2-34 [mE] A refrigerant is compressed by an adiabatic compressor operating at steady state to raise the pressure from 200 kPa to 750 kPa. The following data are supplied for the inlet and exit ports. Inlet: v=0.0835 m3/kg, h=182.1 kJ/kg, V=30 m/s; Exit: v=0.0244 m3/kg, h=205.4 kJ/kg, V=40 m/s. If the volume flow rate at the inlet is 3000 L/min, determine (a) the mass flow rate (b) the volume flow rate at the exit and (c) the compressor power. (d) What-if scenario: How would the power consumption change if the change in kinetic energy was neglected? [Manual Solution] Answers: (a) 0.598 kg/s (b) 876 L/min (c) -14.2 kW (d) -14.0 kW Img. 2-2-34 (click) --------------------------------------------------------------------------------2-2-35 [mE] Two flows of equal mass flow rate, one at state-1 and another at state-2, enter an adiabatic mixing chamber and leaves through a single port at state-3. Obtain an expression for the velocity and specific enthalpy at the exit. Assume negligible changes in ke and pe. [Manual Solution] Answers:(a) v3 = 2mdot/(rho3*A3) (b) h3=0.5*(h1+h2) Anim. 2-2-35 (click) --------------------------------------------------------------------------------2-2-36 Air at 500 kPa, 30oC from a supply line is used to fill an adiabatic tank. At a particular moment during the filling process, the tank contains 0.2 kg of air at 200 kPa and 50oC. If the mass flow rate is 0.1 kg/s, and the specific enthalpy of air at the inlet conditions is 297.2 kJ/kg, determine (a) the rate of flow energy into the tank (b) the rate of increase of internal energy in the tank and (d) the rate of increase of specific internal energy. Assume the tank to be uniform at all time and neglect kinetic and potential energies. [Manual Solution] Answers: (a) 29.72 kW (b) 29.72 kW (c) 5.95 kW/kg Anim. 2-2-36 (click) --------------------------------------------------------------------------------2-2-37 Saturated steam at 200 kPa, which has a specific enthalpy of 2707 kJ/kg, is expelled from a pressure cooker at a rate of 0.1 kg/s. Determine the rate of heat transfer necessary to maintain a constant stored energy E in the cooker. Discuss if it is possible to maintain a constant E in the cooker for an extended period. [Manual Solution] Answers: 270.7 kW Anim. 2-2-37 (click) --------------------------------------------------------------------------------Section-3: Entropy Equation (Recapitulation) --------------------------------------------------------------------------------2-3-1 [mEE] Heat is transferred from a TER at 1500 K to a TER at 300 K at a rate of 10 kW. Determine the rates at which entropy (a) leaves the TER at higher temperature, and (b) enters the TER at lower temperature. (c) How do you explain the discontinuity in the result? [Manual Solution] Answers: (a) 0.0067 kW/K (b) 0.033 kW/K Anim. 2-3-1 (click) --------------------------------------------------------------------------------2-3-2 [mEE] A resistance heater operates inside a tank consuming 0.5 kW of electricity. Due to heat transfer to the ambient atmosphere at 300 K, the tank maintains a steady state. The surface temperature of the tank remains constant at 400 K. Determine the rate at which entropy (a) leaves the tank, and (b) enters the tank's universe. (c) How does the system maintain steady state with regard to entropy? [Manual Solution] Answers: (a) 0.00125 kW/K (b) 0.016 kW/K Anim. 2-3-2 (click) --------------------------------------------------------------------------------2-3-3 [mEE] Heat is conducted through a slab of thickness 2 cm. The temperature vary linearly from 500 K on the left face to 300 K on the right. If the rate of heat transfer is 2 kW, determine the rate of entropy transfer at the (a) left and (b) right faces. (c) Plot how the rate of entropy transfer varies from the left to the right face. [Manual Solution] Answers: (a) 0.04 kW/K (b) 0.0067 kW/K Anim. 2-3-3 (click) --------------------------------------------------------------------------------2-3-4 [mEE] A 30kg aluminum block cools down from its initial temperature of 500 K to the atmospheric temperature of 300 K. Determine the total amount of entropy transfer from the system's universe. Assume the specific internal energy of aluminum (in kJ/kg) to be related to its absolute temperature (K) through u=0.9T. [Manual Solution] Answers: 18 kJ/K Anim. 2-3-4 (click) --------------------------------------------------------------------------------2-3-5 [mEE] Water is heated in a boiler from a source at 1800 K. If the heat transfer rate is 20 kW, (a) determine the rate of entropy transfer into the boiler's universe. (b) Discuss the consequences of reducing the source temperature with respect to the boiler size and entropy transfer. [Manual Solution] Answers: (a) 0.011 kW/K Anim. 2-3-5 (click) --------------------------------------------------------------------------------2-3-6 [mEE] An insulated tank contains 50 kg of water at 30°C, which is stirred by a paddle wheel at 300 rpm while transmitting a torque of 0.2 kNm. Determine (a) the rate of change of temperature (b) the rate of change of total entropy and (c) the rate of generation of entropy within the tank. Assume s=4.2lnT and u=4.2T, where T is in Kelvin. [Manual Solution] Answers: (a) 0.03 K/s (b) 0.0208 kW/K (c) 0.0208 kW/K Anim. 2-3-6 (click) --------------------------------------------------------------------------------2-3-7 [mEE] A rigid insulated tank contains 1 kg of air at 300 K and 100 kPa. A 1 kW internal heater is turned on. Determine the rate of (a) entropy transfer into the tank (b) the rate of change of total entropy of the system and (c) the rate of generation of entropy within the tank. Assume s=lnT and u=T, where T is in Kelvin. [Manual Solution] Answers: (a) 0 (b) 0.033 kW/K --------------------------------------------------------------------------------2-3-8 [TEST] A 10 m3 insulated rigid tank contains 30 kg of wet steam with a quality of 0.9. An internal electric heater is turned on which consumes electric power at a rate of 10 kW. After the heater is on for one minute, determine (a) the change in temperature (b) the change in pressure and (c) the change in entropy of steam. [TEST Solution] Answers: (a) 0.45 oC (b)6.11 kPa (c) 1.02 kJ/K Anim. 2-3-8 (click) --------------------------------------------------------------------------------2-3-9 A 1cm diameter insulated pipe carries a steady flow of water at a velocity of 30 m/s. The temperature increases from 300 K at the inlet to 301 K at the exit due to friction. If the specific entropy of water is related to its absolute temperature through s=4.2lnT, determine the rate of generation of entropy within the pipe. Assume water density to be 1000 kg/m3. [Manual Solution] Answers: 0.033 kW/K Anim. 2-3-9 (click) --------------------------------------------------------------------------------2-3-10 Liquid water (density 997 kg/m3) flows steadily through a pipe with a volume flow rate of 30,000 L/min. Due to viscous friction, the pressure drops from 500 kPa at the inlet to 150 kPa at the exit. If the specific internal energy and specific entropy remain constant along the flow, determine (a) the rate of heat transfer and (b) the rate of entropy generation in and around the pipe. Assume the temperature of the surroundings to be 300 K. [Manual Solution] Answers: (a) -175 kW (b) 0.583 kW/K --------------------------------------------------------------------------------2-3-11 [mE] An electric water heater works by passing electricity through an electrical resistance placed inside the flow of liquid water as shown in the accompanying animation. The specific internal energy and entropy of water are correlated to its absolute temperature through u=4.2T and s=4.2lnT (T in K) respectively. Water enters the heater at 300 K with a flow rate of 5 kg/s. At the exit the temperature is 370 K. Assuming steady state operation, negligible changes in ke and pe, and negligible heat transfer, determine (a) electrical power consumption rate and (b) the rate of entropy generation within the heater. What-if scenario How would the answers change if the exit temperature was reduced by 20 oC? [Manual Solution] Answers: (a) -1470 kW (b) 4.04 kW/K (c) 1050 kW, 3.237 kW/K Anim. 2-3-11 (click) --------------------------------------------------------------------------------2-3-12 [mEE] An open system with only one inlet and one exit operates at steady state. Mass enters the system at a flow rate of 5 kg/s with the following properties: h=3484 kJ/kg, s= 8.0871 kJ/kg.K, and V=20 m/s. At the exit the properties are as follows: h=2611 kJ/kg, s= 8.146 kJ/kg.K, and V=25 m/s. The device produces 4313 kW of shaft work while rejecting some heat to the atmosphere at 25 oC. (a) Do a mass analysis to determine the mass flow rate at the exit. (b) Do an energy analysis to determine the rate of heat transfer (include sign), (c) Do an entropy analysis to evaluate the rate of entropy generation in the system's universe. [Manual Solution] Generic Open System Answers: (a) 5 kg/s (b) -51.4 kW (c) 0.467 kW/K --------------------------------------------------------------------------------2-3-13 [mEE] Steam flows steadily through a work-producing, adiabatic, single-flow device with a flow rate of 7 kg/s. At the inlet h=3589 kJ/kg, s=7.945 kJ/kg.K, and at the exit h=2610 kJ/kg, s=8.042 kJ/kg.K. If changes in ke and pe are negligible, determine (a) the work produced by the device, and (b) the rate of entropy generation within the device. What-if scenario How would the answer in part b change if the device lost 5 kW of heat from its surface at 200 oC? [Manual Solution] Answers: (a) 6853 kW (b) 0.679 kW/K (c)0.69 kW/K Anim. 2-3-13 (click) --------------------------------------------------------------------------------2-3-14 [mEE] The following information are supplied at the inlet and exit of an adiabatic nozzle operating at steady state. Inlet: V=30 m/s, h=976.2 kJ/kg, s=6.149 kJ/kg.K; Exit: h=825.5 kJ/kg. Determine (a) the exit velocity and (b) the minimum specific entropy possible at the exit. [Manual Solution] Answers: (a) 550 m/s (b) 6.149 kJ/kg.K Anim. 2-3-14 (click) --------------------------------------------------------------------------------2-3-15 [mEE] A refrigerant flows steadily through an insulated tube, its entropy increasing from 0.2718 kJ/kg.K at the inlet to 0.3075 kJ/kg.K at the exit. If the mass flow rate of the refrigerant is 0.2 kg/s and the pipe is insulated, determine the rate of entropy generation within the pipe. What-if-scenario: How would the answer change if the mass flow rate doubled? [Manual Solution] Answers: (a) 0.00714 kW/K (b) 0.01428 kW/K Anim. 2-3-15 (click) --------------------------------------------------------------------------------Section-4: Closed Steady Systems Analysis (Generic Closed Steady System) --------------------------------------------------------------------------------2-4-1 [mEE] A tank contains 50 kg of water, which is stirred by a paddle wheel at 300 rpm while transmitting a torque of 0.2 kNm. After the tank achieves steady state, determine (a) the rate of heat transfer (b) the rate of entropy transfer and (c) the rate of generation of entropy in the tank's universe. Assume the atmospheric temperature to be 25oC. [Manual Solution] Answers: (a) -6.28 kW (b) -0.021 kW/K (c) 0.021 kW/K Anim. 2-4-1 (click) --------------------------------------------------------------------------------2-4-2 [mEE] A tank contains 1 kg of air at 500 K and 500 kPa. A 1 kW internal heater operates inside the tank at steady state to make up for the heat lost to the atmosphere which is at 300 K. Determine (a) the rate of entropy transfer into the atmosphere (b) the rate of entropy generation in the system's universe (c) the internal rate of entropy generation and (d) the external rate of entropy generation. [Manual Solution] Answers: (a) -0.0033 kW/K (b) 0.0033 kW/K (c) 0.002 kW/K (d) 0.0013 kW/K Anim. 2-4-2 (click) --------------------------------------------------------------------------------2-4-3 [mEE] A rigid tank contains 1 kg of air initially at 300 K and 100 kPa. A 1 kW internal heater is turned on. After the tank achieves steady state, determine (a) the rate of heat transfer (b) the rate of entropy transfer and (c) the rate of generation of entropy in the tank's universe. Assume the atmospheric temperature to be 0oC. [Manual Solution] Answers: (a) -1 kW (b) -0.003663 kW/K (c) 0.003663 kW/K --------------------------------------------------------------------------------2-4-4 [mEE] A closed chamber containing a gas is at steady state. The shaft transfers power at a rate of 2 kW to the paddle wheel and the electric lamp consumes electricity at a rate of 500 W. Using an energy balance determine (a) the rate of heat transfer. (b) If the surface temperature of the chamber is 400 K, determine the entropy generated within the chamber. (c) What-if Scenario How would the entropy generation within the chamber change if the surface temperature increased to 500 K? [Manual Solution] Answers: (a) -2.5 kW (b) -0.00625 kW/K (c) 0.005 kW/K Anim. 2-4-4 (click) --------------------------------------------------------------------------------2-4-5 [mEE] An electric bulb consumes 500 W of electricity at steady state. The outer surface of the bulb is warmer than the surrounding atmosphere by 75 oC. If the atmospheric temperature is 300 K, determine (a) the rate of heat transfer between the bulb (the system) and the atmosphere. Also determine (b) the entropy generation rate within the bulb (c) in the system's universe and (d) in the immediate surroundings outside the bulb. [Manual Solution] Answers: (a) -0.5 kW (b) 0.00133 kW/K (c) 0.001667 kW/K (d) 0.00033 kW/K Anim. 2-4-5 (click) --------------------------------------------------------------------------------2-4-6 [mEE] An electric adaptor for a notebook computer (converting 110 volts to 19 volts) operates 10oC warmer than the surroundings, which is at 20oC. If the output current is measured at 3 amps and heat is lost from the adapter at a rate of 10 W, determine (a) the energetic efficiency of the device (b) the rate of internal entropy generation and (c) the rate of external entropy generation. [Manual Solution] Answers: (a) 85% (b) 0.000033 kW/K (c) 0.000034 kW/K Figure 2-4-6 --------------------------------------------------------------------------------2-4-7 [mEE] At steady state, the input shaft of a gearbox rotates at 2000 rpm while transmitting a torque of 0.2 kNm. Due to friction, 1 kW of power is dissipated into heat and the rest is delivered to the output shaft. If the atmospheric temperature is 300 K and the surface of the gearbox maintains a constant temperature of 350 K, determine (a) the rate of entropy transfer into the atmosphere (b) the rate of entropy generation in the system's universe (c) the rate of generation of entropy within the gearbox and (d) the rate of entropy generation in the immediate surroundings. [Manual Solution] Answers: (a) 0.0033 kW/K (b) 0.0033 kW/K (c) 0.002857 kW/K (d) 0.000476 kW/K --------------------------------------------------------------------------------2-4-8 [mEE] A closed steady system receives 1000 kW of heat from a reservoir at 1000 K and 2000 kW of heat from a reservoir at 2000 K. Heat is rejected to two reservoirs at 300 K and 3000 K respectively. (a) Determine the maximum amount of heat that can be transferred to the reservoir at 3000 K. (b) The device clearly transfers heat to a high temperature TER without directly consuming external work. Is this a violation of the Clausius statement of the second law of thermodynamics? [Manual Solution] Answers: (a) 2666.67 kW --------------------------------------------------------------------------------Section-5: Overall Analysis of Cycles (Generic Cycles) ----------------------------------------------------------------------------------------------------------------------------------------------------------------• Energy Conversion --------------------------------------------------------------------------------2-5-1 [mE] A steam power plant produces 500 MW of electricity with an overall thermal efficiency of 35%. Determine (a) the rate at which heat is supplied to the boiler and (b) the waste heat that is rejected by the plant. (c) If the heating value of coal (heat that is released when 1 kg of coal is burned) is 30 MJ/kg, determine the rate of consumption of coal in tons(US)/day. Assume that 100% of heat released goes to the cycle. (d) What-if Scenario How would the fuel consumption rate change if the thermal efficiency was to increase to 40%? [Manual Solution] [TEST Solution] Answers: (a) 1429 MW (b) 929 MW (c) 4536.5 tons /day (d) 12.5% reduction Anim. 2-5-1 (click) --------------------------------------------------------------------------------2-5-2 [mE] An utility company charges its residential customers 12 cents/kWh for electricity and \$1.20 per Therm for natural gas. Fed up with the high cost of electricity, a customer decides to generate his own electricity by using a natural gas fired engine that has a thermal efficiency of 35%. Determine the fuel cost per kWh of electricity produced by the customer. Do you think electricity and natural gas are fairly priced by the utility company? [Manual Solution] Answers: 11.7 cents/kWh --------------------------------------------------------------------------------2-5-3 [mE] A sport utility vehicle with a thermal efficiency of 20 percent produces 250 hp of engine output while traveling at a velocity of 80 miles/h. (a) Determine the rate of fuel consumption in kg/s if the heating value of the fuel is 43 MJ/kg. (b) If the density of the fuel is 800 kg/m3, determine the fuel mileage of the vehicle in the unit of miles/gallon. [Manual Solution] Heating Values of Common Fuels Answers: (a) 0.021677 kg/s (b) 3.1 miles/gallon --------------------------------------------------------------------------------2-5-4 [mE] An truck engine consumes diesel at a rate of 30 L/h and delivers 65 kW of power to the wheels. If the fuel has a heating value of 43.5 MJ/kg and a density of 800 kg/m3, determine (a) the thermal efficiency of the engine and (b) the waste heat rejected by the engine. (c) How does the engine discard the waste heat? [Manual Solution] Answers: (a) 22.4% (b) 225 kW --------------------------------------------------------------------------------2-5-5 [mE] Determine the rate of coal consumption by a thermal power plant with a power output of 350 MW in tons/h. The thermal efficiency of the plant is 35% and the heating value of the coal is 30 MJ/kg. [Manual Solution] Answers: 132.275 tons/hr --------------------------------------------------------------------------------2-5-6 [mE] In 2003 the United States generated 3.88 trillion kWh of electricity, 51% of which came from coal-fired power plants. (a) Assuming an average thermal efficiency of 34% and the heating value of coal as 30 MJ/kg, determine the coal consumption in 2003 in tons. (b) What-if Scenario How would the answer change if the average thermal efficiency was 35% instead? [Manual Solution] Energy Use in 2003:USA vs. World Answers: (a) 7.7*108 tons (b) 7.49*108 tons --------------------------------------------------------------------------------• Heat Engine Cycles (Sec. 5 Continued) --------------------------------------------------------------------------------2-5-7 [mE] Determine the fuel cost per kWh of electricity produced by a heat engine with a thermal efficiency of 40% if it uses diesel as the source of heat. The following data is supplied for diesel: price \$2.00 per gallon; heating value 42.8 MJ/kg; density 850 kg/m3. [Manual Solution] Answers: 13.1 cents/kWh Anim. 2-5-7 (click) --------------------------------------------------------------------------------2-5-8 [mE] A gas turbine has a thermal efficiency of 21% and develops a power output of 8 MW. Determine (a) the fuel consumption rate in kg/min if the heating value of the fuel is 50 MJ/kg. (b) If the maximum temperature achieved during the combustion of diesel is 1700 K, determine the maximum thermal efficiency possible. Assume the atmospheric temperature to be 300 K. [Manual Solution] [TEST Solution] Answers: (a) 45.72 kg/s (b) 82.35% --------------------------------------------------------------------------------2-5-9 [mE] Two different fuels are being considered for a 1 MW (net output) heat engine which can operate between the highest temperature produced during the burning of the fuel and the atmospheric temperature of 300 K. Fuel A burns at 2500 K, delivering 50 MJ/kg (heating value) and costs \$2 per kilogram. Fuel B burns at 1500 K, delivering 40 MJ/kg and costs \$1.50 per kilogram. Determine the minimum fuel cost per hour for (a) fuel A, and (b) fuel B. [Manual Solution] Answers: (a) \$163.58 per hour (b) \$180 per hour --------------------------------------------------------------------------------2-5-10 [mEE] A heat engine receives heat from a source at 2000 K at a rate of 500 kW, and rejects the waste heat to a medium at 300 K. The net output from the engine is 300 kW. a) Determine the maximum power that could be generated by the engine, (b) Compare the thermal efficiency of the engine with its maximum possible limit. [Manual Solution] [TEST Solution] Answers: (a) 425 kW (b) 60% vs. 85% -------------------------------------------------------------------------------- Anim. 2-5-10 (click) 2-5-11 [mEE] A Carnot heat engine with an efficiency of 60% receives heat from a source at a rate of 3000 kJ/min, and rejects the waste heat to a medium at 300 K. Determine (a) the power that is generated by the engine, (b) the source temperature. [Manual Solution] [TEST Solution] Answers: (a) 30 kW (b) 750 K --------------------------------------------------------------------------------2-5-12 [mEE] A solar-energy collector produces a maximum temperature of 100oC. The collected energy is used in a cyclic heat engine that operates in a 5oC environment. (a) What is the maximum thermal efficiency? (b) What-if Scenario: How would the answer change if the collector was redesigned to focus the incoming light to enhance the maximum temperature to 400oC? [Manual Solution] [TEST Solution] Answers:(a) 25.5% (b) 58.7% --------------------------------------------------------------------------------2-5-13 [mEE] The Ocean Thermal Energy Conversion (OTEC) system in Hawaii utilizes the surface water and deep water as thermal energy reservoirs. Assuming the ocean temperature at the surface to be 20oC and at some depth to be 5oC, determine the maximum possible thermal efficiency achievable by a heat engine. What-if Scenario: How would the answer change if the surface water temperature increased to 25oC? [Manual Solution] [TEST Solution] Answers: (a) 5.12% (b) 6.71% --------------------------------------------------------------------------------2-5-14 [mEE] You have been hired by a venture capitalist to evaluate a concept engine proposed by an inventor, who claims that the engine consumes 100 MW at a temperature of 500 K, rejects 40 MW at a temperature of 300 K, and delivers 50 MW of mechanical work. What problems, if any, do you identify with this claim? [Manual Solution] [TEST Solution] --------------------------------------------------------------------------------2-5-15 [mEE] A heat engine produces 40 kW of power while consuming 40 kW of heat from a source at 1200 K, 50 kW of heat from a source at 1500 K, and rejecting the waste heat to atmosphere at 300 K. Determine (a) the thermal efficiency of the engine. (b) How would the thermal efficiency improve if all the irreversibilities could be magically eliminated? Assume no change in heat input from the two sources. [Manual Solution] Answers: (a) 44.4% (b) 77.7% --------------------------------------------------------------------------------2-5-16 [mE] Two reversible engines A and B are arranged in series with the waste heat of A used to drive the engine B. Engine A receives 200 MJ from a hot source at a temperature of 420 oC. Engine B is in communication with a heat sink at a temperature of 4.4 oC. If the work output of A is twice that of B, determine (a) the intermediate temperature between A and B, and (b) the thermal efficiency of each engine. [Manual Solution] Answers: (a) 40% (b) 33.33% --------------------------------------------------------------------------------2-5-17 [mEE] A Carnot heat engine receives heat from a TER at TTER through a heat exchanger where the heat transfer rate is proportional to the temperature difference as QdotH=A(TTER-TH). It rejects heat to a cold reservoir at TC. If the heat engine is to maximize the work output, show that the high temperature in the cycle should be selected as TH=sqrt(TTERTC). [Manual Solution] --------------------------------------------------------------------------------2-5-18 [mEE] Two Carnot engines operate in series. The first one receives heat from a TER at 2500 K and rejects the waste heat to another TER at a temperature T. The second engine receives this energy rejected by the first one, converts some of it to work, and rejects the rest to a TER at 300 K. If the thermal efficiency of both the engines are the same, (a) determine the temperature T. What-if Scenario: (b) How would the answer change if the two engines produced the same output instead? [Manual Solution] Answers: (a) 866 K (b) 1400 k --------------------------------------------------------------------------------2-5-19 [mEE] A reversible heat engine operates in outer space. The only way heat can be rejected is by radiation, which is proportional to the fourth power of the temperature and the area of the radiating surface. Show that for a given power output and a given source temperature T1, the area of the radiator is minimized when the radiating surface temperature is T2 =0.75 T1. [Manual Solution*] --------------------------------------------------------------------------------2-5-20 [mEE] A heat engine receives heat at a rate of 3000 kJ/min from a reservoir at 1000 K and rejects the waste heat to the atmosphere at 300 K. If the engine produces 20 kW of power, determine (a) the thermal efficiency and (b) the entropy generated in the engine's universe. [Manual Solution] Answers: (a) 40% (b) 0.05 kW/K --------------------------------------------------------------------------------• Refrigeration Cycles (Sec. 5 Continued) --------------------------------------------------------------------------------2-5-21 [mE] A household freezer operates in a kitchen at 25oC. Heat must be transferred from the cold space at a rate of 2.5 kW to maintain its temperature at -25oC. What is the smallest (power) motor required to operate the freezer. [Manual Solution] [TEST Solution] Answers: 0.5 kW -------------------------------------------------------------------------------- Anim. 2-5-21 (click) 2-5-22 [mEE] To keep a refrigerator in steady state at 2oC, heat has to be removed from it at a rate of 200 kJ/min. If the surrounding air is at 27oC, determine (a) the minimum power input to the refrigerator and (b) the maximum COP. [Manual Solution] [TEST Solution] Answers: (a) 0.3 kW (b) 11 --------------------------------------------------------------------------------2-5-23 [mEE] A Carnot refrigerator consumes 2 kW of power while operating in a room at 20oC. If the food compartment of the refrigerator is to be maintained at 3oC, determine the rate of heat removal in kJ/min from the compartment. [Manual Solution] [TEST Solution] Answers: 1948.2 kJ/min --------------------------------------------------------------------------------2-5-24 [mE] An inventor claims to have developed a refrigerator with a COP of 10 that maintains a cold space temperature of -10oC, while operating in a 25oC kitchen. Is this claim plausible? [Manual Solution] [TEST Solution] Answers: not plausible Anim. 2-5-24 (click) --------------------------------------------------------------------------------2-5-25 [mEE] A refrigeration cycle removes heat at a rate of 250 kJ/min from a cold space maintained at -10oC while rejecting heat to the atmosphere at 25oC. If the power consumption rate is 0.75 kW, determine if the cycle is reversible, irreversible, or impossible. [Manual Solution] [TEST Solution] Answers: Irreversible --------------------------------------------------------------------------------2-5-26 [mEE] A refrigeration cycle removes heat at a rate of 250 kJ/min from a cold space maintained at -10oC while rejecting heat to the atmosphere at 25oC. If the power consumption rate is 1.5 kW, (a) do a first-law analysis to determine the rate of heat rejection to the atmosphere in kW. (b) Do a second-law analysis to determine the entropy generation rate in the refrigerator's universe. [Manual Solution*] [TEST Solution] Answers: (a) 5.667 kW, (b) 0.00317 kW/K --------------------------------------------------------------------------------2-5-27 [mEE] In a cryogenic experiment a container is maintained at -120oC although it gains 200 W due to heat transfer from the surroundings. What is the minimum power of a motor that is needed for a heat pump to absorb heat from the container and reject heat to the room at 25oC? [Manual Solution] [TEST Solution] Answers: 189.542 W --------------------------------------------------------------------------------2-5-28 [mEE] An air-conditioning system maintains a house at a temperature of 20oC while the outside is 40oC. If the cooling load on this house is 10 tons, determine (a) the minimum power requirement, (b) What-if Scenario: How would the answer change if the interior was 5 degrees warmer? [Manual Solution] [TEST Solution] Answers: (a) 2.4 kW (b) 1.77 kW Anim. 2-5-28 (click) --------------------------------------------------------------------------------2-5-29 [mEE] An air-conditioning system is required to transfer heat from a house at a rate of 800 kJ/min to maintain its temperature at 20oC. (a) If the COP of the system is 3.7, determine the power required for air conditioning the house. (b) If the outdoor temperature is 35oC, determine the minimum possible power required. [Manual Solution] [TEST Solution] Image of an Air-conditioning system Answers: (a) 3.6 kW (b) 0.68 kW --------------------------------------------------------------------------------2-5-30 [mEE] A solar-powered refrigeration system receives heat from a solar collector at TH, rejects heat to the atmosphere at T0, and extracts heat from a cold space at TC. The three heat transfer rates are QdotH, Qdot0, and QdotC respectively. Do an energy and entropy analysis of the system to derive an expression for the maximum possible COP, defined as the ratio QdotC/QdotH. [Manual Solution] Answers: (TC(TH-T0))/(TH(T0-TC)) --------------------------------------------------------------------------------2-5-31 [mEE] Assume TH=425 K, T0=298 K, TC=250 K, and QdotC=20 kW in the above system. (a) Determine the maximum COP of the system. (b) If the collector captures 0.2 kW/m2, determine the minimum collector area required. [Manual Solution] Answers: (a) 1.56 (b) 64.3 m2 --------------------------------------------------------------------------------2-5-32 [mEE] A refrigerator with a COP of 2.0 extracts heat from a cold chamber at 0oC at a rate of 400 kJ/min. If the atmospheric temperature is 20oC, determine (a) the power drawn by the refrigerator and (b) the rate of entropy generation in the refrigerator's universe. [Manual Solution] Answers: (a) 3.33 kW (b) 0.00833 kW/K --------------------------------------------------------------------------------• Heat Pump Cycles (Sec. 5 Continued) --------------------------------------------------------------------------------2-5-33 [mEE] On a cold night a house is loosing heat at a rate of 15 kW. A reversible heat pump maintains the house at 20oC, while the outside temperature is 0oC. (a) Determine the heating cost for the night (8 hours). (b) Also determine the heating cost if resistance heating was used instead. Assume the price of electricity to be 15 cents/kWh. [Manual Solution] [TEST Solution] Answers: (a) \$ 1.23 (b) \$ 18.00 Anim. 2-5-33 (click) --------------------------------------------------------------------------------2-5-34 [mEE] On a cold night a house is loosing heat at a rate of 80,000 Btu/h. A reversible heat pump maintains the house at 70oF, while the outside temperature is 30oF. Determine the (a) heating cost for the night (8 hours) assuming the price of 10 cents/kWh for electricity. Also determine (b) the heating cost if resistance heating is used instead. [Manual Solution] Answers: (a) \$ 1.42 (b) \$ 18.75 --------------------------------------------------------------------------------2-5-35 [mEE] A house is maintained at a temperature 20oC by a heat pump pumping heat from the atmosphere. Heat transfer rate through the wall and roof is estimated at 0.6 kW per unit temperature difference between inside and outside. (a) If the atmospheric temperature is -10oC, what is the minimum power required to drive the pump? (b) It is proposed to use the same pump to cool the house in the summer. For the same room temperature, the same heat transfer rate, and the same power input to the pump, determine the maximum permissible atmospheric temperature. (c) What-if Scenario: How would the answer in part (b) change if the heat transfer rate between the house and outside was estimated at 0.7 kW per unit temperature difference? [Manual Solution] [TEST Solution] Answers: (a) 1.84 kW (b) 50oC (c) 50oC --------------------------------------------------------------------------------2-5-36 [mEE] A house is maintained at a temperature TH by a heat pump powered by an electric motor. The outside air at TC is used as the low-temperature TER. Heat loss from the house to the surroundings is directly proportional to the temperature difference and is given by Qdotloss=U(TH -TC). Determine the minimum electric power to drive the heat pump as a function of the given variables. [Manual Solution] Answers: U(TH -TC)^2/TH --------------------------------------------------------------------------------2-5-37 [mEE] A house is maintained at a temperature 25oC by a reversible heat pump powered by an electric motor. The outside air at 10oC is used as the low-temperature TER. Determine the percent savings in electrical power consumption if the house is kept at 20oC instead. Assume that the heat loss from the house to the surroundings is directly proportional to the temperature difference. [Manual Solution] Answers: 54.8% --------------------------------------------------------------------------------2-5-38 [mEE] A house is heated and maintained at 25oC by a heat pump. Determine the maximum possible COP is heat is extracted from the outside atmosphere at (a) 10oC (b) 0oC (c) -10oC and (d) -40oC. (e) Based on these results, would you recommend heat pumps at locations with severe climate? [Manual Solution] Answers: (a) 19.9 (b) 11.92 (c) 8.51 (d) 4.58 --------------------------------------------------------------------------------• Mixed Cycles (Sec. 5 Continued) --------------------------------------------------------------------------------2-5-39 [mEE] A Carnot heat engine receives heat at 800 K and rejects the waste heat to the surroundings at 300 K. The output from the heat engine is used to drive a Carnot refrigerator that removes heat from the cooled space at -20oC at a rate of 400 kJ/min and rejects it to the same surroundings at 300 K. Determine (a) the rate of heat supplied to the heat engine and (b) the total rate of heat rejection to the surroundings. What-if Scenario (c) How would the answer in part (a) change if the temperature of the cooled space was -30oC? [Manual Solution] Answers: (a) 15.3 kJ/min (b) 396.18 kJ/min (c) 150.08 kJ/min --------------------------------------------------------------------------------2-5-40 [mEE] A reversible heat engine is used to drive a reversible heat pump. The power cycle takes in Q1 heat units at T1 and rejects Q2 heat units at T2. The heat pump extracts Q4 from a heat sink at T4 and discharges Q3 at T3. Develop an expression for Q4/Q1 in terms of the four given temperatures. [Manual Solution] Answers: Q4/Q1=T4(T1-T2)/(T1(T3-T4)) Figure 2-5-40 --------------------------------------------------------------------------------2-5-41 [mE] A heat engine with a thermal efficiency of 35% is used to drive a refrigerator having a COP of 4. (a) What is the heat input to the engine for each MJ removed from the cold region by the refrigerator? (b) If the system is used as a heat pump, how many MJ of heat would be available for heating for each MJ of heat input into the engine? [Manual Solution] Answers: (a) 0.714 MJ (b) 1.75 MJ --------------------------------------------------------------------------------2-5-42 [mE] A heat engine operates between two TER's at 1000oC and 20oC respectively. Two-thirds of the work output is used to drive a heat pump that removes heat from the cold surroundings at 0oC and transfers it to a house kept at 20oC. If the house is loosing heat at a rate of 60,000 kJ/h, determine (a) the minimum rate of heat supply to the heat engine. (b) What-if Scenario: How would the answer change if the outside temperature dropped to -10oC? [Manual Solution] Answers: (a) 2.22 kW (b) 3.33 kW --------------------------------------------------------------------------------2-5-43 [mE] A heat engine is used to drive a heat pump. The waste heat from the heat engine and the heat transfer from the heat pump are used to heat the water circulating through the radiator of a building. The thermal efficiency of the heat engine is 30% and the COP of the heat pump is 4.2. Evaluate the COP of the combined system, defined as the ratio of the heat transfer to the circulating water to the heat transfer to the heat engine. [Manual Solution] Answers: 1.96 --------------------------------------------------------------------------------2-5-44 [mE] A furnace delivers heat at a rate QdotH1 at TH1. Instead of directly using this for room heating, it is used to drive a heat engine that rejects the waste heat to atmosphere at T0. The heat engine drives a heat pump that delivers QdotH2 at Troom using the atmosphere as the cold reservoir. Find the ratio QdotH2/QdotH1, the energetic efficiency of the system, as a function of the given temperatures. Why is this a better set-up than direct room heating from the furnace? [Manual Solution] Answers: QdotH2/QdotH1 = ([Troom(TH1-T0))/(TH1(Troom-T0)) --------------------------------------------------------------------------------2-5-45 [mE] A heat pump is used for heating a house in the winter and cooling it in the summer by reversing the flow of the refrigerant. The interior temperature should be 20oC in the winter and 25oC in the summer. Heat transfer through the walls and ceilings is estimated to be 2500 kJ per hour per degree Celsius temperature difference between the inside and outside. (a) If the winter outside temperature is 0oC, what is the minimum power required to drive the heat pump? (b) For the same power input as in part (a), what is the maximum outside summer temperature for which the house can be maintained at 25oC? [Manual Solution] Answers: (a) 0.948 kW (b) 45.16oC    Mass Equation (Recapitulation) PROBLEMS Page 1 of 1
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