Thermo Problems

A Carnot engine receives 250 kJ/sec of heat from a heat source reservoir at 525C and rejects heat to a sink at 50C. What are the power developed and the heat rejected? Ex For 1 kg of an ideal gas, initially at 0C, is heated to 100C by contact with a heat reservoir at 100C, all at constant P. What is the entropy change of the gas? Cp is 4.18 J/g·K and is independent of T. One mole of an ideal gas is compressed isothermally but irreversibly at 130C from 2.5 bar to 6.5 bar in a piston cylinder device. The work required is 30% greater than the work for a reversible, isothermal compression. The heat transferred from the gas during the compression flows to a heat reservoir (ideal gas) at 25C. Calculate the entropy changes of the gas, the heat reservoir and the total. 10 kmol/hr of air is throttled from upstream conditions of 25°C and 10 bar to a downstream pressure of 1.2 bar. Cp = 7/2R. Find the downstream temperature, the entropy change of the air and the rate of entropy generation. Water at 93.3C is pumped from a storage tank at a rate of 189 L/min. The motor of the pump supplies work at a rate of 1.5 kW. The water goes through a heat exchanger, giving up heat at a rate of 42,201 kJ/min and is delivered to a second tank at an elevation 15 m above the first tank. What is the temperature of the water delivered to the second tank? Assume steady state and no fluid velocity changes. Ex. Steam at 1,100 kPa is throttled to 101.33 kPa and 105C. Throttles are considered isenthalpic. What is the quality of the steam prior to the throttle? Ex. In the process described below (water as the fluid), the feed stream to the turbine originates from a moisture separator. Stream 1 is at 361.3 kPa with a mass flow rate of 63 kg/s and an enthalpy of 2000 kJ/kg. Assume the moisture separator is completely efficient in removing liquid, stream 2. The exit stream from the turbine is at 35°C and has a quality of 0.90. What is the quality of the stream entering the moisture separator, the specific enthalpy of the stream entering the turbine and the work done by the turbine?

Ex. Two streams of liquid water are mixed to form the feed to a boiler. The process data provided are: a. Feed stream 1: 120 kg/min at 30C b. Feed stream 2: 175 kg/min at 65C c. The boiler pressure is 1500 kPa d. The outlet stream from the boiler is a saturated vapor at the boiler pressure e. The outlet pipe is 6 cm ID and

the velocity of the two inlet streams can be considered negligible and there are no potential energy changes Find the heat added to the boiler. 1. A compressor receives 0.1 kg/s R-134a at 150 kPa, -10C and delivers it at 1000 kPa, 40C. The power input is measured to be 3 kW. The compressor has heat transfer to air at 100 kPa coming in at 20C and leaving at 30C (i.e. like a heat exchanger). What is the mass flow rate of air? Ans: 0.043 kg/s. 2. A certain industrial process requires a steady 0.75 kg/s supply of compressed air at 500 kPa, at a maximum temperature of 30C. This air is to be supplied by installing a compressor and aftercooler. Local ambient conditions are 100 kPa, 20C. Using a reversible compressor, determine the power required to drive the compressor and the rate of heat rejection in the aftercooler. Ans: Wc = -129 kW 3. For a Carnot cycle, with water as the operating fluid, the following steps are known to occur. The water changes from saturated liquid to saturated vapor at 200°C as heat is added. Heat is rejected at constant pressure at 20 kPa. This heat engine is powering a Carnot cycle refrigerator, that operates between -15 and 20°C. Find the heat added per kilogram of water in the heat engine cycle. Also, how much heat would need to be added so the refrigerator can remove 1 kJ from the cooled space? Ans: 1941 kJ/kg and 0.46 kJ. 4. A turbo charger boosts the inlet air pressure to an automobile engine. It consists of an exhaust gas driven turbine directly connected to an air compressor, as shown below. For a certain engine load the conditions are given in the figure. Assume that both the turbine and the compressor are reversible and adiabatic. Calculate the turbine exit temperature and power output. Find also the compressor exit pressure and temperature. Ans: my answers on this may differ from yours – there are at least two approaches. Turbine out T = 793K (also saw 680K) and the work is 13 kW (will differ based on the different T – if you use mine you should get this value). P compressor out = 349 kPa.

Redo this problem, assuming that the compressor and turbocharger both have isentropic efficiencies of 85%. Ans Compressor out T = 414K, P = 295 kPa

Calculate ΔU and ΔH for 1 kg of H2O when it is vaporized at a constant temperature of 100°C and a constant pressure of 101.33 kPa. For this change 2,256.9 kJ of heat was added. How many grams of O2 are there in a 50.0 L tank at 21C and 15.7 atm? What is the volume of oxygen in a tank at 1 atm and 21°C? What about at 1 atm and 28°C? What is the density of oxygen at 1.0 atm and 25°C? One mole of an ideal gas, with Cp = (7/2)R and Cv = (5/2)R, expands from 8 bar and 600K to 1 bar by three different paths in a closed system A) Constant volume B) Constant temperature C) Adiabatically Calculate W, Q, ΔH and ΔU for each process. Ex: One mole of an ideal gas, with Cp = (7/2)R and Cv = (5/2)R expands from 8 bar and 600K to 1 bar by three paths in a closed system A. Constant volume B. Constant temperature C. Adiabatically Assuming mechanical reversibility, calculate W, Q, ΔH and ΔU for each process. We did parts A and B last lecture – so lets do part C Ex: One m3 of an ideal gas at 600K and 1000 kPa expands to 5 times its initial volume through a mechanically reversible, adiabatic process. If Cp = (7/2)R and is temperature independent, find the final temperature, pressure and work done by the gas. A piston cylinder setup contains 0.5 kg of air at 20C and 100 kPa. The pressure is then increased to a final pressure of 600 kPa during which the air goes through a polytropic process with exponent n = 1.3 Find the initial volume, final temperature and the work in the process. A piston/cylinder contains 1 kg of methane gas at 100 kPa and 300K. The gas is then compressed reversibly to a pressure of 800 kPa. Find the work if the process is adiabatic. Ex. Air undergoes the following sequence of mechanically reversible steps in a closed system. A) From an initial state of 70°C and 1 bar it is compressed adiabatically to 150°C. B) It is then cooled from 150 to 70°C at constant pressure. C) Finally it is expanded isothermally to its original state. Calculate W, Q, ∆U and ∆H for each of the three steps and for the entire cycle.

Example: 5 kg of liquid CCl4 undergo a mechanically reversible, isobaric change of state at 1 bar, during which the temperature changes from 0 to 20°C. Determine the total volume change, the work, heat, enthalpy and internal energy changes associated with this process. Given properties at 1 bar and 0°C may be assumed independent of temperature: = 1.2 x 10-3 K, Cp = 0.84 kJ/kg/K, density = 1,590 kg/m3. 1. A car engine running at 65 horsepower has a thermal efficiency of 24%. What is the fuel consumption for this car, if the fuel has a heating value of 19,000 BTU/lbm? 2. An air conditioner is rated for 5000 BTU/hr of cooling and requires 500 W of power. Find the rate of heat rejected and the coefficient of performance. 3. Air is heated from 300 to 350C, what is the entropy change (assume no pressure change)? What is the entropy change is it is instead heated from 1300 to 1350C? Explain/discuss. 4. A tank contains 1 kg of methane at 500 K and 1500 kPa. The methane is cooled to 300K, what was the heat transferred and entropy change? 5. What is the entropy change when water is heated from 150 to 1200C at a constant pressure of 400 kPa? 6. Argon in a bulb (light bulb) goes from 20 to 60C when the light is turned on. Before being turned on, the pressure in the bulb is 90 kPa. What was the entropy change when the light was turned on and 60C was finally reached?

Calculate the compressibility factor for ethylene at 25C and 12 bar. 1. Using the Lee Kesler (Pitzer) correlation tables 2. Using a second virial coefficient value of -140 cm3/mol 3. Using the Pitzer correlation approach and calculating the B values In an air standard Brayton cycle, the air enters the compressor at 0.1 MPa and 15C. The pressure leaving the compressor is 1.0 MPa and the maximum temperature in the cycle is 1100C. Find the pressure and temperature at each point in the cycle, the compressor work, turbine work and cycle efficiency. Assume ideal gas, with constant heat capacities and isentropic expansion and compression. Find

the

coefficient

of

performance,

for

a

condensation

temperature of 60F in a vapor compression cycle with Freon 11 as

the refrigerant and an evaporation temperature of 10F. The compressor has an efficiency of 75%.

A power plant operates using a regenerative Rankine cycle with one open feedwater heater. Steam enters the turbine at 15 MPa and 600°C and is condensed in the condenser at 10 kPa. Some steam leaves the turbine at 1.2 MPa into the open feedwater heater, with the condenser effluent pumped to this pressure also. Find the fraction of steam extracted and the thermal efficiency of the plant. Assume the pumps and turbines are adiabatic and reversible.

A steam power plant following the Rankine cycle has the following conditions: the pump outlet pressure is 10 MPa, the boiler outlet temperature is 600C, the turbine outlet pressure is 10 kPa, the turbine efficiency is 80%, the pump efficiency is 75% and the power rating is 80,000 kW. Find the heat transfer rate in the boiler and condenser, the water flow rate and the thermal efficiency of the plant. Consider a steam power plant that operates on a regenerative Rankine cycle (has one open feedwater heater) and has a net power output of 150 MW. Steam enters the turbine at 10 MPa and 500°C and the condenser at 10 kPa. The isentropic efficiency of the turbine is 80% and that of the pumps is 95%. Steam is extracted from the turbine at 0.5 MPa to heat the feedwater heater in an open feedwater heater. Water leaves the feedwater heater as a saturated liquid. Show this cycle on a TS diagram, find the mass flowrate of steam through the boiler, and the thermal efficiency of the cycle.

More random extras 1. A rigid, insulated vessel contains superheated vapor steam at 3 MPa, 400C. A valve on the vessel is opened, allowing steam to escape. The overall process is irreversible, but the steam remaining inside the vessel goes through a reversible adiabatic expansion. Determine the fraction of steam that has escaped, when the final state inside is saturated vapor Ans: fraction = 0.80. 2. Water at 100 kPa, 25C is brought to the boiling point in a piston/cylinder with an isobaric process. The heat is supplied by a heat pump with the cold side at the ambient temperature of 25C. Assume that the whole process is reversible and find the work input to the heat pump per kg of water Ans: work = 33.74 kJ/kg.

3. Consider the air conditioning of a house through use of solar energy. At a particular location, experiment has shown that solar radiation allows a large tank of pressurized water to be maintained at 175°C. During a particular time interval, heat in the amount of 1,500 kJ must be extracted from the house to maintain its temperature at 24°C when the surroundings temperature is 33°C. Treating the tank of water, the house, and the surroundings as heat reservoirs, determine the minimum amount of heat that must be extracted from the tank of water by any device built to accomplish the required cooling of the house. No other sources of energy are available. Q = 144 kJ.

Consider a steam power plant operating on a Rankine cycle. Steam enters the turbine at 3 MPa and 350°C and is condensed in the condenser at a pressure of 75 kPa. Find the thermal efficiency.

1. A reversible steady state device receives a flow of 1 kg/s air at 400 K, 450 kPa and the air leaves at 600 K, 100 kPa. Heat transfer of 900 kW is added from a 1000 K reservoir, 50 kW rejected at 350 K and some heat transfer takes place at 500 K. Find the heat transferred at 500 K and the rate of work produced. 2. A reversible and isothermal compressor is used to continuously compress helium initially at 27C, the same temperature as the surroundings, and 1200 kPa to a final pressure of 18 MPa. Calculate the work required per mole of He and the amount of heat removed. Assume ideal gas. 3. Saturated vapor steam at 100 kPa is compressed adiabatically to 300 kPa. If the compressor is 75% efficient, what is the work required and the temperature of the discharge stream? October 20 – Practice Problems 1. A piston/cylinder device contains refrigerant 134A at 20°C and 1600 kPa. The refrigerant is heated to 60°C by an external heat source that is always at 70°C. The refrigerant heating process is internally reversible and at constant pressure. Find the heat transferred and the entropy generation. 2. Consider a combination of a gas turbine power plant and a steam power plant (work is harnessed from both simultaneously). The gas turbine operates at higher temperatures than the steam power plant, and the heat lost from the gas turbine (QL) equals the heat input to the steam (QH). Assume both cycles have a thermal efficiency of 32%. What is the efficiency of the overall combination? 3. A steam turbine inlet is at 1200 kPa, 400°C. The exit is at 200 kPa, 200°C. What is the isentropic efficiency?

4. A steam turbine with a power outlet 56,400 kW has inlet conditions of 8600 kPa and 500°C, and outlet conditions of 10 kPa. The turbine efficiency is 0.75. What is the quality of the exiting steam and what is the mass flow rate of the water? 5. A stream of an ideal gas at 300C and 45 bar is expanded adiabatically in a turbine to 2 bar. Calculate the isentropic work produced?

Tests on an adiabatic gas compressor yield values of the inlet conditions and outlet conditions as follows: T1 = 300K

T2 = 464K

P1 = 2 bar

P2 = 6 bar

Assuming ideal gases with a constant heat capacity of Cp = 7/2R, find the compressor efficiency. Some temperature changes as examples: steam at 1000 kPa and 300C is throttled to 101 kPa H2 = H1 = 3051.15 kJ/kg from the steam tables. @ P = 101 kPa and H = 3051.15 kJ/kg according to the steam tables 250 < T < 300 interpolation results in T = 288.8C 

the temperature decreased 11.2C with a drop in

pressure from 1000 kPa to 101 kPa.

If we throttle “wet” steam to lower pressures, we can vaporize the liquid phase and possibly end up with superheated steam.

Example, saturated steam at 1000 kPa and a quality of 0.96 is throttled to 101.3 kPa. From the steam table @ 1000 kPa 178 < Tsat < 180 Tsat = 179.9C H2 = H1 = (0.04) (767.6) + 0.96 (2776.2) = 2695.7 kJ/kg Again using the steam tables, with P = 101.3 kPa and H =

2695.7

kJ kg 100 < T < 125 interpolating, 

T  110C

T = ~ 70C

Note: The temperature decreased but now we have superheated steam.

Refrigerant 134a enters the capillary tube of a refrigerator as saturated liquid at 0.8 MPa and is throttled to 0.12 MPa. Find the quality of the refrigerant and the temperature change. October 13 Lecture Problems

1. An inventor claims to have devised a cyclic engine which exchanges heat with reservoirs at 25°C and 250°C, and which produces 0.45 kJ of work for each kJ of heat extracted from the hot reservoir. Is this claim believable? 2. An inventor claims to have invented a process that takes saturated steam at 100°C and through a series of steps makes heat continuously available at a temperature of 200°C with 2000 kJ of heat liberated for every kg of steam into the process, if 0°C cooling water is available. Is this process possible? 3. In a steady state flow process, 1 mol/sec of air at 600K and 1 atm is continuously mixed with 2 mol/sec of air at 450K and 1 atm. The product stream is at 400K and 1 atm. Find the rate of heat transfer to or from the surroundings and the rate of entropy generation for the process. Assume Cp = (7/2)R and the surroundings are at 300K. 4. A rigid storage tank of 1.5 m3 contains 1 kg argon at 30oC. Heat is then transferred to the argon from a furnace operating at 1300oC until the specific entropy of the argon has increased by 0.343 kJ/kg K. Find the final temperature and pressure of the argon, and then find the total heat transfer and entropy generated in the process. Assume ideal gas behavior with constant heat capacities.

Examples 1. A pipe contains steam at 1.4 MPa and 300°C. An initially evacuated tank is attached to this pipe and the valve is opened until the tank pressure reaches 1.4 MPa, adiabatically. What is the final temperature of the steam? 2. In a well-insulated concurrent heat exchanger, 1 kg/s of air at 500K flows into one side ad 2 kg/s air flows into the other channel at 300K. If it is infinitely long, what is the exit temperature? How does T vary with time in the two channels? Neglect KE and PE effects and assume steady state. 3. A flow of 5 kg/s water at 100 kPa, 20°C should be delivered as steam at 1000 kPa, 350C to some application. Consider compressing it to 1000 kPa, 20°C and then heating it at constant a constant pressure of 1000 kPa to 350°C. What kind of devices are needed and find the specific energy transfers in those devices. 4. A tank containing 20 kg of water at 20°C is fitted with a stirrer that delivers work to the water at the rate of 0.25 kW. How long does it take for the temperature of the water to rise to 30°C if no heat loss is lost from the water? 5. A heating coil (electric) is used to heat 20.0 kg of a liquid in a closed adiabatic vessel. The liquid is initially at 25°C and 1 atm. The heating coil delivers a steady 2.00 kW of power to the vessel and its contents. How long will it take the liquid to reach its normal boiling point (say 100°C)? The heat capacity of this liquid is 4.18 kJ/kg/°C. 1. One kg of water at 25°C: a. Experiences a temperature rise of 1K. What is ΔUt, in kJ? Ans: 4.18 kJ

b. Experiences a change in elevation of Δz. The change in potential energy is the same as ΔUt for part a. What is Δz, in meters? Ans: 427m c. Is accelerated from rest to a final velocity of u. The change in kinetic energy is the same as ΔUt for part a. What is u, in m/s? Ans: 91.4 m/s 2. A piston cylinder contains 0.5 kg air at 1000K and 2000 kPa. There is a constant force on the piston and the minimum volume is 30 L if the piston is at the stops. Now the air cools by heat transfer to a final temperature of 500K. Find the final volume and the work during the process. 3. An adiabatic steam turbine in a power plant receives 5 kg/s steam at 3000 kPa, 500°C. Twenty percent of the flow is extracted at 1000 kPa, 350°C to a feedwater heater, and the remainder flows out at 200 kPa, 200°C. Find the turbine power output. 4. A steam turbine receives steam from two boilers. One flow is 5 kg/s at 3 MPa, 700°C and the other flow is 10 kg/s at 800 kPa, 500°C. The exit state is 10 kPa, with a quality of 96%. Find the total power out of the adiabatic turbine. 5. Steam enters a heat exchanger operating at steady state at 250 kPa and a quality of 90%, and exits as saturated liquid at the same pressure. A separate stream of oil with a mass flow rate of 30 kg/sec enters at 20°C and exits at 100°C with no significant change in pressure. The specific heat of the oil is 2.0 kJ/kg/K. If heat transfer from the heat exchanger to its surroundings is 10% of the energy required to increase the temperature of the oil, determine the steam mass flow rate.

Lecture 7 – Example Problems 1. Carbon dioxide is throttled from 20°C, 2 MPa to 800 kPa. Assuming ideal gas behavior, what is the exit temperature? 2. A piston-cylinder with a 5 cm ID contains 1.4 grams of nitrogen. The mass of the piston itself is 5.00 kg and has a 20.00 kg weight on top of it. The gas temperature is 30°C and the pressure outside the cylinder is 1.00 atm. a. Find the pressure in the cylinder and volume of gas in the cylinder. b. Suppose the 20.00 kg weight is abruptly removed and the piston rises to a new equilibrium position through a two step process. In the first step a negligible amount of heat is exchanged with the surroundings. In the second step the gas returns to 30°C. Find Q and W. 3. The density of oxygen vapor at 150K and 41.64 atm is measured and is found to be 213.5 g/L and the specific internal energy is 1706 J/mol at this state.

a. Calculate the specific volume of O2 at these conditions using (1) the ideal gas law and (2) the given density. What kind of error is there between these and why? b. Calculate the specific enthalpy under these same conditions. 4. An initially empty 0.2 m3 canister is filled with CO2 from a line that is at 800 kPa, 400 K. Assume the process runs until it stops by itself and it is adiabatic. Use constant heat capacity to find the final temperature in the canister. 5. A tank initially contains 1 m3 of air at 100 kPa and 300K. A pipe flowing air at 1000 kPa and 300K is connected to the tank and the tank is filled slowly to 1000 kPa. Find the heat transfer required to maintain the tank temperature at 300K. 1.1 It is desired to boil water in a pot at 220F How heavy in ibm a lid should be put on the 6-inch diameter pot. 1.13 A .5m^3 rigid tancontaining hydrogen at 20C and 600kPa is connected by a valve to another .5m^3 ridgid tank that holds hydrogen at 30C and 150kPa. Now the valve is opened and the system

is allowed

to

reach

thermal

equilibrium

with

the

surroundings, which are 15C. Determine the final pressure in the tanks. 2.1 2.3 2.4