Chapter_5.pdf

5.A-1 A well-insulated cylindrical tank of water, 3 m in diameter is filled to a depth of 6 m. It is desired to lower the temperature of the water in the tank from 20°C to 2°C in two hours. Using a heat pump, heat is to be rejected to the atmospheric at 38°C. What is the minimum average power that must be supplied to achieve this cooling. State all assumptions that you employ.

© S.A. Klein and G.F. Nellis Cambridge University Press, 2011

5.A-2 After graduation, you take a job with the Acme energy services company. Your first job is to purchase high efficiency heat pumps. The heat pumps employ mechanical power and a heat engine to provide heat at TH = 100°C. A low temperature heat sink at TC = 20ºC is also available. A salesman shows you two models. Performance data for each model operating at steady state are provided in Figure 5.A-2.

Model A

TH = 100°C

TC = 20°C

500 W 1000 W

100 W 500 W

externalmechanicalpower

Model B

TH = 100°C

TC = 20°C

500 W 750 W

400 W 550 W

externalmechanicalpower

Figure 5.A-2: High efficiency heat pumps; Model A is on left and Model B is on the right.

a.) What is the mechanical power that is needed to operate Model A? b.) What is the mechanical power needed to operate Model B? c.) What is the Coefficient of Performance (COP) for heating for Model A? The COP for

heating is defined as the ratio of the net heat provided to TH to the mechanical power required.

d.) What is the COP for heating for Model B? e.) What is the maximum possible heating COP that can be obtained under these conditions? f.) Which heat pump system would you recommend that your company purchase?


5.A-3 A Carnot refrigeration cycle uses water contained in a piston-cylinder device. Process 1-2 is a reversible isothermal compression in which the water is transformed from saturated vapor to saturated liquid at a pressure of P1 = 250 kPa. Process 2-3 is a reversible adiabatic expansion that proceeds until the water reaches T3 = 7ºC. The quality at state 3 is x3 = 0.169. Process 3-4 is a reversible isothermal expansion that proceeds until the quality reaches x4 = 0.784. Process 4-1 is a reversible adiabatic compression. The mass of water in the cylinder is m = 1 kg and the period of the cycle is time = 20 s. a.) Complete the table below.

State Pressure

(kPa) Temperature

(ºC) Specific internal

energy (J/kg) Quality Specific volume

(m3/kg) 1 250 0 2 250 1 3 7 0.169 4 7 0.784

b.) Determine the Coefficient of Performance (COP) for refrigeration for this cycle. c.) What is the refrigeration effect, in kW? d.) What is the power input required to operate this cycle, in hp? e.) Overlay the state points for the cycle on a P-v diagram for water.


5.A-4 The refrigeration system shown in Figure 5.A-4 transfers energy from cylinder A to cylinder B. Cylinder A contains mA = 1.75 kg of R-134a, which is initially saturated vapor at a pressure of PA = 280 kPa. Cylinder B contains mB = 0.5 kg of water at TB = 45°C and an initial quality of xB,1 = 0.25. The pressures in cylinders A and B remain constant during the process in which the R134a changes from saturated vapor to saturated liquid.

Ac,A = 0.002 m2 R134amA = 1.75 kg

PA = 280 kPa

QC QH

W

watermB = 0.5 kg

TB = 45°C

Patm = 100 kPa

Figure 5.A-4: Refrigeration cycle.

a.) The refrigeration cycle operates with a coefficient of performance, COP = 1.5. The COP is

defined as the ratio of the refrigeration provided to the work required. Determine the required work input.

b.) The cross-sectional area of the piston for cylinder A is Ac,A = 0.002 m2. Determine the required mass of the piston. The atmospheric pressure is Patm = 100 kPa.

c.) Determine the work done on or by the water during this process. d.) What is the lower limit on the work input required for this process, assuming that all

irreversible processes could be replaced with reversible processes?


5.A-5 A way to use heat to produce refrigeration is shown in Figure 5.A-5. A heat engine operating between thermal reservoirs at 300°F and 85°F (the environmental temperature) produces power with an efficiency of 20%. This power is used to drive a refrigeration cycle that removes 16,000 Btu/hr of thermal energy from a space at 40°F and rejects thermal energy to the 85°F environment. The overall coefficient of performance, defined as the ratio of the heat removed from the 40°F space to the heat supplied at 300°F, is 0.90.

300°F

40°F

85°Fenvironment

Heat Engine

Heat Pump

16,000Btu/hr

Figure 5.A-5: Heat fired refrigeration system

a.) Determine the power produced by the heat engine. b.) Determine the total rate of heat rejection to the 85°F environment. c) Refrigeration used to be supplied by ice that was cut from a lake in winter and stored

for summer use. What is the mass of an ice block that would provide the same cooling as this system for a 24 hour period?

d.) If both the heat engine and the heat pump operated reversibly, what would the COP of the overall process be?


5.A-6 A refrigeration system for freezing freshly-baked bread is under design. The system must be able to continuously cool N = 4,250 loaves of bread per 8-hour working day from T1 = 25°C (the ambient temperature) to T2 = −10°C. Tests show that the average bread loaf has a mass of m = 450 g and a specific heat capacity of c = 2.93 kJ/kg-K. In addition, when the bread is cooled below Tf = 0°C, the water in the bread freezes releasing an additional ΔHfs = 109.3 kJ per loaf. The proposed design circulates air past cooling coils that cools the air to Tair = -20°C. The cold air is then circulated past the bread and returned to the cooling coils. a.) As the engineer in charge of the project, determine the required steady rate of cooling

capacity for this process b.) To maintain consistent product, the change in temperature of the air blown by the bread must

be less than ΔTair = 8°C. Determine the minimum required volumetric flow rate of air (assume that the air is at atmospheric pressure, Patm = 100 kPa).

c.) A refrigeration contractor has indicated that they can supply refrigeration equipment for the bread cooling project that has a COP = 2.1. Note that COP is defined as the ratio of the refrigeration provided to the power required. Determine the electrical service that will be needed for the motor used to operate the cycle.

d.) Compare the COP with the maximum possible COP for this application.


5.A-7 A heat pump is to be used to heat a house in the winter and then reversed to cool the house in the summer. The interior house temperature is to be maintained at Tin = 70°F during both summer and winter. Heat transfer through the roof, walls, and floor is estimated to occur at a rate of UA = 870 Btu/hr per degree temperature difference between inside and outside. a.) Determine the required cooling capacity to maintain the interior temperature on a summer

day in which the outdoor temperature is Tout = 95°F b.) Determine the minimum electrical power required to provide the cooling capacity determined

in part (a). c.) Now consider a heating application with the heat pump operating with the same power

determined in part (b). For this heating application, determine the lowest outdoor temperature for which the interior temperature could be maintained at Tin.

d.) Compare the actual costs of heating for a 24 hour period using a gas furnace and a heat pump when the outdoor temperature is Tout = 20°F. The gas furnace operates at ηf = 92% efficiency based on the energy content of the gas, which his usually expressed in therms. (1 therm=105 Btu.) The heat pump has a coefficient of performance that is 35% of the Carnot limit. Obtain the unit cost of gas and electricity from your local utility.


5.A-8 A series configuration solar heat pump system is used for a house heating application. In January, the energy required to maintain the house at 75°F is 20E6 Btu and the incident radiation on the solar collectors that are mounted on the roof of the building is 15E6 Btu. The monthly average collector efficiency is t0.7 0.01[1/R] outT T where Tout is

the average outdoor temperature which is 10°F for January and Tt is the average temperature in the storage tank. A heat pump moves energy from the storage tank into the building with a COP that is equal to ½ of the Carnot COP for the same temperature limits. The portion of the load that is not met by the heat pump is provided by electrical resistance heaters that have a COP of 1.0. a.) Determine the amount of electricity required to heat the building in January if a solar

energy system is not used. b.) Prepare a plot of the electrical energy required as a function of the tank temperature

for tank temperatures between 10°F and 60°F. c.) What is your recommendation for the operation of this system?


5.B-1“Please excuse the intrusion. My research has indicated that you are a leader in the development of alternative power. My name is XXXX and I am the director of research for gravitational systems (GS), USA. I would like to introduce myself and our technology for sustainable renewable power. GS is the patent holder and developer of gravitational energy capture systems which collects energy from the movement of significant objects, such as motor vehicles, or livestock, across energy taps. The technology is surprisingly controversial, but technically sound. My mission is to ensure that industry leaders, such as yourself, are aware of the technology, and to answer any questions that you may have relative to it. Gravitational energy capture can, in my opinion, supply a significant portion of the energy needs of urban and rural communities. The system is easily implemented, as either pumps or electric generators. Gravitational energy is clean, renewable, and only dependent upon traffic. Please, at your convenience, take a look at our website: http://www.gravitationalsystems.org/about.htm. Your comments will be appreciated.”

Please evaluate this technology indicating whether you believe that it can work as described. If you do believe it can work, please indicate the advantages and disadvantages. If you do not believe that it is possible, please indicate what fundamental laws it attempts to violate. Feel free to use any available resources to help in your investigation including the company’s website.


5.B-2 You have probably seen the ‘drinking bird’ toy (http://en.wikipedia.org/wiki/Drinking_bird). If examined closely, we could see that this toy is simply two hollow glass bulbs that are separated by a tube and mounted on a swivel joint. The lower bulb is partially filled with a volatile liquid such as isopentane. a.) Explain how the drinking bird works. b.) The drinking bird can be viewed as a machine that converts thermal energy into mechanical

energy. Do you agree with this characterization? If so, determine the maximum efficiency of the bird. Please indicate the assumptions employed in your assessment.

c.) Is the efficiency an important performance indicator for this device? Please provide a justification for your answer.

d.) During its operation, the bird raises the working fluid from the lower bulb to the upper bulb. It is possible for the bird to produce more work than required to raise the working fluid? Do you see any practical potential for useful power production with devices of this nature?


5.B-3 Shown below is a question received from a practicing engineer. Please determine the heat transfer for the process between points A and B and the efficiency of the cycle and prepare a response. I am wondering how much energy it takes to heat R-13 from point A to point B as defined below. I framed the question as a closed heat engine cycle with R-13 as the working fluid and point A to point B as the region of heat input. This heat input is the cycle’s Q1. The cycle operates near the critical locus. There appears to be a discrepancy between how the chemist and engineer determine the cycle’s Q1.

Q1 is the total heat input to the cycle, Q2 is the total heat exhausted from the cycle, U is internal energy, H is enthalpy, and work is the external mechanical work done by or to the R-13. At point A the R-13 is a saturated liquid at 97% of its critical temperature. Heat energy is added between points A and B with the R-13 held at a constant volume. Isentropic expansion occurs from B to C. At point C the R-13 is at a 30% evaporated condition. Isobaric, isothermal condensation occurs from C to A.

In applying the First Law for this closed cycle chemists calculate cycle efficiency from changes in U. For the chemist, Q1 is the U for the constant volume heat input from A to B and the cycle’s gross mechanical work output is considered to be the U from B to C. This method ignores the PV from B to C as part of the gross mechanical work output.

Both chemist and engineer agree that the cycle’s Q2 exhaust heat is the H from C to A, but engineers considered the gross mechanical work output of isentropic expansion from B to C to be the H, not the U.

The net mechanical work output for this cycle in either case should be found by subtracting from the gross mechanical work output, the mechanical work that is reintroduced to the R-13 as PV energy during the constant pressure condensation from C to A. The net mechanical work output plus the Q2 heat exhausted from C to A must equal the cycle’s Q1 heat input.

So what is the gross work output from B to C? Is the chemist correct in saying the gross work output from reversible isentropic expansion is the 1.27 BTUs U or is the engineer correct in saying it is the 3.05 BTUs H?

The chemist subtracts the 1.12 BTUs reintroduced to the R-13 as PV energy from C to A and exhausted as heat from a 1.27 BTUs (U) gross work output leaving .15 BTU as the net work output per cycle. The engineer similarly should subtract the 1.12 BTUs reintroduced from a gross work output of 3.05 BTUs (H), leaving 1.93 BTUs as net work output.

There can be only one value for the amount of energy it takes to heat R-13 from point A to point B, so how can the different disciplines use different values for the cycle’s Q1 and which value is correct, the 7.57 BTUs U, the 9.35 BTUs calculated value, or the 10.50 BTUs H? Any clarification you can offer would be greatly appreciated. Thank you.


5.B-4 Dr. Alan Williams describes a constant volume power cycle that he claims can produce mechanical power with an efficiency that approaches 100%. The analysis and below was excerptws from the website prepared by Dr. Williams:

from: http://www.globalwarmingsolutions.co.uk/energy_cycles_at_constant_volume.htm

Consider the energy cycle depicted in the pressure-volume diagram shown in Figure 5.B-4 that consists of two processes at constant volume and two processes at constant pressure.

Pressure

Volume

1

2 3

4

V V’

P

P’

Figure 5.B-4: Pressure-volume diagram.

At state 1 the system consists of a fixed mass of gas at volume V, pressure P, and temperature T1. The gas then undergoes the following four processes:

State 1 to 2: The gas is heated at constant volume to temperature T2 raising its pressure to P'. The heat input required per unit mass for this process is cv (T2 - T1).

State 2 to 3: The gas is heated at constant pressure to temperature T3 and volume V'. The heat input required per unit mass for this process is cp (T3 - T2).

State 3 to 4: Work is done by the gas at constant volume so that its pressure falls to P and its temperature falls to T4. There is no heat input for this process. An energy balance is therefore: Q = ΔU + W = 0 and the work done by the gas during this process per unit mass is cv (T3 – T4)

State 4 to 1: Heat is transferred out of the gas to an energy sink during a constant pressure process so that the volume returns to V and the temperature to T1.


Efficiency = (useful work achieved) / (total heat input) = [cv (T3 - T4)] / [cv (T2 - T1) + cp (T3 - T2)]

Now consider the energy cycle portrayed approximating towards a constant volume heat engine. As V′ → V, T4 → T1, T3 → T2 and (T3 – T2) → 0, therefore, efficiency →[cv (T2 - T1)] / [cv (T2 - T1)] →100%.

Therefore - if an energy cycle can be devised with the entire heat input and extraction of work at constant volume it will have a maximum theoretical efficiency of 100%.

Please review the analysis above and then answer the following questions. a.) Calculate the efficiency of the cycle with 1 kg of air as the working fluid for the following

conditions: T1 = 300 K, P = 100 kPa, P' =2 P, v3 = 1.5 v1 b.) What is the Carnot efficiency for the conditions of part (a)? Compare the efficiencies. c.) Do you agree with the analysis presented by Dr. Williams? If your answer is no, indicate

where you see a flaw in his logic or a violation of a physical law – other than the Second Law of Thermodynamics. Dr. Williams does not believe that the Second Law is applicable to this process.


5.B-5 The attached pages describe an invention referred to as “The Williams Engine” by its inventor, Alan Williams. The information presented here was taken directly from the inventor’s website, http://www.globalwarmingsolutions.co.uk/the_williams_engine.htm. Mr Williams claims to have identified a “new set of concepts” that will lead to more efficient conversion of heat into electrical power in steady-flow devices. Please read the claims carefully and answer the following questions. a.) Is the concept described by Mr. Williams new? Are you aware of similar energy

recovery systems? If so, please elaborate.

b.) Is Mr. Williams correct in his assertion that the Carnot efficiency is applicable only to a closed system? Please provide an explanation supporting your response.

c.) Are the efficiencies quoted by Mr. Williams for power cycles with various energy sources reasonable?

from: http://www.globalwarmingsolutions.co.uk/the_williams_engine.html

Heat is converted into electricity with about 40% efficiency in power stations. Motor vehicles have efficiencies only ~20%. Solar energy conversion is typically below 10% using the most sophisticated and expensive technologies. A new set of concepts is described below for the conversion of solar energy or combustion heat into electricity. No practical work has been carried out and no quantitative or theoretical calculation. The Williams engine is a design based purely on my understanding of heat and the kinetic theory of gases.

There are three unusual features:

1. The system is driven by convection currents alone. There is no pump. Combustion and generation are at normal atmospheric pressure.

2. The turbine produces electricity from both incoming air and outgoing hot gases.


3. Efficient heat recovery from the gases leaving the turbine repeatedly recycles waste heat to prewarm incoming air.

It has been put to me that the second law of thermodynamics applies to all energy conversion imposing a maximum efficiency of T/T. I maintain that the Second law applies to a closed system as in the Carnot cycle but that in the apparatus described the turbine is simply driven by airflow. The maximum efficiency is then around 60% as in wind turbines or up to 80% as in the turbine component of the solar chimney. The Wells turbine achieves even higher efficiency.

If the Second law does apply then for combustion gases at 500C the maximum efficiency would be about 480/773 about 60%. But residual heat energy in the effluent gases from the turbine will be absorbed in the heat exchanger by incoming air. Such energy recovery systems can be over 95% efficient so that waste heat will be recycled a second, third .. time to achieve an eventual over 80% overall efficiency.

If the above configuration was used for solar energy conversion the temperature involved could be 20 C for ambient air and 30 C after solar absorption. If the second law applies this would give a maximum efficiency of 3%. But any heat rejected by the turbine will be reabsorbed in the heat exchanger and recycled into the incoming air. This will drive up the temperature of air entering the turbine imposing higher efficiency. Heat rejected at first presentation to the turbine will be reabsorbed a second, third .. 10,20 .. 100 .. times until air leaving the apparatus is reduced to just above ambient temperatures. By this repeated recycling of rejected energy, a cumulative efficiency of over 80% should be achievable. Indeed the only energy loss is in the warm air leaving.

Thus, whether the Second law of thermodynamics applies or not, efficient recycling of waste heat should allow conversion of heat into electricity with over 80% efficiency. It may be that the above ideas are fundamentally flawed and that the Williams engine will produce very little electricity and only lots of hot air.

It is my contention however that the principles described will allow conversion efficiencies of over 80% heat into mechanical or electrical energy. The source of heat can be fossil fuels or solar energy. The principles should apply to motor vehicle engines, power stations, roof solar electricity and large scale solar farms. If that is indeed the case then that is global warming solved.

I would suggest that the apparatus outlined be called the Williams engine. If there are problems of copyright versus Formula One (!) then that it be called ECHECCRRWH – the Efficient Conversion of Heat into Electricity using Convection Currents and Repeated Recycling of Waste Heat – or more simply, the Dream Machine.


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