Mechanical Engineering Handbook - Thermodynamic Cycles

Cook, W. J. “Thermodynamic Cycles” The Engineering Handbook. Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

© 1998 by CRC PRESS LLC

48Thermodynamic Cycles

48.1 Power Cycles48.2 Refrigeration Cycles

William J. CookIowa State University

A thermodynamic cycle is a continuous series of thermodynamic processes that periodicallyreturns the working fluid of the cycle to a given state. Although cycles can be executed in closedsystems, the focus here is on the cycles most frequently encountered in practice: steady-flowcycles, cycles in which each process occurs in a steady-flow manner. Practical cycles can beclassified into two groups: power-producing cycles (power cycles) and power-consuming cycles(refrigeration cycles). The working fluid typically undergoes phase changes during either a powercycle or a refrigeration cycle. Devices that operate on thermodynamic cycles are widely used inenergy conversion and utilization processes since such devices operate continuously as theworking fluid undergoes repeated thermodynamic cycles.

The fundamentals of cycle analysis begin with the first law of thermodynamics. Since eachprocess is a steady-flow process, only the first law as it applies to steady-flow processes will beconsidered. For a steady-flow process occurring in a control volume with multiple inflows andoutflows, the first law is written on a time rate basis as

_Q +X

[ _m(h + V 2=2 + gz)]in = _W +X

[ _m(h + V 2=2 + gz)]out (48:1)

For a single-stream process between states i and j, Eq. (48.1) on a unit mass basis becomes

iqj + hi + V 2i =2 + gzi = iwj + hj + V 2

j =2 + gzj (48:2)

where iqj = i_Qj = _m , iwj = i

_Wj = _m , and _m is the mass rate of flow. See Van Wylen et al.[1994].In processes involved with the cycles considered here, changes in kinetic and potential energies(V 2=2 and gz terms, respectively) are small and are neglected. Power _W is considered positivewhen it is transferred out of the control volume, and heat transfer rate _Q is considered positivewhen heat transfer is to the control volume. In the figures herein that describe the transfer of powerand heat energy to and from cycles, arrows indicate direction and take the place of signs. Theaccompanying _W or _Q is then an absolute quantity. Where confusion might arise, absolute valuesigns are used. Only power transfer and heat transfer occur across a closed boundary that enclosesthe complete cycle. For such a boundary,

hX_Q =

X_Wi

cycle(48:3)


48.1 Power CyclesThe purpose of a power cycle is to produce a net power output on a continuous basis from heatenergy supplied to it from a high-temperature energy source. The device in which the power cycleis executed is sometimes referred to as a heat engine. Gas-turbine engines and reciprocatinginternal combustion engines are used widely to produce power. Strictly speaking, these engines arenot classified as power cycles because their working fluids do not undergo thermodynamic cycles.Figure 48.1(a) shows a heat engine that receives heat energy at the rate _QH from ahigh-temperature energy source and produces net power _Wnet . As a consequence of its operationit rejects heat energy to the lower-temperature surroundings at the rate _QL . A widely usedperformance parameter for a power cycle is ´ , the cycle thermal efficiency, defined as

´ = _Wnet = _QH (48:4)

Second law considerations for thermodynamic power cycles restrict ´ to a value less than unity.Thus, _Wnet in Fig. 48.1(a) is less than _QH . By Eq. (48.3), the rate at which heat energy is rejectedto the surroundings is

j _QL j = _QH ¡ _Wnet (48:5)

Figure 48.1 Descriptions of power cycles: (a) Power cycle operation. (b) The simple vapor power cycle.


where the temperatures are on an absolute scale [Wark, 1983].Figure 48.1(b) illustrates a simple vapor power cycle. Each component operates in a steady-flow

manner. The vapor generator delivers high-pressure high-temperature vapor at state 1 to theturbine. The vapor flows through the turbine to the turbine exit state, state 2, and produces power_Wturbine at the turbine output shaft. The vapor is condensed to liquid, state 3, as it passes through

the condenser, which is typically cooled by a water supply at a temperature near that of thesurroundings. The pump, which consumes power _Wpump , compresses the liquid from state 3 tostate 4, the state at which it enters the vapor generator. Heat energy at the rate _QH is supplied tothe vapor generator from the energy source to produce vapor at state 1. Thus, the working fluidexecutes a cycle, in that an element of the working fluid initially at state 1 is periodically returnedto that state through the series of thermodynamic processes as it flows through the varioushardware components. The net power produced, _Wnet , is the algebraic sum of the positive turbinepower _Wturbine and the negative pump power _Wpump .

Example 48.1. Consider the power cycle shown in Fig. 48.1(b) and let water be the workingfluid. The mass flow rate _m through each component is 100 kg/h, the turbine inlet pressure P1 is1000 kPa, and turbine inlet temperature T1 is 480±C . The condenser pressure is 7 kPa andsaturated liquid leaves the condenser. The processes through the turbine and the pump areisentropic (adiabatic and reversible, constant entropy). The pressure drop in the flow direction isassumed to be negligible in both the steam generator and the condenser as well as in theconnecting lines. Compute _Wnet , _QH , ´ , and _QL .Solution. Table 48.1 lists the properties at each state and Fig. 48.2 shows the temperature (T )versus entropy (s) diagram for the cycle. Property values were obtained from Steam Tables byKeenan et al. [1978]. Evaluation of properties using such tables is covered in most basic textbookson engineering thermodynamics, for example, Wark [1983] and Van Wylen et al. [1994].Properties at the various states were established as follows. State 1 is in the superheat region andthe values of entropy and enthalpy were obtained from the superheat table of Steam Tables at thenoted values of P1 and T1 . Also, since s2 is equal to s1 ,

s2 = sf + x2(sg ¡ sf ) = 7:7055 = 0:5592 + x2(8:2758¡ 0:5592)

This yields the value for the quality x2 as 0:9224 and allows h2 to be calculated as

h2 = hf + x2(hg ¡ hf ) = 163:4 + 0:9261(2572:5¡ 163:4) = 2394:4 kJ=kg

In these equations, quantities with f and g subscripts were obtained from the saturation table ofSteam Tables at P2 . The value of enthalpy at state 4 was determined by first computing 3w4 , thework per unit mass for the process through the pump, using the expression for the reversiblesteady-flow work with negligible kinetic and potential energy changes [Wark, 1983]:

It is useful to consider cycles for which the energy source and the surroundingtemperaturesdenoted respectively as TH and TL are uniform. The maximum thermal efficiencyany power cycle can have while operating between a source and its surroundings, each at auniform temperature, is that for a totally reversible thermodynamic cycle (a Carnot cycle, forexample) and is given by the expression

´max = (TH ¡ TL)=TH (48:6)


where the specific volume v is assumed constant at v3 since a liquid is pumped. With v3 obtainedfrom Steam Tables as vf at P3 ,

3w4 = ¡0:001 007(1000¡ 7) = ¡1:00 kJ=kg

Writing Eq. (48.2) for the adiabatic process from state 3 to state 4,

h4 = h3 ¡ 3w4 = 163:39 ¡ (¡1:00) = 164:4 kJ=kg

Proceeding with the solution for _Wnet ,

_Wnet = _Wturbine + _Wpump = _m1w2 + _m3w4 = _m(h1 ¡ h2) + _m3w4

= 100:0(3435:2¡ 2394:4) + 100:0(¡1:00) = 103 980 kW

where 1w2 was obtained by writing Eq. (48.2) between states 1 and 2. Next, _QH is determined bywriting Eq. (48.1) for a control volume enclosing the steam generator and noting that there is nopower transmitted across its surface. Equation (48.1) reduces to

_QH = _mh1 ¡ _mh4 = 100:0(3435:2 ¡ 164:4) = 327 080 kW

To find ´ , substitution into Eq. (48.4) yields

´ = 103 980=327 080 = 0:318 or 31:8%

The solution for _QL can be obtained by either of two approaches. First, by Eq. (48.5),

j _QL j = _QH ¡ _Wnet = 327 080 ¡ 103 980 = 223 100 kW

The solution is also obtained by writing the first law for the process between state 2 and state 3.The result is

j _QL j = j _m(h2 ¡ h3)j = j100(163:4¡ 2394:4)j = 223 100 kW

The cycle in this example is known as the Rankine cycle with superheat. Modified forms of thiscycle are widely used to provide shaft power to drive electric generators in steam-electric powerplants and other power applications.


2 7 39 0.9261 7.7055 2394.4 Liquid-vapor mixture3 7 39 0 163.4 Saturated liquid

4 1000 * 164.4 Subcooled liquid

*Not applicable

Figure 48.2 Temperature-entropy diagram for the steam power cycle in Example 48.1.

Example 48.2. Let _QH in Example 48.1 be supplied from a high-temperature source at a fixedtemperature of 500±C and let the surrounding temperature be 20±C . Find the maximum thermalefficiency a cycle could have while operating between these regions and compare this value with ´calculated in Example 48.1.Solution. Equation (48.6) gives the expression for maximum thermal efficiency:

´max = (TH ¡ TL)=TH = [(500 + 273) ¡ (20 + 273)]=[500 + 273] = 0:621 or 62:1%

State Pressure,kPa

Temperature,°C

Quality,kg/kg

Entropy,kJ/kg K

Enthalpy,kJ/kg

Condition

1 1000 480 * 7.7055 3435.2 Superheated vapor

Table 48.1 Properties at Cycle States for Example 48.1


compared to 31.8% for Example 48.1. The maximum value of cycle thermal efficiency was notrealized because of the inherent irreversibilities associated with heat transfer across finitetemperature differences in the heat reception and heat rejection processes for the cycle.

48.2 Refrigeration CyclesThe function of a refrigeration cycle is to cause heat energy to continuously flow from alow-temperature region to a region at a higher temperature. The operation of a refrigeration cycle isillustrated in Fig. 48.3(a), in which heat energy flows at the rate _QL from the low-temperaturerefrigerated region, heat is rejected at the rate _QH to the higher-temperature surroundings, andpower _Wnet is required. From Eq. (48.3) these are related as

j _QH j = _QL + j _Wnet j (48:7)

The performance parameter for conventional refrigeration cycles is termed coefficient ofperformance and is defined as

¯ = _QL =j _Wnet j (48:8)

The maximum value ¯ can have when regions at uniform temperature TH and TL are considered isagain derived from consideration of totally reversible cycles [Wark, 1983]. The expression, interms of absolute temperatures, is

¯max = TL=(TH ¡ TL) (48:9)

Figure 48.3 Descriptions of refrigeration cycles: (a) Refrigeration cycle operation. (b) The simple vaporcompression refrigeration cycle.


Figure 48.3(b) illustrates a simple vapor-compression refrigeration cycle. The compressorreceives the refrigerant (working fluid) in the vapor phase at low pressure, state 1, and compressesit to state 2, where P2 > P1 . Cooling at the condenser by means of a liquid or air coolant causesthe vapor to condense to a liquid, state 3, after which it passes through a throttling device to theevaporator pressure. The refrigerant is a mixture of saturated liquid and saturated vapor at state 4.The liquid in the evaporator undergoes a phase change to vapor that is caused by the transfer ofheat energy from the refrigerated region. The refrigerant leaves the evaporator as vapor at state 1,completing its thermodynamic cycle. The cycle illustrated in Fig. 48.3(b) is the basis for practicalrefrigeration cycles.

Example 48.3. A simple vapor compression refrigeration cycle, Fig. 48.3(b), has a refrigeratingcapacity of three tons (36 000 Btu=h ) and operates with R134a as the refrigerant. The temperatureof the refrigerated region is 15±F and the surroundings are at 90±F . Saturated vapor leaves theevaporator at 5±F and is compressed isentropically by the compressor to 150 psia. The refrigerantleaves the condenser as saturated liquid at 150 psia and flows through the throttling device to thecondenser, in which the temperature is uniform at 5±F . Determine _Wnet , ¯ , and the maximumcoefficient of performance a refrigerator could have while operating between the refrigeratedregion and the surroundings.Solution. Figure 48.4 shows the T-s diagram for the cycle and the temperatures of the two regions.Table 48.2 lists values for the various properties obtained for R134a from the ASHRAE Handbook[1993] at the four states. The mass rate of flow is determined by applying Eq. (48.1) to theevaporator. The result is

_Q + _mh1 = _mh2 = 36 000 + _m(46:78) = _m(103:745)

_m = 632:0 lbm=h

To find _Wnet , application of Eq. (48.1) to the compressor yields

_Wnet = 1_W2 = _m(h1 ¡ h2) = 632:0(103:745 ¡ 120:3) = ¡10 460 Btu=h

To find ¯ , Eq. (48.8) yields

¯ = 36 000=10 460 = 3:44

The solution for maximum coefficient of performance is obtained by applying Eq. (48.9) asfollows:

¯max = [15 + 460]=[(90 + 460) ¡ (15 + 460)] = 6:33

Irreversibilities present due to finite temperature differences associated with the heat transferprocesses and the irreversibility related to the throttling process cause ¯ to be less than ¯max .


Table 48.2 Properties at Cycle States for Example 48.3

State Pressure,psia

Temperature, °F Quality,lbm/lbm

Entropy,Btu/lbm R

Enthalpy,Btu/lbm

Condition

1 23.767 5 1.00 0.22470 103.745 Saturated vapor2 150 118.7 * 0.22470 120.3 Superheated vapor3 150 105.17 0.0 0.09464 46.78 Saturated liquid4 23.767 5 0.368 0.10210 46.78 Liquid-vapor mixture

*Not applicable

Defining Terms

Control volume: A region specified by a control surface through which mass flows.First law of thermodynamics: An empirical law that in its simplest form states that energy in its

Figure 48.4 Temperature-entropy diagram for the refrigeration cycle in Example 48.3.


various forms must be conserved.Heat energy: Energy that is transferred across a control surface solely because of a temperature

difference between the control volume and its surroundings. This form of energy transfer isfrequently referred to simply as heat transfer.

Irreversibilities: Undesirable phenomena that reduce the work potential of heat energy. Suchphenomena include friction, unrestrained expansions, and heat transfer across a finitetemperature difference.

Steady flow: A condition that prevails in a flow process after all time transients related to theprocess have died out.

Working fluid: The substance that is contained within the apparatus in which the cycle isexecuted. The substance undergoes the series of processes that constitute the cycle.

References

ASHRAE. 1993. Refrigeration systems and applications. ASHRAE Handbook, I-P Edition.American Society of Heating, Refrigeration and Air-Conditioning Engineers, Atlanta, GA.

Keenan, J. H., Keys, F. G., Hill, P. G., and Moore, J. G. 1978. Steam Tables, SI Units. John Wiley& Sons, New York.

Van Wylen, G., Sonntag, R., and Borgnakke, C. 1994. Fundamentals of ClassicalThermodynamics, 4th ed. John Wiley & Sons, New York.

Wark, K. 1983. Thermodynamics, 4th ed. McGraw-Hill, New York.

Further Information

Proceedings of the American Power Conference, Illinois Institute of Technology, Chicago, IL.Published annually.

Stoecker, W. F. and Jones, J. W. 1982. Refrigeration and Air Conditioning, 2nd ed. McGraw-Hill,New York.

Threlkeld, J. L. 1970. Mechanical vapor compression refrigeration cycles (chapter 3). ThermalEnvironmental Engineering, 2nd ed. Prentice Hall, Englewood Cliffs, NJ.


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Mechanical Engineering Handbook - Thermodynamic Cycles