Moran M.J., Shapiro H.N. Fundamentals of Engineering Thermodynamics

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10.24 A window-mounted air conditioner supplies 19 m

/min of

air at 158C, 1 bar to a room. Air returns from the room to the

evaporator of the unit at 228C. The air conditioner operates

at steady state on a vapor-compression refrigeration cycle

with Refrigerant-22 entering the compressor at 4 bar, 108C.

Saturated liquid refrigerant at 9 bar leaves the condenser. The

compressor has an isentropic efficiency of 70%, and refrigerant

exits the compressor at 9 bar. Determine the compressor

power, in kW, the refrigeration capacity, in tons, and the

coefficient of performance.

Fig. P10.24

10.25 A vapor-compression refrigeration system for a

household refrigerator has a refrigerating capacity of 1000

Btu/h. Refrigerant enters the evaporator at 2108F and exits

at 08F. The isentropic compressor efficiency is 80%. The

refrigerant condenses at 958F and exits the condenser

subcooled at 908F. There are no significant pressure drops in

the flows through the evaporator and condenser. Determine

the evaporator and condenser pressures, each in lbf/in.

, the

mass flow rate of refrigerant, in lb/min, the compressor

power input, in horsepower, and the coefficient of

performance for (a) Refrigerant 134a and (b) propane as the

working fluid.

10.26 A vapor-compression air conditioning system operates

at steady state as shown in Fig. P10.26. The system maintains

a cool region at 608F and discharges energy by heat transfer

to the surroundings at 908F. Refrigerant 134a enters the

compressor as a saturated vapor at 408F and is compressed

adiabatically to 160 lbf/in.

The isentropic compressor

efficiency is 80%. Refrigerant exits the condenser as a

saturated liquid at 160 lbf/in.

The mass flow rate of the

refrigerant is 0.15 lb/s. Kinetic and potential energy changes

are negligible as are changes in pressure for flow through

the evaporator and condenser. Determine

(a) the power required by the compressor, in Btu/s.

(b) the coefficient of performance.

expansion valve, each in Btu/s.

(d) the rates of exergy destruction and exergy transfer

accompanying heat transfer, each in Btu/s, for a control volume

comprising the evaporator and a portion of the cool region

such that heat transfer takes place at T

5 5208R (608F).

(e) the rates of exergy destruction and exergy transfer

accompanying heat transfer, each in Btu/s, for a control volume

enclosing the condenser and a portion of the surroundings

such that heat transfer takes place at T

5 5508R (908F).

Let T

5 5508R.

10.27 A vapor-compression refrigeration cycle with Refrigerant

134a as the working fluid operates with an evaporator

temperature of 508F and a condenser pressure of 180 lbf/in.

Saturated vapor enters the compressor. Refrigerant enters

the condenser at 1408F and exits as saturated liquid. The

cycle has a refrigeration capacity of 5 tons. Determine

(a) the refrigerant mass flow rate, in lb/min.

(b) the compressor isentropic efficiency.

(d) the coefficient of performance.

Plot each of the quantities calculated in parts (b) through

(d) for compressor exit temperatures varying from 1308F

to 1408F.

Cascade and Multistage Systems

10.28 A vapor-compression refrigeration system operates

with the cascade arrangement of Fig. 10.9. Refrigerant 22

is the working fluid in the high-temperature cycle and

Refrigerant 134a is used in the low-temperature cycle. For

the Refrigerant 134a cycle, the working fluid enters the

compressor as saturated vapor at 2308F and is compressed

isentropically to 50 lbf/in.

Saturated liquid leaves the

intermediate heat exchanger at 50 lbf/in.

and enters the

expansion valve. For the Refrigerant 22 cycle, the working

fluid enters the compressor as saturated vapor at a

temperature 58F below that of the condensing temperature

of the Refrigerant 134a in the intermediate heat exchanger.

The Refrigerant 22 is compressed isentropically to 250 lbf/

in.

Saturated liquid then enters the expansion valve at 250

lbf/in.

The refrigerating capacity of the cascade system is

20 tons. Determine

(a) the power input to each compressor, in Btu/min.

(b) the overall coefficient of performance of the cascade cycle.

exchanger, in Btu/min. Let T

5 808F, p

5 14.7 lbf/in.

Condenser

Evaporator

Compressor

out

Surroundings T

= 550°R (90°F)

Cool region T

= 520°R (60°F)

= 80%

Saturated vapor

= 40°F

= p

= 160 lbf/in.

Saturated

liquid

= p

Expansion

valve

Fig. P10.26

Problems: Developing Engineering Skills 623

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624 Chapter 10

Refrigeration and Heat Pump Systems

10.29 A vapor-compression refrigeration system uses the

arrangement shown in Fig. 10.10 for two-stage compression

with intercooling between the stages. Refrigerant 134a is

the working fluid. Saturated vapor at 2308C enters the

first compressor stage. The flash chamber and direct

contact heat exchanger operate at 4 bar, and the condenser

pressure is 12 bar. Saturated liquid streams at 12 and 4 bar

enter the high- and low-pressure expansion valves,

respectively. If each compressor operates isentropically

and the refrigerating capacity of the system is 10 tons,

determine

(a) the power input to each compressor, in kW.

(b) the coefficient of performance.

10.30 Figure P10.30 shows a two-stage vapor-compression

refrigeration system with ammonia as the working fluid. The

system uses a direct contact heat exchanger to achieve

intercooling. The evaporator has a refrigerating capacity of

30 tons and produces 2208F saturated vapor at its exit. In

the first compressor stage, the refrigerant is compressed

adiabatically to 80 lbf/in.

, which is the pressure in the direct

contact heat exchanger. Saturated vapor at 80 lbf/in.

enters

the second compressor stage and is compressed adiabatically

to 250 lbf/in.

Each compressor stage has an isentropic

efficiency of 85%. There are no significant pressure drops as

the refrigerant passes through the heat exchangers. Saturated

liquid enters each expansion valve. Determine

(a) the ratio of mass flow rates, m

(b) the power input to each compressor stage, in horsepower.

(d) Plot each of the quantities calculated in parts (a)–(c)

versus the direct-contact heat exchanger pressure ranging

from 20 to 200 lbf/in.

Discuss.

Condenser

Evaporator

Expansion

valve

Direct contact

heat exchanger

Expansion

valve

= 30 tons

out

Comp

Fig. P10.30

Condenser

Evaporator

Direct

contact

heat

exchanger

Evaporator

out

Expansion

valve

Expansion

valve

in,2

= 10 tons

in,1

= 5 tons

Compressor

c,2

Compressor

c,1

Fig. P10.31

1 enters compressor 1 at 18 lbf/in.

and exits at 70 lbf/in.

Evaporator 2 operates at 70 lbf/in.

, with saturated vapor

exiting at state 8. The condenser pressure is 200 lbf/in.

, and

saturated liquid refrigerant exits the condenser. Each

compressor stage has an isentropic efficiency of 80%. The

refrigeration capacity of each evaporator is shown on the

figure. Sketch the T–s diagram of the cycle and determine

(a) the temperatures, in 8F, of the refrigerant in each

evaporator.

(b) the power input to each compressor stage, in horsepower.

10.32 Figure P10.32 shows the schematic diagram of a vapor-

compression refrigeration system with two evaporators using

Refrigerant 134a as the working fluid. This arrangement is

used to achieve refrigeration at two different temperatures

with a single compressor and a single condenser. The low-

temperature evaporator operates at 2188C with saturated

vapor at its exit and has a refrigerating capacity of 3 tons. The

higher-temperature evaporator produces saturated vapor at

3.2 bar at its exit and has a refrigerating capacity of 2 tons.

Compression is isentropic to the condenser pressure of 10 bar.

There are no significant pressure drops in the flows through

the condenser and the two evaporators, and the refrigerant

leaves the condenser as saturated liquid at 10 bar. Calculate

(a) the mass flow rate of refrigerant through each evaporator,

in kg/min.

(b) the compressor power input, in kW.

through the condenser, in kW.

10.31 Figure P10.31 shows a two-stage, vapor-compression

refrigeration system with two evaporators and a direct contact

heat exchanger. Saturated vapor ammonia from evaporator

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Condenser

Evaporator

out

Expansion

valve

Expansion

valve

Expansion

valve

in,2

= 2 tons

in,1

= 3 tons

Compressor

Fig. P10.32

exchanger, entering the expansion valve at 18 bar. If the mass

flow rate of refrigerant is 12 kg/min, determine

(a) the refrigeration capacity, in tons of refrigeration.

(b) the compressor power input, in kW.

Discuss possible advantages and disadvantages of this

arrangement.

Vapor-Compression Heat Pump Systems

10.34 Figure P10.34 gives data for an ideal vapor-compression

heat pump cycle operating at steady state with Refrigerant

134a as the working fluid. The heat pump provides heating

at a rate of 15 kW to maintain the interior of a building at

208C when the outside temperature is 58C. Sketch the T–s

diagram for the cycle and determine the

(a) temperatures at the principal states of the cycle, each in 8C.

(b) the power input to the compressor, in kW.

(d) the coefficient of performance for a Carnot heat pump

cycle operating between reservoirs at the building interior and

outside temperatures, respectively.

Compare the coefficients of performance determined in (c)

and (d). Discuss

10.33 An ideal vapor-compression refrigeration cycle is modified

to include a counterflow heat exchanger, as shown in Fig.

P10.33. Ammonia leaves the evaporator as saturated vapor at

1.0 bar and is heated at constant pressure to 58C before

entering the compressor. Following isentropic compression to

18 bar, the refrigerant passes through the condenser, exiting

at 408C, 18 bar. The liquid then passes through the heat

Condenser

Evaporator

Compressor

Heat exchanger

out

Expansion

valve

Fig. P10.33

Problems: Developing Engineering Skills 625

Condenser

Evaporator

Compressor

Saturated liquid

Saturated vapor

out

= 15 kW

Building interior T

= 20°C = 293 K

= 5°C = 278 K

Outside

Expansion

valve

4 1

State

244.09

268.97

93.42

h (kJ/kg)p (bar)

2.4

Fig. P10.34

10.35 Refrigerant 134a is the working fluid in a vapor-compression

heat pump system with a heating capacity of 60,000 Btu/h. The

condenser operates at 200 lbf/in.

, and the evaporator temperature

is 08F. The refrigerant is a saturated vapor at the evaporator exit

and a liquid at 1108F at the condenser exit. Pressure drops in the

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626 Chapter 10

Refrigeration and Heat Pump Systems

flows through the evaporator and condenser are negligible. The

compression process is adiabatic, and the temperature at the

compressor exit is 1808F. Determine

(a) the mass flow rate of refrigerant, in lb/min.

(b) the compressor power input, in horsepower.

(d) the coefficient of performance.

10.36 Refrigerant 134a is the working fluid in a vapor-

compression heat pump that provides 35 kW to heat a dwelling

on a day when the outside temperature is below freezing.

Saturated vapor enters the compressor at 1.6 bar, and saturated

liquid exits the condenser, which operates at 8 bar. Determine,

for isentropic compression

(a) the refrigerant mass flow rate, in kg/s.

(b) the compressor power, in kW.

Recalculate the quantities in parts (b) and (c) for an

isentropic compressor efficiency of 75%.

10.37 An office building requires a heat transfer rate of 20

kW to maintain the inside temperature at 218C when the

outside temperature is 08C. A vapor-compression heat pump

with Refrigerant 134a as the working fluid is to be used to

provide the necessary heating. The compressor operates

adiabatically with an isentropic efficiency of 82%. Specify

appropriate evaporator and condenser pressures of a cycle

for this purpose assuming DT

cond

5 DT

evap

5 108C, as shown

in Figure P10.37. The states are numbered as in Fig. 10.13.

The refrigerant exits the evaporator as saturated vapor and

exits the condenser as saturated liquid at the respective

pressures. Determine the

(a) mass flow rate of refrigerant, in kg/s.

(b) compressor power, in kW.

coefficient of performance for a Carnot heat pump cycle

operating between reservoirs at the inside and outside

temperatures, respectively.

compression heat pump be used to develop the process

heating using a wastewater stream at 1258F as the lower-

temperature source. Figure P10.39 provides data for this

cycle operating at steady state. The compressor isentropic

efficiency is 80%. Sketch the T–s diagram for the cycle and

determine the

(a) specific enthalpy at the compressor exit, in Btu/lb.

(b) temperatures at each of the principal states, in 8F.

(d) compressor power, in Btu/h.

(e) coefficient of performance and compare with the

coefficient of performance for a Carnot heat pump cycle

operating between reservoirs at the process temperature and

the wastewater temperature, respectively.

21°C

0°C

⌬T

cond

⌬T

evap

Fig. P10.37

Condenser

Evaporator

Compressor

␩

= 80%

out

= 3 ⫻ 10

Btu/h

Process T

= 170°F

= 125°F

Wastewater

Expansion

valve

4 1

State

116.74

76.11

180

400

180

(lbf/in.

)

(Btu/lb)

Fig. P10.39

10.38 Repeat the calculations of Problem 10.37 for Refrigerant

22 as the working fluid. Compare the results with those of

Problem 10.37 and discuss.

10.39 A process requires a heat transfer rate of 3 3 10

Btu/h

at 1708F. It is proposed that a Refrigerant 134a vapor-

10.40 A vapor-compression heat pump with a heating capacity

of 500 kJ/min is driven by a power cycle with a thermal

efficiency of 25%. For the heat pump, Refrigerant 134a is

compressed from saturated vapor at 2108C to the condenser

pressure of 10 bar. The isentropic compressor efficiency is

80%. Liquid enters the expansion valve at 9.6 bar, 348C. For

the power cycle, 80% of the heat rejected is transferred to

the heated space.

(a) Determine the power input to the heat pump compressor,

in kW.

(b) Evaluate the ratio of the total rate that heat is delivered

to the heated space to the rate of heat input to the power

cycle. Discuss.

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10.41 Refrigerant 134a enters the compressor of a vapor-

compression heat pump at 15 lbf/in.

, 08F and is compressed

adiabatically to 160 lbf/in.

, 1608F. Liquid enters the expansion

valve at 160 lbf/in.

, 958F. At the valve exit, the pressure is

15 lbf/in.

(a) Determine the isentropic compressor efficiency.

(b) Determine the coefficient of performance.

power input, in Btu per lb of refrigerant flowing. Discuss.

Let T

5 4808R.

10.42 A geothermal heat pump operating at steady state with

Refrigerant-22 as the working fluid is shown schematically in Fig.

P10.42. The heat pump uses 558F water from wells as the thermal

source. Operating data are shown on the figure for a day in which

the outside air temperature is 208F. Assume adiabatic operation

of the compressor. For the heat pump, determine

(a) the volumetric flow rate of heated air to the house, in

/min.

(b) the isentropic compressor efficiency.

(d) the coefficient of performance.

(e) the volumetric flow rate of water from the geothermal

wells, in gal/min.

For T

5 208F, perform a full exergy accounting of the

compressor power input, and devise and evaluate a second

law efficiency for the heat pump system.

10.44 Air enters the compressor of a Brayton refrigeration

cycle at 100 kPa, 270 K. The compressor pressure ratio is 3,

and the temperature at the turbine inlet is 315 K. The

compressor and turbine have isentropic efficiencies of 82%

and 85%, respectively. Determine the

(a) net work input, per unit mass of air flow, in kJ/kg.

(b) exergy accounting of the net power input, in kJ per kg

of air flowing. Discuss.

Let T

5 315 K.

10.45 Plot the quantities calculated in parts (a) through (c) of

Problem 10.43 versus the compressor pressure ratio ranging

from 3 to 6. Repeat for compressor and turbine isentropic

efficiencies of 90%, 85%, and 80%.

10.46 An ideal Brayton refrigeration cycle has a compressor

pressure ratio of 6. At the compressor inlet, the pressure and

temperature of the entering air are 20 lbf/in.

and 4608R. The

temperature at the inlet of the turbine is 7008R. For a

refrigerating capacity of 15 tons, determine

(a) the mass flow rate, in lb/min.

(b) the net power input, in Btu/min.

10.47 Reconsider Problem 10.46, but include in the analysis

that the compressor and turbine have isentropic efficiencies

of 78% and 92%, respectively.

10.48 The table below provides steady-state operating data

for an ideal Brayton refrigeration cycle with air as the working

fluid. The principal states are numbered as in Fig. 10.15. The

volumetric flow rate at the turbine inlet is 0.4 m

/s. Sketch

the T–s diagram for the cycle and determine the

(a) specific enthalpy, in kJ/kg, at the turbine exit.

(b) mass flow rate, in kg/s.

(d) refrigeration capacity, in kW.

(e) coefficient of performance.

State p (kPa) T (K) h (kJ/kg) p

1 140 270 270.11 0.9590

2 420 — 370.10 2.877

3 420 320 320.29 1.7375

4 140 — ? —

10.49 Air enters the compressor of a Brayton refrigeration

cycle at 100 kPa, 260 K, and is compressed adiabatically to

300 kPa. Air enters the turbine at 300 kPa, 300 K, and

expands adiabatically to 100 kPa. For the cycle

(a) determine the net work per unit mass of air flow, in kJ/kg,

and the coefficient of performance if the compressor and

turbine isentropic efficiencies are both 100%.

(b) plot the net work per unit mass of air flow, in kJ/kg, and

the coefficient of performance for equal compressor and

turbine isentropic efficiencies ranging from 80 to 100%.

10.50 The Brayton refrigeration cycle of Problem 10.43 is

modified by the introduction of a regenerative heat

exchanger. In the modified cycle, compressed air enters

the regenerative heat exchanger at 350 K and is cooled

Condenser

Return air

from house

Heated air

to house

70°F

Evaporator

Condenser

out

= 4.2 tons

Water from

geothermal wells

enters at 55°F

Water

discharged

at 45°F

Expansion

valve

110°F

= 70 lbf/in.

= 40°F

= 70 lbf/in.

= 180 lbf/in.

= 140°F

= 180 lbf/in.

= 75 °F

Fig. P10.42

Problems: Developing Engineering Skills 627

Gas Refrigeration Systems

10.43 Air enters the compressor of an ideal Brayton

refrigeration cycle at 100 kPa, 300 K. The compressor pressure

ratio is 3.75, and the temperature at the turbine inlet is 350 K.

Determine the

(a) net work input, per unit mass of air flow, in kJ/kg.

(b) refrigeration capacity, per unit mass of air flow, in kJ/kg.

(d) coefficient of performance of a Carnot refrigeration

cycle operating between thermal reservoirs at T

5 300 K

and T

5 350 K, respectively.

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628 Chapter 10 Refrigeration and Heat Pump Systems

to 320 K before entering the turbine. Determine, for the

modified cycle,

(a) the lowest temperature, in K.

(b) the net work input per unit mass of air flow, in kJ/kg.

(d) the coefficient of performance.

10.51 Reconsider Problem 10.50, but include in the analysis

that the compressor and turbine have isentropic efficiencies

of 85 and 88% respectively. Answer the same questions as

in Problem 10.50.

10.52 Plot the quantities calculated in parts (a) through (d) of

Problem 10.50 versus the compressor pressure ratio ranging

from 4 to 7. Repeat for equal compressor and turbine isentropic

efficiencies of 95%, 90%, and 80%.

10.53 Consider a Brayton refrigeration cycle with a regenerative

heat exchanger. Air enters the compressor at 4808R, 15 lbf/in.

and is compressed isentropically to 40 lbf/in.

Compressed

air enters the regenerative heat exchanger at 5408R and is

cooled to 4808R before entering the turbine. The expansion

through the turbine is isentropic. If the refrigeration capacity

is 15 tons, calculate

(a) the volumetric flow rate at the compressor inlet, in ft

/min.

(b) the coefficient of performance.

10.54 Reconsider Problem 10.53, but include in the analysis

that the compressor and turbine each have isentropic

efficiencies of 88%. Answer the same questions for the

modified cycle as in Problem 10.53.

10.55 Air at 2 bar, 380 K is extracted from a main jet engine

compressor for cabin cooling. The extracted air enters a heat

exchanger where it is cooled at constant pressure to 320 K

through heat transfer with the ambient. It then expands

adiabatically to 0.95 bar through a turbine and is discharged

into the cabin. The turbine has an isentropic efficiency of

75%. If the mass flow rate of the air is 1.0 kg/s, determine

(a) the power developed by the turbine, in kW.

(b) the rate of heat transfer from the air to the ambient, in kW.

10.56 Air at 32 lbf/in.

, 6808R is extracted from a main jet

engine compressor for cabin cooling. The extracted air enters

a heat exchanger where it is cooled at constant pressure to

600°R through heat transfer with the ambient. It then

expands adiabatically to 14 lbf/in.

through a turbine and is

discharged into the cabin at 5008R with a mass flow rate of

200 lb/min. Determine

(a) the power developed by the turbine, in horsepower.

(b) the isentropic turbine efficiency.

Btu/min.

10.57 Air within a piston–cylinder assembly undergoes a Stirling

refrigeration cycle, which is the reverse of the Stirling power

cycle introduced in Sec. 9.8.4. At the beginning of the isothermal

compression, the pressure and temperature are 100 kPa and

350 K, respectively. The compression ratio is 7, and the temperature

during the isothermal expansion is 150 K. Determine the

(a) heat transfer for the isothermal compression, in kJ per kg

of air.

(b) net work for the cycle, in kJ per kg of air.

10.58 Air undergoes an Ericsson refrigeration cycle, which is the

reverse of the Ericsson power cycle introduced in Sec. 9.8.4.

Figure P10.58 provides data for the cycle operating at steady

state. Sketch the p–y diagram for the cycle and determine the

(a) heat transfer for the isothermal expansion, per unit mass

of air flow, in kJ/kg.

(b) net work, per unit mass of air flow, in kJ/kg.

Ideal regenerator

Air

Turbine Compressor

out

= 3.5

State

310

270

T (K)p (kPa)

100

350

100

Fig. P10.58

c DESIGN & OPEN-ENDED PROBLEMS: EXPLORING ENGINEERING PRACTICE

10.1D Children may wonder how a household refrigerator

works to keep food cold in a warm kitchen. Prepare a 20-

minute presentation suitable for an elementary school

science class to explain the principles of operation of a

refrigerator. Include instructional aids to enhance your

presentation.

10.2D The object of this project is to select a compact

thermoelectric refrigerator to be shared by you and at least

two other students living in the same residence as you do.

Survey the other students to determine their needs in order

to size the unit. Critically evaluate competing brands. What

type and number of thermoelectric modules are used in the

unit selected, and what is its power requirement? Summarize

your findings in a memorandum.

10.3D In cases involving cardiac arrest, stroke, heart attack, and

hyperthermia, hospital medical staff must move quickly to

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Design & Open-Ended Problems: Exploring Engineering Practice 629

reduce the patient’s body temperature by several degrees. A

system for this purpose featuring a disposable plastic body suit

is described in BIOCONNECTIONS in Sec. 4.9.1. Conduct a

search of the patent literature for alternative ways to achieve

cooling of medically distressed individuals. Consider patents

both granted and pending. Critically evaluate two different

methods found in your search relative to each other and the

body suit approach. Write a report including at least three

references.

10.4D Identify and visit a local facility that uses cold thermal

storage. Conduct a forensic study to determine if the cold

storage system is well-suited for the given application today.

Consider costs, effectiveness in providing the desired cooling,

contribution to global climate change, and other pertinent

issues. If the cold storage system is well suited for the

application, document that. If the cold storage system is not

well suited, recommend system upgrades or an alternative

approach for obtaining the desired cooling. Prepare a

PowerPoint presentation of your findings.

10.5D A vertical, closed-loop geothermal heat pump is under

consideration for a new 50,000-ft

school building. The design

capacity is 100 tons for both heating and cooling. The local

water table is 150 ft, and the ground water temperature is

558F. Specify a ground source heat pump as well as the

number and depth of wells for this application and develop

a layout of vertical wells and piping required by the system.

10.6D Investigate the economic feasibility of using a waste

heat-recovery heat pump for domestic water heating that

employs ventilation air being discharged from a dwelling as

the source. Assume typical hot water use of a family of four

living in a 2200-ft

single-family dwelling in your locale.

Write a report of your findings.

10.7D Food poisoning is on the rise and can be fatal. Many of

those affected have eaten recently at a restaurant, café, or

fast-food outlet serving food that has not been cooled properly

by the food supplier or restaurant food-handlers. To be safe,

foods should not be allowed to remain in the temperature

range where bacteria most quickly multiply. Standard

refrigerators typically do not have the ability to provide the

rapid cooling needed to ensure dangerous levels of bacteria

are not attained. A food processing company supplying a wide

range of fish products to restaurants has requested your

project group to provide advice on how to achieve best cooling

practices in its factory. In particular, you are asked to consider

applicable health regulations, suitable equipment, typical

operating costs, and other pertinent issues. A written report

providing your recommendations is required, including an

annotated list of food-cooling Dos and Don’ts for restaurants

supplied by the company with fish.

10.8D According to researchers, advances in nanomaterial

fabrication are leading to development of tiny thermoelectric

modules that could be used in various applications, including

integrating nanoscale cooling devices within the uniforms

of firefighters, emergency workers, and military personnel;

embedding thermoelectric modules in facades of a building;

and using thermoelectric modules to recover waste heat in

automobiles. Research two applications for this technology

proposed within the past five years. Investigate the technical

readiness and economic feasibility for each concept. Report

your findings in an executive summary and a PowerPoint

presentation with at least three references.

10.9D In induced hypothermia, the temperature of a particular

organ, such as the heart, is lowered to reduce the metabolic

rate during surgery. Cooling is achieved by circulating

blood through a heat exchanger outside the body. When

the cooled blood is reintroduced through blood vessels in

the organ, it cools the organ to the desired temperature.

Develop the preliminary design of a vapor-compression

refrigeration system to cool blood during heart surgery.

Determine the necessary temperature requirements and

specify a refrigerant, operating pressures and temperatures

for the working fluid, and the refrigeration capacity.

10.10D A vapor-compression refrigeration system operating

continuously is being considered to provide a minimum of 80

tons of refrigeration for an industrial refrigerator maintaining

a space at 28C. The surroundings to which the system rejects

energy by heat transfer reach a maximum temperature of 408C.

For effective heat transfer, the system requires a temperature

difference of at least 208C between the condensing refrigerant

and surroundings and between the vaporizing refrigerant and

refrigerated space. The project manager wishes to install a

system that minimizes the annual cost for electricity (monthly

electricity cost is fixed at 5.692 cents for the first 250 kW ? h

and 6.006 cents for any usage above 250 kW ? h). You are

asked to evaluate two alternative designs: a standard vapor-

compression refrigeration cycle and a vapor-compression

refrigeration cycle that employs a power-recovery turbine in

lieu of an expansion valve. For each alternative, consider three

refrigerants: ammonia, Refrigerant 22, and Refrigerant 134a.

Based on electricity cost, recommend the better choice between

the two alternatives and a suitable refrigerant. Other than

electricity cost, what additional factors should the manager

consider in making a final selection? Prepare a written report

including results, conclusions, and recommendations.

10.11D High-performance aircraft increasingly feature

electronics that assist flight crews in performing their duties

and reducing their fatigue. While these electronic devices

improve aircraft performance, they also add greatly to the

thermal load that must be managed within the aircraft.

Cooling technologies currently used on aircraft are

approaching their limits and other means are being considered,

including vapor-compression refrigeration systems. However,

unlike cooling systems used on earth, systems employed on

aircraft must meet rapidly changing conditions. For instance,

as onboard electronic devices switch on and off, the energy

they emit by heat transfer alters the thermal load; additionally,

the temperature of the air outside the aircraft into which such

waste heat is discarded changes with altitude and flight speed.

Accordingly, for vapor-compression systems to be practical

for aircraft use, engineers must determine if the systems can

quickly adapt to rapidly changing thermal loads and

temperatures. The object of this project is to develop the

preliminary design of a bench-top laboratory set-up with

which to evaluate the performance of a vapor-compression

refrigeration system subject to broadly variable thermal

inputs and changing ambient conditions. Document your

design in a report having at least three references.

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630

Systems often involve mixtures of several different entities; mixture rules for gases are

provided in Sec. 11.8.

Image Source/Getty Images, Inc.

ENGINEERING CONTEXT As seen in previous chapters, application of thermodynamic principles to

engineering systems requires data for specific internal energy, enthalpy, entropy, and other properties. The

objective of this chapter is to introduce thermodynamic relations that allow u, h, s, and other thermodynamic

properties of simple compressible systems to be evaluated from data that are more readily measured. Pri-

mary emphasis is on systems involving a single chemical species such as water or a mixture such as air. An

introduction to general property relations for mixtures and solutions is also included.

Means are available for determining pressure, temperature, volume, and mass experimentally. In addition,

the relationships between the specific heats c

and c

and temperature at relatively low pressure are acces-

sible experimentally. Values for certain other thermodynamic properties also can be measured without great

difficulty. However, specific internal energy, enthalpy, and entropy are among those properties that are not

easily obtained experimentally, so we resort to computational procedures to determine values for these.

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Thermodynamic

Relations

When you complete your study of this chapter, you will be able to...

calculate p–y–T data using equations of state involving two or more constants.

demonstrate understanding of exact differentials involving properties and utilize the

property relations developed from the exact differentials summarized in Table 11.1.

evaluate Du, Dh, and Ds, using the Clapeyron equation when considering phase change,

and using equations of state and specific heat relations when considering single

phases.

demonstrate understanding of how tables of thermodynamic properties are constructed.

evaluate Dh and Ds using generalized enthalpy and entropy departure charts.

utilize mixture rules, such as Kay’s Rule, to relate pressure, volume, and temperature of

mixtures.

apply thermodynamic relations to multicomponent systems.

LEARNING OUTCOMES

631

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632 Chapter 11

Thermodynamic Relations

11.1 Using Equations of State

An essential ingredient for the calculation of properties such as the specific internal

energy, enthalpy, and entropy of a substance is an accurate representation of the

relationship among pressure, specific volume, and temperature. The p–y–T relation-

ship can be expressed alternatively: There are tabular representations, as exempli-

fied by the steam tables. The relationship also can be expressed graphically, as in

the p–y–T surface and compressibility factor charts. Analytical formulations, called

equations of state, constitute a third general way of expressing the p–y–T relation-

ship. Computer software such as Interactive Thermodynamics: IT also can be used

to retrieve p–y–T data.

The virial equation and the ideal gas equation are examples of analytical equations

of state introduced in previous sections of the book. Analytical formulations of the

p–y–T relationship are particularly convenient for performing the mathematical oper-

ations required to calculate u, h, s, and other thermodynamic properties. The object

of the present section is to expand on the discussion of p–y–T relations for simple

compressible substances presented in Chap. 3 by introducing some commonly used

equations of state.

11.1.1

Getting Started

Recall from Sec. 3.11 that the virial equation of state can be derived from the principles

of statistical mechanics to relate the p–y–T behavior of a gas to the forces between

molecules. In one form, the compressibility factor Z is expanded in inverse powers

of specific volume as

Z 5 1 1

B1T

C1T

D1T

. . .

(11.1)

The coefficients B, C, D, etc. are called, respectively, the second, third, fourth, etc.

virial coefficients. Each virial coefficient is a function of temperature alone. In prin-

ciple, the virial coefficients are calculable if a suitable model for describing the forces

of interaction between the molecules of the gas under consideration is known. Future

advances in refining the theory of molecular interactions may allow the virial coef-

ficients to be predicted with considerable accuracy from the fundamental properties

of the molecules involved. However, at present, just the first few coefficients can be

calculated and only for gases consisting of relatively simple molecules. Equation 11.1

also can be used in an empirical fashion in which the coefficients become parameters

whose magnitudes are determined by fitting p–y–T data in particular realms of inter-

est. Only a few coefficients can be found this way, and the result is a truncated equa-

tion valid only for certain states.

In the limiting case where the gas molecules are assumed not to interact in any

way, the second, third, and higher terms of Eq. 11.1 vanish and the equation reduces

to Z 5 1. Since Z 5 py

RT, this gives the ideal gas equation of state py 5 R

T. The

ideal gas equation of state provides an acceptable approximation at many states,

including but not limited to states where the pressure is low relative to the critical

pressure and/or the temperature is high relative to the critical temperature of the

substance under consideration. At many other states, however, the ideal gas equation

of state provides a poor approximation.

Over 100 equations of state have been developed in an attempt to improve on

the ideal gas equation of state and yet avoid the complexities inherent in a full

virial series. In general, these equations exhibit little in the way of fundamental

physical significance and are mainly empirical in character. Most are developed

TAKE NOTE...

Using the generalized com-

pressibility chart, virial equa-

tions of state and the ideal

gas model are introduced

in Chap. 3. See Secs. 3.11

and 3.12.

equations of state

virial equation

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