Massoud M. Engineering Thermofluids: Thermodynamics, Fluid Mechanics, and Heat Transfer

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Questions and Problems 131

54. Use the Maxwell relations to show that, for an isentropic process of an ideal

gas, Pv

= constant where

is given as

= c

. [Hint: Use ds = (c

dT/T) + R

(dv/v) = 0 and ds = (c

dT/T) - R (dP/P) = 0. Cancel dT/T and integrate.]

55. Use the equation of state for ideal gases in conjunction with the Maxwell rela-

tions to derive an alternative equation to Equation IIa.3.6 (calculation of the

change in the entropy of a system in a reversible process. [Ans.:

()

21 21

/ln/

scdTTRPP−= −

56. A tank at atmospheric pressure contains two inlets and one outlet port. The

first inlet port has a flow area of 0.05 ft

and the flow area of the second inlet port

is 0.025 ft

. Water enters the first inlet port at 5 ft/s and 100 F. Water enters the

second inlet port at 8 ft/s and 175 F. Water leaves the tank at a rate of 2 lbm/s.

Find the rate of change of the tank water level.

57. We compress air at 1 MPa and 150 C to a pressure of 5 MPa in an isentropic

process. Treat air as an ideal gas and find its temperature at this pressure. [Ans.:

≅ 240 C].

58. A mixture of water and steam is contained in a cylinder equipped with a well-

fitted leak-tight piston of cross sectional area A. At state 1, the mixture is at pres-

sure P

having a steam quality of x

and a volume of V

. Heat is added to the cyl-

inder until the piston just touches the spring. At this stage, the volume of the cyl-

inder content is V

= V

+ ∆V. We keep adding heat to the cylinder. The piston

would travel further to the right and start compressing the spring. We terminate

the heating process when the pressure of the cylinder content reaches P

. Write a

procedure from which P

, T

, and T

can be determined. Assume a linear spring

(F = k

x) with known k

State 1

State 2

State 3

59. Show that for steady-flow, steady state isentropic process of an ideal gas the

work from going from state 1 to state 2 is found from:

])(1[

−

60. Flow enters a heat exchanger at a rate of 25E6 lbm/h and a density of 1.5

slugs/ft

. There are 9000 tubes in this heat exchanger. If flow is distributed

evenly among the tubes, find the flow velocity in each tube. Use a diameter of

0.63 in for all the tubes. [Ans.: 7.38 ft/s].

61. Water at 20 C enters the pipe of Figure (a) at a velocity of 2.5 m/s. The pipe

has an inside diameter of 5 cm. Find the mass flow rate, mass flux, and volumet-

ric flow rate. [Ans. 4.9 kg/s, 2497.5 kg/m

⋅s, 4.9E-3 m

/s].

132 IIa. Thermodynamics: Fundamentals

(a) (b) (c)

62. Show that for the pipe of Figure (b) flow velocity at the exit is given by V

)

63. For the pipe in Figure (c), find mass flow rate (



) and velocity

()V at the

exit of the pipe.

64. The mass flow rate through the core of a 2 × 4 PWR (i.e., 2 hot legs and four

cold legs) is 62.82E6 kg/h. The inside dimeter (I.D.) of the hot leg is 1 m. Esti-

mate the I.D. of the cold leg. Data: T

= 320 C, T

= 288 C, and P = 15.5 MPa.

[Hint: V

= V

65. The core of a nuclear reactor produces 2772 MWth at 2155 psia. The volu-

metric flow rate through the core is given as 122.71E6 GPM. The core outlet

temperature is 604 F. Find the core inlet temperature. [Hint. Guess T

, find T

avg

find

avg

(P, T

avg

), find m



, update T

avg

and continue iteration]. [Ans.: 549 F].

66. Consider the pressurizer of a PWR, having an internal volume of 1500 ft

The pressurizer contains 750 ft

of water and steam at 2000 psia at equilibrium.

Due to a turbine trip, an in-surge of 100 GPM and 600 F enters the pressurizer for

5 minutes. Find the temperature of the water region after termination of the in-

surge. Assume perfect mixing between the in-surge and water in the water region.

Ignore work due to boundary change and heat transfer with the steam region.

[Ans. T

= 662.5 F].

67. The pressurizer of a PWR is at 2250 psia. Water through the letdown line

leaves the pressurizer at a rate of 44 GPM and enters the volume control tank

(VCT) for 30 minutes. If no other process has taken place in either tank, use the

data below and find the change in water level in the two tanks. Assume instanta-

neous and perfect mixing in VCT. Data: V

Pressurizer

= 1500 ft

, (V

water

)

Pressurizer

750 ft

, A

Pressurizer

= 50 ft

, V

VCT

= 1000 ft

, (V

water

)

VCT

= 385 ft

, A

VCT

= 44.3 ft

VCT

= 150 F.

68. A rigid vessel is filled with saturated water and steam. In the left figure, wa-

ter and steam constitute two separate regions while in the right figure the water

and steam make a homogenous mixture. Under what condition can the right fig-

ure represent the left figure?

Questions and Problems 133

Saturated

Water

Saturated

Steam

Saturated

Steam

Saturated

Water

69. We plan to design a 1 MWth solar power station, as shown in the figure. So-

lar radiation heats up the circulating water in the solar collectors, which is then

transferred to a heat reservoir to produce steam. The reservoir is maintained at 1

MPa. Dry, saturated steam, after expansion in the turbine, is cooled down in the

condenser and is pumped back to the reservoir. Find the steady state mass flow

rate of feedwater and of steam. The steam enthalpy at the exit of the turbine is

2300 kJ/kg. List the advantages and drawbacks of this design. Changes in the ki-

netic and potential energies are negligible. [Ans.: 2 kg/s].

Incident

Solar

Radiation

Pump

Heat

Reservoir

Flow

Loop

Condenser

Turbine

Steam

Feedwater

Work

Water

70. Total heat in the primary side of a PWR, as shown in the figure, is 2700

MWth. This amount of heat consists of the fission heat produced in the core and

the reactor coolant pump (RCP) heat. The steam generator is maintained at a pres-

sure of 900 psia. The enthalpy of feedwater entering the steam generator is 430

Btu/lbm. Find the steady state mass flow of the dry saturated steam leaving the

steam generator towards the turbine. [Ans.: 11.7E6 lbm/s].

RPV

RCP

Steam

Core

RCP

RPV

: Cold Leg

: Feedwater

: Hot Leg

: Steam Generator

: Reactor Coolant Pump

: Reactor Pressure Vessel

71. Shown in Figure (a) is the flow path of a BWR vessel. Find the steam mass

flow rate (



) in terms of total reactor core power (



), feedwater enthalpy (h

and thermodynamic properties at vessel pressure.

134 IIa. Thermodynamics: Fundamentals

Downcomer

Lower

Plenum

Core

Separator

Steam

Dome

Feedwater

Steam flow

Recirculatio



Downcomer

Tubesheet

Separator

Steam Dome

Feedwater flow

Steam flow

Recirculation

flow

Tube Bundle

(a) (a and b) (b)

Figure for Problem 71 Figure for Problem 73

[Ans:

)/(

dgg

hhQm −=



72. In Problem 71, defining the recirculation flow rate as

mmr



/= , show that

r = (1 – x

)/x

where x

is the steam quality at the core exit.

73. Shown in Figure (b), is the flow path in the secondary side of a PWR steam

generator. For the steady state operation, find the steam mass flow rate in terms of

the following parameters:

: water enthalpy at the inlet to the tube bundle region

: saturated water enthalpy at the steam generator pressure

: latent heat of vaporization at the steam generator pressure

Core



: total rate of heat transfer in the core,



: mass flow rate in the tube bundle region,

N : number of steam generators (

N )

[Ans.:

{}

fginfiSGCores

hhhmNQm /)()/( −−=



74. A PWR steam generator at steady state operation produces dry, saturated

steam. Pressure in the tube bundle region is 6 MPa (875 psia). The PWR power

plant is equipped with two steam generators and is producing a total electric

power of 810 MWe at a thermal efficiency of 30%. Use a feedwater enthalpy of

1007 kJ/kg (433 Btu/lbm), a recirculation ratio of 3.3 to find:

a) the feedwater flow rate entering the steam generator downcomer, b) the steam

flow rate c) the flow rate entering the tube bundle region, d) the recirculation flow

rate entering the downcomer region, e) water enthalpy entering the tube bundle

region, f) mixture enthalpy at the exit of the tube bundle region, g) the degree of

subcooling at the inlet to the tube bundle region.

75. Start with Equation IIa.6.3 and show that, for rigid control volumes with no

internal heat generation, the first law for control volumes simplifies to:

++ Z

hmWQZ

222



IIa.6.3

Questions and Problems 135

76. Saturated steam enters a turbine at 7 MPa and a rate of 6E3 kg/h. Steam

leaves turbine at 7 bar and x

= 85%. There is a total of 20 kW heat loss from the

turbine. Find the power developed by this turbine. [Ans.: 0.93 MW].

77. Hot water enters the steam generator tubes of a PWR at a rate of 138.5E6

lbm/h, pressure of 2250 psia and temperature of 600 F. Water leaves the tubes at

550 F. Steam is produced in the secondary side at 1000 psia. Find the steam mass

flow rate. [Ans.: 7E6 lbm/h].

78. Find the rate of steam produced in a BWR operating at 1,600 MWth. Water

enters the core from the lower plenum at a rate of 50E6 lbm/h and a temperature

of 526 F. Reactor pressure is 1050 psia. [Ans.: 6E6 lb/h].

79. A high temperature gas-cooled reactor (HTGR) is designed to operate at 330

MWe with a

= 39.23%. Helium enters the reactor at a pressure of 710 psia and

temperature 760 F and leaves at 1,430 F. Find the He flow rate through the core.

= 1.24 Btu/lbm·F. [Ans.: 3.455E6 lbm/h]

80.

For a PWR steam generator, we define the recirculation ratio as



where



is the steam mass flow rate and



is the recirculation mass flow rate. Ex-

press

is terms of core exit average quality (



). [Ans.:

/(1 –

)].

81. Obtain an analytical solution in closed form for the set of mass and energy

equations in Example IIa.8.2. [Hint, since the inlet mass flow rates and enthalpies

as well as the outlet mass flow rate are all uniform with time, the rate of change of

mass in the tank is constant, hence, water level is a linear function of time].

[Ans.: Mass of water in the tank as a function of time is found from

=+−=

)0()()(

tmmmtm



and water temperature in the tank from



³³

−

Cdtetqe

dttpdttp )()(

)(

where p(t) and q(t) are obtained from

)(/)(

tmcmcmmtp

ive

−=

¦¦



, )(/)(

tmhmtq



, and C is

found from the initial condition for water temperature.]

82. Find an analytical solution for Example IIa.8.2 if heat is also added to the tank

at a constant rate of



Btu/s. [Ans.: The only modification is in q(t) which be-

comes

)(/)(

tmhmQtq



83. If in Example IIa.8.2 heat is added to the water at a rate of 2000 Btu/s, find

the time it takes for water to reach saturation at atmospheric pressure.

136 IIa. Thermodynamics: Fundamentals

84. A tank contains 5 ft

of water at 100 F and 1 atm. Heat, at a constant rate of

1,000 Btu/s, is added to the tank in an isobaric parocess. Find the time it takes for

the last drop of water to evaporate. Properties of subcooled water at 1 atm and

100 F are v = 0.01613 ft

/lbm and h = 68.04 Btu/lbm. [Ans.: 4.6 min.]

85. A tank contains 5 ft

of water at 100 F. Heat, at a constant rate of 1,000 Btu/s

is added to the tank in an isobaric process. Find the time it takes for the last drop

of water to evaporate. Solve this problem for three cases. In case 1, the tank pres-

sure is 300 psia. In case 2, the tank pressure is 1200 psia. In case 3, the tank pres-

sure is 2500 psia. a) What conclusion do you reach from this study? b) Assume a

tank cross sectional area of 1 ft

and plot water level as a function of time for all

three cases. [Ans.: a) As shown in Figures IIa.3.1(c) and IIa.3.4, latent heat of

vaporization decreases as pressure increases. Hence, the tank loses water faster at

higher pressures. b) the plot should have the trend shown below:

100

Normalized Water Level

Time

2500 psia

1200 psia

300 psia

86. A tank of 8000 ft

(225 m

) contains a two-phase mixture of water and steam

at P

= 1000 psia (7 MPa) and x

= 28.233%. Heat is now added to the tank,

treated as a single control volume at a rate of:

eQtQ



=)(

where



= 34.4 MW and

= 18 h

–1

. Find the time it takes for pressure to reach

3000 psia (21 MPa). [Hint: At state 2, we have P

and v

= v

]. [Ans.: 5 min].

87. A cylindrical tank has a base diameter of D = 3 ft and a height of H = 7 ft.

The tank contains air at 100 psia and 100 F. The intake valve is now opened to al-

low 100 F water at a constant rate of 10 lbm/s to enter the tank. Assume that no

air is dissolved in the water and no heat transfer takes place at the air - water and

air - wall interfaces. Find air pressure in the tank 62 seconds after the intake valve

is opened. Treat air as an ideal gas and the compression of air as an isentropic

process. The vent valve remains closed. [Ans.: 137.2 psia].

Intake Valve

Vent Valve

Questions and Problems 137

88. A rigid vessel having a volume of V contains air, initially at a pressure of P

and a temperature of T

. An intake valve is now opened to allow pressurized air at

a temperature of T

to enter the vessel. The intake valve is closed when pressure in

the vessel reaches P

. Use the conservation equations of mass and energy as well

as the equation of state to derive a relation for the final temperature. Consider the

process adiabatic and neglect any storage of heat in the tank wall. [Ans.:

)}/()(/{

111222

TTcPPPcTcPT

ipvip

+−= ].

89. A rigid tank has a volume of V = 1.0 ft

and contains air at P

= 14.7 psia and

= 70 F. An admission valve is now opened to allow pressurized air at P

= 100

psia and T

= 70 F enter the tank. The valve is closed when P

= 30 psia. Find T

[Ans.: T

= 161 F].

90. The volume of the water in the secondary side of a PWR steam generator is

130 m

. The power deposited to the water ten minutes after the reactor is shut-

down is 58 MW. If there is no feedwater delivered to the steam generator, find the

time to boil the steam generator dry. The secondary side pressure is 9 MPa.

91. A tank with a volume of V = 2 m

contains air at 3 MPa and 200 C. The vent

valve is now opened. Find the tank pressure and temperature when 1/3 of the air

escapes through the vent valve. The intake valve remains closed.

92. A tank with a volume of 10 m

contains air at 0.1 MPa and 15 C. The intake

valve is opened to allow pressurized air enter the tank at an average mass flow rate

of 0.5 kg/s and temperature of 100 C. If the tank is fully insulated, find the tank

pressure after 60 seconds. The vent valve remains closed. [Ans.: P = 0.55 MPa,

T = 182 C].

93. A tank with a volume of V = 10 m

contains air at 0.1 MPa and 15 C. The in-

take valve is opened to allow pressurized air enter the tank at an average mass

flow rate of 0.5 kg/s and temperature of 100 C. During the charging process, heat

is transferred to the atmosphere at a rate of 0.01(T - T

) Btu/s where T is the air

temperature in the tank and T

= 10 C is temperature of the surroundings. Find the

tank pressure after 60 seconds. The vent valve remains closed. [Ans.: P = 0.53

MPa, T = 168 C].

94. A pressurized rigid vessel having a volume of 10 ft

is filled with air to 600

psia and 185 F. We want to vent this tank so that the final pressure drops to at-

mospheric pressure (14.7 psia). However, we would like to maintain the air tem-

perature in the tank at 185 F throughout the venting process. Find the amount of

heat necessary to accomplish this task. Treat air as an ideal gas with constant spe-

cific heat and ignore changes in the air kinetic and potential energies compared to

its internal energy. [Ans.: 898 Btu].

95. A rigid vessel is depressurized from 3 MPa to 101 kPa through a small vent.

Heat is added in this process to maintain temperature at its initial value. The tank

has a volume of 5 m

and its inventory is an ideal gas. Find the amount of heat re-

quired for accomplishing this task. [Ans.: 14,495 kJ].

138 IIa. Thermodynamics: Fundamentals

96. A cylindrical pressurized vessel is filled with air to 500 psia at an initial tem-

perature of 250 F. The vessel has a diameter of D = 4 ft and total volume of V =

200 ft

. A small, 1 inch vent valve is now opened. Find pressure (P) and tempera-

ture (T) of the gas in the vessel 20 seconds into the venting event. Treat air as an

ideal gas. The tank is fully insulated. The valve discharge coefficient is 0.65.

The intake valve remains closed [Hint: Flow rate through the value should be mul-

tiplied by the specified discharge coefficient]. [Ans.: P = 133 psia, T = 10 F].

97. A cylindrical pressurized vessel is filled with air to 500 psia at an initial tem-

perature of 250 F. The vessel has a diameter of D = 4 ft and total volume of V =

200 ft

. A small, 1 inch vent valve is now opened. Find pressure (P) and tempera-

ture (T) of the gas in the vessel 20 seconds into the venting event. Treat air as an

ideal gas. The rate of heat transfer from the tank to the surroundings is estimated

at 0.008(T - T

) Btu/s where the surrounding is at a temperature of T

= 35 F. The

valve discharge coefficient is 0.65. The intake valve remains closed. [Ans.: P =

132 psia, T = 6 F].

98. The pressurizer of a PWR is a cylindrical tank having a volume of 1500 ft

Initially, the tank is full of a mixture of water and steam. Consider this saturated

mixture to be distributed uniformly in the tank at P

= 2250 psia. The initial steam

quality is 70%. We now start heating the tank but would like to keep pressure at

its initial value of 2250 psia. To achieve this goal, we must simultaneously remove

mass from this tank. If only steam is removed by a valve at the top of the tank and

the kinetic and potential energies are negligible, find the amount of the mass re-

moved and heat added when the last drop of water boils and becomes steam.

99. A small amount of leakeage exists in the steam generator of a PWR operating

at the rated power of

100



. A noble gas escapes the primary side and enters the

secondary side at a fixed rate of



. Find the density of the gas in the secondary

side of the steam generator versus time. The volume of the secondary side is V.

Secondary-side

of the steam

generator

Influx

of Gas

Steam

Efflux

of Gas

Feedwater

Steam

Gas

Density

Control

Volume



[Hint: Since the leak is small, we treat gas as a component. Find

100

(t), density

of the gas at 100% power from V(d

100

/dt) =



–

100

Steam



where

Steam



the steam volumetric flow rate at 100% with

100

(0) = 0].

[Ans.:

(

)

αρ

−

−= 1)(

100

where a =



Steam



and

Steam



/V].

100. A small amount of leak exists in the steam generator of a PWR operating at

the rated power of

100



. A noble gas escapes the primary side and enters the sec-

Questions and Problems 139

secondary side at a fixed rate of



. At

seconds into the operation at nominal

power, we reduce power to 20% of nominal. a) If the primary side and the secon-

dary side pressure remain about the same value as at nominal power, find the

steam flow rate (



) in terms of

100



, and steam mass flow rate at full

power (

100



). b) Find the partial density of the noble gas in the secondary side

(

), at the reduced power of 20%, versus time.

[Ans.:

()

20 100

(') ( ) '

tte

θα

−

=+ =−′ where t’ = t –

′



Steam

′



and

′

Steam

′



/V].

[Note, the above answers assume that the noble gas is stable. If radioactive gases

such as Xe-135 are involved, the Xe buildup in the primary side and decay in the

secondary side must be factored in.]

101. Consider a mixing tank that contains m

kg of water intitially at T

C. This

tank is fed by N feed lines carrying water at various temperatures. The mass flow

rate and enthalpy of water in the feed lines are known functions of time. There are

mixers and N

heaters. The tank is poorly insualted. The rate of heat loss is

given as

C.V.

– T

) where

has units of W/C and T

is temperature of the sur-

roundings. Both

and T

are known functions of time. There are N inlet ports and

M outlet ports. Assume instantaneous and perfect mixing so that water in the tank

can be represented by one control volume.

Instantaneous & Perfect Mixing

Control Volume

loss



iNiN



eeN



a) Use other simplifying assumptions, as used in Section 8, and obtain the govern-

ing differential equations for the control volume mass and enthalpy. b) Assume a

constant specific heat and obtain an analytical solution for the water temperature

leaving the tank in terms of the specified forcing functions. c) Use the definitions

in Chapter VIIe and obtain the solution in explicit, semi-implicit, and fully-

implicit numerical schemes.

Section IIa.9 through IIa.11

102. Is the motion of the sphere a reversible process in the absence of any air re-

sistance and friction?

140 IIa. Thermodynamics: Fundamentals

103. A smooth pipe equipped with an isolation valve connects two tanks contain-

ing air. When the valve is closed P

> P

. We now open the valve until both

tanks reach equilibrium. Is this a reversible process?

104. A pendulum, placed in an enclosure, is operating in a vacuum. The connect-

ing rod is attached to a frictionless joint. Is the motion of this pendulum a reversi-

ble process?

105. An adiabatic and reversible process is an isentropic process. Can an irre-

versible process in which heat is allowed to transfer have no change in entropy?

Clarify your answer. [Ans.: Yes].

106. Consider two heat engines operating between the same heat source and heat

sink in the Carnot cycle. One heat engine uses gas and the other uses water as

working fluid. Which heat engine would have higher thermal efficiency?

107. We want to heat up the contents of a closed system. We may use a paddle

wheel or a heat reservoir. Thermodynamically, which method is preferred?

108. A cylinder, fitted with a frictionless piston, contains saturated steam at a

specified pressure. We now let heat transfer take place from the cylinder to the

surroundings until saturated steam becomes saturated water. Does this constitute a

reversible process? Does this constitute an isentropic process?

109. Heat is added to a cylinder containing air. The cylinder has a volume of 0.12

and initially is at P

= 1 MPa and T

= 50 C. Find a) the air pressure when the

air temperature reaches 150 C, b) the amount of heat added to the cylinder, and c)

the change in the air entropy. [Ans.: 1.31 MPa, 94 kJ, 0.252 kJ/C].

110. A cylinder contains 3 kg of air at P

= 1 bar and T

= 27 C. In a polytropic

compression, the pressure and temperature of the air are raised to P

= 15 bar and

= 227 C. Find the polytropic exponent, the final volume, the amount of com-

pression work delivered to the system, and the amount of heat rejected to the sur-

roundings. [Ans.: 1.23, 0.287 m

, –763 kJ, and –316 kJ].

111. Thermal efficiency of a power plant is calculated as 30%. The electrical out-

put of the plant is 1000 MWe. How much heat is transferred in the heat source to

the working fluid (i.e., MWth)?

112. In a 1000 MW power plant, steam at 1000 F enters the turbine. Pressure in

the primary side of the condenser is 4 psia. Find the least possible amount of heat

rejected to the surroundings.

113. A power plant operates at a thermal efficiency of 32%. The rate of heat

transfer in the heat source is 2700 MW. Find the power produced by the turbine.