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Problems
Chapter 1
0
0
1.1: Two kg of water at 80 C are mixed with 3 (three) kg of water at 30 C in a
constant pressure of 1 atmosphere. Find the increase in the entropy of the total mass
of water due to mixing process ( C p for water = 4.187 kJ/kg-K).
Solution:
Let T f = Final temperature of the mixture.
Since the mixing occurs adiabatically, no heat transfer occurs across the system
boundary. However, heat transfer takes place between the fluid streams only due to
the temperature difference between the fluid streams.
∴ m1 C p (T1 − T f ) = m 2 C p (T f − T2 )
.
.
.
T1 +
m2
T2
.
(1)
m1
⇒ Tf =
.
1+
m2
.
m1
Here,
.
m1 = 2 kg/s
.
m2 = 3 kg/s
C p ( water ) = 4.187 kJ/kg.K
T1 = 80 + 273 = 353K
T2 = 30 + 273 = 303K
From Eq. (1),
T f = 323K
.
Now, entropy change of water with m1 = 2 kg/s at 800 C to attain 323 K is
.
.
T
∆ s1 = m1 C p ln f = −0.7437 kJ / kg − K − s
T1
.
Entropy change of water with m2 = 3 kg/s at 300 C to attain 323 K is
177
.
.
∆ s2 = m2 C p ln
Tf
T2
= 0.8029 kJ / kg - K - s
.
.
Entropy of the total mass of water= ∆ s1 + ∆ s2 = 0.05919 kJ/kg-K-s
1.2: Two identical bodies of constant heat capacity are at the initial temperature
Ti . A
refrigerator operates between these two bodies until one body is cooled to
temperature T2 . If the bodies remain at constant pressure and undergo no change of
phase, show that the minimum amount of work needed to do this is
Wmin
⎡ Ti 2
⎤
= C p ⎢ + T2 − 2Ti ⎥
⎣ T2
⎦
1.3: A reversible heat engine shown in Fig. 1.10, during a cycle of operation draws
5 MJ from 400 K reservoir and does 840 kJ of work. Find the amount and
direction of heat interaction with other reservoirs.
3
2
200K
1
400 K
300K
Q1=5 MJ
Q2
Q3
W= 840 KJ
Fig. 1.10: Figure for problem 1.3
1.4: Show that if two bodies of thermal capacity
are brought to the same temperature
T
ln T =
c1
and
c2
at temperatures
T1
and
T2
by means of a reversible heat engine, then
c1 ln T1 + c2 ln T2
c1 + c2
1.5: An inventor claims to have developed an engine that takes in
107 J
at a
temperature of 400 K , rejects 4 × 10 J at a temperature of 200 K and delivers
3.6 × 106 J of mechanical work. Would you advice-investing money to put the engine
on the market?
6
178
1.6: A mass
m of
a liquid at a temperature
same liquid at temperature
T2 .
T1
is mixed with an equal mass of the
The system is thermally insulated. Show that the
change of the universe is
(T + T ) / 2
1
2mC p ln
2
TT
1 2
Also prove that this is necessarily positive.
C p and at a temperature T1 is put in contact
with a reservoir at a higher temperature T f . The pressure remains constant while the
1.7: A body of constant heat capacity
body comes to equilibrium with the reservoir. Show that the entropy change of the
universe is equal to
⎡ T − Tf
⎛ T − Tf
Cp ⎢ i
− ln ⎜1 + i
Tf
⎢⎣ Tf
⎝
⎞⎤
⎟⎥
⎠ ⎥⎦
Prove that this entropy change is positive.
1.8: An adiabatic vessel contains
2 kg of
250 C . By paddle wheel work
300 C . If the specific heat of water
water at
transfer, the temperature of water is increased to
is assumed constant at 4.187 kJ/kg-K , find the entropy change of the
universe.[Ans: 0.139 kJ/kg]
0
1.9: A resistor of 30 ohms is maintained at a constant temperature of 27 C while a
current of 10 amperes is allowed to flow for 1second. Determine the entropy change
of the resistor and the universe. [ Ans: entropy change of universe=10 J/K]
0
1.10: If the resistor ( See Problem 1.9) initially at 27 C is now insulated and the
same current is passed for same time, determine the entropy change of the resistor
and the universe. The specific heat of resistor is 0.9 kJ/kg-K and mass of resistor is
10 gm .
1.11: The value of
Cp
of a certain substance can be represented by
C p = a + bT .
Determine the heat absorbed and increase in entropy of a mass m of the substance
when its temperature is increased at constant pressure from T1 to T2 .
179
1.12: The heat transfer rate from a very long fin of constant cross section is given by
(T
.
Q = h P kt A
f
− T∞ )
where h is the convective heat transfer coefficient, P is the perimeter of the fin in a
plane normal to its axis, k t is the thermal conductivity of the fin, and A is the crosssectional area of the fin (again in a plane normal to its axis). T∞ is the temperature of
the fin’s surrounding (measured far away from the fin itself) and T f is the temperature
of the base of the fin.
Fin of thermal conductivity kt
Perimeter P
Tf
Area A
x
TH
L
Fig. 1.11: A long fin ( Problem 1.12)
The temperature profile along the fin is given by
T ( x ) = T∞ + (T f − T∞ ) e-mx
0.5
⎛ hP ⎞
⎟⎟
where m = ⎜⎜
⎝ kt A ⎠
The fin is attached to an engine whose surface temperature is 950 C. Determine the
entropy generation rate for the fin if it is a very long square aluminium fin 0.01 m on a
side in air at 200 C. The thermal conductivity of aluminium is 204 W/m.K and the
convective heat transfer coefficient of the fin is 3.5 W/m2. K. [Ans: 0.0013 W/K]
180
1.13: At a certain location the temperature of the water supply is 288 K. Ice is to be
made from this water supply by the process shown in Fig. 1.12. The final temperature
of the ice is 263 K, and the final temperature of the water that is used as cooling water
in the condenser is 303 K. What is the minimum work required to produce 1000 kg of
ice?
Water at 30C
Q1
R
Water at
15C
W rev
Q2
Ice at 10C
Fig. 1.12
1.14: The velocity profile in the steady laminar flow of an incompressible Newtonian
fluid contained between concentric cylinders in which the inner cylinder is rotating and
the outer cylinder is stationary is given by
V=
ω R12 ⎛ R22
⎞
⎜ − x⎟
R2 − R1 ⎝ x
⎠
2
2
for R1 ≤ x ≤ R2
where ω is the angular velocity of the inner cylinder and x is measured radially
outward. Determine the rate of entropy production due to laminar viscous losses for
0
engine oil at 20 C in the gap between cylinders of radii 0.060 and 0.065 m when
the inner cylinder is rotating at 1000 rev / min . The viscosity of the oil is
0.8 N.s/m 2 and the length of the cylinder is 0.250 m .
1.15: A single tube, single pass heat exchanger is used to cool a compressed air flow
0
0
of 0.2 kg/s from 95 C to 80 C . The cooling fluid is liquid water that enters the
0
0
heat exchanger at 20 C and leaves at 45 C . If the overall heat transfer coefficient
2
is 160 W / m K and all flow streams have negligible pressure drop, determine the
required heat exchanger area and entropy production rate for
(a) Parallel flow, and,
(b) Counter flow.
Assume the compressed air behaves as an ideal gas with constant specific heat.
181
Chapter 2
2.1: Prove that the specific heat at constant pressure, C p for an ideal gas is a function
of temperature only.
Solution:
_
n RT
For an ideal gas, V =
p
_
nR
⎛ ∂V ⎞
⎜
⎟ =
p
⎝ ∂T ⎠ p
⎛ ∂ 2V ⎞
⎜ 2 ⎟ =0
⎝ ∂T ⎠ p
⎛ ∂C p ⎞
⎜
⎟ =0
⎝ ∂p ⎠T
Hence, C p is a function of temperature alone.
∴
2.2: Show that
⎛ ∂v ⎞
⎛ ∂u ⎞
⎛ ∂v ⎞
⎜⎜ ⎟⎟ = - T ⎜
⎟ − p ⎜⎜ ⎟⎟
⎝ ∂T ⎠ p
⎝ ∂p ⎠ T
⎝ ∂p ⎠ T
Solution:
du = Tds − pdv
or,
⎛ ∂u ⎞
⎛ ∂s ⎞
⎛ ∂v ⎞
⎜ ⎟ = T ⎜ ⎟ − p⎜ ⎟
⎝ ∂p ⎠T
⎝ ∂p ⎠T
⎝ ∂p ⎠T
From Maxwell’s relations,
⎛ ∂s ⎞
⎛ ∂v ⎞
⎜ ⎟ = −⎜
⎟
⎝ ∂T ⎠ p
⎝ ∂p ⎠T
Hence,
⎛ ∂v ⎞
⎛ ∂u ⎞
⎛ ∂v ⎞
⎜⎜ ⎟⎟ = - T ⎜
⎟ − p ⎜⎜ ⎟⎟
⎝ ∂T ⎠ p
⎝ ∂p ⎠ T
⎝ ∂p ⎠ T
182
2.3: Show that for a van der Waal’s gas
⎛ ∂C ⎞
(a) ⎜ v ⎟ = 0
⎝ ∂v ⎠ T
(b)
(s 2 − s1 )T
= R ln
( c ) T (v − b )
(d) C p − C v =
R / Cv
v2 − b
v1 − b
= constant (for isentropic condition)
R 2 T v3
1 − 2 a (v − b )
2
⎛1 1⎞
(e) (h2 − h1 )T = ( p 2 v 2 − p1 v1 ) + a ⎜⎜ − ⎟⎟
⎝ v1 v 2 ⎠
2.4: Derive the third T dS equation
⎛ ∂T ⎞
⎟⎟ dp + C p
T dS = C v ⎜⎜
⎝ ∂p ⎠ v
⎛ ∂T ⎞
⎟ dV
⎜
⎝ ∂V ⎠ p
and show that the three T dS equations can be written as
βT
dV
κ
T dS = C p dT − V β T dp
(a) T dS = C v dT +
(b)
Cp
Cv
dV
β
βV
2.5: (a) Derive the equation
(c) T dS =
κ dp +
⎛ ∂2 p ⎞
⎛ ∂C v ⎞
⎟
⎜
⎟ = T ⎜⎜
2 ⎟
⎝ ∂v ⎠ T
⎝ ∂ T ⎠v
(b) In the case of a gas obeying the equation of state
pv
B"
=1+
RT
v
"
where B is a function of T only. Show that
where (C v )o
(
)
RT d 2
B " T + (C v )o
v dT 2
is the value at very large volumes.
Cv = −
183
Cp
⎛ ∂s ⎞
2.6: Show that T ⎜
⎟ =
⎝ ∂T ⎠ h 1 − β T
⎛ ∂T ⎞ ⎛ ∂T ⎞
v
⎟⎟ − ⎜⎜
⎟⎟ = −
2.7: Show that ⎜⎜
Cp
⎝ ∂p ⎠ h ⎝ ∂p ⎠ s
⎛ ∂h ⎞
⎛ ∂u ⎞
2.8: A substance has the properties that ⎜ ⎟ = 0 and ⎜⎜ ⎟⎟ = 0 . Show that the
⎝ ∂v ⎠ T
⎝ ∂p ⎠ T
equation of state must be T = A p v where A is a constant. What additional
information is necessary to specify the entropy of the substance?
2.9: The equation of state of a certain gas is ( p + b ) v = R T . Find C p − C v and entropy
change in an isothermal process.
2.10: Find the difference C p − C v for mercury at a temperature 00 C and a pressure of
1 atm taking the value of β =16 and κ = 37 .
2.11: Joule-Kelvin coefficient µ J is a measure of the temperature change during a
throttling process. A similar measure of temperature change produced is by an
⎛ ∂T ⎞
⎟⎟ . Prove that
isentropic change of pressure by the coefficient µ s where µ s = ⎜⎜
⎝ ∂p ⎠ s
V
.
Cp
2.12: Estimate the maximum inversion temperature of hydrogen if it is assumed to
obey the equation of state p V = R T + B1 p + B2 p 2 + B3 p 3 + ...................... . For
c
hydrogen, B1 × 10 5 = a + 10 − 2 b T + 10 2 .
T
Where a = 166 , b = − 7.66 and c = − 172.33 .
µs − µJ =
2.13: Compute µ J for a gas whose equation of state is p (v − b ) R T .
⎡⎛
⎤
a⎞
2.14: Show that for a van der Waals gas ⎢⎜ p + 2 ⎟ (v − b ) = R T ⎥ , the Joule-Kelvin
v ⎠
⎣⎝
⎦
coefficient is given by
2
v ⎡ 2 a (v − b ) − R T b v 2 ⎤
µJ =
⎢
⎥
C p ⎣ R T v 3 − 2 a (v − b )2 ⎦
2.15: Calculate the maximum inversion temperature of helium.
2.16: Assuming that helium obeys the van der Waals equation of state, determine the
change in temperature when 1 kilomole of helium gas undergoes a Joule expansion at
20 K to atmospheric pressure. The initial volume of helium is 0.12 m3. Describe
approximation.
184
Chapter 3
3.1: A gas is flowing through a pipe at the rate of 2 Kg/s . Because of
inadequate
insulation; the gas temperature decreases from 8000C to 7900 C between two sections
of the pipe. Neglecting pressure losses; determine the rate of energy dessipitation
(degration) due to this heat loss.
Take To= 300 k
Cp=1.1kJ/kg
For the same temperature drop of 10oC, when the gas cool from 800C to 700C, due to
heat loss , what would have been the rate of energy degradation. What interference
you can draw from this.
3.2: An ideal gas is flowing through an insulated pipe at the rate of 3.0 kg/s. There is a
10% pressure drop from inlet to outlet of the pipe. Find the energy loss. Take R=
0.287 kJ/kgK, T0= 300 K.
3.3: Water at 900C is flowing at the rate of 1Kg/s. Estimate the rate of entropy
generation & rate of energy loss due to mixing. Take T0= 300K
3.4: An air pre-heater is used to cool the product of combustion, the rate of air flow of
products is 12.5 kg/s & products cool from 3000C to 2000C. Rate of air flow is 11.5
kg/s. Initial air temperature is 400C.
Find
(a) Initial and final availability of the products.
(b) What is the energy loss of the process?
(c) If the heat transfer from the products took place reversibility through heat
engines what would be the final temperature of the air?
What power will be developed by the heat engine.
Take T0=300 K, Cpg=1.09kJ/kgK, Cpa=1.005 kJ/kg K.
Neglect pressure drop for both fluids &heat transfer to the surroundings.
3.5: Air is flowing through an insulated duct, the pressure & temperature
measurements of the air at two stations P&Q are given below.
Station P Station
Q
Pressure
1.30
k 100 k
Pa
Pa
0
Temperature 50 C
130 C
Establish the direction of flow air in the duct. Assume air as and ideal gas having Cp=
1.005 & R=0.287kJ/kg.
3.6: What is the maximum useful work, which can be obtained when 100kJ is
extracted from a heat reservoir at 675 K In an environment at 288 K? What is the loss
of useful work or availability if,
(a) Temperature drop of 500C is introduced between the source and heat engine &
heat engine & sink on the other?
(b) The source temperature drops by 500C & sink temperature by 500C(rise) during
the heat transfer process according to the law dQ/dT= Constant.
185
Chapter 4
4.1: In the vicinity of the triple point, the vapor pressure of liquid ammonia (in
atmospheres) is represented by
3063
T
This is the liquid-vapor boundary curve in a p − T diagram. Similarly, the vapor
pressure of solid ammonia is
ln p = 15.16 −
3754
T
(a) What is the temperature and pressure at the triple point?
(b) What are the latent heat of sublimation and vaporization?
(c) What is the latent heat of fusion at the triple point?
ln p = 18.70 −
4.2: It is found that a certain liquid boils at a temperature of 950 C at the top of a hill,
whereas it boils at a temperature of 1050 C at the bottom. The latent heat is 4.187 kJ/g
mol. What is the approximate height of the hill? Assume T0 = 300 K .
4.3: (a) Establish the condition of equilibrium of a closed composite system consisting
of two simple systems separated by a movable diathermal wall that is impervious to
the flow of matter.
(b) If the walls are rigid and diathermal, permeable to one type of material and
impermeable to all others, state the condition of equilibrium of the composite system.
© Two particular systems have the following equations of state
1 3 − N1
= R
,
T1 2 U 1
p1 N 1 _
=
R
T1 V1
_
N
p
1 3 _ N2
= R
, 2 =R 2
V2
T2 2 U 2
T2
_
where R = 8.3143 kJ/kg mol - K , and the subscripts indicate systems 1 and 2. The mole
number of the first system is N 1 = 0.5, and that of the second is N 2 = 0.75. The two
systems are contained in a closed adiabatic cylinder, separated by a movable
diathermal piston. The initial temperatures are T1 = 200 K and T2 = 300 K and the total
volume is 0.02 m3. What is the energy and volume of each system in equilibrium?
What is the pressure and temperature?
186
4.4: Two particular systems have the following equations of state
1
3 _ N (1)
1
5 _ N (2 )
R
R
=
and
=
T (1) 2 U (1)
T (2 ) 2 U (2 )
_
where R = 8.3143 kJ/kg mol - K . The mole number of the first system is N (1) = 2 , and
that of the second is N (2 ) = 3 . The two systems are separated by a diathermal wall,
and the total energy of the composite system is 25.120 kJ. What is the internal energy
of each system in equilibrium?
For the same system, for initial temperatures T (1) = 250 K and T (2 ) = 350 K, what are
the values of U (1) , U (2 ) and equilibrium temperature?
4.5: Show that
− S dT + V dp − ∑ ni dµ i = 0
i
4.6: For a 2-component system show that
⎛ ∂µ ⎞
⎛ ∂µ ⎞
x ⎜ a ⎟ + (1 − x ) ⎜ b ⎟ = 0
⎝ ∂x ⎠ T , p
⎝ ∂x ⎠ T , p
na
n a + nb
4.7: To make baking soda ( NaHCO3 ), a concentrated solution of Na 2 CO3 is saturated
where x =
in CO2 . The reaction is given by
2 Na + + CO3
thus
Na + ions, CO3
−2
−2
+ H 2 O + CO2 ↔ 2 NaHCO3
ions, H 2 O , CO2 and NaHCO3 are present in arbitrary
amounts, except that all the Na + and CO3
degree of freedom.
−2
187
are from Na 2 CO3 . Find the number of
4.8: Determine the number of degree of freedom for the system at each lettered point
and state the variables for a cadmium-bismuth system.
3210C
0
271 C
A
Liquid solution
+ solid Cd
T
B
Liquid solution
+ solid Cd
Liquid
Solution +
solid Bi
C
1440C
D
Liquid Cd + solid Bi
E
20
40
60
80
100
Weight
(Cd %)
Chapter 5
5.1: Butane is burned with air and a volumetric analysis of the combustion products on
a dry basis yields the following composition:
CO2 = 7.8%
CO = 1.1%
O2 = 8.2%
N 2 = 82.9%
Determine the percentage of theoretical air used in this combustion process.
5.2: Starting with n0 moles of NO , which dissociates according to the equation
NO ⇔
1
1
N2 + O2
2
2
Show that at equilibrium
K=
1 εe
2 1- εe
188
5.3: A mixture of n0 ν 1 moles of A1 and n0 ν 2 moles of A2 at temperature T and
pressure p occupies a volume V0 . When the reaction
ν 1 A1 + ν 2 A 2 ⇔ ν 3 A3 + ν 4 A4
has come to equilibrium at the same T and p , the volume is Ve . Show that
Ve - V0
ν1 +ν 2
.
V0 ν 3 + ν 4 + ν 1 − ν 2
: At high temperature the potassium atom is ionized according to the equation
K ⇔ K + + e − . The values of the equilibrium constant at 3000 K and 3500 K are
8.33 x 10 −6 and 1.33 x 10 −4 respectively. Compute the average heat of reaction in the
given temperature range.
εe =
5.4: When 1 kgmol of HI dissociates according to the reaction
1
1
HI ⇔ H 2 + I 2
2
2
⎛ ∂ε ⎞
at T = 675 K , K = 0.0174 and ∆H = 5910 kJ/kgmol. Calculate ⎜ e ⎟ at this
⎝ ∂T ⎠ p
temperature.
5.5: Liquid hydrazine (N 2 H 4 ) and oxygen gas, both at 250 C, 0.1 Mpa are fed to a
rocket combustion chamber in the ratio of 0.5 kg O2 /kg N 2 H 4 . The heat transfer from
the chamber to the surroundings is estimated to be 100 kJ/kg N 2 H 4 . Determine the
temperature of the products, assuming only H 2 O , H 2 and N 2 to be present. The
enthalpy of formation of (N 2 H 4 ) is +50,417 kJ/kgmol.
5.6: Propane is reacted with air in such a ratio that an analysis of the products of combustion
gives CO2 11.5%, O2 2.7%, and CO 0.7%. What is theoretical air used during the test?
5.7: Gaseous butane at 250 C is mixed with air at 400 K and burned with 400% theoretical air.
Determine the adiabatic flame temperature.
5.8: Starting with n0 moles of water vapour which dissociates according to the equation
1
H 2 O ⇔ H 2 + O2 , show that at equilibrium
2
K=
3
2
e
1
2
ε
(2 + ε e ) (1 − ε e )
p
1
2
At an average temperature of 1900 K, the slope of the graph of log K against 1/T for the
dissociation of water vapour is found to be –13,000. Determine the heat of dissociation. Is it
exothermic or endothermic?
5.9: Design a burner for a furnace that will produce 8400 Kcal/hr, 8160 C. Choose the fuel,
flow rates, air fuel ratio, burner material, burner geometry, flow controls, etc.
189
5.10: An inventor claims to have perfected a hydrogen-oxygen fuel cell
H 2 + 0.5 O2 → H 2 O(l ) , that will produce 300 MJ per kgmole of hydrogen consumed at 250 C
and 0.1 MPa. Is it possible? If not, what is the maximum possible power output? Assume each
component in the reaction is at 250 C and 0.1 MPa.
Chapter 6
6.1 Calculate the rms speed for oxygen at 300 K. What is the mean translational
kinetic energy of a molecule of xygen?
Solution:
Mass of an oxygen molecule,
M
N0
where, M = molecular weight= 32 kg/kgmol
N 0 = Avogadro’s number= 5.32 ×10−26 kg
m=
vrms =
3kT
= 483 m/s
m
1 _ 3
KE(mean)= m v 2 = kT = 6.21×10−21 J/molecule
2
2
6.2: Compute the most probable speed, the mean speed and root mean square speed
for hydrogen at 273 K.
6.3: Calculate the collision rate of oxygen molecules on the wall per unit area at 1 atm
and 273 K.
6.4: Prove that
_
vmp : v : vrms = 1:1.128 :1.224
190
Chapter 7
7.1: For a particle in a cubical box of side L, find the number of quantum states at
each of the following energy levels:
h2
8mL2
h2
(b) 25
8mL2
(a) 12
h2
(c) 36
8mL2
[Ans: (a)1, (b) 9, (c ) 6]
7.2: If a particle has a translational energy 3
h2
, what are the possible directions for
8mL2
its velocity ? [ Ans:8]
7.3: Calculate the number of ways of arranging six indishtinguishable particles in
either distinguishable boxes with no more than one particle per box.
7.4: Compare the Fermi-Dirac, Bose-Einstein, and Maxwell-Boltmann statistics when
four particles are arranged in two energy levels. Three particles are at energy level ε1
having a degeneracy g1 = 4 and one particle at energy level ε 2 having a degeneracy
g2 = 2 .
7.5: Consider a cubical box of edge 10 cm, containing gaseous helium at 300 K.
Evaluate the energy ε x and its corresponding nx . [Ans: ε x = 20.7 × 10−22 J/molecule,
nx = 15.8 ×1031 ]
191