Rac lecture 5

1
Lesson
5
Review of fundamental
principles –
Thermodynamics : Part II
Version 1 ME, IIT Kharagpur 1

2
.The specific objectives are to:
1. State principles of evaluating thermodynamic properties of pure substances using:
a) Equations of State (Section 5.2)
b) Thermodynamic charts (Section 5.2)
c) Thermodynamic tables (Section 5.2)
2. Derive expressions for heat and work transfer in important thermodynamic processes
such as:
a) Isochoric process (Section 5.3)
b) Isobaric process (Section 5.3)
c) Isothermal process (Section 5.3)
d) Isentropic process (Section 5.3)
e) Isenthalpic process etc. (Section 5.3)
At the end of the lesson the student should be able to:
1. Evaluate thermodynamic properties using equations of state, tables and charts
2. Identify various regimes on T-s and P-h charts
3. Estimate heat and work transferred in various thermodynamic processes
5.1. Thermodynamic relations
There are some general thermodynamic relations, which are useful for determination of several
thermodynamic properties from measured data on a few properties. The following relationships
are generally used for the evaluation of entropy change. These are called T ds equations. They
are obtained by applying first and second laws of thermodynamics
first equation
second equation
T ds du p dv T ds
T ds dh v dP T ds
= +
= −
(5.1)
Two more fundamental thermodynamic relations can be obtained by defining two new
properties called Gibbs and Helmholtz functions.
5.2. Evaluation of thermodynamic properties
In order to perform thermodynamic calculations, one has to know various thermodynamic
properties of the system. Properties such as internal energy, enthalpy and entropy cannot be
measured directly. Thermodynamics gives mathematical relations using which one can obtain
properties, which cannot be measured directly in terms of the measurable properties such as
pressure, temperature, volume, specific heat etc.

3
In general thermodynamic properties can be evaluated from:
1. Thermodynamic equations of state
2. Thermodynamic tables
3. Thermodynamic charts
4. Direct experimental results, and
5. The formulae of statistical thermodynamics
An equation of state (EOS) is a fundamental equation, which expresses the relationship between
pressure, specific volume and temperature. The simplest equation of state is that for an
incompressible substance (e.g. solids and liquids), which states that the specific volume is
constant. The next simplest EOS is that for an ideal gas.
Ideal (perfect) gas equation is a special equation of state, which is applicable to ideal gases. The
molecular forces of attraction between gas molecules are small compared to those in liquids. In
the limit when these forces are zero, a gas is called a perfect gas. In addition the volume of the
molecules should be negligible compared to total volume for a perfect gas. The perfect or ideal
gas equation of state is given by:
RTPv = (5.2)
Where P = Absolute pressure
v = Specific volume
R = Gas constant
T = Absolute temperature
The gas constant R is given by:
M/RR u= (5.3)
Where Ru = Universal gas constant
M = Molecular weight
The ideal gas equation is satisfactory for low molecular mass, real gases at relatively high
temperatures and low pressures. Ideal gas equation can be used for evaluating properties of
moist air used in air conditioning applications without significant error.
For ideal gases, the change in internal energy and enthalpy are sole functions of temperature.
Assuming constant specific heats ( pc , ) in the temperature range Tvc 1 to T2, for ideal gases one
can write the change in internal energy (u), enthalpy (h) and entropy (s) as:
Rcc
P
P
lnR
T
T
lncss
v
v
lnR
T
T
lncss
)TT(chh
)TT(cuu
vp
1
2
1
2
p12
1
2
1
2
v12
12p12
12v12
=−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=−
−=−
−=−
(5.4)

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The study of the properties of moist air is known as psychrometry. The psychrometric properties
(temperature, humidity ratio, relative humidity, enthalpy etc.) are normally available in the form
of charts, known as psychrometric charts. The psychrometric properties will be discussed in
later chapters.
For gases with complex molecular structure or for real gases at high pressure and low
temperatures or for gases approaching the saturated vapour region, the use of Ideal gas equation
results in significant errors. Hence more complex but more realistic equations of states have to
be applied. The accuracy of these EOS depend on the nature of the gas. Some of these EOSs are
given below:
van der Waals equation:
RT)bv)(
v
a
P( 2
=−+ (5.5)
where a and b are constants that account for the intermolecular forces and volume of the gas
molecules respectively.
Redlich-Kwong equation:
)bv(vT
a
bv
RT
P
+
−
−
= (5.6)
A virial equation is more generalized form of equation of state. It is written as:
......
v
C
v
B
v
A
RTPv 32
++++= (5.7)
where A,B,C,… are all empirically determined functions of temperature and are called as virial
coefficients.
5.2.1. Properties Of Pure Substance
A pure substance is one whose chemical composition does not change during thermodynamic
processes. Water and refrigerants are examples of pure substances. These days emphasis is on
the use mixture of refrigerants. The properties of mixtures also require understanding of the
properties of pure substances.
Water is a substance of prime importance in refrigeration and air-conditioning. It exists in three
states namely, solid ice, liquid water and water vapour and undergoes transformation from one
state to another. Steam and hot water are used for heating of buildings while chilled water is
used for cooling of buildings. Hence, an understanding of its properties is essential for air
conditioning calculations. Substances, which absorb heat from other substances or space, are
called refrigerants. These substances also exist in three states. These also undergo
transformations usually from liquid to vapour and vice-versa during heat absorption and
rejection respectively. Hence, it is important to understand their properties also.
If a liquid (pure substance) is heated at constant pressure, the temperature at which it boils is
called saturation temperature. This temperature will remain constant during heating until all the

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liquid boils off. At this temperature, the liquid and the associated vapour at same temperature are
in equilibrium and are called saturated liquid and vapour respectively. The saturation
temperature of a pure substance is a function of pressure only. At atmospheric pressure, the
saturation temperature is called normal boiling point. Similarly, if the vapour of a pure
substance is cooled at constant pressure, the temperature at which the condensation starts, is
called dew point temperature. For a pure substance, dew point and boiling point are same at a
given pressure.
Similarly, when a solid is heated at constant, it melts at a definite temperature called melting
point. Similarly cooling of a liquid causes freezing at the freezing point. The melting point and
freezing point are same at same pressure for a pure substance and the solid and liquid are in
equilibrium at this temperature.
For all pure substances there is a temperature at which all the three phases exist in equilibrium.
This is called triple point.
The liquid-vapour phase diagram of pure substance is conveniently shown in temperature-
entropy diagram or pressure-enthalpy diagram or p-v diagram. Sometimes, three dimensional p-
v-t diagrams are also drawn to show the phase transformation. In most of the refrigeration
applications except dry ice manufacture, we encounter liquid and vapour phases only.
Thermodynamic properties of various pure substances are available in the form of charts and
tables. Thermodynamic property charts such as Temperature-entropy (T-s) charts, pressure-
enthalpy (P-h) charts are very useful in evaluating properties of substances and also for
representing the thermodynamic processes and cycles. Figures 5.1 and 5.2 show the P-h and T-s
diagrams for pure substances.
Fig. 5.1. P-h diagram for a pure substance

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Fig. 5.2. T-s diagram for a pure substance
Critical point :
Figures 5.1 and 5.2 show the critical point. The temperature, pressure and specific volume at
critical point are denoted by Tc, Pc and vc, respectively. A liquid below the critical pressure when
heated first becomes a mixture of liquid and vapour and then becomes saturated vapour and
finally a superheated vapour. At critical point there is no distinction between liquid state and
vapour state; these two merge together. At constant pressure greater than critical pressure, PC
when liquid is heated in supercritical region, there is no distinction between liquid and vapour;
as a result if heating is done in a transparent tube, the meniscus of liquid and vapour does not
appear as transformation from liquid to vapour takes place. At pressures below critical pressure,
when a liquid is heated there is a clear-cut meniscus between liquid and vapour, until all the
liquid evaporates.
For water: Triple point: 0.1 o
C, 0.006112 bar
Critical point: 221.2 bar, 647.3K and 0.00317 m3
/kg
For Dry Ice (CO2): Triple point: 5.18 bar, -56.6 o
C
Critical point: 73.8 bar, 31o
C
T-s and p-h diagrams for liquid-vapour regime
These are of great importance in refrigeration cycle calculations. Figure 5.3 and 5.4 show typical
T-s diagram and p-h (Mollier) diagrams, respectively for a pure refrigerant. The T-s diagram

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shows two constant pressure lines for pressures P1and P2 where P1 > P2. The constant pressure
line 1-2-3-4 is for pressure P1. The portion 1-2 is in the sub-cooled region, 2-3 is in wet region,
that is mixture of liquid and vapour, and 3-4 is in superheated region. A frequent problem in
refrigeration cycle calculations is to find the properties of sub-cooled liquid at point a shown in
the figure. The liquid at pressure P1and temperature Ta is sub-cooled liquid. The liquid at state
is saturated liquid at lower pressure Pa′ a, but at the same temperature.
Fig. 5.3. T-s diagram of a pure substance
Fig. 5.4. P-h diagram of a pure substance

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From 1st
T ds equation , Eq. (5.1):
dvPdudsT += (5.1a)
If the liquid is assumed to be incompressible then dv = 0 and
dudsT = (5.8)
For liquids, the internal energy may be assumed to be function of temperature alone, that is,
a a' a a' a au = u , because T = T this implies that s = s '
Therefore states a and are coincident.a′
Also from the 2nd
T ds equation, Eq. (5.1)
T ds=dh - vdP (5.1b)
The specific volume v is small for liquids hence v dp is also negligible, therefore ha = ha’, That
is, the enthalpy of sub-cooled liquid is equal to the enthalpy of saturated liquid at liquid
temperature. For all practical purposes the constant pressure lines are assumed to be coincident
with saturated liquid line in the sub-cooled region. This is a very useful concept.
T-s diagram gives a lot of information about the refrigeration cycle. It was observed in Chapter
4 that for a reversible process, the heat transfer is related to the change in entropy given by:
⎟
⎠
⎞
⎜
⎝
⎛
∫=⎟
⎠
⎞
⎜
⎝
⎛
∫
δ
=−
2
1
21
rev
int
2
1
12 ds.TQthatimpliesthis,
T
Q
SS (5.9)
The above equation implies that the heat transferred in a reversible process 1-2 is equal to area
under the line 1-2 on the T-s diagram.
Also, from Eq. (5.1b), T nce for a constant pressure process (dP = 0), therefore,
for a constant pressure process Tds = dh, which means that for an isobaric process the area
under the curve is equal to change in enthalpy on T-s diagram.
ds=dh - vdP , he
Properties at Saturation
The properties of refrigerants and water for saturated states are available in the form of Tables.
The properties along the saturated liquid line are indicated by subscript ‘f’ for example vf, uf, hf
and sf indicate specific volume, internal energy, enthalpy and entropy of saturated liquid
respectively. The corresponding saturated vapour states are indicated by subscript ‘g’ for
example vg, ug, hg and sg respectively. All properties with subscript ‘fg’ are the difference
between saturated vapour and saturated liquid states. For example, hfg = hg - hf , the latent heat
of vaporization.
The specific volume, internal energy, enthalpy and entropy of the mixture in two-phase region
may be found in terms of quality, ‘x’ of the mixture. The quality of the mixture denotes the mass

9
(kg) of the vapour per unit mass (kg) of the mixture. That is there is x kg of vapour and (1-x) kg
of liquid in one kg of the mixture.
Therefore the properties of the liquid-vapour mixture can be obtained by using the following
equations:
fgfgf
fgfgf
fgfgf
fgfgf
s.xss.xs)x1(s
h.xhh.xh)x1(h
u.xuu.xu)x1(u
v.xvv.xv)x1(v
+=+−=
+=+−=
+=+−=
+=+−=
(5.10)
The table of properties at saturation is usually temperature based. For each temperature it lists
the values of saturation pressure (Psat), vf, vg, hf, hg, sf and sg. Two reference states or datum or
used in these tables. In ASHRAE reference hf = 0.0 kJ/kg and sf = 1.0 kJ/kg.K at – 40o
C. In IIR
reference hf = 200.00 kJ/kg and sf = 1.0 kJ/kg-K at 0o
C.
The properties in the superheated region are given in separate tables. The values of v, h and s are
tabulated along constant pressure lines (that is, at saturation pressures corresponding to, say 0o
C,
1o
C, 2o
C etc.) at various values of degree of superheat.
Clapeyron Equation
The Clapeyron equation represents the dependence of saturation pressure on saturation
temperature (boiling point). This is given by,
T)vv(
h
v
s
Td
dP
fg
fg
fg
fgsat
−
== (5.11)
Some useful relations can be derived using Clapeyron equation. The specific volume of liquid is
very small compared to that of vapour, hence it may be neglected and then perfect gas relation
pvg= RT may be used to yield:
2
fgsat
g
fg
fg
fgsat
RT
.hP
Tv
h
T)vv(
h
Td
dP
==
−
= (5.12)
This may be integrated between states 1 to an arbitrary state Psat, T to yield
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=∫=∫
T
1
T
1
R
h
P
P
lnor
T
dT
R
h
P
dP
1
fg
1
sat
T
T
2
fg
p
p sat
sat
11
(5.13)
If P1 is chosen as standard atmospheric pressure of say 1 atm. (ln (P1) = ln (1) = 0), and P is
measured in atmospheres, then T1= Tnb is the normal boiling point of the substance, then from
Eq. (5.13), we obtain:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
T
1
T
1
R
h
)P(ln
nb
fg
sat (5.14)

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Therefore if ln (P) is plotted against 1/T, the saturated vapour line will be a straight line.
Also, it has been observed that for a set of similar substances the product of Mhfg/Tnb called
Trouton number is constant. Here M is the molecular weight of the substance (kg/kmole). If we
denote the Trouton number by Ntrouton , then
R
N
RT
h
pln
or,
R
N
MR
N
RT
h
T
Mh
N
troutonfg
troutontrouton
nb
fg
nb
fg
trouton
+=
==
=
(5.15)
For most of the substances, the Trouton number value is found to be about 85 kJ/kmol.K
5.3. Thermodynamic processes
In most of the refrigeration and air conditioning systems, the mass flow rates do not change with
time or the change is very small, in such cases one can assume the flow to be steady. For such
systems, the energy balance equation (1st
law of thermodynamics) is known as steady-flow
energy equation.
Q
W
m
h2
v2
z2
m
h1
v1
z1
E
Fig. 5.5. Steady flow energy balance on a control volume
For the open system shown in Fig. 5.5, it is given by:
W)gz
2
v
h(mQ)gz
2
v
h(m 2
2
2
21
2
1
1 +++=+++ (5.16)
In many cases, compared to other terms, the changes in kinetic and potential energy terms, i.e.,
(v1
2
-v2
2
)/2 and (gz1-gz2) are negligible.
Heating and cooling: During these processes normally there will be no work done either on the
system or by the system, i.e., W= 0. Hence, the energy equation for cooling/heating becomes:
)hh(mQormhmhQ 1221 −==+ (5.17)
Version 1 ME, IIT Kharagpur10

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Some of the important thermodynamic processes encountered in refrigeration and air
conditioning are discussed below.
Constant volume (isochoric) process: An example of this process is the heating or cooling of a
gas stored in a rigid cylinder. Since the volume of the gas does not change, no external work is
done, and work transferred WΔ zero. Therefore from 1is st
law of thermodynamics for a constant
volume process:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=−
−=−∫ ==
=
1
2
avg,v12
12avg,v12
2
1
21
21
T
T
lnmcSS
)TT(mcUUdUQ
0W
(5.18)
The above equation implies that for a constant volume process in a closed system, the heat
transferred is equal to the change in internal energy of the system. If ‘m’ is the mass of the gas,
Cv is its specific heat at constant volume which remains almost constant in the temperature range
ΔT, and ΔT is the temperature change during the process, then:
T.C.mUQ v Δ=Δ=Δ (5.19)
Constant pressure (isobaric) process: If the temperature of a gas is increased by the addition of
heat while the gas is allowed to expand so that its pressure is kept constant, the volume of the
gas will increase in accordance with Charles law. Since the volume of the gas increases during
the process, work is done by the gas at the same time that its internal energy also changes.
Therefore for constant pressure process, assuming constant specific heats and ideal gas
behaviour,
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=−
−××=−=
−×=∫=
+−=
1
2
avg,p12
12avg,p1221
12
2
1
21
211221
T
T
lnmcSS
)TT(Cm)hh(mQ
)VV(PPdVW
W)UU(Q
(5.20)
Constant temperature (isothermal) process: According to Boyle’s law, when a gas is
compressed or expanded at constant temperature, the pressure will vary inversely with the
volume. Since the gas does work as it expands, if the temperature is to remain constant, energy
to do the work must be supplied from an external source. When a gas is compressed, work is
done on the gas and if the gas is not cooled during the process the internal energy of the gas will
increase by an amount equal to the work of compression. Therefore if the temperature of the gas
is to remain constant during the process gas must reject heat to the surroundings. Since there is
no temperature increase in the system change in internal energy becomes zero. And the amount
of work done will be the amount of heat supplied. So for isothermal process

12
∫=
+−=
2
1
21
211221
dV.PW
W)UU(Q
(5.21)
If the working fluid behaves as an ideal gas and there are no phase changes, then, the work done,
heat transferred and entropy change during the isothermal process are given by:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=−
∫ ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
==
==
2
1
1
2
12
2
1 2
1
1
2
21
2121
P
P
lnmR
v
v
lnmRSS
P
P
lnmRT
v
v
lnmRTdV.PW
))T(fU(WQ ∵
(5.22)
Adiabatic process: An adiabatic process is one in which no heat transfer takes place to or from
the system during the process. For a fluid undergoing an adiabatic process, the pressure and
volume satisfy the following relation:
PV constantk
= (5.23)
where k is the coefficient of adiabatic compression or expansion. For an ideal gas, it can be
shown that:
PV constant, where p
v
C
C
γ
γ= = (5.24)
Applying first law of thermodynamics for an adiabatic process, we get:
( ) )UU(VPVP
1k
k
dV.PW
0W)UU(Q
211122
2
1
21
211221
−=−⎟
⎠
⎞
⎜
⎝
⎛
−
∫ ==
=+−=
(5.25)
If the process is reversible, adiabatic then the process is also isentropic:
21
2
1
21 SS0dS.TQ =⇒∫ == (5.26)
The following P-V-T relationships can be derived for a compressible fluid undergoing an
adiabatic process:
1 (
2 1 2
1 2 1
T V P
T V P
k k− −
⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
1)/k
(5.27)
If the adiabatic process is reversible, then from the definition of entropy, the process becomes an
isentropic process or the entropy of the system does not change during a reversible adiabatic
process. Hence all reversible, adiabatic processes are isentropic processes, however, the
converse is not true, i.e., all isentropic processes need not be reversible, adiabatic processes.

13
Polytropic process: When a gas undergoes a reversible process in which there is heat transfer,
the process frequently takes place in such a way that a plot of log P vs log V is a straightline,
implying that:
PV constantn
= (5.28)
The value of n can vary from −∞ to +∞, depending upon the process. For example:
For an isobaric process, n = 0 and P = constant
For an isothermal process, n = 1 and T = constant
For an isentropic process, n = k and s = constant, and
For an isochoric process, n = −∞ and v = constant
For a polytropic process, expressions for work done, heat transferred can be derived in the same
way as that of a adiabatic process discussed above, i.e.,
( )
( )
∫+∫=−
−
−
+−=
−=−
−
−
=
2
1
2
1
12
11221221
12avg,v12
112221
T
PdV
T
dU
SS
VPVP
)1n(
n
)UU(Q
)TT(mc)UU(
VPVP
)1n(
n
W
(5.29)
The above expressions are valid for all values of n, except n = 1 (isothermal process)
Throttling (Isenthalpic) process: A throttling process occurs when a fluid flowing through a
passage suddenly encounters a restriction in the passage. The restriction could be due to the
presence of an almost completely closed valve or due to sudden and large reduction in flow area
etc. The result of this restriction is a sudden drop in the pressure of the fluid as it is forced to
flow through the restriction. This is a highly irreversible process and is used to reduce the
pressure and temperature of the refrigerant in a refrigeration system. Since generally throttling
occurs in a small area, it may be considered as an adiabatic process (as area available for heat
transfer is negligibly small) also since no external work is done, we can write the 1st
law of
thermodynamics for this process as:
2
V
h
2
V
h
0WQ
2
2
2
2
1
1
..
+=+
==
(5.30)
where V1 and V2 are the inlet and exit velocities of the fluid respectively. The areas of inlet and
outlet of a throttling device are designed in such a way that velocities at inlet and outlet become
almost equal. Then the above equation becomes
1 2h = h (5.31)
Thus throttling process is an isenthalpic process.

14
Though throttling is an expansion process, it is fundamentally different from expansion taking
place in a turbine. The expansion of a fluid in a turbine yields useful work output, and can
approach a reversible process (e.g. isentropic process), whereas expansion by throttling is highly
irreversible. Depending upon the throttling conditions and the nature of the fluid, the exit
temperature may be greater than or equal to or less than the inlet temperature.
Questions:
1. Prove T dS equations starting from basic laws of thermodynamics? (Solution)
2. An interesting feature of the process of cooling the human body by evaporation is that the
heat extracted by the evaporation of a gram of perspiration from the human skin at body
temperature (37°C) is quoted in physiology books as 580 calories/gm rather than the nominal
540 calories/gm at the normal boiling point. Why is it larger at body temperature? (Solution)
3. Find the saturation temperature, the changes in specific volume and entropy during
evaporation, and the latent heat of vaporization of steam at 0.1 MPa ? (Solution)
4. Under what conditions of pressure and temperature does saturated steam have a entropy of
6.4448 kJ/kg K? State the specific volume and entropy under such conditions. (Solution)
5. Find the enthalpy of steam when the pressure is 2 MPa and the specific volume is 0.09 m3
/kg.
(Solution)
6. A gas of mass 4 kg is adiabatically expanded in a cylinder from 0.2 m3
to 0.5 m3
Initial
pressure of the gas is 2 bar, and the gas follows the following pressure-volume relationship
1.4
KPV = (K= constant)
Find the decrease in the temperature of the gas? (CV for the gas = 0.84 kJ/kg-K) (Solution)
7. Air is contained in a vertical cylinder that is fitted with a frictionless piston. A set of stops is
provided 0.5 m below the initial position of the piston. The piston cross-sectional area is 0.5 m2
and the air inside is initially at 100 kPa and 400 K. The air is slowly cooled as a result of heat
transfer to the surroundings.
a) Sketch these two processes on P-V and T-V diagrams
b) What is the temperature of the air inside the cylinder when the piston reaches the stops?

15
c) After the piston hits the stops, the cooling is continued until the temperature reaches 100 K.
What is the pressure at this state?
d) How much work is done by the system in the first cooling process?
e) How much work is done by the system in the second cooling process?
Assume air to be a thermally perfect gas and the first cooling is a quasi-static process. (Solution)
8. Consider a thermodynamic system containing air at V1=1 m3
/kg, P1=100 kPa. The system is
compressed to 0.5 m3
/kg via anyone of three quasi-static processes: isobaric, isothermal, or
adiabatic. Assume that cv = 0.7165 kJ/kg-K, and R = 0.287 kJ/kg-K.
a) Sketch all three processes on the same P-V diagram.
b) For each process determine the pressure and temperature at the final state.
c) For each process determine the work done by the system and the heat transferred to the
system. (Solution)

Rac lecture 5

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