COURSE OF ENERGY CONVERSION
THERMODYNAMIC PROPERTIES OF
FLUIDS
These class notes are for the students of the course "Energy Conversion A" at Politecnico di Milano.
Anyone who finds inaccuracies or, anyhow, wishes to send comments to improve them is invited to
the lecturer (gianluca.valenti@polimi.it), who thanks in advance.
Thermodynamic properties of fluids - Energy Conversion A – Version 7.0
Why this classnotes ............................................................................................................................... 4
1 Introduction ...................................................................................................................................... 5
2 Review of thermodynamics fundamental relations .......................................................................... 7
3 The ideal gas model ........................................................................................................................ 13
3.1 Definition of pure fluid ........................................................................................................... 13
3.2 Thermodynamic behavior of ideal gases ................................................................................. 14
3.3 Thermodynamic properties of ideal gases calculation ............................................................ 17
3.4 Specific heat of ideal gases calculation ................................................................................... 18
3.4.1 Monoatomic molecules ............................................................................................... 22
3.4.2 Diatomic molecules ..................................................................................................... 23
3.4.3 Polyatomic molecules.................................................................................................. 24
3.4.4 Practical examples ....................................................................................................... 25
3.5 Thermodynamic diagrams and transformations ...................................................................... 25
3.5.1 Trend of the isobars of ideal gas in the Ts diagram .................................................... 25
3.5.2 Temperature rise in an isentropic compression ........................................................... 28
3.5.3 Optimum compression ratio in a closed cycle............................................................. 29
3.5.4 Molecular complexity effect on the isentropic compression ratio .............................. 30
3.5.5 Molecular complexity and molecular weight effect on isentropic enthalpy change ... 31
3.5.6 Molecular complexity effect on the volumetric flow rate in a heat exchanger ........... 32
4 The ideal liquid model.................................................................................................................... 34
4.1 Thermodynamic properties of an ideal liquid calculation ....................................................... 35
4.1.1 Internal energy ............................................................................................................. 35
4.1.2 Enthalpy....................................................................................................................... 35
4.1.3 Entropy ........................................................................................................................ 37
5 The real fluid properties ................................................................................................................. 38
5.1 Thermodynamic properties of a real fluid calculation ............................................................ 39
5.1.1 Residual enthalpy ........................................................................................................ 39
5.1.2 Specific heat residual................................................................................................... 45
5.1.3 Residual entropy .......................................................................................................... 48
5.2 Effects on diagram Ts ............................................................................................................. 50
5.2.1 Trend of the isobar curves ........................................................................................... 50
5.2.2 Trend of the isenthalpic curves ................................................................................... 51
6 The real fluid equations of state ..................................................................................................... 53
6.1 A brief history of the equations of State ................................................................................. 53
6.2 Calculation programs .............................................................................................................. 59
6.3 The compressibility factor and the principle of corresponding states ..................................... 60
6.4 Clausius-Clapeyron relation .................................................................................................... 65
6.5 The saturation pressure curve .................................................................................................. 68
6.6 The acentric factor ................................................................................................................... 74
6.7 Residual of enthalpy, entropy and specific heat in reduced terms .......................................... 75
6.7.1 Enthalpy difference between two thermodynamic states calculation ......................... 78
6.7.2 Approximated estimate of enthalpy of vaporization ................................................... 79
6.7.3 Specific heat on a molar-basis variation between saturated liquid and vapor............. 80
6.7.4 Molecular complexity effect of the fluid on the shape of the Ts diagram .................. 81
7 The real liquid behavior ................................................................................................................. 84
7.1 Ideal compression work of a pump ......................................................................................... 84
7.2 Evaluation of the heating caused by an isentropic compression ............................................. 86
8 Solutions and Mixtures................................................................................................................... 89
8.1 Introduction to solutions.......................................................................................................... 89
8.2 Ideal mixtures .......................................................................................................................... 90
8.3 Mixtures in combustion........................................................................................................... 92
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Thermodynamic properties of fluids - Energy Conversion A – Version 7.0
9 Appendices ..................................................................................................................................... 94
9.1 Pressure of radiation: a practical example............................................................................... 94
9.2 Thermodynamic square ........................................................................................................... 94
9.3 Application of the kinetic theory of gases ............................................................................... 96
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Thermodynamic properties of fluids - Energy Conversion A – Version 7.0
WHY THIS CLASSNOTES
Fluids are one of the three cornerstones of energy conversion systems together with cycles and
equipment. Models of the thermodynamic properties of fluids are necessary to study properly energy
conversion systems from both the cycle and the equipment perspectives. In this context, it is
fundamental to understand when it is possible to apply one model depending upon the required
accuracy. Specifically, a distinction between ideal and real fluids and mixtures shall be considered.
The scope of this class note is to provide the students of Energy Conversion with the knowledge and
competence to understand and apply thermodynamic modelling of fluids. The methodology
adopted here is based on a review of fundamental thermodynamic relations first for pure fluids,
developed first of ideal pure fluids and then for real pure fluids. Similarly, fundamentals relations for
solutions are only outlined and detailed for ideal mixtures.
The structure of the present document is as follows.
• Chapter 0 introduces the topic of thermodynamic properties of fluids and their modelling
• Chapter 2 reviews the thermodynamic fundamental relations
• Chapter 3 deals with the ideal gas model
• Chapter 4 deals with the ideal liquid model
• Chapter 5 deals with real fluid properties
• Chapter 6 deals with real fluid equation of states
• Chapter 7 deals with the real liquid model
• Chapter 8 deals with solution and mixture properties
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Thermodynamic properties of fluids - Energy Conversion A – Version 7.0
1
INTRODUCTION
The thermodynamic state of a fluid in equilibrium can be described by a set of parameters called
state variables. These state variables represent the effect of the behavior of atoms and molecules at
the microscopic scale (typical of the Kinetic theory of gases) on the macroscopic scale (of Classical
thermodynamics). The approach here is investigating mathematically matter at the macroscopic scale,
providing the relations of Classic thermodynamics, and analyzing qualitatively at the microscopic
scale, recalling the main outcomes of the Kinetic theory of gases.
For a pure fluid, a thermodynamic state of equilibrium is completely defined when two independent
state variables are known. For example, temperature of a stable equilibrium state can be calculated as
a function of pressure and specific volume. The relation between these three state variables is called
“volumetric Equation of State (EOS)” and can be expressed as:
𝑓(𝑝, 𝑣, 𝑇) = 0 → 𝑝 = 𝑓(𝑇, 𝑣)
(1.1)
It is possible to derive any other state variable and fully characterize the properties of the fluid from
this equation (or from any other Equation of State that relates three thermodynamic state variables)
through differentiation and integration operations of the Equation of State itself.
The Equation of State can take on particularly simple analytical expressions in the case of ideal gases
or ideal liquids, while in other cases its formulation may require a large number of terms. The choice
of the Equation of State that models the properties of the fluid of interests most accurately is mainly a
function of the fluid itself, its thermodynamic conditions and the desired accuracy. The most widely
used fluids in the field of power generation plants are certainly water in steam power plants, air and
exhaust gases in gas turbines. While air and - within certain limits - exhaust gases can be treated as
ideal gases with results of reasonable accuracy in most applications, in the case of water it is usually
necessary to adopt formulations that refer to the real fluid behavior. Many other fluids, such as
hydrocarbons, can be used in specific applications; on top of these, if inverse cycles also are considered
for refrigerant systems, the database becomes even broader. Since these fluids cannot be considered
ideal gases, the availability of accurate Equations of State is essential to model any process and to
design, optimize and ultimately manufacture every single plant component with a good degree of
confidence. Thus, the need arises to have a sufficiently precise calculation method for determining the
thermodynamic properties of fluids via an Equations of State able to describe the real fluid behavior.
As a first example, consider an ideal turbine that expands a fluid through an isentropic process from
an initial state 0 (characterized by 𝑇0 and 𝑝0 ) to a final state 1 (characterized by 𝑝1 and 𝑠1 ). The work
obtainable via an adiabatic fluid machine, neglecting the kinetic and potential energy terms, is:
𝑤 = ℎ0 (𝑇0 , 𝑝0 ) − ℎ1 (𝑝1 , 𝑠0 (𝑇0 , 𝑝0 ))
(1.2)
As shown in Section 3, in the case of ideal gas, enthalpy depends only on temperature. For a real fluid,
enthalpy is instead a function of both temperature and pressure. Underestimating the dependency of
enthalpy on pressure can lead in general to severe errors for some fluids in certain conditions.
Therefore, an accurate calculation of the work obtainable from a fluid machine cannot disregard an
accurate calculation of the thermodynamic properties of the working fluid. Likewise, when using
sophisticated design methods for turbomachines based on fluid dynamics, adopting accurate
thermodynamic properties to determine the velocity field of the fluid, the presence of supersonic flows,
as well as other phenomena, is a crucial step for reliable results.
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Thermodynamic properties of fluids - Energy Conversion A – Version 7.0
The second example, regarding gas cycles, highlights the importance of an accurate calculation of the
thermodynamic properties of the working fluid. Consider the regeneration in a Brayton closed cycle
operating respectively with an ideal gas and a real fluid (cycles with helium, He, and carbon dioxide,
CO2, respectively in Fig. 1.1 left and right).
3
a)
6
2
3
b)
4
4
6
2
5
5
1
1
CO2
He
Fig. 1.1 - Closed gas cycle operating between the same extreme temperatures: in the Helium cycle (a),
the fluid can be considered an ideal gas at each point, but in the CO2 cycle (b) the fluid close to the
saturation dome shows important effects of real fluid.
In the cycle with helium (Fig. 1.1a), which can be considered an ideal gas in these conditions because
it operates at temperatures much higher than its critical temperature, an infinite surface regenerator
would give rise to a reversible process, resulting in heat transfer under infinitesimal temperature
differences, leading to 𝑇2 = 𝑇5 and 𝑇6 = 𝑇4 . On the contrary, in the CO2 cycle (Fig. 1.1b) an ideal
regeneration involves an irreversibility in heat transmission, hence 𝑇2 = 𝑇5 but 𝑇6 < 𝑇4, because of
the real fluid effect of pressure on the specific heat along the high pressure isobar, as explained later.
In the following chapters, the equations for calculating the properties of the ideal pure gases as well
as ideal pure liquid are described; then, the corrections due to the real fluid behavior are analyzed.
Subsequently, the Principle of Corresponding States and a brief history of the evolution of the
Equations of State is discussed, in addition to the theoretical description of the trend of the
thermodynamic quantities in the 𝑇 − 𝑠 diagram. Ultimately, concepts for solutions of fluids are
provided, focusing specifically on idea mixtures.
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2
REVIEW OF THERMODYNAMICS FUNDAMENTAL RELATIONS
This chapter deals with a review of the thermodynamics fundamental relations for the calculation of
fluid properties. For a single-phase multi-component homogeneous system in stable equilibrium,
the internal energy 𝑈 can be expressed as follows:
(2.1)
𝑈 = 𝑈(𝑆, 𝑉, 𝒏)
which is called the fundamental relation in the energy form and where 𝑆 is the entropy of the system,
𝑉 is the volume and 𝒏 is the vector of the number of moles of each component. By differentiating:
𝜕𝑈
𝜕𝑈
𝜕𝑈
𝑑𝑈 = ( )
𝑑𝑆 + ( ) 𝑑𝑉 + ∑ ( )
𝜕𝑆 𝑉,𝑛𝑖
𝜕𝑉 𝑆,𝑛𝑖
𝜕𝑛𝑖 𝑆,𝑉,𝑛
𝑖
𝑑𝑛𝑖
(2.2)
𝑗≠𝑖
where 𝑛𝑖 is the number of moles of the i-th component and in which the partial derivatives have a
physical meaning:
𝜕𝑈
( )
≡ 𝑇 (temperature)
𝜕𝑆 𝑉,𝑛𝑖
𝜕𝑈
−( )
≡ 𝑝 (pressure)
𝜕𝑉 𝑆,𝑛𝑖
𝜕𝑈
( )
𝜕𝑛𝑖 𝑆,𝑉,𝑛
≡ 𝜇𝑖 (chemical potential)
(2.3)
(2.4)
(2.5)
𝑗≠𝑖
Consequently, Eq. (2.2) can be rewritten as:
𝑑𝑈 = 𝑇𝑑𝑆 − 𝑝𝑑𝑉 + ∑ 𝜇𝑖 𝑑𝑛𝑖
(2.6)
𝑖
which in the case of a closed system turns to be:
𝑑𝑈 = 𝑇𝑑𝑆 − 𝑝𝑑𝑉
𝑇𝑑𝑆 = 𝑑𝑈 + 𝑝𝑑𝑉
(2.7)
Eq. (2.7) is called the Gibbs equation and defines the relations between states of stable equilibrium:
evolving from one condition of stable equilibrium to another of stable equilibrium, the state variables
(𝑈, 𝑆, 𝑉) must satisfy the Gibbs equation. Otherwise, the final state is no longer of stable equilibrium.
The same result can also be obtained by rearranging the energy balance of a closed-flow system. In
fact, consider a cylinder-piston system in which a gas at temperature 𝑇 and pressure 𝑝 receives
reversibly energy by heat interaction at constant temperature and expands moving the piston against
an environment at same pressure (Fig. 2.1a).
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Thermodynamic properties of fluids - Energy Conversion A – Version 7.0
Fig. 2.1 - Reference system formed by a frictionless piston in which both the expansion and the
introduction of energy by heat interaction take place reversibly (a) and the case in which the expansion
is not reversible (b).
The energy balance for a closed-flow system and a reversible transformation is:
𝑑𝑄 = 𝑑𝑈 + 𝑑𝑊
(2.8)
where 𝑄 is the energy transfer by heat interation (commonly said exchanged heat) and 𝑊 is the energy
transfer by work interaction (exchanged work). For a reversible process, entropy can be computed as:
𝑑𝑄
𝑇
(2.9)
𝑑𝑊 = 𝑝𝑑𝑉
(2.10)
𝑑𝑆 =
while the energy transfer by work interaction is:
Replacing Eqs. (2.9) and (2.10) in Eq. (2.8) and referring to a unit weight:
𝑑𝑢 = 𝑇𝑑𝑠 − 𝑝𝑑𝑣
(2.11)
This relation may seem to be valid only along reversible transformations because of the way it has
been derive. Its validity applies to the general case, as it relates changes in state quantities. This
statement may also be proven by examining the terms of an irreversible transformation, considering a
pressure inside the cylinder higher than the external one, 𝑝 > 𝑝𝑎𝑚𝑏 .
The first principle can thus be formulated as follows:
𝛿𝑞 = 𝑑𝑢 + 𝛿𝑤 = 𝑑𝑢 + 𝑝𝑎𝑚𝑏 𝑑𝑣
(2.12)
in which the work actually exchanged with the outside environment is expressed as the product of the
external pressure (not internal!) by the increase in volume.
The derivative of entropy for an irreversible transformation can be rewritten as:
𝑑𝑠 =
𝛿𝑞
+ 𝑑𝑠𝑖𝑟𝑟
𝑇
(2.13)
which says that the change in entropy of the system comprises two terms: the first term takes into
account the change in entropy due to the exchanged heat, 𝛿𝑞 ⁄𝑇; the second term corresponds to the
internal generation of entropy due to the irreversibility, 𝑑𝑠𝑖𝑟𝑟 . Rearranging:
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Thermodynamic properties of fluids - Energy Conversion A – Version 7.0
𝛿𝑞 = 𝑇(𝑑𝑠 − 𝑑𝑠𝑖𝑟𝑟 )
(2.14)
𝑇(𝑑𝑠 − 𝑑𝑠𝑖𝑟𝑟 ) = 𝑑𝑢 + 𝑝𝑎𝑚𝑏 𝑑𝑣
(2.15)
𝑑𝑢 = 𝑇𝑑𝑠 − (𝑇𝑑𝑠𝑖𝑟𝑟 + 𝑝𝑎𝑚𝑏 𝑑𝑣)
(2.16)
that replaced in (2.12) becomes:
or:
Now let us calculate the production of entropy, 𝑑𝑠𝑖𝑟𝑟 . The wasted work in the irreversible
transformation is the difference between the internal work of the expansion, 𝑝𝑑𝑣, and the work that
the external environment can receive, 𝑝𝑑𝑣 . In general, this difference will transform into kinetic
energy of the particles of the fluid. In accordance with the fact that the transformation is irreversible,
the kinetic energy degrades into internal energy. The dissipation leads to an increase in entropy given
by:
(𝑝𝑑𝑣 − 𝑝𝑎𝑚𝑏 𝑑𝑣)
𝑇
(2.17)
𝑇𝑑𝑠𝑖𝑟𝑟 = (𝑝𝑑𝑣 − 𝑝𝑎𝑚𝑏 𝑑𝑣)
(2.18)
𝑑𝑠𝑖𝑟𝑟 =
from which:
that replaced in (2.16) still supplies the (2.11) whose overall validity is proven. Therefore, it can be
considered a thermodynamic identity valid for a general case.
𝑑𝑢 = 𝑇𝑑𝑠 − 𝑝𝑑𝑣
(2.19)
Let us introduce the thermodynamic state variable enthalpy, ℎ, and its differential, 𝑑ℎ:
ℎ = 𝑢 + 𝑝𝑣
𝑑ℎ = 𝑑𝑢 + 𝑝𝑑𝑣 + 𝑣𝑑𝑝
(2.20)
(2.21)
𝑑ℎ = 𝑇𝑑𝑠 − 𝑝𝑑𝑣 + 𝑝𝑑𝑣 + 𝑣𝑑𝑝 = 𝑇𝑑𝑠 + 𝑣𝑑𝑝
(2.22)
Replacing in (2.11):
Now it is possible to introduce the following definitions:
𝜕𝑢
) specific heat at constant volume
𝜕𝑇 𝑣
𝜕ℎ
𝑐𝑝 ≝ ( ) specific heat at constant pressure
𝜕𝑇 𝑝
1 𝜕𝑣
𝛼𝑝 ≝ ( ) isobaric expansion coefficient
𝑣 𝜕𝑇 𝑝
1 𝜕𝑣
𝑘𝑇 ≝ − ( ) isothermal compressibility coefficient
𝑣 𝜕𝑝 𝑇
𝑐𝑣 ≝ (
9
(2.23)
(2.24)
(2.25)
(2.26)
Thermodynamic properties of fluids - Energy Conversion A – Version 7.0
It is also possible to define two other state variables, i.e. the specific Helmholtz free energy 𝑎 and the
Gibbs free energy.
𝑎 = 𝑢 − 𝑇𝑠
𝑔 = ℎ − 𝑇𝑠
(2.27)
(2.28)
Upon differentiating, 𝑎 and 𝑔 become:
𝑑𝑎 = 𝑇𝑑𝑠 − 𝑝𝑑𝑣 − 𝑇𝑑𝑠 − 𝑠𝑑𝑇 = −𝑝𝑑𝑣 − 𝑠𝑑𝑇
𝑑𝑔 = 𝑇𝑑𝑠 + 𝑣𝑑𝑝 − 𝑇𝑑𝑠 − 𝑠𝑑𝑇 = 𝑣𝑑𝑝 − 𝑠𝑑𝑇
(2.29)
(2.30)
which are thermodynamic identities that will be widely used in the following discussion.
It is now possible to derive the first two Maxwell’s relations, which relate the thermal behavior to
the volumetric behavior of the fluid, from the derivatives of 𝑎 and 𝑔.
Helmholtz free energy
Gibbs free energy
𝑎 = 𝑢 − 𝑇𝑠
𝑔 = ℎ − 𝑇𝑠
By deriving the expression of 𝑎:
By deriving the expression of 𝐺:
𝑑𝑎 = −𝑝𝑑𝑣 − 𝑠𝑑𝑇
Hence, considering
constant volume:
(
a
𝑑𝑔 = 𝑣𝑑𝑝 − 𝑠𝑑𝑇
transformation
𝜕𝑎
) = −𝑠
𝜕𝑇 𝑣
at
Hence, considering
constant pressure:
𝜕 2𝑎
𝜕𝑠
) = −( )
𝜕𝑇𝜕𝑣 𝑣,𝑇
𝜕𝑣 𝑇
𝜕 2𝑎
𝜕𝑝
(
) = −( )
𝜕𝑣𝜕𝑇 𝑇,𝑣
𝜕𝑇 𝑣
𝜕𝑔
( ) =𝑣
𝜕𝑝 𝑇
(2.32)
(2.36)
(2.37)
Again deriving (2.36) and (2.37):
𝜕 2𝑔
𝜕𝑠
(
) = −( )
𝜕𝑇𝜕𝑝 𝑝,𝑇
𝜕𝑝 𝑇
(2.33)
𝜕 2𝑔
𝜕𝑣
(
) =( )
𝜕𝑝𝜕𝑇 𝑇,𝑝
𝜕𝑇 𝑝
(2.34)
(2.38)
(2.39)
Since 𝑔 is a state function, Schwarz’s theorem
on the symmetry of second order derivatives
applies, making (2.38) equal to (2.39):
Since 𝑎 is a state function, Schwarz’s theorem
on the symmetry of second order derivatives
applies, making (2.33) equal to (2.34):
𝜕𝑠
𝜕𝑝
( ) =( )
𝜕𝑣 𝑇
𝜕𝑇 𝑣
which is the first Maxwell’s relation.
at
and a transformation at constant temperature:
Again deriving (2.31) and (2.32):
(
transformation
𝜕𝑔
( ) = −𝑠
𝜕𝑇 𝑝
(2.31)
and a transformation at constant temperature:
𝜕𝑎
( ) = −𝑝
𝜕𝑣 𝑇
a
−(
(2.35)
𝜕𝑠
𝜕𝑣
) =( )
𝜕𝑝 𝑇
𝜕𝑇 𝑝
(2.40)
which is the second Maxwell’s relation.
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Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
These relations are of fundamental importance for the analysis of properties of real fluids. They are
useful to get the general definition of the total state function differentials such as specific volume,
enthalpy and entropy as a function of 𝑑𝑇, 𝑑𝑝 and volumetric behavior of the fluid. One practical
example is shown in Appendix 9.1.
The differential of specific volume as a function of temperature and pressure, 𝑣 = 𝑣(𝑇, 𝑝), is:
𝑑𝑣 = (
𝜕𝑣
𝜕𝑣
) 𝑑𝑇 + ( ) 𝑑𝑝
𝜕𝑇 𝑝
𝜕𝑝 𝑇
(2.41)
or:
𝑑𝑣 = 𝑣𝛼𝑝 𝑑𝑇 − 𝑣𝑘𝑇 𝑑𝑝
(2.42)
From the formula of the differential of 𝑣 – see Eq. (2.42) – by setting 𝑑𝑣 = 0 it is possible to obtain:
𝛼𝑝
𝜕𝑝
( ) =
𝜕𝑇 𝑣 𝑘𝑇
(2.43)
Moreover, the differential of entropy as a function of temperature and pressure, 𝑠 = 𝑠(𝑇, 𝑣), is:
𝜕𝑠
𝜕𝑠
𝑑𝑠 = ( ) 𝑑𝑇 + ( ) 𝑑𝑣
𝜕𝑇 𝑣
𝜕𝑣 𝑇
(2.44)
that substituted into the thermodynamic relation 𝑑𝑢 = 𝑇𝑑𝑠 − 𝑝𝑑𝑣 gives:
𝜕𝑠
𝜕𝑠
𝜕𝑠
𝜕𝑠
𝑑𝑢 = 𝑇 ( ) 𝑑𝑇 + 𝑇 ( ) 𝑑𝑣 − 𝑝𝑑𝑣 = 𝑇 ( ) 𝑑𝑇 + [𝑇 ( ) − 𝑝] 𝑑𝑣
𝜕𝑇 𝑣
𝜕𝑣 𝑇
𝜕𝑇 𝑣
𝜕𝑣 𝑇
𝜕𝑠
𝜕𝑝
(2.45)
𝜕𝑠
where by replacing the first Maxwell relation (𝜕𝑣) = (𝜕𝑇) and considering that 𝑇 (𝜕𝑇) = 𝑐𝑣 :
𝑇
𝑣
𝑣
𝜕𝑝
𝑑𝑢 = 𝑐𝑣 𝑑𝑇 + [𝑇 ( ) − 𝑝] 𝑑𝑣
𝜕𝑇 𝑣
(2.46)
Replacing Eq. (2.43) into Eq. (2.46):
𝛼𝑝
− 𝑝) (𝛼𝑝 𝑑𝑇 − 𝑘𝑇 𝑑𝑝)
𝑘𝑇
𝛼𝑝
= [𝑐𝑣 − 𝑣 (𝑝𝑘𝑇 − 𝑇𝛼𝑝 )] 𝑑𝑇 + 𝑣(𝑝𝑘𝑇 − 𝑇𝛼𝑝 )𝑑𝑝
𝑘𝑇
𝑑𝑢 = 𝑐𝑣 𝑑𝑇 + 𝑣 (𝑇
(2.47)
that is the differential of the state function 𝑢(𝑇, 𝑝).
A similar procedure can be followed for enthalpy, 𝑑ℎ = 𝑇𝑑𝑠 + 𝑣𝑑𝑝, differentiating 𝑠(𝑇, 𝑝):
𝜕𝑠
𝜕𝑠
𝑑ℎ = 𝑇 ( ) 𝑑𝑇 + [𝑇 ( ) + 𝑣] 𝑑𝑝
𝜕𝑇 𝑝
𝜕𝑝 𝑇
11
(2.48)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝜕𝑠
𝜕𝑣
𝜕𝑠
For the second Maxwell relation − (𝜕𝑝) = (𝜕𝑇) and considering that 𝑇 (𝜕𝑇
) = 𝑐𝑝 :
𝑇
𝑝
𝑝
𝜕𝑣
𝑑ℎ = 𝑐𝑝 𝑑𝑇 + [−𝑇 ( ) + 𝑣] 𝑑𝑝
𝜕𝑇 𝑝
(2.49)
1 𝜕𝑣
Where the isobaric expansion coefficient is is again 𝛼𝑝 = 𝑣 (𝜕𝑇) :
𝑝
𝑑ℎ = 𝑐𝑝 𝑑𝑇 + 𝑣[1 − 𝑇𝛼𝑝 ]𝑑𝑝
(2.50)
that is the differential of enthalpy as function of temperature and pressure ℎ(𝑇, 𝑝).
From this, by recalling the thermodynamic identity 𝑑ℎ = 𝑇𝑑𝑠 + 𝑣𝑑𝑝, a formulation for the differential
of entropy 𝑑𝑠 is obtained:
𝑑𝑠 =
𝑑ℎ 𝑣𝑑𝑝 𝑐𝑝 𝑑𝑇 𝑣𝑑𝑝 𝑣𝑇α𝑝 𝑑𝑝 𝑣𝑑𝑝 𝑐𝑝 𝑑𝑇
−
=
+
−
−
=
− 𝑣𝛼𝑝 𝑑𝑝
𝑇
𝑇
𝑇
𝑇
𝑇
𝑇
𝑇
(2.51)
The last relation to be shown is the link between 𝑐𝑝 and 𝑐𝑣 , called Mayer's relation:
𝛼𝑝 2
𝑐𝑝 − 𝑐𝑣 = 𝑇𝑣
𝑘𝑇
that can be obtained by way of a similar procedure.
12
(2.52)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
3
THE IDEAL GAS MODEL
This chapter deals with the definition of pure fluid and the thermodynamic properties of ideal gases.
Additionally, thermodynamic diagrams and transformations are shown and discussed.
3.1
DEFINITION OF PURE FLUID
An ideal gas is a gas that can be modeled by the following Equation of State:
𝑝𝑣̃ = 𝑅𝑢 𝑇 (𝑚𝑜𝑙𝑎𝑟 − 𝑏𝑎𝑠𝑖𝑠)
𝑝𝑣 = 𝑅𝑔 𝑇 (𝑚𝑎𝑠𝑠 − 𝑏𝑎𝑠𝑖𝑠)
(3.1)
with:
𝑅𝑔 =
𝑅𝑢
𝑀𝑀
(3.2)
where 𝑅𝑢 is the universal constant of the gases (equal to 8314 J/kmol K) and 𝑀𝑀 the molecular mass
of the gas considered in kg/kmol.
The kinetic theory of gases states that the volumetric behavior described by the equation of ideal gases
defines a system characterized by:
• a large number of particles interacting with each other only by elastic collisions
• the absence of interaction between the particles through attraction or repulsion forces
• a proper volume of the particles that is negligible compared to the volume taken up by the fluid
In practical cases, a gas can meet these conditions only if rarefied (low pressures) or if its molecules
are moving at a very high speed (high temperature). In these conditions, the particles are very distant
from each other and the intermolecular forces have no way of diverting them from their trajectory.
The fact that specific heats are constant with temperature and thermal conductivity or the absence of
viscous effects constitute additional hypotheses that from time to time may be useful to introduce in
particular discussions, but which are not in any way necessary for defining the ideal gas.
The ideal gas model cannot be applied anymore when the conditions mentioned above are not met, i.e.
there is a relevant amount of collisions due to high fluid density (high pressure and/or low
temperature), the force fields around the molecules is relevant, and the volume of the particles when
the molecules are close together is not negligible. In practice, the molecules tend to attract and/or repel
each other with a behavior that can considerably deviate from that of ideal gas.
There is also an entire class of gases that do not obey the Equation of State of ideal gases, even in
conditions of rarefaction. This occurs when the number of molecules forming the system varies as the
state parameters vary, and molecules associate and dissociate. In this case the molecular weight and
therefore the factor 𝑅 ∗ do not stay constant and the equation of ideal gases is no longer adequate for
describing the volumetric behavior of the substance. All the gases formed by polyatomic molecules
are subject to dissociating in certain regions of the phase diagram and may be therefore found in the
condition mentioned. When a rarefied gas is partially dissociated (e.g.-𝑂2 ⇔ 2𝑂), every single type
(O2, O) still obeys the equation of ideal gases and the specific global volume can then be calculated
with (3.1) on either a molar- or a mass-basis. A generic transformation that changes the molecular
composition of the mixture by increasing or decreasing the degree of dissociation is therefore a cause
for a change in 𝑅𝑔 (real fluid effect).
13
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
In conclusion, there are know two major effects of real fluid: one caused by the intermolecular forces
at high densities, and the other due to phenomena of dissociation or recombination, and hence to the
variability of the mean molecular weight of the gas. Further ahead in these lecture notes, only the first
effect is considered, which is much more important in the power cycles. The following will be
discussed:
• pure fluids (made up of molecules all having the same chemical formula)
• mixtures of multiple pure fluids that do not alter their chemical composition
In other words, fluids that take part in chemical reactions will not be considered.
3.2
THERMODYNAMIC BEHAVIOR OF IDEAL GASES
The ideal gas does not just display a very simple volumetric behavior, but it also enjoys particular
thermal properties that simplify the usual technical calculations considerably. In fact, it can be
demonstrated that internal energy, enthalpy and specific heats are functions of temperature alone,
without introducing any additional hypotheses or kinetic theory.
When replacing Eq. (3.1) within the definition of the isothermal compressibility coefficient, Eq. (2.25),
and in the definition of isobaric expansion coefficient, Eq. (2.26), it turns out that:
1 𝜕𝑣
1 𝑅𝑔
𝑝 𝑅𝑔 1
( ) =
=
=
𝑣 𝜕𝑇 𝑝
𝑣 𝑝
𝑅𝑔 𝑇 𝑝
𝑇
𝑅𝑔 𝑇
1 𝜕𝑣
1
𝑝 𝑅𝑔 𝑇 1
𝑘𝑇 = − ( ) = − (− 2 ) =
=
𝑣 𝜕𝑝 𝑇
𝑣
𝑝
𝑅𝑔 𝑇 𝑝2
𝑝
𝛼𝑝 =
(3.3)
(3.4)
From the general definition of the differential of internal energy shown in Eq. (2.47):
𝛼𝑝
(𝑝𝑘𝑇 − 𝑇𝛼𝑝 )] 𝑑𝑇 + 𝑣(𝑝𝑘𝑇 − 𝑇𝛼𝑝 )𝑑𝑝
𝑘𝑇
𝑝 1
1
1
1
𝑑𝑢 = [𝑐𝑣 − 𝑣 (𝑝 − 𝑇 )] 𝑑𝑇 + 𝑣 (𝑝 − 𝑇 ) 𝑑𝑝
𝑇 𝑝
𝑇
𝑝
𝑇
𝑑𝑢 = [𝑐𝑣 − 𝑣
that returns
𝑑𝑢 = 𝑐𝑣0 𝑑𝑇
(3.5)
where the prime 0 defines the condition of ideal gas.
The same procedure can be repeated for the enthalpy starting from Eq. (2.50)
𝑑ℎ = 𝑐𝑝 𝑑𝑇 + 𝑣[1 − 𝑇𝛼𝑝 ]𝑑𝑝
1
𝑑ℎ = 𝑐𝑝 𝑑𝑇 + 𝑣 [1 − 𝑇 ] 𝑑𝑝
𝑇
𝑑ℎ = 𝑐𝑝0 𝑑𝑇
As far as entropy is considered, starting from Eq. (2.51)
𝑑𝑠 =
𝑐𝑝 𝑑𝑇
− 𝑣𝛼𝑝 𝑑𝑝
𝑇
14
(3.6)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑐𝑝 𝑑𝑇
1
− 𝑣 𝑑𝑝
𝑇
𝑇
𝑐𝑝0 𝑑𝑇 𝑅𝑢
𝑑𝑠 =
−
𝑑𝑝
𝑇
𝑝
which proves to be dependent not only on the temperature, but on the pressure as well.
Lastly, Mayer's relation turns to be:
𝑑𝑠 =
α𝑝 2
𝑐𝑝 − 𝑐𝑣 = 𝑇𝑣
𝑘𝑇
𝑅𝑢 𝑇 1
𝑐𝑝 − 𝑐𝑣 = 𝑇
𝑝
𝑝 𝑇2
𝑐𝑝0 − 𝑐𝑣0 = 𝑅𝑢
(3.7)
(3.8)
The same result is achieved with the classic Joule's experiment. Let us consider a rigid adiabatic
container divided into two compartments, with vacuum created in one of them while the other is filled
with an ideal gas.
Fig. 3.1 - Joule's experiment.
The two compartments are connected by a valve. When the valve is opened, the pressure becomes
even, and the specific volume of the gas rises. The experiment shows that the temperature is not
changed. Consider internal energy 𝑢 as a function of the temperature and specific volume:
𝑢 = 𝑢(𝑇, 𝑣)
(3.9)
By differentiating it holds true that:
𝑑𝑢 = (
𝜕𝑢
𝜕𝑢
) 𝑑𝑇 + ( ) 𝑑𝑣
𝜕𝑇 𝑣
𝜕𝑣 𝑇
(3.10)
The first law of thermodynamics (𝛿𝑞 = 𝑑𝑢 + 𝛿𝑤) applied to the insulated system under consideration
(in which 𝛿𝑞 = 0 and 𝛿𝑤 = 0) supplies
𝑑𝑢 = 0
while experimentally it is valid that
15
(3.11)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑑𝑇 = 0
(3.12)
from Eqs. (3.11), (3.12) and (3.13) since 𝑑𝑣 ≠ 0, so the result is
𝜕𝑢
( ) =0
𝜕𝑣 𝑇
(3.13)
i.e. the internal energy 𝑢 is not a function of the specific volume.
Similarly, when taking 𝑇 and 𝑝 as variables it would be demonstrated that the internal energy is not a
function of the pressure either. That is to say, ultimately:
𝑢 = 𝑢(𝑇)
(3.14)
That is, an isothermal transformation is also a process at constant internal energy for an ideal gas.
16
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
3.3
THERMODYNAMIC PROPERTIES OF IDEAL GASES CALCULATION
The dependency of internal energy, enthalpy and specific heats on temperature only provides
remarkable simplifications in calculating the thermodynamic properties of an ideal gas. Specific
thermodynamic properties, i.e. referred to the mass unit of the fluid, will be derived hereunder.
The internal energy 𝑢 (kJ/kg) can be immediately obtained integrating the Eq. (3.5):
𝑇
𝑢(𝑇) = 𝑢0 + ∫ 𝑐𝑣0 (𝑇)𝑑𝑇
𝑇0
(3.15)
where 𝑢0 is an arbitrary constant, which is assigned the value of the internal energy of the temperature
(arbitrary as well) 𝑇0 .
The enthalpy ℎ (kJ/kg) can be calculated with the relation:
𝑇
ℎ(𝑇) = ℎ0 + ∫ 𝑐𝑝0 (𝑇)𝑑𝑇
𝑇0
(3.16)
The assumptions of ℎ0 = 0 at 𝑇0 = 25°𝐶 are considered for substances made up of a single atomic
species (e.g. N2, O2, Ar). For non-reacting flows, another common assumption is to set ℎ0 = 0 at the
triple point. For substances resulting from the combination of multiple atomic species, and hence for
reacting systems, it is assumed that ℎ0 corresponds to the heat of formatoin of the molecule (if the
reaction is exothermic, ℎ0 will be negative; if the reaction is endothermic, ℎ0 will be positive).1
The entropy 𝑠 (kJ/kg K) is also a function of the pressure and can be calculated using the following
relation
𝑠(𝑇, 𝑝) = 𝑠0 + ∫
𝑇 𝑐 0 (𝑇)
𝑝
𝑇
𝑇0
𝑠(𝑇, 𝑝) = 𝑠0 + ∫
𝑑𝑇 −
𝑇 𝑐 0 (𝑇)
𝑝
𝑇0
𝑇
𝑅𝑢 𝑝 1
∫ 𝑑𝑝
𝑀𝑀 𝑝0 𝑝
𝑑𝑇 −
𝑅𝑢
𝑝
𝑙𝑛
𝑀𝑀 𝑝0
(3.17)
The value of the constant 𝑠0 may be arbitrarily chosen since it always cancels out when differences of
entropy for non-reacting fluids are considered. In the case of reacting systems, on the other hand, the
entropy of reference must be chosen consistently with the third law of thermodynamics, which
postulates that the entropy at absolute zero for a perfect crystal is zero. It is however observed that by
setting 𝑇0 = 0 𝐾, it is necessary to bear in mind all changes of state from the perfect crystal condition
to the state of ideal gas, imagining that they take place in equilibrium conditions.
1
It is well known that enthalpy is defined minus an arbitrary constant. In the calculations relating to the machines,
differences of enthalpies are always considered, which are therefore entirely independent from the selected constant. Use
of the convention specified above is justified by the simplifications obtained in calculating the combustion reactions. A
detailed analysis of the rationale of this assumption is given in “Thermodynamics: Foundations and Applications” by E.
Gyftopoulos and G.P. Beretta, Chapter 29, Dover Publications, 2005
17
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 3.2 - Qualitative representation of the definition of entropy of reference at Kelvin's Zero.
The prior relation therefore becomes:
𝑠(𝑇, 𝑝) = ∫
𝑇𝑐𝑡 𝑐 𝑠1 (𝑇)
𝑝
𝑇0
𝑇
𝑇𝑓𝑢𝑠 𝑐 𝑠2 (𝑇)
𝑇𝑒𝑣𝑎 𝑐 𝑙 (𝑇)
∆ℎ𝑓𝑢𝑠
∆ℎ𝑐𝑡
𝑝
𝑝
𝑑𝑇 +
+∫
𝑑𝑇 +
+∫
𝑑𝑇
𝑇𝑐𝑡
𝑇
𝑇𝑓𝑢𝑠
𝑇
𝑇𝑐𝑡
𝑇𝑓𝑢𝑠
𝑇 𝑐 𝑔 (𝑇)
∆ℎ𝑒𝑣𝑎
𝑅𝑢
𝑝
𝑝
+
+∫
𝑑𝑇 −
𝑙𝑛 𝑠𝑎𝑡(𝑇 )
𝑒𝑣𝑎
𝑇𝑒𝑣𝑎
𝑇
𝑀𝑀 𝑝
𝑇𝑒𝑣𝑎
(3.18)
where for greater clearness the specific heats of solid, liquid and gas, respectively, are indicated with
𝑐𝑝 𝑠1 , 𝑐𝑝 𝑠2 , 𝑐𝑝 𝑙 , 𝑐𝑝 𝑔 and where at pressure 𝑝0 the saturation pressure corresponding to the change in
liquid-steam in equilibrium conditions𝑝 𝑠𝑎𝑡(𝑇𝑒𝑣𝑎) has been replaced.
The entropy of an ideal gas can also be obtained by means of statistical thermodynamics. In this case
the link to the degree of disorder of the thermodynamic system under consideration is well known.
Similar to what has been seen for specific heats, it is found that the entropy of an ideal gas can be
calculated as the sum of a translation contribution, a rotational contribution and a vibrational
contribution.
In summing up what has been explained up to this point, the behavior of an ideal gas is known when
the following are known:
•
•
the molar mass 𝑀𝑀
the trend of the specific heat depending on the temperature 𝑐𝑣0 (𝑇) or 𝑐𝑝0 (𝑇)
The molar mass and the specific heat are determined experimentally; other thermodynamic properties
are then obtained on the basis of experimental data applying the previously described theory.
3.4
SPECIFIC HEAT OF IDEAL GASES CALCULATION
Each molecule in an ideal gas behaves independently from the others. To calculate the thermodynamic
properties, it is therefore sufficient to study the single molecule. This paragraph focuses on the specific
18
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
heat at constant volume first since the other thermodynamic properties of practical interest are easily
obtained from it.
The specific heat at constant pressure for ideal gases can be calculated according to the relation:
𝑐𝑝0 = 𝑐𝑣0 + 𝑅𝑔 (𝑚𝑎𝑠𝑠 − 𝑏𝑎𝑠𝑖𝑠)
(3.19)
𝑐̃𝑝0 = 𝑐̃𝑣0 + 𝑅𝑢 (𝑚𝑜𝑙𝑎𝑟 − 𝑏𝑎𝑠𝑖𝑠)
The specific heat indicates the capability of a fluid to build up energy. Based on the kinetic gas theory
of gases, specific heat at constant volume 𝑐𝑣0 is obtained as the sum of 𝑛 contributions, corresponding
to the 𝑛 excited degrees of freedom of the molecule. Furthermore, based on the law of equipartition of
energy, the absorbed energy is equally divided between the different degrees of freedom of the
molecules.
The degrees of freedom can be classified as:
•
𝑓𝑡 translational degrees of freedom: each particle is characterized by a speed of translation
along the three coordinates (𝑥, 𝑦, 𝑧) of its center of mass. There are always three translational
degrees of freedom for each molecule, regardless of its structure2.
Fig. 3.3 - Translational degrees of freedom for a molecule.
1
Each of them has a contribution equal to 2 𝑅𝑢 . The root-mean-square translational speed of the
particles in a volume forms the mean translational kinetic energy that is, in turn, a function of
the temperature of the gas through the law:
𝐸𝑘 =
3𝑘𝑇
2
(3.20)
where 𝑘 is the Boltzmann constant (1.38 10-23 J/K).
In conclusion, only the portion of energy absorbed to increase the translational kinetic energy
of the molecules results in an increase of fluid temperature.
For more details on the kinetic theory of gases, please refer to Appendix 9.3.
The translational degrees of freedom are always activated for all temperatures over absolute
zero.
The original images of the molecules provided in these lecture notes are obtained with the Jmol
software [Jmol: an open-source Java viewer for chemical structures in 3D. http://www.jmol.org/] and
subsequently changed graphically.
2
19
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
•
𝑓𝑟 rotational degrees of freedom: every molecule can build up energy by rotating around an
axis (𝑥, 𝑦, 𝑧) with respect to which it has a non-null momentum of inertia due to the distance
of the masses of nuclei (considered point masses) from the axis of rotation.
The number of degrees of freedom depends on the molecular structure and the spatial layout
of the atoms.
o For monatomic molecules 𝑓𝑟 = 0 as they are negligible
o For linear diatomic and triatomic molecules 𝑓𝑟 = 2. The theoretical possibility of
rotation around the axis passing through the center of mass of the two atoms of a
diatomic molecule (or the rotation of a monatomic molecule) is used by the molecule
since this rotation, marked by a very small momentum of inertia, is extremely unlikely
according to the laws of quantum mechanics.
o For non-linear polyatomic molecules 𝑓𝑟 = 3.
The rotational degrees of freedom are activated beyond a given threshold temperature that is
generally much lower than the typical conditions of energy engineering. Each of them has a
1
contribution equal to 2 𝑅𝑢 .
Fig. 3.4 - Rotational degrees of freedom for a linear triatomic molecule and for a branched
polyatomic molecule.
•
𝑓𝑣 vibrational degrees of freedom: each pair of atoms can vibrate with a certain proper
vibrational frequency 𝜐𝑖 (usually between 1012 and 1014 Hz and which is traced using
spectroscopic methods) according to stretching movements (approach and departure of the
atoms - symmetric and asymmetric) and bending movements (change in molecular angles wagging, twisting, rocking, scissoring). This causes a change in the dipole moment of the
molecule. The number of vibrational modes is defined by the number of atoms and the spatial
configuration of the molecule, and each of them is activated once a certain threshold
temperature is reached 𝜃𝜐𝑖 . Even if they have the same frequency, some vibrational modes stay
independent.
20
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 3.5 - Vibrational degrees of freedom of stretching and bending.
Considering each molecule as a harmonic oscillator, it is possible to calculate the total
vibrational energy of the gas (i.e. the sum total of the molecules) as the sum of the energies of
the molecules occupying the various energy levels allowed by quantum mechanics. From this,
the following expression (owed to Einstein) is obtained by differentiation for the vibrational
component of the specific heat on a molar-basis at constant volume:
𝑐̃𝑣0,𝑣
𝑢𝑖 2 𝑒 𝑢𝑖
= 𝑅𝑢 ∑ 𝑢
(𝑒 𝑖 − 1)2
(3.21)
𝑖
𝑢𝑖 =
ℎ𝜐𝑖
𝑘𝑇
(3.22)
in which ℎ is the Planck constant (6.626 10-34 Js), 𝑘 the Boltzmann constant (1.38 10-23 J/K), T
the absolute temperature, 𝜐𝑖 the proper oscillation frequency.
The trend of 𝑐̃𝑣0,𝑣𝑖 as a function of the parameter 𝑢 is shown in Fig. 3.6a, while Fig. 3.6b shows
the trend of specific heat on a molar-basis at constant volume throughout the temperature field
(starting from 0 K).
21
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 3.6 - a) Trend of the vibrational contribution of the specific heat on a molar-basis as a function
of the parameter 𝑢 and hence of the temperature, b) trend of the specific heat on a molar-basis as
the temperature rises from Kelvin zero.
•
An examination of the figures reveals the following.
As the temperature rises (reduction of the parameter 𝑢) the vibrational degree of freedom is
excited in an increasing number of molecules and increases the vibrational contribution to the
specific heat on; at the extremely high temperatures, the parameter 𝑢 tends to zero and the
vibrational contribution is equal to 𝑅𝑢 . It is therefore acknowledged that the vibrational degree
1
of freedom is characterized by the sum of two equal terms, each at 2 𝑅𝑢 . They respectively take
•
•
into account the potential energy and the kinetic energy of the harmonic oscillator.
This is valid as long as the molecule remains intact without sustaining dissociation or changes
its structure.
For a given temperature, the contribution of vibration is more appreciable for the molecules
made up of heavy atoms (low vibrational frequency, small 𝑢 parameter) than for those made
up of light atoms (high frequency, large 𝑢 parameter).
Lastly, there are two other ways of absorbing energy, i.e. electronic excitation and ionization. These
mechanisms however require very high temperatures in order to be activated and their contribution to
molar heat, at the temperatures of interest, is usually negligible.
The hypothesis of 𝑐̃𝑣0 constant with the temperature can be applied only for monatomic ideal gases or
for polyatomic gases in which vibrational degrees of freedom are not activated, or lastly for
transformations that involve an infinitesimal change in temperature.
3.4.1
Monoatomic molecules
The three degrees of freedom of translation equally contribute to the specific heat with constant volume
1
with 2 𝑅𝑢 each
22
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
3
𝑐𝑣0 = 𝑅𝑢
2
(3.23)
No other mechanisms that allow the molecules to store energy outside kinetic energy of translation
exist. Therefore, the monoatomic gases, some of which are of great technical importance (He, Ar),
have specific heats on a molar basis equal and constant with temperature.
Additionally, the metal vapors under conditions of rarefaction (such as mercury, sodium, etc.) belong
to the category of monoatomic gases as well. Despite having the same value of specific heat on a
molar-basis, all these monoatomic gases have a great variability of specific heats on a mass-basis due
to the largely different molar mass.
3.4.2
Diatomic molecules
In the case of diatomic and polyatomic molecules, the specific heat on a molar-basis is not constant
with temperature. This is because of the increased impact of the vibrational degrees of freedom. The
trend of vibrational degrees of freedom depends on temperature according to quantum mechanics as
previously explained.
In addition to the degrees of freedom of translation, diatomic molecules possess two degrees of
freedom of rotation around two perpendicular axes, lying in the normal plane at the line segment
joining the two atoms.
Each degree of freedom of rotation is normally excited already at very low temperatures (lower than
1
those of our field of interest) and thus it contributes to 𝑐̃𝑣0 with 2 𝑅𝑢 each. Therefore, the specific heat
of a diatomic molecule considered as rigid is:
1
1
5
𝑐̃𝑣0 = 3 𝑅𝑢 + 2 𝑅𝑢 = 𝑅𝑢
2
2
2
(3.24)
Oxygen, nitrogen (and therefore air, 𝑀𝑀 ≈ 29.0) for temperatures below ambient have a specific heat
equal to:
5
kJ
𝑐̃𝑣0 = 𝑅𝑢 = 20.785
2
kmolK
7
kJ
𝑐̃𝑝0 = 𝑅𝑢 = 29.099
2
kmolK
7 𝑅𝑢
kJ
𝑐𝑝0 =
= 1.003
2 𝑀𝑀
kgK
As the temperature increases, the distance between the two atoms does not remain unchanged over
time: the degree of freedom of vibration becomes excited. In the temperature range of interest for the
study of turbomachines, the variability of specific heat of the diatomic molecules with temperature is
essentially due to the variability of the contribution of vibrations, since the rotational degrees of
freedom are usually fully excited. For these reasons the specific heat of ideal gas for the exhaust gases
of a gas turbine is expressed as a polynomial function of the temperature, where the term 𝑎0 represents
the contribution of the translational and rotational degrees of freedom to the 𝑐̃𝑣0 of the fluid.
23
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑛
𝑐̃𝑣0 (𝑇)
= 𝑎0 + ∑ 𝑎𝑖 𝑇 𝑖
(3.25)
𝑖=1
𝑐̃𝑣0 (𝑇) =
5
0
𝑅 + 𝑐̃𝑣,𝑣
(𝑇)
2 𝑢
(3.26)
In the figure the trend of the specific heat at constant pressure of some diatomic molecules depending
on the temperature is shown; the term 2 𝑐̃𝑣0 ⁄𝑅𝑢 is given on the second scale, which is the number of
degrees of freedom (if translational and rotational are considered). It can be noted how the number of
degrees of freedom tends to the value 5 for low temperatures, in accordance with what the kinetic
theory of gases establishes (3 translational degrees and 2 rotational degrees). By sake of comparison,
monoatomic molecules have a constant specific heat equal to 𝑐̃𝑝0 =20.785 kJ/(kmol K) or a parameter
2 𝑐̃𝑣0 ⁄𝑅𝑢 = 3.
42
8
40
38
I2
7
36
Cl2
F2
34
HCl
6
H2
32
30
5
28
0
500
1000
1500
2000
(K)
Fig. 3.7 - Trend of the specific heat on a molar-basis for diatomic molecules as function of the
temperature.
3.4.3
Polyatomic molecules
First of all, a differentiation between molecules arranged in a linear structure and molecules arranged
in a 3D structure must be done. For the first (for example, carbon dioxide O=C=O) three degrees of
freedom of translation, two degrees of freedom of rotation and a certain number of vibrational degrees
of freedom contribute to heat on a molar-basis, due to the change of the relative distances between the
various atoms constituting the molecule. Angular and branched molecules have three rotational
degrees of freedom, on top of which the contribution of the vibrational degrees of freedom must always
be added. The greater the number of atoms present in the molecule, called molecular complexity of
the fluid, the greater the number of vibrational modes possible and thus the number of degrees of
freedom that allow energy to be absorbed.
For example, for complex molecules (e.g. hydrocarbons or halocarbons) the large number of possible
vibrating modes causes a very high specific heat at constant volume that can reach values of 240~320
kJ/(kmol K), of which at the most 6/2 𝑅 = 24.9 kJ/(kmol K) are justified by the degrees of freedom of
translation and rotation.
24
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
In conclusion, while for many situations of practical interest in the study of machines the ideal gas
hypothesis is acceptable, the hypothesis of constant specific heats is not as often acceptable (in fact,
only for monoatomic gases).
Fig. 3.8 - Trend of the specific heat on a molar-basis of the polyatomic molecules as function of the
temperature.
3.4.4
Practical examples
A mole of ideal gas is confined in a vessel at constant volume: in supplying a certain amount of heat,
its temperature and pressure grow inversely proportionate to its specific heat, and hence to the
molecular complexity.
• A monoatomic molecule can absorb energy only by increasing the velocity of its molecules
and therefore the temperature and pressure.
• With the heat introduced being equal and based on the energy equipartition theorem, a
polyatomic molecule manifests a smaller increase in temperature since only part of the energy
supplied serves to accelerate the molecules in the components of the translational velocity
while the remaining part serves to excite rotational and translational degrees of freedom that
have no effect on the temperature, and thus on the pressure.
3.5
THERMODYNAMIC DIAGRAMS AND TRANSFORMATIONS
In this section, how thermodynamic diagrams can be built and how thermodynamic transformations
can be calculated by means of a purely theoretical analysis will be shown.
3.5.1
Trend of the isobars of ideal gas in the Ts diagram
Let us consider, for example, the temperature-entropy plane. In the case of monoatomic gases, this
plane is equivalent to an enthalpy-entropy plane since the specific heat is constant; in the case of a
polyatomic ideal gas, the two scales (𝑇 and ℎ) instead are not equivalent in view of the dependence of
the specific heat on temperature. Here reference to the case of gas with 𝑐̃𝑣0 constant is made.
In the fundamental relation the isobar curves are easily obtainable by setting 𝑑𝑝 = 0:
𝑑ℎ = 𝑇𝑑𝑠 + 𝑣𝑑𝑝
25
(3.27)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
that thus becomes:
𝑐𝑝0 𝑑𝑇 = 𝑇𝑑𝑠
(3.28)
𝑠−𝑠0
𝑐𝑝0
(3.29)
from which integrating it holds true that:
𝑇 = 𝑇0 𝑒
with temperature 𝑇[K], entropy s [kJ/(kg K)] and heat capacity 𝑐𝑝0 [kJ/(kg K)].
Once a 𝑇0 and a 𝑠0 are set, the trend of an isobar is exponential with respect to 𝑠, with a gradient equal
to:
𝑑𝑇 𝑇
=
𝑑𝑠 𝑐𝑝0
(3.30)
It is possible to observe that:
• The gradient of the isobar curves grows as the temperature increases for monoatomic gases as
well
• with temperature being equal, the isobars slope is lower the higher the molecular complexity
of the gas is.
• This behavior is also valid for the polyatomic ideal gases, although in a less marked manner
since 𝑐𝑝0 (𝑇) increases less than linearly with temperature.
Once the isobars as a function of temperature and pressure are known, also all the isochoric lines in
the 𝑇𝑠 plane are known. This is because the ideal gas assumption, that relates temperature, pressure
and volume itself can be used..
Now let us consider a point of the isobar at 𝑝0 with temperature 𝑇𝑥 and let's move along an isotherm
up to the pressure 𝑝𝑥 . The entropy of the new point will be equal to:
𝑝𝑥
𝑠(𝑇𝑥 ,𝑝𝑥 ) = 𝑠(𝑇𝑥 ,𝑝0) − ∫
𝑝0
𝑅𝑢 𝑑𝑝
𝑀𝑀 𝑝
(3.31)
or:
∆𝑠(𝑇𝑥 ,𝑝0→𝑝𝑥 ) = −
𝑅𝑢
𝑝𝑥
𝑙𝑛
𝑀𝑀 𝑝0
(3.32)
Since the deviation ∆𝑠 does not depend on the temperature, all the points of the new isobar 𝑝𝑥 are
obtained by translation of the reference isobar 𝑝0 .
The isobars are instead divergent with 𝑠 constant as the temperature and entropy rise. The temperature
difference of an isentropic transformation increases as the initial temperature rises: a peculiarity that
allows gas turbine cycles to have a useful positive work.
26
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
T
s
=
=
<
<
Fig. 3.9 - Trend of the isobars in the temperature-entropy plane for an ideal gas; all the curves are
equal and translated horizontally; the isobars "diverge" in the direction that the temperature
changes obtained between two isobars with isentropic processes increase as the level of temperature
(or of entropy) considered rises.
27
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
3.5.2
Temperature rise in an isentropic compression
It is possible to give an intuitive explanation for the different behaviors of the various fluids
considering 𝑛 moles of a gas that can be assumed ideal in an adiabatic cylinder closed at one end by a
fixed wall and at the other by a frictionless moving piston.
Under these assumptions, a process of compression (or expansion) is isentropic.
Fig. 3.10 - Adiabatic cylinder piston system.
Three cases are considered: in the first case the gas is formed by helium (a monoatomic molecule), in
the second by air (simplified to as a diatomic molecule) and in the last a fluorocarbon (as an example
of a complex molecule).
The change in internal energy and enthalpy, considering 𝑑𝑆 = 0, will be:
𝑑𝑈 = 𝑇𝑑𝑆 − 𝑝𝑑𝑉 = −𝑝𝑑𝑉 = 𝑛𝑐̃𝑣0 (𝑇)𝑑𝑇
𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑝 = 𝑉𝑑𝑝 = 𝑛𝑐̃𝑝0 (𝑇)𝑑𝑇
From which the coefficient 𝛾 called heat capacity ratio or adiabatic index can be defined as:
𝑐̃𝑝0 (𝑇)
𝑑𝑝⁄𝑝 𝑐̃𝑣0 (𝑇) + 𝑅𝑢
𝑅𝑢
𝛾= 0
=−
=
=
1
+
𝑑𝑉 ⁄𝑉
𝑐̃𝑣 (𝑇)
𝑐̃𝑣0 (𝑇)
𝑐̃𝑣0 (𝑇)
(3.33)
The smaller the parameter 𝛾, the more similar the values for the specific heats are for complex
molecules with high 𝑐̃𝑣0 .
When integrating the Eq. (3.33), it holds true that:
𝛾 ln(𝑉) + 𝑐𝑜𝑛𝑠𝑡 = − ln(𝑝) + 𝑐𝑜𝑛𝑠𝑡
𝛾 ln(𝑉) + ln(𝑝) = 𝑐𝑜𝑛𝑠𝑡
ln(𝑝𝑉 𝛾 ) = 𝑐𝑜𝑛𝑠𝑡
𝑝𝑉 𝛾 = 𝑐𝑜𝑛𝑠𝑡
(3.34)
Alternatively, the previous equation can be written as a function of 𝑇, 𝑝 and 𝑣, 𝑇 by replacing the ideal
gas law:
𝑝
𝛾−1
𝛾
𝑇
= 𝑐𝑜𝑛𝑠𝑡
𝑉 𝛾−1 𝑇 = 𝑐𝑜𝑛𝑠𝑡
28
(3.35)
(3.36)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝛾 is a function of the temperature except for monoatomic gases. Hence, the three equations are valid
only for very small changes in temperature; they may, however, be applied with an average value for
𝛾 in the considered temperature range.
Table 3.1 shows the values of 𝛾 of three fluids, in the simplifying hypothesis in which they are constant
with temperature. The number of degrees of freedom and the value of specific heats for the complex
fluid are derived backwards by reversing the equation (3.26) and setting the value of 𝛾. The percentage
change in temperature is obtained by reducing the system volume by 10% (i.e. 𝑉1⁄𝑉0 = 0.9).
Table 3.1 - Characteristics of isentropic compression at same compression ratio for fluids with
different complexities.
Fluid
Degrees of
freedom
𝐶𝑣0
𝐶𝑝0
𝛾
𝑇1 ⁄𝑇0
Helium
Air
Fluorocarbon
3
5
50
3/2 𝑅
5/2 𝑅
1.67
7.31%
5/2 𝑅
7/2 𝑅
1.4
4.30%
25 𝑅
26 𝑅
1.04
0.42%
From a kinetic theory of gases viewpoint, the work done by the piston is justified with the increase in
kinetic energy of the molecules that by knocking against a movable wall are reflected at a velocity
higher than the original one. In a monoatomic gas, the increase in translation energy - even if it is
divided between all the molecules - does not find other ways in which to manifest itself except in an
increase in temperature which gives precisely an average measure of the kinetic energy of translation.
On the contrary, in polyatomic gases an increase in energy of translation of an extent similar to that of
the previous case is then quickly distributed to all the degrees of freedom, and in particular to those of
vibration. In conclusion, the residual increase of the translational energy, and therefore of the
temperature, is greatly reduced.
3.5.3
Optimum compression ratio in a closed cycle
Let's consider making a closed Brayton cycle operating with the three fluids previously considered in
temperature and pressure conditions such that they can be considered ideal gases. Let's consider an
ideal cycle operating between a T1 and a T3 and the optimum compression ratio is set for maximizing
the useful work of the cycle. This hypothesis results in a T2=T4, regardless of the working fluid. Hence,
the percentage temperature increase in the compressor, as well as the percentage temperature drop in
the turbine will have equal values.
T
3
2
4
1
s
29
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 3.11 - Thermodynamic diagram of a closed Brayton cycle with ideal gas at the point of
maximum useful work in the Ts plane.
This design assumption however leads to compression ratio values drastically different when the
molecular complexity of the fluid varies.
From the definition of 𝛾 and remembering that 𝑐̃𝑝0 − 𝑐̃𝑣0 = 𝑅𝑢 the definition of 𝜃 is found as:
𝜃=
𝛾 − 1 𝑅𝑢
= 0
𝛾
𝑐̃𝑝
(3.37)
From Eq. (3.35), the temperature change in an isentropic compression or expansion is calculated by
means of the relation:
𝑇2
𝑝2 𝜃
=( )
𝑇1
𝑝1
(3.38)
In the case of a complex molecule gas, the high heat on a molar basis directly translates into a
peculiarity of behavior in the isentropic transformations; in fact, if 𝑐̃𝑝0 is very large, 𝜃 tends to zero and
the isentropic transformation tends to also become isotherm (if the number of degrees of freedom is
infinite, it is impossible to heat the gas).
The results that can be obtained with a ratio between the set temperature equal to 𝑇2 ⁄𝑇1 =1.25 are
shown in the table:
Table 3.2 - Characteristic indexes of an isentropic expansion for fluids with different complexities.
Fluid
𝛾
𝜃
𝛽
Helium
1.67
0.4
1.75
Air
1.4
0.286
2.18
Fluorocarbon
1.04
0.038
331
It is evident that the three systems, although equivalent from the thermodynamic point of view as they
have the same efficiency and the same useful work, are noticeably different from the technological
and construction viewpoint of the turbo machines.
3.5.4
Molecular complexity effect on the isentropic compression ratio
The same conclusions can be obtained by observing the following figures.
With the growth of the isentropic compression ratio, the ratio between the initial and final temperatures
of the transformation is a function of the molecular complexity and hence of the 𝑐̃𝑝0 and 𝛾 of the fluid.
The relative increase in temperature goes lower as the complexity of the fluid goes higher, and
therefore the degrees of freedom available to store energy, is greater.
Vice versa, with the ratio between the temperatures set, the ratio between the pressures is all the larger
the greater the complexity of the fluid. A similar result can be obtained for the volumetric ratios, which
are an extremely important parameter for the choice of the number of stages of a turbo machine.
30
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 3.12 - Trend of the ratio of temperatures and the volumetric ratio for an isentropic compression
with fluids with different molecular complexities.
3.5.5
Molecular complexity and molecular weight effect on isentropic enthalpy change
The previous procedure is repeated plotting the ∆ℎ on a mass-basis as the expansion ratio changes.This
variable affects the choice of the number of stages of a turbo machine once a maximum peripheral
velocity and a maximum load coefficient are defined.
∆ℎ = 𝑐𝑝0 ∆𝑇 =
𝑅
𝛾
𝑇 (1 − 𝛽 −𝜃 )
𝑀𝑀 𝛾 − 1 1
(3.39)
As it can be observed, it depends not only on the molecular complexity, but also on the molecular
weight. The results at same 𝛽 and fluid inlet temperature are shown in Fig. 3.13:
• At same 𝑀𝑀 the complexity of the fluid has a certain influence, and the lower the complexity
the higher the enthalpy change is.
• With complexities being equal, the enthalpy change is inversely proportional to 𝑀𝑀.
31
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 3.13 - Trend of the enthalpy change for an isentropic expansion for different molecular
complexities as a function of the molecular weight.
3.5.6
Molecular complexity effect on the volumetric flow rate in a heat exchanger
Let us consider a heat exchanger that heats a certain gas flow rate. The available thermal energy, the
initial and final temperatures of the transformation (and therefore the ∆𝑇 carried out by the fluid) are
known, as well as the pressure (low enough to be able to consider the gas an ideal gas) at which the
process considered isobaric takes place.
The flow rate of fluid on a mass-basis can be obtained from the energy balance of the component:
𝑄̇ = 𝑚̇∆ℎ = 𝑚̇𝑐𝑝0 ∆𝑇
(3.40)
𝑄̇
𝑄̇
=
𝑐𝑝0 ∆𝑇
𝑐̃𝑝0
𝑀𝑀 ∆𝑇
(3.41)
𝑚̇ =
𝛾
By replacing the formula of the 𝑐̃𝑝0 as a function of the molecular complexity of the fluid (𝑐̃𝑝0 = 𝛾−1 𝑅𝑢 ),
the following is obtained:
𝑚̇ =
𝑄̇
𝑄̇
𝛾−1
=
0
𝑅𝑢
𝛾
𝑐𝑝 ∆𝑇
𝑀𝑀 ∆𝑇
(3.42)
That is, the circulating flow rate on mass-basis depends both on the molecular complexity and the
molecular weight of the fluid.
For example, He and Hg both have:
32
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑐̃𝑝0 =
5
5
J
kJ
𝑅𝑢 = 8314
= 20.785
2
2
kmolK
kmolK
(3.43)
and, therefore, for He (MM=4) and Hg (MM=200) it is respectively:
𝑐̃𝑝0
0
𝑐𝑝 =
→
𝑀𝑀
kJ
kgK
kJ
= 0.104
kgK
0
𝑐𝑝,𝐻𝑒
= 5.196
0
𝑐𝑝,𝐻𝑔
(3.44)
This means that a mercury heat exchanger at same thermal load and same temperature difference will
have a flow rate on a mass-basis about 50 times larger than the one of a helium exchanger.
For the design of a heat exchanger, as well as for the connection piping between the components, the
real parameter of interest is the volumetric flow rate of the fluid, which gives the real dimensions of
the component in terms of front section (since a nominal design velocity is generally assumed).
The volumetric flow rate expressed under the conditions of average temperature of the process 𝑇̅ does
not depend on the molecular weight of the fluid, but only on its complexity. As the molecular
complexity increases, the volumetric flow rate of the fluid sharply decreases, and therefore the size of
the admission and discharge sections of a heat exchanger and the connection piping, and the overall
cost of these components drastically changes:
𝑉̇ = 𝑚̇𝑣 =
𝑄̇
𝑅𝑢
𝑀𝑀
𝛾 − 1 𝑅𝑢 𝑇̅ 𝛾 − 1 𝑄̇ 𝑇̅
=
𝛾 ∆𝑇 𝑝
∆𝑇 𝛾 𝑀𝑀 𝑝
33
(3.45)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
4
THE IDEAL LIQUID MODEL
This chapter deals with the model of ideal liquids fo the calculation of their thermodynamic properties.
The behavior of an ideal liquid is described by a volumetric Equation of State that has the simple form:
𝑣 = 𝑐𝑜𝑛𝑠𝑡
(4.1)
which mathematically translates the characteristic of the incompressibility of the liquid. As already
seen for the ideal gas, also in this case additional hypotheses are not included, such as, the lack of
viscosity or the constancy of specific heats.
Substituting Eq. (4.1) in the definition of the isobaric expansion coefficient, Eq. (2.25), and isothermal
compressibility coefficient, Eq.(2.26), it holds true that:
1 𝜕𝑣
( ) =0
𝑣 𝜕𝑇 𝑝
1 𝜕𝑣
𝑘𝑇 = − ( ) = 0
𝑣 𝜕𝑝 𝑇
𝛼𝑝 =
(4.2)
(4.3)
As a first observation, the general Mayer's relation (2.52) boils down to:
𝑐𝑝 − 𝑐𝑣 = 𝑇𝑣
𝛼𝑝 2
=0
𝑘𝑇
And hence the specific heats coincide, and it is enough to consider a single value defined as 𝑐 𝑙 .
From equation (4.1) it follows also that since all the processes occur necessarily at specific constant
volume, a single specific heat is defined that coincides with the one at constant volume 𝑑𝑣 = 0:
𝜕𝑢
𝜕𝑢
𝑑𝑢 = ( ) 𝑑𝑇 + ( ) 𝑑𝑣 = 𝑐𝑣 𝑑𝑇 = 𝑐 𝑙 𝑑𝑇
𝜕𝑇 𝑉
𝜕𝑣 𝑇
(4.4)
From the identity related to the internal energy and setting 𝑑𝑣 = 0, it holds true that:
𝑑𝑢 = 𝑇𝑑𝑠 − 𝑝𝑑𝑣
𝑑𝑢 = 𝑇𝑑𝑠 = 𝑐 𝑙 𝑑𝑇
(4.5)
Entropy, internal energy and temperature are uniquely bound and independent from the pressure. From
the previous relation it is possible to observe that, as opposed to a gas, mechanical work 𝑊 = ∫ 𝑝𝑑𝑣
cannot be performed on an ideal liquid and the liquid cannot be heated due to the effect of an isentropic
compression (since 𝑑𝑠 = 0, 𝑑𝑇 = 0).
In the case of an ideal liquid, an increase in temperature can be obtained only at the expense of an
increase in entropy resulting from thermal exchange or irreversibility.
The temperature-entropy diagram of an ideal liquid is thus reduced to a single line representing all the
isobars that "collapse" on the lower limit curve. For an ideal liquid an isentropic compression is also
isotherm. On the 𝑇 − 𝑠 diagram the compression initial and final points belonging to two different
isobars are superposed.
34
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
An improvement in the equation of state (4.1), which considers the specific volume independent from
the other state variables (𝑝, 𝑇) is obtained by re-introducing the dependency of the specific volume on
the temperature, and keeping it independent from the pressure.
𝑣 = 𝑣(𝑇)
(4.6)
The above equation is no longer sufficient at particularly high pressures or in the vicinity of the critical
point, and it is necessary to use a law of dependency of the specific volume on two state variables, for
example, 𝑣 = 𝑣(𝑇, 𝑝).
4.1
THERMODYNAMIC PROPERTIES OF AN IDEAL LIQUID CALCULATION
Similarly to what was done for the ideal gases, it is possible to get the differentials 𝑑𝑢, 𝑑ℎ, 𝑑𝑠 starting
from the general and sound relations for each fluid obtained in function of 𝑑𝑝, 𝑑𝑇 and of the volumetric
behavior.
4.1.1
Internal energy
From the general definition of internal energy shown in Eq. (2.47):
𝑑𝑢 = [𝑐𝑣 − 𝑣
𝛼𝑝
(𝑝𝑘𝑇 − 𝑇𝛼𝑝 )] 𝑑𝑇 + 𝑣(𝑝𝑘𝑇 − 𝑇𝛼𝑝 )𝑑𝑝
𝑘𝑇
it follows that
𝑑𝑢 = 𝑐𝑣 𝑑𝑇 = 𝑐 𝑙 𝑑𝑇
(4.7)
The internal energy of an incompressible liquid in 𝑇 conditions is obtained as:
𝑇
𝑢(𝑇) = 𝑢0 + ∫ 𝑐 𝑙 (𝑇)𝑑𝑇
𝑇0
(4.8)
and, as already pointed out, it is a function only of the temperature.
4.1.2
Enthalpy
From the general definition of enthalpy shown in Eq. (2.50)
𝑑ℎ = 𝑐𝑝 𝑑𝑇 + 𝑣[1 − 𝑇𝛼𝑝 ]𝑑𝑝
From which it holds true that:
𝑑ℎ = 𝑐𝑝 𝑑𝑇 + 𝑣𝑑𝑝 = 𝑐 𝑙 𝑑𝑇 + 𝑣𝑑𝑝
(4.9)
Unlike the ideal gas, the enthalpy also depends on the pressure for an ideal liquid. The enthalpy of an
incompressible liquid in 𝑇, 𝑝 conditions is obtained as:
𝑇
ℎ(𝑇, 𝑝) = ℎ0 + ∫ 𝑐 𝑙 (𝑇)𝑑𝑇 + 𝑣(𝑝 − 𝑝0 )
𝑇0
35
(4.10)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
The reference conditions and the constant ℎ0 can be arbitrarily chosen in advance. However, a
reference that is preferable in practical applications exists. It is based on the knowledge of the
thermodynamic values on the limit curve which generally are set out in tabulation and are available
without difficulty.
To calculate ℎ(𝑇, 𝑝), a point on the limit curve at the temperature 𝑇 is considered as a reference. Thus,
𝑇 = 𝑇 𝑠𝑎𝑡 and 𝑝 = 𝑝 𝑠𝑎𝑡 ; moving along an isothermal transformation, from Eq. (4.10) it holds true that:
ℎ(𝑇, 𝑝) = ℎ 𝑠𝑎𝑡(𝑇) + 𝑣(𝑝 − 𝑝 𝑠𝑎𝑡(𝑇) )
(4.11)
To calculate ℎ(𝑇, 𝑝), a point on the limit curve at the pressure 𝑝 as reference is considered. Thus, 𝑝 =
𝑝 𝑠𝑎𝑡 and 𝑇 = 𝑇 𝑠𝑎𝑡(𝑝). From these values it holds true that:
𝑇
ℎ(𝑇, 𝑝) = ℎ 𝑠𝑎𝑡(𝑝) + ∫
𝑐 𝑙 (𝑇)𝑑𝑇
𝑇 𝑠𝑎𝑡(𝑝)
(4.12)
The point (𝑇, 𝑝) is reached along an isobaric transformation from 𝑇 𝑠𝑎𝑡(𝑝) to 𝑇; in this case, however,
it is necessary to know the trend of the specific heat in the temperature range considered in order to
calculate the heat exchanged along the isobar.
The thermodynamic points of interest for the previous relations in a diagram 𝑇 − 𝑠 and ℎ − 𝑠 are
shown in the figure. Note that the isobars in the 𝑇 − 𝑠 plane are represented as distinct curves for
graphic reasons when actually, for the hypothesis of incompressible liquid, they are all coincident and
superimposed on the saturation curve.
If Eq. (4.12) is used for the calculation of the enthalpy difference between two points 𝐴 and 𝐵 getting:
𝑇𝐵
∆ℎ𝐴→𝐵 = ∫ 𝑐 𝑙 (𝑇)𝑑𝑇 + 𝑣(𝑝𝐵 − 𝑝𝐴 )
𝑇𝐴
When temperature changes are not very small, the first term is usually predominant. For example, the
enthalpy change resulting from 𝛥𝑇 = 10 K for water is higher than that corresponding to 𝑃 =
400 bar. This assures that in the ℎ − 𝑠 diagram the isobars are concentrated in the vicinity of the limit
curve (although without being superimposed as in the case of the diagram 𝑇 − 𝑠).
36
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 4.1 - Enthalpy calculation for a sub-cooled liquid starting from the saturation data moving
along an isobar or an isotherm. a) T-s plane, b) h-s plane.
4.1.3
Entropy
From the general enthalpy definition provided in Eq. (2.51)
𝑑𝑠 =
𝑐𝑝 𝑑𝑇
− 𝑣α𝑝 𝑑𝑝
𝑇
from which it holds true that:
𝑑𝑠 =
𝑐𝑝 𝑑𝑇 𝑐 𝑙 𝑑𝑇
=
𝑇
𝑇
(4.13)
which as mentioned above also depends only on the temperature.
The previous integrated equation calculates the entropy of any point:
𝑇 𝑙
𝑠(𝑇) = 𝑠0 + ∫
𝑇0
37
𝑐 (𝑇)𝑑𝑇
𝑇
(4.14)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
5
THE REAL FLUID PROPERTIES
This chapter deals with the model to describe rea fluids and their properties, starting from the ideal gas
ones. When a fluid is not well described by the ideal gas model, a real fluid hypothesis needs to be
posed (which also includes the two-phase mixtures and liquids). The thermodynamic properties in
these conditions can be calculated starting from the corresponding quantity of ideal gas and applying
a suitable correction of real fluid. In these lecture notes, the real fluid corrections to be applied to
specific volume, specific heat, enthalpy and entropy will be analytically obtained.
A fluid is considered real when the hypotheses underlying ideal gas drop, and it is necessary to take
into account the intermolecular forces and finite dimensions of the molecules. As previously
mentioned, two different real fluid effects are distinguished. The first, which is called conventionally
"at low temperature", results in high densities, especially in the proximity of the limit curve. The
second, which called "at very high temperature", intervenes in conjunction with dissociation or
recombination phenomena. Thermodynamic analysis valid in the more general conditions for any
substance (solid, liquid, etc.) will be investigated. It is applied in practice above all to calculate the
effect of real fluid at low temperature (for example, the water vapor properties tables have been
obtained by following this route). Dissociation of diatomic or polyatomic molecules are taken into
account considering, through the balance constants, the composition at a particular point in the phase
diagram. Given the composition, it is possible to calculate the thermodynamic properties of the
mixture. In fact, the enthalpy of a mixture of O2 and of O will be given by the sum of the energy of
thermal motion of the molecular oxygen plus the energy of thermal motion of the atomic oxygen plus
the heat of dissociation of the dissociated O2 fraction. Only the first effect will be considered in our
discussion.
For a real fluid, introducing heat only partially increases the energy bound to the degrees of freedom
of the molecules since it must also help overcome the forces of intermolecular attraction that occur at
high densities. Therefore, a temperature increase is achieved, heat introduced being equal, that is
smaller than the one obtained with an ideal gas. The real gas has thus a greater specific heat. On the
other hand, such fields of forces also affect the volumetric behavior of the real fluid, which generally
tends to occupy a volume less than the corresponding case of ideal gas due to the attractive forces
between the molecules. What is stated above sets out to provide an intuitive explanation for the close
bond between the volumetric behavior (or equation of state) and thermodynamic behavior of a fluid.
This bond can be obtained analytically, in an entirely rigorous way, as will be shown in the following
paragraphs.
The Maxwell relations that bind volumetric behavior to thermal behavior of a fluid are extremely
useful for obtaining important relations at the root of the thermodynamic behavior of any fluid.
To fully characterize the behavior of a real fluid, three different approaches are possible a priori:
•
•
•
purely experimental model: it must provide the information needed to construct the phase
diagram. It is not a method used in practice because it is not possible to get a coherent picture
if the data come from different experiments;
model based on statistical thermodynamics: the degree of approximation that can be achieved
is not always considered sufficient;
use of the equation of state, possibly in the implicit form, for determining the volumetric
behavior of the fluid theoretically or experimentally: when this is known, it is then possible to
analytically (i.e. strictly) get all the thermodynamic properties of the fluid.
38
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
5.1
THERMODYNAMIC PROPERTIES OF A REAL FLUID CALCULATION
As mentioned before, the thermodynamic properties of real fluids are calculated starting from those of
ideal gases considering appropriate correction terms that take into account the effect of real fluid.
These terms may be calculated when the volumetric behavior of the fluid is known. Since for a real
fluid the thermodynamic properties depend not only on temperature (like in ideal gases) but also on
pressure, it will be always necessary to integrate an expression in which the partial derivative of the
quantity considered with respect to the pressure along an isotherm appears. For a generic property 𝑋:
𝑋(𝑇, 𝑝) = 𝑋(𝑇, 𝑝0 ) + 𝛥𝑋𝑟𝑒𝑠 = 𝑋 0 (𝑇) + 𝛥𝑋𝑟𝑒𝑠
(5.1)
with 𝑋(𝑇, 𝑝0 ) = 𝑋 0 (𝑇) property of ideal gas at temperature 𝑇 and 𝛥𝑋𝑟𝑒𝑠 correction of real fluid at
pressure 𝑝. 𝑝0 is the pressure at which the gas can be considered ideal at the temperature 𝑇.
These corrections will be obtained as integral corrections of a certain function of the specific volume
and its derivatives, so they are valid outside the saturation curve where there are no discontinuities. In
the phase transition, the Clausius-Clapeyron relation will give the additional information required for
the complete characterization of these corrective terms.
5.1.1
Residual enthalpy
Consider a fluid at the temperature 𝑇 and at the pressure 𝑝, for which the correction of the enthalpy
due to the effect of real fluid behavior is calculated. From 𝑑ℎ = 𝑇𝑑𝑠 + 𝑣𝑑𝑝 and considering a
transformation at constant temperature and deriving with respect to pressure:
(
𝜕ℎ
𝜕𝑠
) = 𝑣 +𝑇( )
𝜕𝑝 𝑇
𝜕𝑝 𝑇
(5.2)
In this relation, there is the entropy function, a quantity of the thermal type, for which it is not possible
to take experimental measurements. Applying the second Maxwell’s relation – Eq. (2.40) – the
previous relation becomes:
𝜕ℎ
𝜕𝑣
( ) = 𝑣−𝑇( )
𝜕𝑝 𝑇
𝜕𝑇 𝑝
(5.3)
which is a function of volumetric quantities of which it is possible to take a measurement.
By integrating the previous relation along an isotherm, it holds true that:
𝑝2
ℎ(𝑝2 , 𝑇̂) − ℎ(𝑝1 , 𝑇̂) = ∫ [−𝑇̂ (
𝑝1
𝜕𝑣
) + 𝑣] 𝑑𝑝
𝜕𝑇 𝑝
(5.4)
Adding and subtracting the ideal gas enthalpy at temperature 𝑇̂
𝑝2
𝜕𝑣
ℎ(𝑝2 , 𝑇̂) − ℎ0 (𝑇̂) + ℎ0 (𝑇̂) − ℎ(𝑝1 , 𝑇̂) = ∫ [𝑣 − 𝑇̂ ( ) ] 𝑑𝑝
𝜕𝑇 𝑝
𝑝1
and letting 𝑝1 to 0, it holds true
ℎ0 (𝑇̂) ≡ ℎ(𝑝1 , 𝑇̂)
so that
39
(5.5)
(5.6)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑝2
𝑣
𝜕𝑣
0 ̂
̂
ℎ(𝑝2 , 𝑇) = ℎ (𝑇) + ∫ 𝑇̂ [ − ( ) ] 𝑑𝑝
̂
𝜕𝑇 𝑝
𝑇
0
where ℎ0 (𝑇̂) is, according to usual symbols, the enthalpy of the substance in the state of ideal gas,
ℎ(𝑇̂, 𝑝0 ).
In general, it holds true that
𝑝
𝑣
𝜕𝑣
0 (𝑇)
ℎ(𝑝, 𝑇) = ℎ
+ ∫ 𝑇 [ − ( ) ] 𝑑𝑝
(5.7)
𝑇
𝜕𝑇 𝑝
0
This equation represents the correction of real fluid along an isotherm with increasing pressure and,
as mentioned repeatedly, it links the thermodynamic behavior to the volumetric behavior of the fluid.
This equation can be derived also considering the general definition of 𝑑ℎ obtained in Eq. (2.50) and
recalling the definition of isobaric expansion coefficient 𝛼𝑝 .
𝑑ℎ = 𝑐𝑝 𝑑𝑇 + 𝑣[1 − 𝑇α𝑝 ]𝑑𝑝
(5.8)
in which the enthalpy change along an isotherm depends on a term that really takes into account the
volumetric behavior of the fluid.
Let's check Eq. (5.7) in one of its limits of applicability. A gas that is well described by the ideal gas
model is considered, so:
𝜕(𝑅𝑔 𝑇/𝑝)
𝑅𝑔 𝑇̂
𝜕𝑣
𝑇̂ ( ) = 𝑇̂ (
) =
=𝑣
𝜕𝑇 𝑝
𝜕𝑇
𝑝
𝑝
(5.9)
which replaced in Eq. (5.7) cancels the integrand in the whole range of pressures and thus provides:
ℎ = ℎ0 .
In practice, the procedure to calculate the thermodynamic properties of a real fluid (e.g. a non-rarefied
vapor) is as follows. The specific volume of the substance is experimentally measured and an analytical
expression (equation of state) that correctly interprets the experimental data is investigated. The abovementioned analytical expression of the specific volume is introduced in Eq. (5.7), directly getting the
residual enthalpy. It is possible and recommendable to verify the results with direct (thermal) control
measurements.
To forecast the trend of the correction term, from the theoretical point of view the volumetric behavior
of the fluid is qualitatively modeled and the isobars in the Tv plane are plotted (Fig. 5.1) with the aim
𝜕𝑣
of describing the trend of the term 𝑇̂ (𝜕𝑇) .
𝑝
40
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 5.1 - Trend of the specific volume as a function of the temperature for different subcritical and
supercritical isobar curves.
For an ideal gas:
𝑣=
𝑅𝑔 𝑇
𝑝
(5.10)
and so the isobars are straight lines leaving the origin with the angular coefficient inversely
proportional to the pressure of the isobar and gradient 𝑅𝑔 /𝑝: at temperatures tending to absolute zero
the volume occupied by any gas at any pressure tends to zero.
For pressures below the critical point, the volumetric behavior of the fluid can be described by four
distinct lines:
1) at high temperatures the fluid behaves like an ideal gas; the change in specific volume is linear
with temperature with a gradient inversely proportional to the pressure;
2) when decreasing the temperature, the gas becomes denser and the particles begin to experience
the effects of intermolecular attractive forces. The gas has a specific volume lower than that of
an ideal gas. For very low pressures, the correction of real gas is not relevant until the beginning
of the condensation step;
3) when the fluid reaches the saturation conditions 𝑣 𝑣 it starts to condense, yielding a reduction
of volume at constant temperature up to 𝑣𝑙 ;
4) for lower temperatures, the specific volume of the liquid changes very little and a line almost
horizontal at low temperatures is representative of the liquid state3. In these conditions the
𝜕𝑣
derivative (𝜕𝑇) → 0 and therefore along an isotherm ∆ℎ = 𝑣∆𝑝 like in the hydraulic
𝑝
machines. The correction is small since 𝑣 is small for the liquids.
3
At this time the area with very low temperature, characteristic of the solid state, is not being considered.
41
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
For supercritical pressures, the trend has no vertical line and the behavior of ideal liquid and ideal gas
are connected by a continuous curve. Therefore, for temperatures higher than the temperature of phase
change, the volumetric behavior of the fluid is affected by the intermolecular interaction with a
decrease in specific volume compared to the behavior of ideal gas and that the derivative of specific
volume compared to the temperature is greater than that of ideal gas 𝑅𝑔 /𝑝.
The integrand of Eq. (5.7) is generally negative. The correction term is thus usually less than zero: the
enthalpy of real fluid is therefore generally less than that of ideal gas. In the following, its trend in the
ℎ − 𝑝 plane along isotherm curves are described.
For an ideal gas, the enthalpy is independent from the pressure, so the enthalpy curve is a horizontal
straight line with intercept increasing with the temperature. Fig. 5.2 compares the trend with the one
of real gases:
1) with pressure tending to zero, the behavior of the gas tends to the behavior of the ideal gas and
the enthalpy is equal to that of ideal gas or constant with pressure changes
2) with increasing pressure, the molecules begin to experience the presence of the others, and the
specific volume is less than that of ideal gas, its derivative compared to the temperature along
the isobar is greater than that of ideal gas and the correction is negative
3) when it reaches the saturated vapor point, the fluid condenses, and the enthalpy decreases at
constant pressure. The integrand is discontinuous on this line. The difference between enthalpy
of saturated vapor and saturated liquid can be found with the Clausius-Clapeyron relation.
4) in the liquid zone, the enthalpy starts to increase again due to the effect of the term 𝑣𝑑𝑝: this
is easily understandable considering the liquid as ideal in Eq. (5.7) so the second term of the
𝜕𝑣
integrand is canceled (𝜕𝑇) = 0
𝑝
ℎ(𝑝, 𝑇̂) − ℎ0 (𝑇̂) = ℎ𝑣 (𝑝 𝑠𝑎𝑡 , 𝑇̂) − ℎ0 (𝑇̂) − ∆ℎ𝑒𝑣𝑎 (𝑇̂) + ℎ(𝑝, 𝑇̂) − ℎ𝑙 (𝑝 𝑠𝑎𝑡 , 𝑇̂)
𝑝𝑠𝑎𝑡
𝑝
𝜕𝑣
𝜕𝑣
0
ℎ(𝑝, 𝑇̂) − ℎ (𝑇̂) = ∫
[𝑣 − 𝑇̂ ( ) ] 𝑑𝑝 − ∆ℎ𝑒𝑣𝑎 (𝑇̂) + ∫ [𝑣 − 𝑇̂ ( ) ] 𝑑𝑝
𝜕𝑇 𝑝
𝜕𝑇 𝑝
𝑝0
𝑝𝑠𝑎𝑡
𝑝𝑠𝑎𝑡
ℎ(𝑝, 𝑇̂) − ℎ (𝑇̂) = ∫
0
𝑝0
𝜕𝑣
[𝑣 − 𝑇̂ ( ) ] 𝑑𝑝 − ∆ℎ𝑒𝑣𝑎 (𝑇̂) + 𝑣(𝑝 − 𝑝 𝑠𝑎𝑡 )
𝜕𝑇 𝑝
42
(5.11)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 5.2 - Trend of the enthalpy of real fluid versus pressure for different subcritical and supercritical
isotherm curves.
Fig. 5.3 shows the qualitative trend of enthalpy as a function of temperature along an isobaric
transformation. For an ideal gas, the enthalpy does not depend on pressure and the trend of ℎ(𝑇) is
linear for monoatomic gases, while it has a positive second derivative for ideal polyatomic gases as
𝑐𝑝0 (𝑇) increases with the temperature. It can be stated that:
1) At low temperatures the fluid is in liquid phase and its volume is almost constant. Enthalpy
increases due to the increase in temperature.
2) Where the phase changes there is a vertical line, from the enthalpy of saturated liquid to that
of saturated vapor.
3) At very high temperatures, ideal gas behavior can be considered and the enthalpy does not
depend on pressure, but only on temperature. The gradient is inversely proportional to the
specific heat 𝑐𝑝0 (𝑇). The real fluid effects (4) generate a gap from the ideal gas curve.
43
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 5.3 - Trend of the enthalpy of real fluid as a function of the temperature for different subcritical
and supercritical isobar curves.
An example related to carbon dioxide is shown in figures Fig. 5.4 and Fig. 5.5: the results are obtained
by using an equation of state. The first figure shows the volumetric behavior; the second figure shows
the trend of the residual enthalpy term as a function of pressure for different temperatures. Consistently
with what has been discussed, this correction is negative (that is, the enthalpy of the real fluid is less
than that of the ideal gas).
Fig. 5.4 - Trend of the specific volume of CO2 calculated with REFPROP.
44
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
300
, kJ/kgK
250
200
150
100
50
0
0
50
100
150
200
250
300
, bar
Fig. 5.5 - Trend of the residual enthalpy of CO2 calculated with REFPROP.
5.1.2
Specific heat residual
For real fluids specific heat depends also on pressure: this means that the relations previously obtained
for ideal gases are no longer valid. Let's consider once again the correction of the 𝑐𝑝 along a
transformation at constant temperature and derive it with respect to 𝑝.
From the definition of 𝑐𝑝 :
𝑐𝑝 = (
𝜕ℎ
)
𝜕𝑇 𝑝
(5.12)
considering its derivative compared to the pressure at constant temperature 𝑇̂ and applying Schwarz’s
theorem:
𝜕𝑐𝑝
𝜕 𝜕ℎ
𝜕 𝜕ℎ
( ) =( ( ) ) =( ( ) )
𝜕𝑝 𝑇
𝜕𝑝 𝜕𝑇 𝑝 𝑇
𝜕𝑇 𝜕𝑝 𝑇
𝜕ℎ
𝜕𝑣
𝜕𝑝 𝑇
𝜕𝑇 𝑝
(5.13)
𝑝
where ( ) = 𝑣 − 𝑇̂ ( ) and so:
𝜕𝑐𝑝
𝜕 𝜕ℎ
𝜕
𝜕𝑣
( ) = ( ( ) ) = ( [𝑣 − 𝑇̂ ( ) ])
𝜕𝑝 𝑇
𝜕𝑇 𝜕𝑝 𝑇
𝜕𝑇
𝜕𝑇 𝑝 𝑝
𝑝
𝜕𝑐𝑝
𝜕 𝜕ℎ
𝜕𝑣
𝜕 2𝑣
𝜕𝑣
̂
( ) = ( ( ) ) = ( ) − 𝑇 ( 2) − ( )
𝜕𝑝 𝑇
𝜕𝑇 𝜕𝑝 𝑇
𝜕𝑇 𝑝
𝜕𝑇 𝑝
𝜕𝑇 𝑝
(5.14)
𝜕𝑐𝑝
𝜕 2𝑣
( ) = −𝑇̂ ( 2 )
𝜕𝑝 𝑇̂
𝜕𝑇 𝑝
(5.15)
𝑝
or finally:
𝑥
45
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
that integrated along an isotherm, using the same schemes seen in the case of the enthalpy, it provides:
𝑝2
𝑝2
𝜕 2𝑣
∫ 𝑑𝑐𝑝 = ∫ −𝑇̂ ( 2 ) 𝑑𝑝
𝜕𝑇 𝑝
𝑝1
𝑝1
(5.16)
Hence, eq. 6.17 becomes:
𝑝2
𝑐𝑝 (𝑝2 , 𝑇̂) − 𝑐𝑝 (𝑝1 , 𝑇̂) = − ∫ 𝑇̂ (
𝑝1
𝜕 2𝑣
) 𝑑𝑝
𝜕𝑇 2 𝑝
(5.17)
Adding and subtracting the ideal gas specific heat at 𝑇̂
𝑝2
𝑐𝑝 (𝑝2 , 𝑇̂) − 𝑐𝑝0 (𝑇̂) + 𝑐𝑝0 (𝑇̂) − 𝑐𝑝 (𝑝1 , 𝑇̂) = − ∫ 𝑇̂ (
𝑝1
and letting 𝑝1 to 0, it holds true
𝜕 2𝑣
) 𝑑𝑝
𝜕𝑇 2 𝑝
(5.18)
𝑐𝑝0 (𝑇̂) ≡ 𝑐𝑝 (𝑝1 , 𝑇̂)
(5.19)
𝑝2
𝜕 2𝑣
0 ̂
̂
̂
𝑐𝑝 (𝑝2 , 𝑇) = 𝑐𝑝 (𝑇) − ∫ 𝑇 ( 2 ) 𝑑𝑝
𝜕𝑇 𝑝
0
(5.20)
so that
In general, it holds true that
𝑝
𝑐𝑝 (𝑝, 𝑇) = 𝑐𝑝0 (𝑇) − ∫ 𝑇 (
0
𝜕 2𝑣
) 𝑑𝑝
𝜕𝑇 2 𝑝
(5.21)
This equation allows us to calculate the deviation of the specific heat of a fluid in general conditions
from that of the same fluid in the state of ideal gas once the second derivative of the specific volume
with respect to the temperature is known. The numerical calculation shows that even a small
uncertainty on the specific volume leads to a much more serious error in the second derivative. Because
of this and considering that the correction term may be large compared to the specific heat of ideal gas
(for example, it is the case of water, where ∆𝑐𝑝 > 𝑐𝑝 ) in many circumstances a direct measurement of
the specific heat is preferable for evaluating the correction due to the effect of the real fluid than the
use of relation (5.17).
As already done for enthalpy, it is easy to verify that by applying the relation obtained to an ideal gas,
the integrand cancels out for all the pressures and therefore 𝑐𝑝 (𝑝, 𝑇) − 𝑐𝑝0 (𝑇) = 0.
𝜕2 𝑣
Recalling the trend of the isobars in the 𝑣 − 𝑇 diagram (Fig. 5.1), it is observed that the term (𝜕𝑇 2)
𝑝
is usually negative and therefore the correction term of the specific heat is normally positive. In
practice, for a real gas a portion of the energy absorbed is spent to overcome the intermolecular bonds
instead of exciting the remaining degrees of freedom of the fluid.
The trend of 𝑐𝑝 as a function of pressure along the isotherms is shown in Fig. 5.6. The following applies
to a subcritical isotherm:
1) At low pressures the gas behaves like an ideal gas and 𝑐𝑝 is increasing with the temperature,
but independent from the pressure
46
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
2) As the pressure increases, the intermolecular forces become appreciable and the fluid shows a
specific volume lower than that of ideal gas, the concavity of 𝑣 compared to the temperature is
negative and the correction on 𝑐𝑝 is positive.
3) The specific heat in the phase change tends to infinity: the energy transferred to the fluid is no
longer distributed between all the molecules, but is absorbed by a single molecule which uses
it to overcoming the field of forces of attraction due to the other molecules. The absorption of
heat takes place at a constant temperature.
4) At higher pressures, the fluid behaves like a liquid and the 𝑐𝑝 is constant
By increasing the temperature, a greater value of the intercept 𝑐𝑝0 (except in the case of pure monatomic
fluids) and a greater 𝑐𝑝 of saturated vapor since a denser vapor corresponds to higher pressures. The
correction of ∆𝑐𝑝 between liquid and saturated vapor decreases because the 𝑐𝑝 of liquid instead is
nearly constant with the temperature.
Along a supercritical isobar the discontinuity is no longer present but the specific heat, however, shows
a peak in the proximity of the critical temperature. For this reason, it is not important to increase too
much the pressure in a supercritical plant since most of the heat is in any case introduced in conditions
next to the critical point.
Fig. 5.6 - Trend of the specific heat when pressure changes for different subcritical and supercritical
isotherm curves.
A qualitative example
This result can be obtained also with another qualitative line of reasoning. With reference to Fig. 5.7,
it is observed that for absolute temperatures in the order of 3-4 times the critical temperature, regardless
of the value of the pressure, the gases behave like ideal gases and therefore ℎ4 = ℎ5 . Considering two
different isobars, one subcritical (points 2-1-5) and the other supercritical (points 3-4), neglecting the
difference in enthalpy of the liquid between points 2 and 3 (approximation) due to the term 𝑣∆𝑝, it is
found that ℎ3 ≈ ℎ2 = ℎ1 − ∆ℎ𝑒𝑣𝑎 that is ℎ3 < ℎ1 . This is confirmed by the relations regarding the
47
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
residual enthalpy, which give that the enthalpy of the liquid is always lower than the enthalpy of ideal
gas, at same temperature and pressure. It is also possible to conclude that ℎ4 − ℎ3 ≈ ℎ5 − ℎ3 ≈ ℎ5 −
ℎ1 + ∆ℎ𝑐𝑜𝑛𝑑 , which means at same temperature differences, the specific heat between 4-3 is greater
than that between 5-1, as defined by the correction of the specific heat for real fluids.
4
3
5
1
2
Fig. 5.7 - Trend in the Ts plane of two isobars: one supercritical (3-4) and one subcritical (2-1-5).
5.1.3
Residual entropy
By applying the method used so far for calculating the residual entropy of a real gas at temperature 𝑇
𝜕𝑠
𝜕𝑣
and pressure 𝑝, integrating along an isotherm and using the second Maxwell relation − (𝜕𝑝) = (𝜕𝑇) ,
𝑇
𝑝
it turns out that:
𝑝
𝑝
𝜕𝑠
𝜕𝑣
𝑠(𝑝, 𝑇̂) − 𝑠 0 (𝑇̂) = ∫ ( ) 𝑑𝑝 = − ∫ ( ) 𝑑𝑝
𝑝0 𝜕𝑝 𝑇
𝑝0 𝜕𝑇 𝑝
(5.22)
where 𝑠(𝑝, 𝑇̂) is the entropy at pressure 𝑝 and temperature 𝑇̂, and 𝑠 0 (𝑇̂) is the entropy of the ideal gas
at temperature 𝑇̂ and pressure of rarefaction 𝑝0 .
Unlike for enthalpy and specific heat, the first and therefore also the second member of Eq. (5.22)
diverge for an ideal gas: at zero pressure the entropy tends to infinity, and the correction term written
in this way is not calculable
𝑅𝑔
𝜕𝑣
( ) =
→ ∞
𝜕𝑇 𝑝
𝑝 𝑝→0
(5.23)
This is due to the fact that the entropy of the ideal gas depends on pressure and it is therefore necessary
to redefine the residual entropy as the correction between the condition of ideal gas and that of real
fluid in the same conditions of 𝑇̂ and 𝑝:
𝑠(𝑝, 𝑇̂) − 𝑠 0 (𝑝, 𝑇̂)
48
(5.24)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Differentiating the Maxwell relation on both sides at any 𝑇̂ with respect to 𝑝, it holds true that:
𝑝2
𝑝2
𝜕𝑣
∫ 𝑑𝑠 = ∫ − ( ) 𝑑𝑝
𝜕𝑇 𝑝
𝑝1
𝑝1
𝑝2
𝜕𝑣
̂
̂
𝑠(𝑝2 , 𝑇) − 𝑠(𝑝1 , 𝑇) = − ∫ ( ) 𝑑𝑝
𝑝1 𝜕𝑇 𝑝
Adding and subtracting the ideal gas property at 𝑇̂ and 𝑝1:
(5.25)
𝑝2
𝑠(𝑝2 , 𝑇̂) − 𝑠 0 (𝑝1 , 𝑇̂) + 𝑠 0 (𝑝1 , 𝑇̂) − 𝑠(𝑝1 , 𝑇̂) = − ∫ (
𝑝1
and observing that:
𝑠 0 (𝑝2 , 𝑇̂) = 𝑠 0 (𝑝1 , 𝑇̂) − 𝑅𝑔 ∫
𝑝2
𝑝1
it turns out that:
𝑠(𝑝2 , 𝑇̂) − 𝑠 0 (𝑝2 , 𝑇̂) − 𝑅𝑔 ∫
𝑝2
𝑝1
𝜕𝑣
) 𝑑𝑝
𝜕𝑇 𝑝
(5.26)
𝑑𝑝
𝑝
(5.27)
𝑝2
𝑑𝑝
𝜕𝑣
+ 𝑠 0 (𝑝1 , 𝑇̂) − 𝑠(𝑝1 , 𝑇̂) = − ∫ ( ) 𝑑𝑝
𝑝
𝑝1 𝜕𝑇 𝑝
(5.28)
or similarly:
𝑝2 𝑅
𝜕𝑣
𝑔
𝑠(𝑝2 , 𝑇̂) − 𝑠 0 (𝑝2 , 𝑇̂) + 𝑠 0 (𝑝1 , 𝑇̂) − 𝑠(𝑝1 , 𝑇̂) = ∫ [ − ( ) ] 𝑑𝑝
(5.29)
𝑝
𝜕𝑇 𝑝
𝑝1
Letting 𝑝1 to 0 it holds true:
𝑠 0 (𝑝1 , 𝑇̂) ≡ 𝑠(𝑝1 , 𝑇̂)
(5.30)
so that
𝑝2 𝑅
𝜕𝑣
𝑔
𝑠(𝑝2 , 𝑇̂) = 𝑠 0 (𝑝2 , 𝑇̂) + ∫ [ − ( ) ] 𝑑𝑝
(5.31)
𝑝
𝜕𝑇 𝑝
0
In general, it holds true that:
𝑝 𝑅
𝜕𝑣
𝑔
𝑠(𝑝, 𝑇) = 𝑠 0 (𝑝, 𝑇) + ∫ [ − ( ) ] 𝑑𝑝
(5.32)
𝑝
𝜕𝑇 𝑝
0
where the first term refers to the behavior of ideal gas and the second is the term related with real fluid
deriving from the Maxwell relations. It is possible to check the relation thus obtained for the conditions
𝑅𝑔
𝜕𝑣
of ideal gas 𝑝 → 0: the term ( ) = and the correction of real fluid tends to zero.
𝜕𝑇 𝑝
𝑝
𝜕𝑣
The residual entropy is always negative since the gradient of the isobar of real fluid (𝜕𝑇) is greater
𝑝
than the gradient of the isobar of ideal gas in the 𝑣 − 𝑇 plane (Fig. 5.1) and therefore a real fluid always
has a lower entropy than an ideal gas. The entropy change due to the phase transition can be calculated
∆ℎ
as ∆𝑠 = 𝑇 𝑒𝑣𝑎 from the definition of entropy applied to an isothermal/baric transformation.
𝑒𝑣𝑎
The residual entropy for the incompressible liquid can be written as
𝑠(𝑝, 𝑇) − 𝑠 0 (𝑝, 𝑇) = (𝑠 𝑣 (𝑝, 𝑇) − 𝑠 0 (𝑝, 𝑇) ) − ∆𝑠𝑒𝑣𝑎 (𝑇) + (𝑠(𝑝, 𝑇) − 𝑠 𝑙 (𝑝, 𝑇))
𝑝𝑠𝑎𝑡 𝑅
𝑝
𝑅𝑔
𝜕𝑣
𝜕𝑣
𝑔
0
𝑠(𝑝, 𝑇) − 𝑠 (𝑝, 𝑇) = ∫
[ − ( ) ] 𝑑𝑝 − ∆𝑠𝑒𝑣𝑎 (𝑇) + ∫ [ − ( ) ] 𝑑𝑝
𝑝
𝜕𝑇 𝑝
𝜕𝑇 𝑝
𝑝0
𝑝𝑠𝑎𝑡 𝑝
49
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑝𝑠𝑎𝑡
𝑙
0
𝑠 (𝑝, 𝑇) − 𝑠 (𝑝, 𝑇) = ∫
𝑝0
5.2
[
𝑅𝑔
𝜕𝑣
∆ℎ𝑒𝑣𝑎 (𝑇)
𝑝
− ( ) ] 𝑑𝑝 −
+ 𝑅𝑔 𝑙𝑛 ( 𝑠𝑎𝑡 )
𝑝
𝜕𝑇 𝑝
𝑇
𝑝
(5.33)
EFFECTS ON DIAGRAM TS
The integral correction terms of real fluid represent residuals, i.e. often small quantities (if the behavior
of the real fluid is not very distant from that of ideal gas) on which a relatively large error is tolerated,
at least in the cases where the correction represents only a small percentage of the real value of the
quantity. When plotting thermodynamic diagrams, it is important to take into account the relations
obtained hereabove: these diagrams will therefore be modified by comparing them to those of the ideal
gas, with a trend of the curves in first approximation predictable.
The practical importance of knowing the effects of real fluid can be illustrated, for example, by
obtaining the expression which provides the work of an isentropic infinitesimal expansion in a
turbomachine:(𝑑ℎ)𝑠 = 𝑣𝑑𝑝 which is linearly dependent on the specific volume of the fluid at the same
pressure change.
5.2.1
Trend of the isobar curves
The increase in specific heat represents a greater capacity of the fluid to absorb energy. This is reflected
on the form of the isobars in the Ts plane which an exponential trend with derivative (Eq. (3.43)):
𝑑𝑇
𝑇
= 0
𝑑𝑠 𝑐𝑝 (𝑇)
For real gases, along an isobar at low temperatures, the 𝑐𝑝 increases and the gradient of the isobars
decreases up to the saturated vapor point where, during phase transition, 𝑐𝑝 tends to infinity and the
derivative of 𝑇 with respect to 𝑠 cancels out. In fact, the phase transition is isothermal/baric. For real
gases, it is possible to write that:
𝑑𝑇
𝑇
= 0
𝑑𝑠 𝑐𝑝 (𝑇) + ∆𝑐𝑝
For very low pressures, effects of real fluid do not appear before the beginning of the condensation,
while for supercritical transitions the trend shows no discontinuity. However, in this latter case there
is a very strong reduction in gradient of the isobars in the vicinity of the critical point where the specific
heat shows a peak.
50
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
s
Fig. 5.8 - Trend in the Ts plane of the isobars for real fluid (solid) and their deviation from those of
perfect gas (dotted).
The trend of the isobars is also confirmed by the fact that the entropy of real fluid is always less than
that of a perfect gas.
5.2.2
Trend of the isenthalpic curves
It is now possible to describe the trend of the isenthalpic curves in the Ts plane. For low pressures
and/or high temperatures, the gas behaves like an ideal gas and the isenthalpic curves are also isotherm
curves. They are therefore horizontal lines.
By moving along an isotherm and increasing the pressure, the saturation curve is approached and the
specific volume of real fluid is less than that of perfect gas. The 𝑐𝑝 increases and a negative residual
enthalpy that at the same temperature has a lower value is found:
𝜕ℎ
𝜕𝑠
< 0.
In the two-phase field, the enthalpy decreases as it moves from the point of saturated vapor to that of
saturated liquid. The difference between these two values is the enthalpy of evaporation that can be
calculated with the Clausius-Clapeyron relation. In liquid field, the enthalpy starts to increase with the
pressure due to the term 𝑣𝑑𝑝. In supercritical field, the behavior is similar: as the pressure decreases
and moves along an isotherm, the enthalpy first decreases, reaches a minimum and then resumes its
rise. Consequently, the isenthalpic curves present a maximum value which defines the locus of the
Joule Thompson points. For very high temperatures (𝑇 = 3 ÷ 4𝑇𝑐𝑟 ), there is not a maximum value and
the enthalpy increases at an increasingly lower gradient until it reaches the horizontal trend of ideal
gas.
For water, this curve is coincident with the curve of saturated liquid at the low temperatures to then
deviate at higher temperatures. The thermodynamic plane is divided into two zones: to the left when
the fluid is throttled, there is heating, to the right cooling. The exception is the field of ideal gas in
which an isenthalpic lamination is also isotherm. The diagram Ts of the water is shown below.
51
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 5.9 - Trend in the Ts plane of the isobars and isenthalpic curves for water calculated with
REFPROP.
52
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
6
THE REAL FLUID EQUATIONS OF STATE
The knowledge of 𝑐𝑝0 (𝑇) for calculating the properties of ideal gas (ℎ0 (𝑇), 𝑠 0 (𝑇)) is relatively simple
and numerous databases of the coefficients of the polynomial as a function of temperature are available
in literature. Knowing an equation of state for the volumetric behavior of the fluid that is used to
calculate the residuals of real fluid properties is much more difficult.
The accurate description of the volumetric behavior of a fluid by means of a single equation is,
however, very difficult since the behavior of real fluid has areas in which the behavior of the fluid is
very different, two-phase areas (and points of discontinuity) and areas where the properties change
very quickly (critical point).
6.1
A BRIEF HISTORY OF THE EQUATIONS OF STATE
The bond 𝑓(𝑝, 𝑣, 𝑇) for a fluid, called equation of state (EoS), can be obtained from more or less
complex analytical formulations. Due to their empirical nature, all equations of state currently
available contain parameters that must be calibrated on the basis of experimental data. By increasing
the degree of precision to be obtained, it is necessary to adopt equations with an increasing number of
coefficients whose value should be optimized by properly interpolating the experimental data. This
entails the need to have (i) models that, if optimized, are able to reproduce the thermodynamic behavior
of the fluids, (ii) numerous (in type and number) experimental data (terns of values 𝑝, 𝑣, 𝑇 and
saturation data 𝑝, 𝑇) that have to be sufficiently accurate and (iii) optimization algorithms able to
address the problem. In fact, data in the type, number and levels of accuracy is sufficient to properly
optimize the equations of state for only a few fluids (water, CO2, methane, etc.).
The history of equations of state has covered the last 150 years of progress in the thermodynamic
description of fluids commonly used in engineering. During that time dozens of mathematical
formulations have been proposed, improved and then partly abandoned since they have been surpassed
by more advanced ones. Depending on their degree of accuracy in representing the thermodynamics
of fluids, the degree of empiricism underlying their formulation and the complexity of their
implementation, the "best" equations of state have been consolidated, improved and complemented by
models that have always been able to increasingly combine these three aspects.
In general, there is no equation of state that is able to accurately represent all types of fluids. For this
reason, selecting the appropriate equation of state for the system studied is necessary.
The concurrence in highly reliable equation calculation software and often obsolete EoS often creates
confusion and sometimes disastrous results when an outdated but more familiar state function is
preferred over one that is state of the art. Therefore, very different results may be found in simulations
of components and plants based on the same basic assumptions. (Please note that the degree of
uncertainty with which the volumetric properties are calculated is amplified by switching to enthalpy,
to entropy and even more to the specific heats that are their derivative, as already discussed in previous
sections.)
1873: Van der Walls cubic equation
The first attempt to represent the volumetric behavior of fluids is due to Van der Walls, who formulated
the first cubic equation of state in which pressure is linked to the specific volume to the third power.
𝑝=
𝑅𝑇
𝑎
− 2
𝑣 −𝑏 𝑣
53
(6.1)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
200
200
150
150
, bar
, bar
Where the parameter 𝑏 is the covolume of the molecules and the parameter 𝑎 is the interaction between
the particles.
The choice of this functional form was essentially due to the basic observation that in a diagram p-v
(Fig. 6.1):
• Pressure that tends to zero regardless of the temperature, since for large specific volumes on a
molar-basis 𝑣 the gas should behave like an ideal gas;
• The critical isotherm must have an inflection to a horizontal tangency at the critical point;
• All isotherms must tend to infinity for small specific volumes, that is in the field of
incompressible fluid
100
100
50
50
0
0.01
0.1
0
1
10
0.01
0.1
1
-50
10
-50
, m3/kmol
, m3/kmol
Fig. 6.1 - a) Trend on the plan pv of the Van der Waals isotherms, b) principle of the equal area rule
applied to a subcritical isotherm.
The functional form chosen by VdW also reflects the theoretical basis according to which a fluid would
behave like an ideal gas if it were possible to consider the volume of the particles zero and if the forces
of molecular interaction were negligible.
Both coefficients are positive since in the presence of intermolecular forces the pressure on the walls
of the system is less than that of ideal gas and the equation for 𝑏 → 0 and 𝑎 → 0 must coincide with
the equation of ideal gases. For 𝑉 → ∞ the behavior is that of ideal gas, while for 𝑉 → 0 the first and
the second term increase very quickly but with a different sign: the first predominates over the second
and the pressure tends to infinity like for an incompressible liquid.
The additional condition of horizontal inflection for a critical isotherm at the critical point allows us
to define the parameters 𝑎 and 𝑏 as a function of the critical properties only.
2
𝜕𝑝
27𝑅 2 𝑇𝑐𝑟
( ) =0
𝑎=
𝜕𝑣 𝑇
64𝑝𝑐𝑟
→
𝜕 2𝑝
𝑅𝑇𝑐𝑟
( 2) = 0
𝑏=
𝜕𝑣 𝑇
8𝑝𝑐𝑟
} {
(6.2)
And so the volumetric and thermodynamic behavior of the fluid is defined when the critical properties
and the molecular weight are known. It can be also obtained:
𝑝𝑐𝑟 =
𝑎
8𝑎
; 𝑣𝑐𝑟 = 3𝑏; 𝑇𝑐𝑟 =
2
27 𝑏
27 𝑏𝑅
54
(6.3)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
The determination of the phase transition is not immediate (as for every functional form for the
equations of state) but, in the case of VdW's equation, an intuitive graphic representation is possible.
Each isotherm lower than the critical one presents a minimum value at first and then a maximum value
as the specific volume increases when instead it is well known that the phase transition occurs at
constant temperature and pressure. Along this curve, it is possible to identify the points of liquid (to
the left of the minimum) and vapor (to the right of the maximum) that under saturation conditions must
have not only the same pressure and temperature, but also the same Gibbs free energy as a condition
in order for the two phases to coexist. The points of VdW's isotherms between the saturation points
also have a physical meaning: the locus of the maxima and minima are defined by the spinodal curves
which define the zones of metastable equilibrium (subcooled liquid and superheated vapor) and the
condition of unstable equilibrium (between the minimum and the maximum).
From this observation Maxwell derived the equal area rule principle, observing that in the 𝑝 − 𝑣
diagram the area corresponds to the reversible work done: in plotting a horizontal line (isobar) that
links two points on an isotherm curve and that defines two equal areas, this basically means performing
two works of the same magnitude, one positive and one negative (Fig. 6.1b). The extreme points must
then have the same Gibbs free energy since a net work is not globally achieved, and they are therefore
those of liquid and saturated vapor, also having the same temperature and pressure. It is therefore
possible to get an estimate of the saturation pressure at a given temperature from VdW equation
graphically. However, the results are not accurate due to the low precision with which the data are
represented outside the saturation curve, mainly caused by the extremely limited number of parameters
in the equation.
Therefore, the importance of this relation is more theoretical than practical since the accuracy is low
especially in the field of subcooled liquid and around the critical point. Therefore, it is not used in
engineering for practical purposes.
1901: Onnes virial equation
The virial equation introduced by Heike Kamerlingh Onnes in 1901 is a generalization of the equation
of ideal gases. The equation of state for the volumetric behavior of fluid is represented as an expansion
of the terms in series.
𝑝𝑣
𝐵(𝑇) 𝐶(𝑇) 𝐷(𝑇)
=1+
+ 2 + 3 +⋯
𝑅𝑇
𝑣
𝑣
𝑣
(6.4)
For a specific volume tending to infinity it rightly coincides with the equation of ideal gases. The virial
coefficients B, C, D, etc. are a function of the temperature with equations, generally high degree
polynomial, able to globally represent the phase diagram. Their parameters must be appropriately
calibrated on the basis of experimental data with high computational requirements when using a high
number of virial coefficients (usually at least 8) with many parameters each. Although these virial
coefficients are the result of a purely empirical description, it is possible to give them - on the basis of
statistical mechanics - a physical meaning and one of their formulations can be obtained theoretically.
For example, parameter B represents the interaction between pairs of molecules, parameter C between
three particles, and so on.
A final important aspect of the virial equation is that from it a new parameter is defined and called
compressibility factor 𝑍, that is the ratio between the specific volume of real fluid and that of ideal gas
under the same temperature and pressure conditions. This parameter lets us define the principle of
corresponding states.
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Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
1940-1973: Benedict-Webb-Rubin (BWR-1970), BWR and Starling (BWRS-1973)
With the aim of improving the accuracy in the description of the volumetric and thermodynamic
behavior of fluids, a new class of equations of state was developed by Benedict-Webb-Rubin in 1940
and subsequently improved by Starling. BWR represents the first example of multi-parameter equation
of state and it was defined explicit with respect to the pressure as a corrective term to be applied to the
pressure of ideal gas. This equation is thus an improvement in terms of accuracy of the virial equation4
through the use of a larger number of terms and exponential terms.
𝑝 = 𝜌𝑅𝑇 (1 + ∑ 𝑛𝑖 𝜏 𝑡𝑖 𝛿 𝑑𝑖 + ∑ 𝑛𝑖 𝜏 𝑡𝑖 𝛿 𝑑𝑖 exp(−𝛾𝑖 𝛿 2 ))
𝑖
(6.5)
𝑖
where 𝜏 = 𝑇𝑐𝑟 /𝑇 and 𝛿 = 𝜌/𝜌𝑐𝑟 .
This equation has been widely used in spite of the difficulties in the calibration of the coefficients 𝑛𝑖
which may fluctuate in a very wide field of orders of magnitude. They may change radically from fluid
to fluid and make their determination a numerically difficult problem. Furthermore, the BWR equation
has two other drawbacks: the first is that the subcritical isotherms, to have a trend practically vertical
in the field of liquid, present dramatic variations in the two-phase field with maximum and minimum
multiples, with consequent difficulties in the vapor-liquid equilibrium calculation. The second
drawback is the use of transcendental terms that make difficult their analytical integration for
calculating properties such as, for example, the Helmholtz free energy. The increasing computational
velocity in the 60s allowed these limits to be partially overcome.
1949-1976: Cubic equations of Redlich-Kwong (1949) Soave- Redlich-Kwong (1972) and PengRobinson (1976)
All these equations represent a substantial step forward in the representation of the behavior of a real
fluid and are still widely used even though they are often characterized by inaccuracy and limitations,
especially in defining the liquid-vapor equilibrium and the density of liquid, in particular if few
experimental data are available. The two equations are substantially equivalent and the latest comes
from the need of the US to use its own equation of state and not scientifically depend on Europe.
The important conceptual leap of the last two equations of state consists of introducing the acentric
factor 𝜔 defined by Pitzer in 1955 and which characterizes the spatial form of the molecule and, in a
certain way, how it interacts with the others. The parameter 𝜔 tends to zero for spherical monatomic
molecules while for complex fluids it takes higher values: for example, decane has the value 𝜔 =
0.484.
In addition, the coefficient 𝛼 depends on the temperature with a further possibility of reducing the
average errors on the experimental data. The functional form in which it intervenes in the cubic
equation depends on a number of parameters that must be suitably fitted and that were calibrated to
characterize the hydrocarbons or substantially non-polar and complex molecules.
SRK (6.6)
PR (6.7)
𝑅𝑇
𝑎𝛼
𝑅𝑇
𝑎𝛼
𝑝=
−
𝑝=
− 2
𝑉 − 𝑏 𝑉(𝑉 + 𝑏)
𝑉 − 𝑏 𝑉 + 2𝑏𝑉 − 𝑏 2
𝛼 = 𝛼(𝜔, 𝑇, 𝑇𝑐 )
4
The functional form of reference simply uses mole density instead of mole volume
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Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
‘70s: Multi-property fitting - Bender (1970), Wagner (1970), Jacobsen and Stewart (1973)
A substantial improvement in the calibration of equations of state was proposed independently by
Bender and Wagner at the beginning of the 70s when for the first time not only pvT data were used,
but also data at saturation and the condition of phase equilibrium, as well as properties such as the
speed of sound, specific heats, latent heats of phase transition, etc.
Especially, the equation of Bender was calibrated for 19 substances and certain mixtures (although
with less success) and in fact was the first technical equation of wide use and fast implementation.
The equation of state MBWR (1973) allowed a growing number of fluids of interest to be described
with sufficient accuracy, with an adequate description even of the field of liquid, by increasing the
number of terms and using exponential functions. This equation was the first equation of state used as
a thermodynamic reference in the creation of the IUPAC nitrogen tables (1979) with a precision equal
to that of the experimental measurements upon which it was calibrated.
1974: The Stepwise regression and Wagner
In the late 70s Ahrendts and Baehr (1979-1981) formally defined the foundations of the theory of
multi-property regression and the use of non-linear algorithms for the definition of the functional form.
Up to this point the choice of the functional form, number of terms and exponents was made a priori
before the optimization of the coefficients with a process of trial and error based on the experience and
sensitivity of the co-examiner. The possibility of including the choice of the equation terms, and hence
of the functional form, within the optimization routine led to a further increase in the accuracy of the
equations of state.
For this purpose it is necessary to create an extremely extended bank of terms and then leave the task
of selecting those best indicated for representing the thermodynamic and volumetric behavior of the
fluid from these to the algorithm. The stepwise regression applied to the saturation curve of different
fluids by Wagner (1974) was the first practical example of this approach, which however returned
discouraging results once applied to calibration of the equation of state. The cause is once again to be
sought in the insufficiency of computational power. If a bank of 20 terms is sufficient for calibrating
the saturation curve, for calibrating equations of state it is necessary to have at least a hundred terms
which, in turn, require a sufficiently large number of experimental data. The strong non-linearity and
discontinuity of the objective function was a major obstacle to the numerical optimizers available in
those years. Another limiting aspect was the choice to still use pressure explicit equations of state with
an additional difficulty given by the numerical integration required for regressing on thermodynamic
properties of a higher order.
From the 80s to today: New optimization algorithms and explicit functional forms in the Helmholtz
energy
The 80s saw on the one hand an improvement in optimization algorithms available and, on the other,
the transition of the formulations of the equations of state from pressure explicit to Helmholtz explicit.
Among the new optimization algorithms able to escape the local minima and manage the presence of
discontinuity, evolutionary genetic algorithms and particles swarm methods became popular. In the
first, a population of better and better solutions was gradually selected, reproduced and changed to
converge into the optimum solution. In the latter, with every iteration a swarm of solutions moves in
the direction of the optimum, increasingly exploring the neighborhood of the current optimum
solution.
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Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
On the other hand, the formulation of the equation in the form of Helmholtz offered greater flexibility
in the choice of the functional form. Unlike the explicit equations in pressure or in the compressibility
coefficient, the fundamental equation of state lets us obtain any other state variable of state only with
derivation operations, and thus it allows us to calculate any property analytically, greatly facilitating
the calibration of multi-parameter equations.
The fundamental equation of state 𝑎 = 𝑎(𝜏, 𝛿) is the sum of two terms: one related to ideal gas 𝑎0 (𝜏, 𝛿)
and one to real fluid 𝑎𝑟 (𝜏, 𝛿). The first is the Helmholtz free energy. The second is instead represented
by the volumetric behavior of the fluid. The first formulation in this form was proposed by Keenan
(1969), but only with Schmidt and Wagner (1985) it reached the general form in which for the first
time it was possible to use large exponents for the exponential terms in density.
𝐴
= 𝑎0 (𝜏, 𝛿) + 𝑎𝑟 (𝜏, 𝛿)
𝑅𝑇
𝑇 𝑐 0 (𝑇)
𝜌𝑇
ℎ00 𝑠00 1 𝑇 𝑐𝑝0 (𝑇)
𝑝
𝑎0 (𝜏, 𝛿) = −1 + ln
+
− + ∫
𝑑𝑇 − ∫
𝑑𝑇
𝜌0 𝑇0 𝑅𝑇 𝑅 𝑇 𝑇0 𝑅
𝑇0 𝑅𝑇
𝑎=
(6.8)
(6.9)
𝑎𝑟 (𝜏, 𝛿) = ∑ 𝑛𝑖 𝜏 𝑡𝑖 𝛿 𝑑𝑖 + ∑ 𝑛𝑖 𝜏 𝑡𝑖 𝛿 𝑑𝑖 exp(−𝛾𝑖 𝛿 𝑙𝑖 )
𝑖
𝑖
𝑡𝑖 𝑑𝑖
+ ∑ 𝑛𝑖 𝜏 𝛿 exp(−𝑎𝑖 (𝛿 − 𝜖𝑖 )2 − 𝑏𝑖 (𝜏 − 𝛽𝑖 )2 ) .
(6.10)
𝑖
The choice of the best terms is to be sought with a stepwise regression for each fluid: for fluids very
difficult to represent, such as water, equations with a very high number of parameters (more than 50)
are required, while for the others a dozen parameters are sufficient. Span and Wagner (2003) addressed
this problem by performing the contemporaneous calibration of many fluids with the object to obtain
a robust and valid formulation regardless of the fluid. The functional form thus obtained has 12
parameters and is an excellent compromise between accuracy and computational requirements, and it
is widely used as a basic equation for describing organic fluids.
The greater freedom of choice of the functional form also offers the advantage that, with suitable
constraints, the absence of multiple roots in phase transition can be obtained, with greater ease in
detecting their conditions of saturation.
This formulation allows us to initialize the coefficients to be calibrated at a reasonable value that allows
execution of the optimization algorithm to be accelerated and, maintaining the same functional form,
the coefficients change in a few orders of magnitude, going from one fluid to another.
The newcomer
Nowadays no equation of state can exceed in accuracy the equations in fundamental form with a large
number of parameters. However, these equations require a large number of experimental data and their
nature is purely numerical and empirical, that is, the value of the parameters is obtained only as a result
of a regression and it has no physical or theoretical meaning. The PC-SAFT equations of state, whose
formulation was proposed by Gross and Sadowski (2001), try to fill this gap by proposing a
formulation of the residual 𝑎𝑟 (𝑇, 𝜌) as a function of specific and measurable properties of the
compound. The PC-SAFT Perturbed Chain Statistical Association Fluid Theory is based on molecular
theory. These equations are suggested and advisable when there are few experimental data and they
have been effectively used for modeling pure compounds and mixtures. The functional form of the
residual part of Helmholtz energy is expressed by just three terms:
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Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑎𝑟 = 𝑎𝑠𝑒𝑔 + 𝑎𝑐ℎ𝑎𝑖𝑛 + 𝑎𝑎𝑠𝑠
(6.11)
Fig. 6.2 - Graphic representation of the correction terms for a PC-SAFT equation of state.
They can be totally defined once the 5 characteristic terms of the molecular structure and molecular
interaction forces are known:
• 𝑚 = number of the segments in a chain
• 𝑠 = diameter of the segments (in Angstroms)
• 𝑒 = energy of the segments (in Joules)
• 𝑘 = association volume
• 𝜀 = association energy
Although studied to a large degree, these equations still do not allow the thermodynamic properties of
a fluid to be described better than complex multi-parameter empirical equations, but they play a major
role in the description of little known fluids or mixtures never tested experimentally.
6.2
CALCULATION PROGRAMS
The equations described so far are currently implemented in the leading engineering software
programs, where they are used for the simulations of industrial processes and power plants, for CFD
simulations and for the design of components. Many programs, such as Aspen and Thermoflex, allow
even outdated equations to be used, often leading to inaccurate results. One example is that for a PengRobinson or SRK equation, which make high level errors in the area of liquid, the condition of
perfectly incompressible fluid is extended up to the critical point with an absolutely inaccurate
representation of the thermodynamic properties of the fluid over the entire thermodynamic plane.
The choice of one equation or another however depends on the field in which it is used: a simple
equation may be adjusted for real vapor not too close to the saturation dome, while more complex
equations are required for supercritical cycles or detailed calculations. For this purpose, even small
errors on the volumetric behavior of fluids are amplified when switching to the calculation of quantities
obtained by derivative. The following figure is useful to highlight what has been said: by using the
two different and more accurate volumetric equations of state for water (absolutely the fluid studied
the most), both widely used in calculation software near very small variations in density calculation
~0.1% (left figure), significant differences up to 70% (right scale) are obtained for the specific heat,
which is related to the second derivatives of the specific volume.
59
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 6.3 - Comparison between the percentage differences in terms of specific volume and specific
heat for two very accurate equations of state.
It is advisable to always rely on state of the art equations, and in particular the REFPROP program
made by NIST that makes equations of state in the fundamental form available for an increasing
number of fluids (to date more than 110) with frequent recalibrations based on new experimental data.
6.3
THE COMPRESSIBILITY FACTOR AND THE PRINCIPLE OF CORRESPONDING STATES
The principle of corresponding states is extremely useful for characterizing, approximately but
relatively reliably, the thermodynamic properties of any fluid once the volumetric behavior of another
fluid is known. This principle requires definition of the parameter 𝒁, called compressibility factor,
that specifies the ratio between the real specific volume of the real fluid and the corresponding volume
in the hypothesis of ideal gas. The parameter 𝑍 obviously depends on the thermodynamic state of the
fluid. In particular, if it can be considered an ideal gas, the index will tend to the unit. If there are real
fluid effects, it will generally be lower due to the less volume occupied by the molecules due to the
intermolecular forces.
𝑍(𝑇, 𝑝) =
𝑣(𝑇, 𝑝)
𝑣 0 (𝑇, 𝑝)
(6.12)
A formulation for the volumetric equation of state of a real fluid can be obtained using 𝑍:
𝑣 = 𝑍(𝑇, 𝑝)
𝑅𝑢 𝑇
𝑀𝑀𝑝
(6.13)
Only in the case of supercritical liquids and fluids at high pressure 𝑍 > 1, namely in those conditions
in which the molecules cannot be considered as point masses and the real molecule volume is not
negligible compared to the volume occupied by the fluid.
Naturally, 𝑍(𝑇, 𝑝) is generally linked to the state variables by a highly complex function, depending
on the nature of the fluid in question and that can be expressed with the virial equation or an MBWR.
Let’s consider the trend of specific volume in a 𝑣 − 𝑝 diagram. For a subcritical isotherm when the
pressure increases the fluid behaves like an ideal gas (1), then it begins to experience the intermolecular
60
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
forces that bring a reduction of specific reduced volume (2), and then it condensates (3). In the subcooled liquid field, the specific volume stays more or less constant with the temperature and pressure
(4).
Fig. 6.4 - Trend in the 𝑣𝑝 plane of the subcritical and supercritical isotherm curves.
The ratio between the specific reduced volume and the specific reduced volume of ideal gas returns
the diagram of 𝑍 (Fig. 6.5).
Fig. 6.5 - Trend of 𝑍 as a function of 𝑝𝑟 for isotherm curves with different 𝑇𝑟 both subcritical and
supercritical.
1) for 𝑝 tending to zero, the behavior at any temperature is that of ideal gas, and therefore 𝑍 tends
to 1
61
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
2) 𝑍 then decreases when 𝑝 increases (manifestation of the intermolecular forces) for which the
real fluid occupies a volume lower than that of ideal gas
3) when 𝑝 reaches the value corresponding to the saturation pressure, 𝑍 suddenly drops (transition
from the saturated vapor conditions to those of saturated liquid takes place at constant pressure)
4) For high values of 𝑝 the curve of 𝑍 increases until it intersects and overcomes (for very high
values of 𝑝) the line 𝑍 = 1; in the intersection point, the equation of state of ideal gases is
fulfilled, however without being able to associate a physical meaning with this occurrence
since the fluid is in the state of sub-cooled liquid.
While the decrease of 𝑍 with reduced pressure is attributable to the forces of intermolecular
attraction, which carry the molecules of a fluid to occupy a smaller space than that required by
an ideal gas, the increase of 𝑍 at high pressures is given by the existence of a covolume. While
the law of ideal gas leads to cancellation of the specific volume for pressures tending to infinity,
this is not true for a real fluid, wherein the volume has a lower limit, linked to the intrinsic
volume of the molecules, determined by the repulsive interactions which occur when their
mutual distance tends to cancel out.
The trend of 𝑍 is clearly dependent on the fluid considered when it is expressed in terms of temperature
and absolute pressure. For example, once a temperature and a pressure for a certain fluid is set, fluid
conditions could be very far from the saturation dome, with little effects of real fluid; whereas for
another they might be in the vicinity of the dome or even in a two-phase field or in a sub-cooled liquid
field.
Table 6.1 shows the temperature, critical pressure and specific critical volume values for some fluids
where it is possible to observe their enormous variability.
Table 6.1 - Temperature and critical pressure for some fluids.
Fluid
Helium
Hydrogen
Nitrogen
Oxygen
Methane
Carbon dioxide
Tetrafluoroethane
(R134a)
Ammonia
Isobutane
Isopentane
Water
Air
Tcr (°C)
-267.9
-239.9
-146.9
-118.6
-82.5
31.1
101.1
pcr (bar)
2.28
13.15
33.96
50.43
46.38
73.72
40.67
vcr (l/kg)
14.43
33.21
3.18
2.29
6.18
2.13
1.81
133.0
135.0
187.4
374.0
-140.6
114.17
36.45
33.71
220.64
37.64
4.24
4.53
4.20
3.11
3.31
The previous equations can be rewritten in terms of reduced thermodynamic properties such as the
ratio between the absolute quantity and the value it takes on in the critical point, for example:
𝑇𝑟 =
𝑇
𝑝
; 𝑝𝑟 =
𝑇𝑐𝑟
𝑝𝑐𝑟
62
(6.14)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Substituting these equations in VdW equation of state and assuming the horizontal inflection point at
8𝑇
3
the critical point, an equation of state as function of reduced variables only, i.e. 𝑝𝑟 = 3 𝑣 𝑟−1 − 𝑣2 , can
𝑟
𝑟
be obtained. Thus, it holds true that
𝑍(𝑇𝑟 , 𝑝𝑟 )
(6.15)
The trend of 𝑍 as a function of the reduced pressure, using the reduced temperature as a parameter,
can be obtained graphically from the trend of the isotherm curves of real fluid in the 𝑣𝑟 𝑝𝑟 plane (Fig.
6.4). In this plane the ideal gas isotherms have also a hyperbolic trend that when reported in reduced
terms gives:
𝑣=
𝑅𝑇
𝑝
𝑇
𝑅𝑇 𝑇𝑐𝑟 1
𝑣
𝑅𝑇𝑟 𝑇𝑐𝑟
= 𝑝 𝑐𝑟
→ 𝑣𝑟 =
𝑣𝑐𝑟
𝑝𝑟 𝑝𝑐𝑟 𝑣𝑐𝑟
𝑝 𝑝𝑐𝑟 𝑣𝑐𝑟
𝑐𝑟
(6.16)
(6.17)
That is, given the critical parameters of the fluid, specific volume in reduced terms is inversely
proportional to the reduced pressure and directly proportional to the reduced temperature.
Fig. 6.6 shows the trend of 𝑍 for water for pressures up to 𝑝𝑟 = 10.
Fig. 6.6 - Trend of 𝑧 of water as a function of 𝑝𝑟 for isotherm curves up to 𝑇𝑟 = 4 calculated with
REFPROP.
For supercritical isotherms with 1 < 𝑇𝑟 < 3 the curve has a minimum but does not show
discontinuities as there is no phase change; for 𝑇𝑟 ≈ 4 ÷ 5 the fluid behaves for a large field of reduced
pressures like an ideal gas with 𝑍 = 1. For very large 𝑇𝑟 , 𝑍 > 1 in the whole field of reduced pressures
with an ever-increasing trend.
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Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
The general observation is that the parameter 𝑍 depends only on the position of the thermodynamic
state of the fluid in relation to the position of the saturation dome and of the critical point (i.e. the
reduced variables).
From this observation arises the principle of corresponding states that can be stated as follows:
“the existing bond between the compressibility factor 𝑧 and the pressure and temperature variables
of state, expressed in reduced terms, is independent from the nature of the fluid considered”.
This principle is not completely rigorous, but with good approximation is valid for all fluids, and it is,
within certain limits, derivable theoretically with the help of statistical mechanics. The principle of
corresponding states is not applied for the saturated liquid because 𝑍 is very small and varies with the
pressure in a different way from fluid to fluid. Therefore, this approximation is useless and it is
sufficient to use the incompressible liquid hypothesis with the value of the specific volume at ambient
temperature (data of easy experimental determination); possibly it could be corrected by means of the
coefficients of compressibility and expansion (see the final part of the lecture note).
As proof of what is stated above, the trend of the parameter 𝑍 for different substances (H2O, N2, CO2),
very different chemically, is provided. In fact, the density of a fluid can be determined with good
precision (order of a few %). The errors are most relevant in the vicinity of the critical point, which is
the major limitation of this approach.
Fig. 6.7 - Comparison of the values of 𝑍 for three different fluids calculated with REFPROP.
An examination of the curves shown in Fig. 6.7 suggests the following considerations:
• there is a field of pressures and temperatures in which the compressibility factor can be
considered constant and unitary for all fluids; in this case, the equation of state in question is
reduced to the equation of state of ideal gases
• outside of this field, especially for high reduced pressures and/or low reduced temperatures, the
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Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
•
•
equation of state of ideal gases is no longer valid, with all the thermodynamic implications that
follow. The indiscriminate use of the hypothesis of ideal gas is therefore totally arbitrary, and
a source of significant errors.
a fluid may be considered an ideal gas when its reduced temperature is high and/or when its
reduced pressure is low. The absolute values are of no interest, only the reduced ones are. For
example, at ambient temperature and pressure air can be considered an ideal gas because 𝑇𝑟 is
very high, while water is a real fluid because 𝑇𝑟 is very low.
an important implication of what is stated above is the fact that for 𝑇𝑟 relatively low (e.g. <0.7),
the reduced vapor pressure is very small (𝑝𝑟 < 0.05) and the saturated vapor has a volumetric
behavior very close to ideal gas (𝑍 ≅ 1)
The usefulness of the principle of corresponding states is obvious: once the volumetric behavior of a
sample fluid is known, it is sufficient to know the molecular weight and the critical parameters of any
other fluid to predict, with good accuracy, its volumetric behavior by using the curves (or the
corresponding equations) of variation of the compressibility factor 𝑍. In many applications this makes
it possible to compensate for the lack of availability of thermodynamic tables or diagrams often found
in technical literature.
This approach has limitations when comparing fluids with very different spatial molecular geometries
or polar fluids.
The principle of corresponding states can be applied in the field of real gases close to the saturation
curve and for supercritical isotherms with 𝑇𝑟 > 1, while its use is not correct for subcooled liquids
where it is suggested the choice of other models.
Lastly, the method is also applicable to mixtures, taking care to use in determining 𝑍 the pseudo critical
temperature and pressure values defined starting from the corresponding critical values of the single
components.
𝑇𝑐𝑟 = ∑ 𝑥𝑖 𝑇𝑐𝑟,𝑖 ; 𝑝𝑐𝑟 = ∑ 𝑥𝑖 𝑝𝑐𝑟,𝑖
𝑖
6.4
(6.18)
𝑖
CLAUSIUS-CLAPEYRON RELATION
The first relation example of the close link between volumetric and thermal behavior of fluids is
the Clausius-Clapeyron relation. This relation links volumetric behavior and heat of vaporization of
𝜕𝑠
𝜕𝑝
fluids. It is derivable in purely theoretical form from the first Maxwell relation (𝜕𝑣) = (𝜕𝑇) applied
𝑇
𝑣
to a liquid-vapor phase transition at temperature 𝑇.
At saturation, the pressure is a function of temperature only, and so the second term is independent
from the specific volume at which it is calculated (𝑣 𝑣 , 𝑣 𝑙 or any intermediate value). Since the process
𝑑ℎ
of phase change is isothermal/baric, it results that 𝑑𝑠𝑒𝑣𝑎 = 𝑇 𝑒𝑣𝑎
𝑒𝑣𝑎
∆ℎ𝑒𝑣𝑎
𝜕𝑠
𝜕𝑝
𝑑𝑝 𝑠𝑎𝑡
(
)
(
)
=
=
=
(𝑣 𝑣 − 𝑣 𝑙 )𝑇
𝜕𝑣 𝑇
𝜕𝑇 𝑣
𝑑𝑇
from which:
65
(6.19)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
∆ℎ𝑒𝑣𝑎
𝑑𝑝 𝑠𝑎𝑡 𝑣
(𝑣 − 𝑣 𝑙 )𝑇
=
𝑑𝑇
(6.20)
that is, it is sufficient to know the specific volume of the liquid 𝑣 𝑙 , the specific vapor volume 𝑣 𝑣 and
𝑑𝑝𝑠𝑎𝑡
the gradient of the saturation pressure curve 𝑑𝑇 to strictly calculate the evaporation enthalpy.
If the saturation pressure is low (lower than the atmospheric value) in the region of the limit curve
considered, it is usually reasonable to calculate the volume of the saturated vapor by means of the
equation of ideal gases. Under these conditions the volume of the liquid can be considered constant
and much lower than the vapor one (resulting negligible in limit conditions)
𝑣𝑣 =
𝑅 𝑇
≫ 𝑣𝑙
𝑀𝑀 𝑝
(6.21)
In practice, the heat of vaporization of the various liquids as tabulated in the collections of
thermodynamic data is calculated as follows. The saturation pressure and the density of the liquid and
vapor are determined experimentally and the experimental data are interpreted analytically by means
of appropriate equations5, and are introduced in Eq. (6.19), reaching an analytical expression of the
heat of vaporization as a function of the temperature. The importance of this procedure lies in the fact
that it is possible to calculate a thermal quantity (∆ℎ𝑒𝑣𝑎 ) through measurements of the volumetric
behavior of the fluid; the whole procedure does not require, in theory, any direct measurements of heat
quantity (calorimetric measurements). In practical reality, such measurements are still needed to
validate the values thus obtained by the indirect method.
Three observations:
• A thermal quantity (heat of vaporization) is calculated only from volumetric quantities (𝑝, 𝑇, 𝑣)
• The knowledge of the volumetric behavior is necessary not only for the point of interest, but
also in its neighborhood (to calculate the derivative)
• The thermal quantity requires the calculation of derivatives, for which a possible error on the
specific volume is amplified.
The trend of the ∆ℎ𝑒𝑣𝑎 of water as function of the temperature is shown in the Fig. 6.8. Values are
calculated with a refined equation of state for the volumetric behavior of the fluid and for the saturation
curve (C0). The other curves represent the result of the Clausius-Clapeyron relation in the case where
different approximations are introduced
• C1 – the hypothesis of incompressible liquid is introduced (𝜌 = 1000kg/s) and the specific
volume of vapor is reliably calculated. The deviations are minimal, and an appreciable
difference is recorded only for temperatures close to the critical one where the fluid can no
longer be considered an ideal liquid.
5
The experimental data are inevitably affected by error: this is reflected in the numerical determination of the heat of phase
change. The effect is even more sensitive in the calculation of the derivative of the saturation pressure compared to the
temperature. This is why the data are usually processed using the method of least squares instead of interpolation of all
experimental points.
66
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
•
•
•
C2 – the specific volume of the liquid is neglected, and the specific volume of the vapor is
reliably calculated. The deviations are greater, but sufficient accuracy is obtained for a
relatively extended range of temperatures.
C3 – the density of the vapor under the hypothesis of ideal gas and that of liquid with the
accurate model are calculated. The deviation is evident, and it is possible to state that the
hypothesis of ideal gas for the vapor is valid only for low temperatures (<100°C) (and therefore
low pressures).
C4 – the density of the vapor with hypothesis of ideal gas is calculated and the specific liquid
volume is neglected: the results are valid only at the low pressures and temperatures
3000
2500
C4
2000
C3
1500
1000
C2
C0 C1
500
0
0
100
200
300
400
Fig. 6.8 - Trend of the enthalpy of vaporization as a function of the saturation temperature for water
and the trend of the results obtainable with the approximated Clausius-Clapeyron relation.
Watson’s correlation allows computing the enthalpy of vaporization, Δℎ𝑒𝑣𝑎 (𝑇𝑟2 ) at any reduced
temperature, 𝑇𝑟2, which can be even close to the critical point, known the enthalpy of vaporization,
Δℎ𝑒𝑣𝑎 (𝑇𝑟2 ), at a given reduced temperature, 𝑇𝑟1:
1 − 𝑇𝑟2 0.38
𝑇1
𝑇2
)
Δℎ𝑒𝑣𝑎 (𝑇𝑟2 ) = Δℎ𝑒𝑣𝑎 (𝑇𝑟1 ) (
; 𝑇𝑟1 =
; 𝑇𝑟2 =
;
1 − 𝑇𝑟1
𝑇𝑐𝑟
𝑇𝑐𝑟
The amount of the vaporization heat of the working fluid has considerable influence on the technical
parameters of the Rankine cycle power plants, such as the sizing of the components (exchangers and
turbomachines). Now let us consider a simplified Rankine cycle (without superheating, ideal
turbomachinery and without flow resistances) in which the fluid evolves between 𝑇𝑐𝑜𝑛𝑑 and 𝑇𝑒𝑣𝑎 . Let's
also consider molecules with similar complexity (it is the condition necessary to have similar
saturation domes) and with a critical temperature sufficiently higher than that of evaporation (such as
to consider the effects of real gas having little influence on the vapor conditions). The two
thermodynamic cycles are similar in the Ts plane and will therefore have comparable efficiencies.
Therefore, the work will be a certain fixed amount of the heat introduced or transferred to the cold
well. Moreover, with the same heat introduced, it will produce the same power and will transfer the
67
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
same thermal energy to the environment. In terms relative to the mass, the useful work is linked to the
latent heat of evaporation or condensation.
𝜂
𝑄
1−𝜂 2
𝜂
𝑤 = 𝜂∆ℎ𝑒𝑣𝑎 ; 𝑤 =
∆ℎ
1 − 𝜂 𝑐𝑜𝑛𝑑
𝑊 = 𝜂𝑄1 ; 𝑊 =
From Eqq. (6.20) and (6.21), it is possible to see that the heat of vaporization is great for the liquids
𝑑𝑝
with low molecular weight which, when vaporizing, take on large specific volumes (the factor 𝑑𝑇𝑠,
variable from fluid to fluid, does not change the previous consideration qualitatively). It will be natural
to expect for the water vapor (𝑀𝑀 = 18) a vaporization heat much higher than that, for example, of
mercury (𝑀𝑀 = 200), as actually occurs.
Imagining sizing a water or mercury heat exchanger in which both fluids undergo phase change, a
mercury flow rate about 10 times greater than that of water given the same amount of exchanged heat
is found.
6.5
THE SATURATION PRESSURE CURVE
Starting from the Clapeyron equation, it is possible to obtain an equation that relates the saturation
pressure with the temperature. The following statements have to be assumed:
• specific volume of the saturated liquid is negligible
• specific volume of the saturated vapor can be calculated according to the ideal gas model
• evaporation enthalpy is constant with temperature in the range of interest
These assumptions are valid only in a very small range of temperatures and far from the critical point;
anyway, the resulting relation can be extended in a wider range of temperatures because the errors
compensate each other6. The resulting formulation is:
∆ℎ𝑒𝑣𝑎 (𝑇)
𝑑𝑝 𝑠𝑎𝑡
∆ℎ𝑒𝑣𝑎 𝑑𝑝 𝑠𝑎𝑡 𝑑𝑝 𝑠𝑎𝑡 ∆ℎ𝑒𝑣𝑎 𝑑𝑇
=
→
=
→
=
(𝑣 𝑣 − 𝑣 𝑙 )𝑇
𝑅𝑔 𝑇 2
𝑑𝑇
𝑑𝑇
𝑝
𝑅𝑔 𝑇 2
𝑝
From which, by integration, it can be obtained:
𝑙𝑛(𝑝 𝑠𝑎𝑡 ) = −
∆ℎ𝑒𝑣𝑎 1
+ 𝑐𝑜𝑛𝑠𝑡
𝑅𝑔 𝑇
(6.22)
(6.23)
This extension of the equation is known as Antoine equation and allows to express, with good
approximation, the saturation pressure curve for every fluid, i.e. the exponential bond between
temperature and pressure. The general form of this equation is:
𝑙𝑛(𝑝 𝑠𝑎𝑡 ) = 𝐴 −
6
𝐵
𝑇
(6.24)
In particular, the extension towards high temperatures yields an overestimation of both evaporation enthalpy and specific
volume of the saturated vapor.
68
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
with 𝐴 and 𝐵 constants characteristic of the fluid that have to be determined experimentally; the
constant term can be assumed equal to zero, as a first approximation.
Fig. 6.9 - Saturation pressure and temperature bond for acetone.
1
For different fluids the saturation curves in the 𝑇 plane, 𝑙𝑛(𝑝) have different trends not directly
attributable to an overall behavior regardless of the fluid.
6
200
H2O
180
CO2
4
160
CH4
Ne
2
120
H2O
100
)
Pressure, bar
140
0
CO2
80
60
0
Ne
C5H12
20
-4
MM
0
0
200
0.02
0.03
0.04
-2
CH4
40
0.01
400
600
-6
MM
C5H12
Temperature, K
Fig. 6.10 - Comparison between the bond of saturation for different fluids on p-T and ln(p)-T-1 scale.
On the basis of the principle of corresponding states, however, the equation of vapor pressure as a
function of the temperature, expressed in reduced terms, must be the same for all fluids: as it is a line
of discontinuity of the volumetric behavior of the fluids, this line must be constant for all fluids, in
reduced terms.
Remembering the definitions of reduced quantities, it holds true that:
69
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑙𝑛(𝑝𝑐𝑟 𝑝𝑟 ) = 𝐴 −
𝐵
𝑇𝑐𝑟 𝑇𝑟
(6.25)
from which:
𝑙𝑛(𝑝𝑟 ) = 𝐴 − ln (𝑝𝑐𝑟 ) −
𝐵
𝑇𝑐𝑟 𝑇𝑟
(6.26)
𝐵
𝑇𝑐𝑟
(6.27)
Defining:
𝐴∗ = 𝐴 − ln(𝑝𝑐𝑟 ) ;
𝐵∗ =
it holds true that:
𝑙𝑛(𝑝𝑟 ) = 𝐴∗ −
𝐵∗
𝑇𝑟
(6.28)
where 𝐴∗ and 𝐵 ∗ do not depend on the nature of the fluid.
Practically, 𝐴∗ and 𝐵 ∗ , in view of the not absolute validity of the principle of corresponding states, are
not strictly equal for all the substances, but take on different values for different classes of fluids.
In setting the validity of the curve by the critical point (𝑇𝑟 = 1, 𝑝𝑟 = 1), it can be obtained
𝑙𝑛(1) = 𝐴∗ −
𝐵∗
→ 𝐴∗ = 𝐵 ∗
1
(6.29)
It therefore suffices to define the value of a single constant to get a fairly precise idea of the bond
between the vapor pressure and the temperature for any fluid with known critical properties.
1
The figure shows the saturation curves in the 𝑇 - 𝑙𝑛(𝑝𝑟 ) plane for different fluids. The curves do not
𝑟
coincide but give rise to a range of lines.
0
-2
1
1.5
2
2.5
3
Ne
-4
)
-6
CH4
MM
-8
H2O
-10
C5H12
-12
-14
Fig. 6.11 - Trend of the saturation pressure and temperature bond in reduced terms for different
fluids.
By interpolating the data with the equation
70
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑙𝑛(𝑝𝑟 ) = 𝐴∗ −
𝐴∗
𝑇𝑟
(6.30)
different values of the parameter 𝐴∗ are obtained.
The results for the 117 fluids in the REFPROP program are provided in the figure together with the
average of the values.
The following figure also shows the frequency distribution for different classes of the parameter 𝐴∗ ,
the classes with greater frequency fall between the values 6 and 8 and generally a value of 7 allows us
to get a sufficiently accurate estimate of the saturation curve of any fluid.
12
10
8
6
5050
4040
4
3030
2020
1010
2
00
0
acetone
benze…
co2
c2but…
cyclop…
d6
dmc
ebenz…
ethyle…
d2o
hexane
ioctane
ihexane
md2m
mdm
mlinol…
mpal…
mm
neope…
n2o
octane
pentane
propa…
propyne
r113
r116
r123
r1234ze
r13
r141b
r152a
r218
r23
r245ca
r365mfc
re143a
re347…
toluene
c11
4 4 5 5 6 6 7 7 8 8 9 9 1010111112121313
Fig. 6.12 - Calibrated value of A* for 117 fluids in the REFPROP program and their frequency
distribution by classes.
Two important observations can be made:
• At the same phase change temperature, the saturation pressure is lower the higher the critical
temperature is. The effect of the critical pressure is, on the other hand, of an order of magnitude
lower since the dependency is linear and the 𝑝𝑐𝑟 for the different substances vary in a relatively
reduced field. This implies that working at a condensing temperature close to the ambient
temperature with fluids at a high critical temperature (water, for example) involves very low
condensing pressures with problems of the condenser holding the vacuum and infiltration of
incondensable gases.
• The principle of corresponding states establishes the impossibility of existence of a fluid which,
when there is a high difference in temperature between two points in saturation conditions,
presents a small difference in pressure. Let's imagine to operate a saturated vapor cycle, which
more than any other single cycle resembles the Carnot cycle, between two sources of
predetermined temperature, between which there is a high difference in temperature. The ratio
of evaporating pressure and condensing pressure can be approximately obtained as:
71
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝛽=
1
1
1
1
𝑝𝑒𝑣𝑎
)
𝐴∗ (
−
𝐴∗ 𝑇 (
−
)
= 𝑒 𝑇𝑟,𝑐𝑜𝑛𝑑 𝑇𝑟,𝑒𝑣𝑎 = 𝑒 𝑐𝑟 𝑇𝑐𝑜𝑛𝑑 𝑇𝑒𝑣𝑎
𝑝𝑐𝑜𝑛𝑑
(6.31)
therefore, it is possible to understand how this ratio is extremely high and substantially
independent from the fluid. The huge variation in pressure, and thus in volumetric flow rate,
which may be encountered on the basis of this solution may discourage or even prohibit the
use of a saturated vapor cycle which extends through the entire temperature change available.
This would not be true if the parameter 𝐴∗ were small (e.g. 2) and could exploit high
temperature changes with a small 𝛽.
It is necessary to decouple the pressure variation from that of the temperature to avoid the
technological disadvantages involved in the previous relation. For example, this is achieved in
plants with superheated vapor and in particular in super critical plants, which allow an
introduction of heat at a higher average temperature than that of the subcritical cycles. Another
possibility is to use cycles in cascade, where the total temperature difference is divided between
two saturated cycles and the fluids are chosen to work between reduced temperatures of
condensation and evaporation such that the single expansion ratios are not too high.
For example, let's imagine exploiting a source at a constant temperature of 300 °C with
condensation at a temperature of 15 °C. To build a saturated cycle it is necessary to use a fluid
with a critical temperature sufficiently greater than that of the source; a fluid with an average
complex molecule with 𝛾 = 1.3 and 𝑇𝑐𝑟 = 400°C is found.
Assuming a 𝐴∗ = 7, the resulting expansion ratio is equal to about 3400.
By implementing a cascade cycle using the same fluid for the Topping cycle, a fluid with 𝑇𝑐𝑟 =
200°C for the Bottoming cycle and a temperature of intermediate condensation-evaporation of
150°C, a 𝛽𝑇𝑜𝑝 =18.4 and a 𝛽𝐵𝑜𝑡 =39.1 is found. Their product (722) is significantly lower than
that of a single cycle.
72
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 6.13 - Representation in the Ts plane of a saturated Rankine cycle and of a plant consisting of
two saturated cascade cycles.
73
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
6.6
THE ACENTRIC FACTOR
The extension of the principle of corresponding states in fluids chemically very different (for example,
to evaluate the properties of a heavy hydrocarbon starting from the volumetric behavior of helium)
leads to gradually greater errors. It is therefore a good idea to use the Principle of Corresponding States
among similar fluids or at least belonging to the same chemical class.
Various attempts have been made to improve this aspect, by trying to extend the adherence of the
principle of corresponding states to the actual behavior of liquids and gases for very different fluids.
One approach has been to divide the substances into classes, within each of which the compressibility
factor is a function of only 𝑝𝑟 and 𝑇𝑟 .
Other proposed methods are based on looking for a further parameter in addition to 𝑇𝑟 and 𝑝𝑟 that
would allow considering the different chemical characteristics of the fluids. Some authors have singled
out this parameter from the critical density, others, and in particular Pitzer (1955), from the vapor
pressure curve, introducing the acentric factor 𝜔 which takes account of the spatial geometry of the
molecule and its non-sphericity.
The definition of this parameter comes from the observation that by bringing back the Antoine curves
in reduced terms, a range of lines with different angular gradients is created: especially the noble gases
(Ne, Xe, Kr, Ar) follow the same trend while the molecules with a greater number of atoms have a
gradually increased gradient. The bond between spatial configuration of the molecules and
thermodynamic properties is then defined by the acentric factor that should be precisely an evaluation
of the non-sphericity of the molecule and can be calculated with the following formula:
𝜔 = −𝑙𝑜𝑔10
𝑝𝑇𝑠𝑎𝑡
𝑟 =0.7
−1
𝑝𝑐𝑟
(6.32)
That is, it depends on the base 10 logarithm of the saturation pressure at a reduced temperature equal
to 0.7 that for noble gases is equal to -1.
Therefore, the acentric factor is equal to 0 for all monatomic fluids while for polyatomic fluids
distinguished by a non-spherical molecule this parameter assumes positive values.
The use of the parameter 𝜔 for defining the saturation curve in reduced terms and for defining the
compressibility factor in reduced terms leads to a large improvement in the quality of the principle of
corresponding states with the possibility of extending it to fluids that are very different chemically.
𝑍 = 𝑍(𝑇𝑟 , 𝑝𝑟 , 𝜔)
(6.33)
𝜕𝑍
𝑍 = 𝑍0 (𝑇𝑟 , 𝑝𝑟 ) + 𝜔 ( )
𝜕𝜔 𝑇𝑟 ,𝑝𝑟
(6.34)
In particular, it holds true that:
where 𝑍0 is the compressibility factor for acentric factor equal to zero; there are diagrams, tables and
equations for the derivative term that provide its trend as a function of the reduced quantities.
The previous relation lets us calculate the volumetric behavior of molecules with weak polarity with
sufficient approximation, but it also gives approximate results in the case of molecules with strong
polarity or strong asymmetries. The acentric factor can be calculated also for mixtures with an
74
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
appropriate weighted average. If the approximations obtained with the principle of corresponding
states is not sufficient, a customized equation of state for the fluid in question must be considered.
6.7
RESIDUAL OF ENTHALPY, ENTROPY AND SPECIFIC HEAT IN REDUCED TERMS
The residual terms of real fluid for enthalpy, specific heat and entropy can also be written as a function
of the reduced parameters. By somehow processing the expressions that result from them, the
following conclusions are drawn, valid within the limits previously clarified:
• the residual of the enthalpy, expressed on a molar-basis and divided by the critical
̃
∆ℎ
temperature (𝑇 ) is independent from the nature of the fluid with the same 𝑝𝑟 and 𝑇𝑟 ;
𝑐𝑟
•
the residual of the specific heat ∆𝑐̃𝑝 and that of the entropy ∆𝑠̃ , expressed on a molar-basis,
are independent from the nature of the fluid with the same 𝑝𝑟 and 𝑇𝑟 .
For example, in order to calculate the residual enthalpy, the specific volume on a mass-basis can be
expressed with the principle of corresponding states:
𝑝𝑥
𝜕𝑣
∆ℎ𝑟𝑒𝑠 = ℎ(𝑇𝑥 , 𝑝𝑥 ) − ℎ (𝑇𝑥 , 𝑝0 ) = ∫ [𝑣 − 𝑇𝑥 ( ) ] 𝑑𝑝
𝜕𝑇 𝑝
𝑝0
𝑅𝑢 𝑇
𝑣 = 𝑍(𝑇𝑟 , 𝑝𝑟 )
𝑀𝑀𝑝
0
(6.35)
(6.36)
and replace it according to the reduced parameters
𝑇 = 𝑇𝑐𝑟 𝑇𝑟 ; 𝑝 = 𝑝𝑐𝑟 𝑝𝑟 ;
𝑑𝑇 = 𝑇𝑐𝑟 𝑑𝑇𝑟 ; 𝑝 = 𝑝𝑐𝑟 𝑑𝑝𝑟 ;
(6.37)
thus:
𝑅𝑢 𝑇𝑐𝑟 𝑇𝑟
𝑀𝑀 𝑝𝑐𝑟 𝑝𝑟
𝑇𝑐𝑟 𝑇𝑟
𝜕𝑣
𝑅𝑢 𝜕 [𝑍(𝑇𝑟 , 𝑝𝑟 ) 𝑝𝑐𝑟 𝑝𝑟 ]
( ) =
(
)
𝜕𝑇 𝑝 𝑀𝑀
𝜕𝑇𝑟 𝑇𝑐𝑟
𝑣 = 𝑍(𝑇𝑟 , 𝑝𝑟 )
(6.38)
𝑝𝑟
And the term of residual enthalpy is:
∆ℎ𝑟𝑒𝑠 = ∫
𝑝𝑟,𝑥
𝑍(𝑇𝑟 , 𝑝𝑟 )
𝑝𝑟,0
𝑅𝑢 𝑇𝑐𝑟 𝑇𝑟
𝑀𝑀 𝑝𝑐𝑟 𝑝
𝑟
− 𝑇𝑐𝑟 𝑇𝑟
𝜕 [𝑍(𝑇𝑟 , 𝑝𝑟 )
𝑅𝑢
𝑀𝑀
𝑟
𝜕𝑇𝑟 𝑇𝑐𝑟
(
[
𝑇𝑐𝑟 𝑇𝑟
]
𝑝𝑐𝑟 𝑝
𝑝𝑐𝑟 𝑑𝑝
𝑟
(6.39)
)𝑝𝑟 ]
𝑅
𝑢
By simplifying 𝑝𝑐𝑟 and 𝑇𝑐𝑟 and extracting 𝑇𝑐𝑟 and 𝑀𝑀
, it holds true that:
𝑇
𝜕 [𝑍(𝑇𝑟 , 𝑝𝑟 ) 𝑝𝑟 ]
𝑝𝑟,𝑥
∆ℎ𝑟𝑒𝑠
𝑇𝑟
𝑟
) 𝑑𝑝𝑟
=∫
𝑍(𝑇𝑟 , 𝑝𝑟 ) − 𝑇𝑟 (
𝑅𝑢
𝑝
𝜕𝑇
𝑟
𝑟
𝑝
𝑟,0
𝑀𝑀 𝑇𝑐𝑟
[
𝑝𝑟 ]
75
(6.40)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑝𝑟,𝑥
∆ℎ𝑟𝑒𝑠
𝑇𝑟 𝑇𝑟 𝜕𝑍(𝑇𝑟 , 𝑝𝑟 )
= ∫ [𝑍(𝑇𝑟 , 𝑝𝑟 ) − [(
) 𝑇𝑟 + 𝑍(𝑇𝑟 , 𝑝𝑟 )]] 𝑑𝑝𝑟
𝑅𝑢
𝑝𝑟 𝑝𝑟
𝜕𝑇𝑟
𝑝𝑟,0
𝑇
𝑝𝑟
𝑀𝑀 𝑐𝑟
(6.41)
Therefore, it is recognized that for the principle of corresponding states, the residual enthalpy term
expressed in mass-basis terms divided by the critical temperature and compared to the molecular
weight is constant for all fluids since the integral is a function of only the reduced quantities.
The same procedure can be applied to the other residuals and briefly for the mass-basis and molarbasis residual terms, obtaining:
∆ℎ𝑟𝑒𝑠
= 𝑓1 (𝑇𝑟 , 𝑝𝑟 );
𝑅𝑢
𝑀𝑀 𝑇𝑐𝑟
∆𝑠𝑟𝑒𝑠
= 𝑓2 (𝑇𝑟 , 𝑝𝑟 );
𝑅𝑢
𝑀𝑀
∆𝑐𝑝,𝑟𝑒𝑠
= 𝑓3 (𝑇𝑟 , 𝑝𝑟 )
𝑅𝑢
𝑀𝑀
(6.42)
where 𝑓1 , 𝑓2 and 𝑓3 are independent from the nature of the fluid. Calculating the thermodynamic
properties of a generic fluid is therefore possible by using the curves shown in Fig. 6.14, Fig. 6.15 and
Fig. 6.16.
Fig. 6.14 - Dimensionless residual enthalpy due to real fluid effects valid for all fluids in the
approximation of the principle of corresponding states.
76
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 6.15 - Dimensionless residual entropy due to real fluid effects valid for all fluids in the
approximation of the principle of corresponding states.
Fig. 6.16 - Dimensionless specific heat due to real fluid effects valid for all fluids in the
approximation of the principle of corresponding states.
77
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
6.7.1
Enthalpy difference between two thermodynamic states calculation
As an example, the enthalpy difference between two different thermodynamic states for water is
evaluated as application of the principle of corresponding states.
A
B
ℎ𝐵 − ℎ𝐴 = (ℎ0𝑇𝐵 + ∆ℎ 𝑇𝐵 ,𝑝𝐵 ) − (ℎ0𝑇𝐴 + ∆ℎ 𝑇𝐴,𝑝𝐴 )
ℎ𝐵 − ℎ𝐴 = (ℎ0𝑇𝐵 − ℎ0𝑇𝐴 ) + (∆ℎ 𝑇𝐵 ,𝑝𝐵 − ∆ℎ 𝑇𝐴,𝑝𝐴 )
Numerically, it is found that:
𝑀𝑀 = 18
kg
,
kmol
𝑇𝑐𝑟 = 373.95 °C , 𝑝𝑐𝑟 = 220.64 bar
Then, it is possible to derive the "reduced" conditions of points A and B, which included in the figure
of residual enthalpy in reduced form provide the residuals based on the principle of corresponding
states:
A
B
𝑇 °C
𝑝 bar
𝑇𝑐𝑟
𝑝𝑐𝑟
𝑍
𝑐𝑝0
400
450
50
200
1.0402
1.1175
0.227
0.906
0.931
0.762
2.0635
2.0968
∆ℎ
𝑅
𝑇
𝑀𝑀 𝑐𝑟
-0.1
-0.9
∆ℎ
(kJ/kg)
-29.888
-269.00
Considering the specific heat varies little and by using its average value 𝑐̅𝑝0 = 2080.15
true:
ℎ𝐵 − ℎ𝐴 = 𝑐̅𝑝0 (𝑇𝐵 − 𝑇𝐴 ) + (∆ℎ𝑇𝐵 ,𝑝𝐵 − ∆ℎ 𝑇𝐴,𝑝𝐴 )
78
J
kgK
, it holds
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Which numerically means:
ℎ𝐵 − ℎ𝐴 = 2.080.15 ∙ 50 + (−269.00 + 29.888) = −135.1 J/kg
From the Mollier diagram it is read: ℎ𝐴 = 3196.7 kJ/kg and ℎ𝐵 = 3061.7 kJ/kg that is ℎ𝐵 − ℎ𝐴 =
−135 kJ/kg value close to the result obtained above. When thermodynamic tables or Mollier
diagrams are available (as is the case for water vapor and many organic fluids), it is of course easier
and more accurate to directly use these thermodynamic tools instead of resorting to the principle of
corresponding states.
6.7.2
Approximated estimate of enthalpy of vaporization
The enthalpy of vaporization of a substance can be interpreted as the residual enthalpy (isothermal)
between two points of the saturation isobar (saturated liquid = real fluid, saturated vapor ≅ ideal gas).
It is therefore subject to the rules given above. Rewriting the Clausius-Clapeyron relation as a function
of the reduced parameters:
∆ℎ𝑒𝑣𝑎(𝑇)
𝑑𝑝 𝑠𝑎𝑡
=
(𝑣 𝑣 − 𝑣 𝑙 )𝑇
𝑑𝑇
∆ℎ𝑒𝑣𝑎(𝑇)
𝑑𝑝 𝑠𝑎𝑡
=
0
0
𝑑𝑇
(𝑍 𝑣 (𝑇𝑟 , 𝑝𝑟 )𝑣(𝑇,𝑝)
− 𝑍𝑙 (𝑇𝑟 , 𝑝𝑟 )𝑣(𝑇,𝑝)
)𝑇
𝑠𝑎𝑡
∆ℎ𝑒𝑣𝑎(𝑇)
𝑑𝑝
=
𝑅𝑢 𝑇
𝑑𝑇
(𝑍 𝑣 (𝑇𝑟 , 𝑝𝑟 ) − 𝑍𝑙 (𝑇𝑟 , 𝑝𝑟 ))𝑇 𝑀𝑀
𝑝
∆ℎ𝑒𝑣𝑎(𝑇)
𝑑𝑝𝑟 𝑝𝑐𝑟
=
𝑅𝑢 𝑇𝑐𝑟 𝑇𝑟 𝑑𝑇𝑟 𝑇𝑐𝑟
(𝑍 𝑣 (𝑇𝑟 , 𝑝𝑟 ) − 𝑍𝑙 (𝑇𝑟 , 𝑝𝑟 )) 𝑇𝑐𝑟 𝑇𝑟 𝑀𝑀
𝑝𝑐𝑟 𝑝𝑟
(6.43)
(6.44)
(6.45)
(6.46)
By simplifying 𝑇𝑐𝑟 and 𝑝𝑐𝑟 :
∆ℎ𝑒𝑣𝑎(𝑇) 𝑑𝑝𝑟 𝑣
𝑇𝑟 2
𝑙
=
(𝑍 (𝑇𝑟 , 𝑝𝑟 ) − 𝑍 (𝑇𝑟 , 𝑝𝑟 ))
𝑅𝑢
𝑑𝑇𝑟
𝑝𝑟
𝑇
𝑀𝑀 𝑐𝑟
(6.47)
that once again depends only on the reduced conditions of the fluid.
Considering fluids with similar critical temperatures, it is found out that the enthalpy of vaporization
at a given temperature is inversely proportional to the molecular weight. This is the same result already
found for the correction of enthalpy in reduced terms. Light fluids such as water, ammonia and sodium
have enormous enthalpies of vaporization. Heavy fluids such as some halogenated hydrocarbons or
mercury have very small enthalpies of vaporization.
By applying this relation and neglecting the specific volume of the liquid, it is found out that:
∆ℎ𝑒𝑣𝑎(𝑇) 𝑑𝑝𝑟 𝑣
𝑇𝑟 2
=
𝑍 (𝑇𝑟 , 𝑝𝑟 )
𝑅𝑢
𝑑𝑇𝑟
𝑝𝑟
𝑇
𝑀𝑀 𝑐𝑟
where the term
𝑑𝑝𝑟
𝑑𝑇𝑟
can be expressed starting from the Antoine equation in reduced form as
79
(6.48)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑑𝑝𝑟 𝐴∗ 𝑝𝑟
= 2
𝑑𝑇𝑟
𝑇𝑟
(6.49)
∆ℎ𝑒𝑣𝑎(𝑇)
= 𝑍 𝑣 (𝑇𝑟 , 𝑝𝑟 )𝐴∗
𝑅𝑢
𝑀𝑀 𝑇𝑐𝑟
(6.50)
and therefore
In the event the effects of real fluid for the vapor in conditions of saturation can be neglected,
𝑍 𝑣 (𝑇𝑟 , 𝑝𝑟 ) = 1:
∆ℎ𝑒𝑣𝑎(𝑇)
= 𝐴∗
𝑅𝑢
𝑀𝑀 𝑇𝑐𝑟
(6.51)
That is to say, for a phase transition at very low pressures in which the hypotheses above apply, the
∆ℎ𝑒𝑣𝑎(𝑇) on a mass-basis only depends on the molecular weight and the critical temperature of the
fluid.
6.7.3
Specific heat on a molar-basis variation between saturated liquid and vapor
Let's consider, for example, specific heat at constant pressure. The following is also valid for all other
thermodynamic properties. The application of the principle of corresponding states to the residual of
the specific heat on a molar-basis has shown how this residual for a certain 𝑝𝑟 and 𝑇𝑟 is independent
from the nature of the fluid. Moreover, it is a positive and constant correction in absolute terms. In
relative terms, however, the correction of the 𝑐̃𝑝 switching from ideal gas to liquid can be large or
small, depending on whether the molecule considered in the state of ideal gas has small or large heat
capacity on molar-basis.
Remembering that the 𝑐̃𝑝0 depends substantially on the complexity of the molecular structure, switching
from a condition of ideal gas to a sub-cooled liquid:
• simple molecules experience major changes in the specific heat and the ∆𝑐̃𝑝 may be even
greater than 𝑐̃𝑝0 so
•
𝑐̃𝑝𝑙
𝑐̃𝑝0
>1
complex molecules have, instead, the 𝑐𝑝,𝐿 ~𝑐𝑝0 , in view of the small percentage contribution of
𝑐̃ 𝑙
the correction of real fluid, and 𝑐̃𝑝0 → 1.
𝑝
For example, for water at ambient temperature, the specific heat of the liquid is equal to about
4.2 kJ/(kg K), while the specific heat of low pressure vapor is equal to about 1.8 kJ/kg; therefore, the
residual is greater than the value of the ideal gas. This difference is even more significant for
monatomic molecules, while for very complex molecules the percentage difference becomes almost
zero.
80
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
pr
Fig. 6.17 - Trend of the specific heat on a molar-basis as a function of the reduced pressure for a
supercritical reduced temperature. The different weight of the residual term ∆𝑐̃𝑝 compared to the
specific heat as the molecular complexity changes is clear.
6.7.4
Molecular complexity effect of the fluid on the shape of the Ts diagram
Let's look for a method that allows us to qualitatively predict what shape the saturation curve of a
generic substance will be.
That said, two points A and B on the saturated vapor curve of a generic substance are considered.
When moving from A to B, both pressure and temperature increase simultaneously.
The increase in temperature results in an increase in entropy, while the pressure increase produces a
decrease in the same state function. Depending on whether the influence of temperature or that of the
pressure overrides, the saturated vapor curve slopes positively or negatively in the 𝑇 − 𝑠 plane.
Suppose that the pressure in A is sufficiently low because the vapor behaves like an ideal gas. An
infinitesimal increase in temperature therefore corresponds to a change in entropy given by:
𝑑𝑠̃ = 𝑐̃𝑝0
𝑑𝑇
𝑑𝑝
− 𝑅𝑢
𝑇
𝑝
(6.52)
where 𝑠̃ and 𝑐̃𝑝 represent quantities on a molar-basis and 𝑅𝑢 is the universal constant of gases. Then,
it is clear that a temperature increase (and therefore of pressure due to the link between the two
properties in saturation conditions) generates a variation in specific entropy (molar-basis) given by the
sum between two opposite effects. On one hand, the higher temperature increases the entropy; on the
other hand, the higher pressure reduces it. Depending on the dominant effect, the molar entropy could
increase or decrease with saturation temperature (and pressure). It is possible to write, dividing by 𝑑𝑇:
𝑑𝑠̃ 𝑠𝑎𝑡 𝑐̃𝑝0 𝑅𝑢 𝑑𝑝
= −
𝑑𝑇
𝑇 𝑑𝑇 𝑝
81
(6.53)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝐵
Differentiating both terms of the Antoine equation 𝑙𝑛(𝑝 𝑠𝑎𝑡 ) = 𝐴 − 𝑇 , it holds true that:
𝑑𝑝
𝐵
𝑑𝑝
𝐵
= 2 𝑑𝑇 →
= 2
𝑝
𝑇
𝑝𝑑𝑇 𝑇
(6.54)
which replaced in the previous relation:
𝑑𝑠̃ 𝑠𝑎𝑡 1 0 𝑅𝑢 𝐵
)
= (𝑐̃𝑝 −
𝑑𝑇
𝑇
𝑇
(6.55)
𝐵
Recalling that 𝐵 ∗ = 𝑇 = 𝐴∗ and the definition of 𝑇𝑟 it holds true that:
𝑐𝑟
𝑑𝑠̃ 𝑠𝑎𝑡 1 0 𝑅𝑢 𝐴∗
)
= (𝑐̃𝑝 −
𝑑𝑇
𝑇
𝑇𝑟
when the reduced temperature is such that 𝑐̃𝑝0 =
𝑅𝑢 𝐵∗
𝑇𝑟
(6.56)
the curve in the Ts plane is vertical.
In general, assigning 𝐵 ∗ an average value of 7 (see Fig. 6.12) and assuming a fixed reduced
temperature, for example 𝑇𝑟 = 0.5, it is found that for values of the specific heat on a molar-basis
greater than about 116 kJ/(kmol K) the saturation curve slope is positive in the T-s plane; for lower
values it instead slopes negatively. The saturation curve of complex fluids is called retrograde.
Anyway, for low temperatures, the second term becomes predominant and the curve has a negative
gradient, independently from the fluid. The same phenomenon can be observed for temperatures next
to the critical point, because of the exponential increase of the pressure and therefore of the negative
𝑅 𝑑𝑝
contribution of the term − 𝑑𝑇 𝑝 .
3
5
In substances with a not very complex molecule (or even monatomic in which 𝑐̃𝑝0 = 2 𝑅𝑢 + 𝑅𝑢 = 2 𝑅𝑢 ,
𝑐̃𝑝0 = 20.785 kJ/(kmol K) such as the rare gases, metals, water, ammonia, etc. the increase of entropy
due to the temperature increase is more than offset by the decrease in entropy due to the increased
pressure. On the contrary, in complex molecule substances, a prevalence of the thermal effect over the
effect of pressure due to the large value of the heat on a molar-basis is found. As an example, the limit
curves of three fluids are plotted (Fig. 6.18): water, R11 (CCl3F1) and isopentane.
Fig. 6.18 - Shape of the saturation dome with increasing molecular complexity: a) simple fluid, b)
isentropic fluid, c) high complexity fluid (retrograde dome).
82
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
In the first case, with low molecular complexity, the curve is the "dome" type; in the second case, the
curve has a vertical section; in the third case, with high molecular complexity fluid, the term
independent from the molecular complexity (Δ𝑐̃𝑝 ) can reach the point of being negligible compared to
the specific heat of ideal gas. In the latter case, since the gradient of the isobars is inversely proportional
to the specific heat at the same temperature, and since the specific heats of the liquid and the steam are
not very different, it if found that the isobars tend to be substantially parallel and lean against the limit
curve in the vicinity of this one. Moreover, an expansion from the point of saturated vapor for a
complex fluid takes place entirely in the field of superheated vapor.
Fig. 6.19 - Trend of the isenthalpic curves as the molecular complexity changes.
Simple molecules
The limit curve has a “dome” shape
The isobars have gradients that are proportional
𝑇
to the ratio 𝑐 , very different for the liquid and
Complex molecules
The limit curve has a similar gradient for liquid
and vapor for an extensive range of reduced
temperatures, and the curve is retrograde
The isobars have gradients very similar for the
liquid and the vapor
𝑝
the vapor
The critical isobar has a very variable gradient.
In the vicinity of the critical point, it has an
almost horizontal trend, peaks of high specific
heat and gradients in the vapor phase much
higher than those of the liquid
The critical isobar has a slightly variable
gradient. In the vicinity of the critical point, it
rests for a modest range, and then resumes a
gradient close to that of the liquid
83
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
7
THE REAL LIQUID BEHAVIOR
The relations obtained in the foregoing paragraphs are able to represent the volumetric and
thermodynamic behavior of any fluid, from the conditions of ideal gas up to those of incompressible
liquid. Any other consideration about the behavior of fluids would therefore be unnecessary, but it is
worth analyzing one last case, namely that of the real liquids for which the approximation of the
incompressible liquid is no longer acceptable.
This is the case of liquids near the critical point in which the properties are certainly not those of the
ideal fluid and compressibility effects begin to have a significant weight on the thermodynamic
behavior of the fluid.
For example, in the case of the supercritical water cycle vapor, which has now become widespread in
power plants having enormous electricity production potential, the positioning of the feed pumps
makes it essential to use relations more sophisticated than those introduced up to now.
To adequately characterize the volumetric behavior of a compressible liquid, in addition to the
coefficients 𝑘𝑇 and 𝛼𝑝 , the isentropic coefficient of compressibility 𝑘𝑠 is introduced. This parameter
is directly linked to the coefficient of isothermal compressibility 𝑘𝑇
1 𝜕𝑣
𝑘𝑠 = − ( )
𝑣 𝜕𝑝 𝑠
𝑘𝑇 𝑐𝑝
=
𝑘𝑠 𝑐𝑣
(7.1)
(7.2)
If 𝑘𝑇 , 𝑐𝑝 and 𝑐𝑣 were not known, they could be estimated with a good degree of approximation by
means of the theorem of corresponding states.
Let's now try to strictly calculate, as an example of application, the compression work of a pump and
the heating of the liquid due to the effect an isentropic compression in general pressure and temperature
conditions.
The following analysis is very useful when the temperature of the liquid is close to critical point. In
this circumstance, the mechanical compressibility and thermal expansion coefficient assume high
values and cannot be neglected in the calculations.
7.1
IDEAL COMPRESSION WORK OF A PUMP
Let's consider a volumetric pump that has to supply a certain volume of fluid 𝑉 with a certain increase
in pressure from the saturated inlet liquid conditions. In the 𝑝 − 𝑣 diagram the pump cycle is defined
by three processes:
• 1-2 suction: in which the fluid is sucked, the work is carried out by the environment on the
system
• 2-3 compression: in which the pressure of the fluid increases up to the discharge conditions
• 3-4 discharge: in which the fluid is discharged, the work is carried out by the system on the
environment
The technical work of compression is the sum of the thermodynamic work required by the compression
phase and the works of pulsion and suction, which are simply equal to 𝑝𝑑𝑉 because they take place at
constant pressure.
Let's compare two cases: in the first (A) the starting pressure is very low and the liquid can be
considered almost incompressible while in the second (B), closer to the critical point, the fluid has a
84
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
certain degree of compressibility. In both cases the increase in pressure equal to 200 bar and the
volumetric flow rate of liquid discharged at high pressure 1 m3/s is set.
Water is adopted as working fluid and begin compressing from two points of saturated liquid, the first
at 5 °C and the other at 360 °C.
The table below shows the results of the isentropic compressions.
Table 7.1 - Characteristics of water at the beginning and end of compression starting from a point
at low temperature (A) and from a point near the critical point (B).
A
B
𝑝, bar
0.0087
200.0087
186.6601
386.6601
𝑇, °C
s, kJ/(kg K) 𝑣, m3/kg 𝑢, kJ/kg
ℎ, kJ/kg
𝑉, m3
𝑚, kg/s
5.0000
0.0763
0.0010
21.0191
21.0200 1.0096
1009.5
5.0604
0.0763
0.0010
21.1133
40.9253 1.0000
360.0000
3.9167
0.0019 1726.2850 1761.6646 1.1026
581.74
382.1292
3.9167
0.0017 1731.0625 1797.5277 1.0000
Note that in case B the increase in pressure results in a 9.3% reduction of specific volume, while in
case A the variation is 0.9% as would be expected in the case of an almost incompressible liquid. The
volumetric flow rate discharged being equal, greater aspirated volume will be necessary and a
significantly lower mass flow rate will be processed since the specific volume in the case at high
pressure is about double.
The suction and discharge works can be calculated as 𝑚𝑝𝑑𝑣, while the compression work can be
estimated from the change in internal energy of the fluid since the process regards a closed system.
The values for the three different works (in kW) are shown in Table 7.2:
Table 7.2 - Total compression works for two compressions that start from a point at low
temperature (A) and from a point near the critical point (B).
suction
compression
discharge
Total work
Total isentropic work, kJ/kg
A
-0.881
95.13
20001
20095.1
19.9
B
-20582
2779.34
38666
20863.3
35.86
The total work of compression is greater for compressible fluid B in the absolute sense and is almost
double that of case A when referring to the flow rate on a mass-basis. In addition, the real compression
work of the fluid in case A is practically negligible (0.47%) compared to the total work, while in case
B it reaches a figure equal to 13.3% due to the compressibility of the fluid.
This shows that extending the hypothesis of incompressible fluid in the vicinity of the critical point
can result in significant errors. In practice, this is valid especially in the supercritical cycles where,
given the high admission pressure in the turbine, the compression phase of the liquid by means of the
feed pumps takes on remarkable importance in the global mechanical and thermal balance of the plant.
The graphic representation in the plane ∆𝑝 − 𝑉 (Fig. 7.1) in fact shows that the work done in the case
of incompressible fluid, i.e. the area under the chart, is greater for the compressible fluid.
85
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 7.1 - Trend in the p-V plane of the compression of an incompressible liquid and a real liquid.
The same considerations can be reached also taking the purely theoretical route.
Starting from the definition of specific volume as a function of pressure and entropy and calculating
its total differentia, it is obtained that:
𝑑𝑣 = (
𝜕𝑣
𝜕𝑣
) 𝑑𝑝 + ( ) 𝑑𝑠
𝜕𝑝 𝑠
𝜕𝑠 𝑝
(7.3)
𝜕𝑣
that, for a transformation at constant entropy, is reduced to (𝑑𝑣)𝑠 = (𝜕𝑝) 𝑑𝑝 from which with the
𝑠
coefficient of isentropic compressibility it holds true:
(𝑑𝑣)𝑠 = −𝑣𝑘𝑠 𝑑𝑝 = −𝑣
𝑐𝑣
𝑘 𝑑𝑝
𝑐𝑝 𝑇
(7.4)
replacing this relation in the expression for calculating the ideal work of compression and integrating:
3
3
𝑊2−3 = − ∫ 𝑝𝑑𝑣 = ∫ 𝑝𝑣
2
2
𝑐𝑣
𝑘 𝑑𝑝
𝑐𝑝 𝑇
(7.5)
The specific volume, the specific heats and the coefficient 𝑘𝑇 depend little on the pressure; considering
their appropriate average values, the following final expression holds true:
𝑊2−3 = 𝑣̅
𝑐̅𝑣 𝑝3 2 − 𝑝2 2
̅̅̅
𝑘
𝑐̅𝑝 𝑇
2
(7.6)
This work is negligible compared to the pulsion work when the liquid temperature is much lower than
̅̅̅𝑇 → 0); otherwise, it has to be taken into consideration for a correct
the critical temperature (𝑘
calculation.
7.2
EVALUATION OF THE HEATING CAUSED BY AN ISENTROPIC COMPRESSION
Let's consider the entropy as a function of pressure and temperature, and find its differential:
𝑑𝑠 = (
𝜕𝑠
𝜕𝑠
) 𝑑𝑝 + ( ) 𝑑𝑇
𝜕𝑝 𝑇
𝜕𝑇 𝑝
86
(7.7)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Moreover, the increase in entropy along an isobar is given by:
(𝜕𝑠)𝑝 =
𝑐𝑝 𝑑𝑇
𝑇
(7.8)
from which:
𝑐𝑝
𝜕𝑠
( ) =
𝜕𝑇 𝑝
𝑇
(7.9)
𝜕𝑠
𝜕𝑣
𝜕𝑝 𝑇
𝜕𝑇 𝑝
Furthermore, for the second Maxwell equation − ( ) = ( ) :
𝑐𝑝
𝜕𝑣
𝑑𝑠 = − ( ) 𝑑𝑝 + 𝑑𝑇
𝜕𝑇 𝑝
𝑇
(7.10)
Recognizing that the first term on the second member can be expressed by the isobaric cubic expansion
coefficient 𝛼𝑝 7:
𝑑𝑠 = −𝑣𝛼𝑝 𝑑𝑝 +
𝑐𝑝
𝑑𝑇
𝑇
(7.11)
where, considering an isentropic transformation:
𝑑𝑇 =
𝑇𝑣𝛼𝑝
𝑑𝑝
𝑐𝑝
(7.12)
This relation can be integrated taking into account the circumstance that 𝑇, 𝑣, 𝛼𝑝 and 𝑐𝑝 vary little
during compression, therefore, considering appropriate average values, it holds true that:
∆𝑇 = ∫
𝑝𝑚𝑎𝑥 𝑇𝑣𝛼
𝑝𝑚𝑖𝑛
𝑝
𝑐𝑝
𝑑𝑝 =
𝑇̅𝑣̅ 𝛼
̅̅̅
𝑝
(𝑝𝑚𝑎𝑥 − 𝑝𝑚𝑖𝑛 )
𝑐̅𝑝
(7.13)
which provides the answer to our problem. Also in this case the heating takes on small values at the
low reduced temperatures, but it cannot be neglected in the vicinity of the critical region due to the
deviation of the isobar curves from the lower limit curve. The numerical result is shown in the previous
table (Table 7.1), where it is highlighted that for the same pressure increase in case A, a negligible
temperature increase (0.06°C) is obtained, instead in case B of 22.12°C.
7
If the volumetric behavior of the fluid is known (assumption of the whole discussion on real fluids in the previous
paragraph), the increase in temperature can be calculated directly without passing through the isobaric expansion
coefficient 𝛼𝑝 . Therefore, the discussion here is compatible with the previous discussion on real fluids, of which, as
mentioned, the liquids form a particular case.
87
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Fig. 7.2 - Temperature increase for an isentropic compression for an incompressible and a real
liquid.
88
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
8
SOLUTIONS AND MIXTURES
This chapter deals with the investigation of solutions and mixtures. First, an overall introduction to
solutions is performed. Then, the calculation of the properties of mixtures focusing on the case of ideal
gas mixtures is studied.
8.1
INTRODUCTION TO SOLUTIONS
Solutions can be classified in:
• non-reacting, when composition does NOT change
• reacting, when composition does change
Among solutions, a mixture is a solution of gases that do not alter their chemical composition. Among
reacting systems, combustion is studied with the following simplifications:
• Reactant and products as ideal gases
• Complete combustion
To study the properties of solutions, it is fundamental to define the molar fraction of component 𝑖, 𝑧̃𝑖 ,
the liquid molar fraction, 𝑥̃𝑖 , the gas molar fraction, 𝑦̃𝑖 , the liquid molal fraction, 𝛼, and the gas molal
fraction, 𝛽 as follows
𝑛𝑖𝑠𝑜𝑙
𝑛𝑇𝑠𝑜𝑙
𝑛𝑖𝑙
𝑥̃𝑖 = 𝑙
𝑛𝑇
𝑛𝑖𝑣
𝑦̃𝑖 = 𝑣
𝑛𝑇
𝑛𝑙𝑇
𝛼 = 𝑠𝑜𝑙
𝑛𝑇
𝑛𝑣𝑇
𝛽 = 𝑠𝑜𝑙
𝑛𝑇
𝑧̃𝑖 =
(8.1)
(8.2)
(8.3)
(8.4)
(8.5)
where the superscripts ‘sol’ stands for solution. Additionally, the following properties are valid:
∑ 𝑧̃𝑖 = 1
(8.6)
∑ 𝑥̃𝑖 = 1
(8.7)
∑ 𝑦̃𝑖 = 1
(8.8)
𝛼+𝛽 =1
(8.9)
In general, a distinction between the definition of intensive and extensive properties is considered. An
intensive solution property, 𝜋̃ 𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑝, 𝜒⃗), can be defined as:
𝑝ℎ𝑎𝑠𝑒 ̂ 𝑝ℎ𝑎𝑠𝑒
𝜋̃𝑖
𝜋̃ 𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑝, 𝜒⃗) = ∑ 𝜒̃𝑖
𝑖
89
(𝑇, 𝑝, 𝜒⃗)
(8.10)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
where 𝜒̃𝑖𝑝ℎ𝑎𝑠𝑒 can be either 𝑥̃𝑖 or 𝑦̃𝑖 , and 𝜋̂̃ 𝑖𝑝ℎ𝑎𝑠𝑒 is the mixture property of the 𝑖 component. For an
extensive property, Π𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑝, 𝜒⃗), it holds true:
̂ 𝑖𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑝)
Π𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑝, 𝜒⃗) = ∑ Π
(8.11)
𝑖
In the case of ideal solutions, the relation for an intensive property is simplified as:
𝜋̃ 𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑝, 𝜒⃗) = ∑ 𝜒̃𝑖𝑝ℎ𝑎𝑠𝑒 [𝜋̃𝑖𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑝) + ∆𝑠𝑜𝑙 𝜋̃𝑖𝑝ℎ𝑎𝑠𝑒 (𝜒⃗)]
(8.12)
𝑖
where 𝜋̃𝑖𝑝ℎ𝑎𝑠𝑒 is the property of the 𝑖th component a pure fluid condition, and ∆𝑠𝑜𝑙 𝜋̃𝑖𝑝ℎ𝑎𝑠𝑒 the deviation
of the property of the species in the mixture. This allows to decouple the dependance on the properties
of the species from the composition of the mixture. The first term, i.e. 𝜋̃𝑖 𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑃) is the value of the
property species 𝑖 would have were it a pure fluid. The second term, Δ𝑠𝑜𝑙 𝜋̃𝑖 𝑝ℎ𝑎𝑠𝑒 , is the term that relates
the property of the species to the phase composition.
For ideal solutions, ∆𝑠𝑜𝑙 𝜋̃𝑖 can be simplified as:
0 𝑓𝑜𝑟 𝑣̃, 𝑐̃𝑝 , 𝑐̃𝑣 , ℎ̃
∆𝑠𝑜𝑙 𝜋̃𝑖𝑝ℎ𝑎𝑠𝑒 (𝜒⃗) { −𝑅𝑢 𝑙𝑛𝜒̃𝑖𝑝ℎ𝑎𝑠𝑒 𝑓𝑜𝑟 𝑠̃
(8.13)
𝑅𝑢 𝑇 𝑙𝑛𝜒̃𝑖𝑝ℎ𝑎𝑠𝑒 𝑓𝑜𝑟 𝑔̃
This means that the volume of species 𝑖 𝑣̃𝑖 does not depend on the composition, while the entropy of
species 𝑖 𝑠̃𝑖 does, as the term −𝑅𝑢 ln 𝑥̃𝑖 𝑝ℎ𝑎𝑠𝑒 needs to be added to the pure fluid property. The volume
of the species 𝑣̃𝑖 in this condition does not change from the pure fluid condition, while 𝑠̃ 𝑖 does.
For an extensive property it holds true:
Π𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑝, 𝜒⃗) = ∑[Π𝑖𝑝ℎ𝑎𝑠𝑒 (𝑇, 𝑝) + ∆𝑠𝑜𝑙 Π𝑖𝑝ℎ𝑎𝑠𝑒 ]
(8.14)
𝑖
Finally, real solution properties are computed from ideal solution properties adding excess solution
properties.
8.2
IDEAL MIXTURES
It is possible to prove that a mixture of ideal gases is an ideal solution. Defining 𝑦̃𝑖 =
𝑚𝑖
𝑚𝑇
𝑛𝑖
𝑛𝑇
and 𝑦𝑖 =
, it holds true:
0
𝑣̃ 𝑚𝑖𝑥 = ∑ 𝑦̃𝑣
̃𝑖
𝑖 ̃𝑖 = ∑ 𝑦
𝑖
𝑖
𝑅𝑢 𝑇 𝑅𝑢 𝑇
𝑅𝑢 𝑇
=
∑ 𝑦̃𝑖 =
𝑝
𝑝
𝑝
(8.15)
𝑅𝑢 𝑇 𝑛𝑇 𝑅𝑢 𝑇
=
𝑝
𝑝
(8.16)
𝑖
𝑉̃ 𝑚𝑖𝑥 = ∑ 𝑉̃𝑖0 = ∑ 𝑛𝑖 𝑣̃𝑖0 = ∑ 𝑛𝑖
𝑖
𝑖
𝑖
This represents the demonstration that a mixture of ideal gases is an ideal gas itself.
90
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Also, the following relations for the molar mass of the mixture, 𝑀𝑀𝑇 , hold true:
𝑚 𝑇 ∑𝑖 𝑛𝑖 𝑀𝑀𝑖
𝑀𝑀𝑇 =
=
= ∑ 𝑦̃𝑀𝑀
𝑖
𝑖
𝑛𝑇
𝑛𝑇
𝑖
𝑚𝑇
𝑚𝑇
𝑦𝑖
)
𝑀𝑀𝑇 =
=
= (∑
∑𝑖 𝑚𝑖 /𝑀𝑀𝑖
𝑛𝑇
𝑀𝑀𝑖
𝑖
𝑀𝑀𝑇
𝑦𝑖 = 𝑦̃𝑖
𝑀𝑀𝑖
(8.17)
−1
(8.18)
(8.19)
Moreover, the total pressure of the mixture can be expressed as function of the partial pressures of the
components in the mixture as follows:
𝑝 = ∑ 𝑝𝑖 = ∑ 𝑦̃𝑖 𝑝
𝑖
(8.20)
𝑖
Additionally, the following relations for an ideal mixture hold true:
𝑣̃ 𝑚𝑖𝑥 (𝑇, 𝑝, 𝑦⃗) = ∑ 𝑦̃𝑣̃𝑖0 (𝑇, 𝑝)
(8.21)
0
𝑐̃ 𝑚𝑖𝑥 (𝑇, 𝑝, 𝑦⃗) = ∑ 𝑦̃𝑐̃
𝑖 𝑖 (𝑇)
(8.22)
ℎ̃𝑚𝑖𝑥 (𝑇, 𝑝, 𝑦) = ∑ 𝑦̃𝑖 ℎ̃𝑖0 (𝑇)
(8.23)
0
𝑠̃ 𝑚𝑖𝑥 (𝑇, 𝑝, 𝑦⃗) = ∑ 𝑦̃[𝑠̃
̃]
𝑖 𝑖 (𝑇, 𝑝) − 𝑅𝑢 𝑙𝑛𝑦
𝑖
𝑇
=
∑ 𝑦̃𝑖 [𝑠̃𝑖0 (𝑇𝑟𝑒𝑓 , 𝑝𝑟𝑒𝑓 )
𝑐̃𝑝,𝑖 (𝑇)
𝑝
𝑑𝑇 − 𝑅𝑢 𝑙𝑛
− 𝑅𝑢 𝑙𝑛𝑦̃𝑖 ]
𝑇
𝑝𝑟𝑒𝑓
𝑇𝑟𝑒𝑓
+∫
(8.24)
𝑇
𝑐̃𝑝,𝑖 (𝑇)
𝑝𝑖
0
𝑑𝑇 − 𝑅𝑢 𝑙𝑛
] = ∑ 𝑦̃𝑠̃
𝑖 𝑖 (𝑇, 𝑝𝑖 )
𝑇
𝑝
𝑟𝑒𝑓
𝑇𝑟𝑒𝑓
= ∑ 𝑦̃𝑖 [𝑠̃𝑖0 (𝑇𝑟𝑒𝑓 , 𝑝𝑟𝑒𝑓 ) + ∫
𝑇𝑐𝑟 = ∑ 𝑦̃𝑖 𝑇𝑐𝑟,𝑖
(8.25)
𝑖
𝑝𝑐𝑟 = ∑ 𝑦̃𝑖 𝑝𝑐𝑟,𝑖
(8.26)
𝑖
From the previous relations, the Gibb’s theorem holds true, stating that all properties (other than
specific volume) in an ideal gas mixture are determined as:
0
𝜋̃ 𝑚𝑖𝑥 (𝑇, 𝑝, 𝑦⃗) = ∑ 𝑦̃𝜋
𝑖 ̃ 𝑖 (𝑇, 𝑝𝑖 )
(8.27)
𝑖
In practice, the enthalpy of eq. (8.23) can be expressed in a simplified form as follows (on a mass
basis)
𝑇
ℎ
𝑚𝑖𝑥
(𝑇, 𝑦⃗) =
∑ 𝑦𝑖 ℎ𝑖0 (𝑇)
𝑖
= ∑ 𝑦𝑖 [ℎ𝑖 (𝑇𝑟𝑒𝑓 ) + ∫
𝑇𝑟𝑒𝑓
𝑖
0
𝑐𝑝,𝑖
(𝑇)𝑑𝑇]
𝑇
ℎ𝑚𝑖𝑥 (𝑇, 𝑦⃗) = ℎ0 + ∑ 𝑦𝑖 ∫
𝑖
91
𝑇𝑟𝑒𝑓
0
𝑐𝑝,𝑖
(𝑇)𝑑𝑇
(8.28)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
It is also possible to simplify the entropy relation in eq. (8.24) as follows (on a mass basis):
𝑅𝑢
𝑠 𝑚𝑖𝑥 (𝑇, 𝑝, 𝑦⃗) = ∑ 𝑦𝑖 [𝑠𝑖0 (𝑇, 𝑝) −
𝑙𝑛𝑦̃]
𝑖
𝑀𝑀𝑖
𝑇 𝑐 (𝑇)
𝑅𝑢
𝑝𝑖
𝑝,𝑖
0
= ∑ 𝑦𝑖 [𝑠𝑖 (𝑇𝑟𝑒𝑓 , 𝑝𝑟𝑒𝑓 ) + ∫
𝑑𝑇 −
𝑙𝑛
]
𝑇
𝑀𝑀𝑖 𝑝𝑟𝑒𝑓
𝑇𝑟𝑒𝑓
(8.29)
by introducing the following rearrangements:
− ∑ 𝑦𝑖
𝑖
− ∑ 𝑦𝑖
𝑖
𝑅𝑢
𝑝𝑖
𝑅𝑢
𝑝𝑖 𝑝
𝑙𝑛
= − ∑ 𝑦𝑖
𝑙𝑛 (
)
𝑀𝑀𝑖 𝑝𝑟𝑒𝑓
𝑀𝑀𝑖
𝑝 𝑝𝑟𝑒𝑓
𝑖
𝑅𝑢
𝑝𝑖
𝑅𝑢
𝑅𝑢
𝑝
𝑙𝑛
= − ∑ 𝑦𝑖
𝑙𝑛𝑦̃𝑖 − ∑ 𝑦𝑖
𝑙𝑛
𝑀𝑀𝑖 𝑝𝑟𝑒𝑓
𝑀𝑀𝑖
𝑀𝑀𝑖 𝑝𝑟𝑒𝑓
𝑖
(8.30)
𝑖
Thus, to make the pressure of the mixture 𝑝 appear in the expression of entropy instead of partial
𝑅
pressures, it is necessary to add the term − ∑𝑖 𝑦𝑖 𝑀𝑀𝑢 𝑙𝑛𝑦̃𝑖 . It takes the name of entropy of mixing and
𝑖
it accounts for the irreversibility of the mixing gases of different chemical composition. This term can
also be rewritten remembering that:
𝑦𝑖 =
− ∑ 𝑦𝑖
𝑖
8.3
𝑚𝑖
𝑛𝑖 𝑀𝑀𝑖
𝑀𝑀𝑖
=
= 𝑦̃𝑖
̅̅̅̅̅
̅̅̅̅̅
𝑚 𝑇𝑂𝑇 𝑛𝑇𝑂𝑇 𝑀𝑀
𝑀𝑀
𝑅𝑢
𝑀𝑀𝑖 𝑅𝑢
𝑅𝑢
𝑙𝑛𝑦̃𝑖 = − ∑ 𝑦̃𝑖
𝑙𝑛𝑦̃𝑖 = −
∑ 𝑦̃𝑖 𝑙𝑛𝑦̃𝑖
̅̅̅̅̅ 𝑀𝑀𝑖
̅̅̅̅̅
𝑀𝑀𝑖
𝑀𝑀
𝑀𝑀
𝑖
(8.31)
𝑖
MIXTURES IN COMBUSTION
From Gibbs’ theorem, before and after a reaction, it holds true for all the properties except the specific
volume that:
𝜋̃ 𝑚𝑖𝑥 (𝑇, 𝑝, 𝑥⃗) = ∑ 𝑥𝑖 𝜋̃𝑖0 (𝑇, 𝑝𝑖 )
(8.32)
𝑖
𝜋 𝑚𝑖𝑥 (𝑇, 𝑝, 𝑥⃗) = ∑ 𝑦𝑖 𝜋𝑖0 (𝑇, 𝑝𝑖 )
(8.33)
𝑖
Thus, the calculation of the properties is simplified, but attention must be paid to the standard
reference state. This is because when reactions occur the formation properties (enthalpy, entropy etc.)
do not cancel out in energy and entropy balances as the molecules change during the reaction.
Formation properties are a function of the reference state, and it is thus necessary to define it properly.
As an example, the enthalpy and entropy for a combustion mixture are presented as follows:
̃ 𝒎𝒊𝒙 (𝑻, 𝒚
⃗⃗) = ∑ 𝑦̃𝑖 ℎ̃𝑖0 (𝑇) = ∑ 𝑦̃𝑖 [ℎ̃𝑖 (𝑇𝑟𝑒𝑓 ) + ∫
𝒉
𝒔̃𝒎𝒊𝒙 (𝑻, 𝒑, 𝒚) = ∑ 𝑦̃𝑖 𝑠̃𝑖0 (𝑇, 𝑝𝑖 ) = ∑ 𝑦̃𝑖 [𝑠̃𝑖0 (𝑇𝑟𝑒𝑓 , 𝑝𝑟𝑒𝑓 ) + ∫
𝑇
𝑇𝑟𝑒𝑓
𝑇 𝑐̃ 0
𝑝,𝑖
𝑇𝑟𝑒𝑓
92
0
𝑐̃𝑝,𝑖
𝑑𝑇]
𝑇
𝑑𝑇 − 𝑅𝑢 𝑙𝑛
(8.34)
𝑝𝑖
]
𝑝𝑟𝑒𝑓
(8.35)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
where the differences in the formation properties determine the energy balance of the combustion.
93
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
9
9.1
APPENDICES
PRESSURE OF RADIATION: A PRACTICAL EXAMPLE
Let's illustrate with an example how general the starting assumptions from which the Maxwell
relations were obtained were. More specifically, during the proof of the first Maxwell equation
𝜕𝑠
𝜕𝑝
(𝜕𝑣) = (𝜕𝑇) , it was unnecessary to specify the nature of the substance to which the same equation
𝑇
𝑣
can be applied.
In particular, a finite region of space is examined, enclosed by material walls, as illustrated in the
figure:
The first Maxwell relation is applied to the vacuum space forming the cavity, which it can be imagined
filled with an extremely rarefied fluid.
Moving the piston outwards, the volume increases by a quantity 𝑑𝑉. Let's make the transformation
occur isothermally. Since radiant energy is exchanged in the cavity (its walls continuously exchange
thermal energy), heat from the outside must be supplied to keep the temperature constant. This
necessarily leads to an increase in entropy of the system in question, that is, the Ist member of the
Maxwell relation is positive. Since the second member has to be positive as well, it ensures that the
pressure that prevails in the cavity increases with the temperature. In general, a pressure not zero on
the wall is found, and the pressure the greater the higher the temperature of the system is. In this way,
the existence of a radiation pressure rising with the temperature is envisaged in purely thermal terms.
9.2
THERMODYNAMIC SQUARE
The thermodynamic square is useful for getting the thermodynamic identities, the meaning of the
partial derivatives of the variables of state and the Maxwell relations.
94
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
V
A
U
S
T
G
H
p
Thermodynamic identities referring to one of the U, H, A, G quantities.
• The quantities next to 𝑋1 and 𝑋2 are the variables for which we are differentiating
• The quantities in the opposite corner 𝑌1 and 𝑌2 are the variables by which they are multiplied
• The direction of the arrow gives the sign of the product: toward the variable 𝑋 positive sign,
otherwise negative
V
A
T
U
G
dS
dH
dp
V
A
dT
U
S
𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑝
dG
H
𝑑𝐺 = 𝑉𝑑𝑝 − 𝑆𝑑𝑇
dp
Partial derivatives referring to one of the U, H, A, G quantities.
• One of the quantities next to 𝑋1 is the variable for which we are differentiating
• The quantity next to 𝑋1 , on the other hand, is the variable that is kept constant
• The quantities in the corner 𝑌1 at the opposite corner is the result
• The direction of the arrow gives the sign of the product: toward the variable 𝑌 positive sign,
otherwise negative
V
A
U
𝜕𝑈
( ) ≡𝑇
𝜕𝑆 𝑉
G
S
H
V
A
U
S
T
p
T
𝜕𝑈
( ) ≡ −𝑝
𝜕𝑉 𝑆
G
H
p
The Maxwell relations:
• Start from a corner and move in a clockwise or counterclockwise direction using only the
variables in the corners.
𝜕𝑋
• The first differential is constituted in the order in which the first three variables are met. (𝜕𝑋1)
2
95
𝑋3
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
•
•
𝜕𝑌
Continuing up to the next corner, the direction changes toward (𝜕𝑌1 )
2
If the arrows point at the same direction with respect to 𝑋1 and 𝑌1 , the sign is positive,
otherwise negative
V
A
U
S
H
V
A
S
T
𝜕𝑆
𝜕𝑝
( ) =( )
𝜕𝑉 𝑇
𝜕𝑇 𝑉
G
U
9.3
𝑌3
p
T
G
H
(
𝜕𝑆
𝜕𝑉
) = −( )
𝜕𝑝 𝑉
𝜕𝑇 𝑝
p
APPLICATION OF THE KINETIC THEORY OF GASES
With the hypothesis of ideal gas molecules can be represented as dot-like bodies moving in space with
a certain velocity, whose trajectory is not disturbed by the presence of attractive or repulsive forces
between the molecules. The collisions between the molecules or between the molecules and the walls
of the system are elastic and do not change the kinetic energy of the molecules.
Let's consider a system at constant volume consisting of a cube of side 𝐿 and volume 𝑉, containing a
number 𝑁 of particles with mass 𝑚 in random motion without a preferential direction.
Each particle is in motion with a velocity equal to:
𝑣𝑖 = 𝑣𝑥,𝑖 𝑢𝑥 + 𝑣𝑦,𝑖 𝑢𝑦 + 𝑣𝑧,𝑖 𝑢𝑧
(9.1)
With 𝑣𝑥,𝑖 , 𝑣𝑦,𝑖 and 𝑣𝑧,𝑖 , the projections of the velocity of the i-th particle along the three orthogonal
directrices 𝑥,𝑦 and 𝑧.
96
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
Considering the elastic collision of a particle with the wall 𝑦𝑧, the only component of the velocity that
is changed is 𝑣𝑥,𝑖 𝑢𝑥 , which becomes −𝑣𝑥,𝑖 𝑢𝑥 while the other components remain unchanged in the
absence of friction.
The change in momentum is −2𝑚𝑣𝑥,𝑖 𝑢𝑥 . In every elastic collision, the change in momentum of the
particle will be so much greater the greater the mass of the particle and its velocity are. The change of
the momentum of the particle discharges a pulse of equal entity on the wall of the system 𝑦𝑧.
Now let's consider only those particles 𝑁𝑖 which in volume 𝑉 have a velocity component in the
direction 𝑥 equal to 𝑣𝑥,𝑖 . Amongst these, the particles 𝑁𝑖,𝑘 that could hit the wall 𝑦𝑧 in time 𝑑𝑡 are
those within a distance equal to 𝑣𝑥,𝑖 𝑑𝑡 from the same wall:
𝑁𝑖,𝑘 = 𝑁𝑖
𝐿2 𝑣𝑥,𝑖 𝑑𝑡
𝑉
(9.2)
Since there is no a preferential direction of the particles, statistically half of those present in the volume
𝐿2 𝑣𝑥,𝑖 𝑑𝑡 will move in the opposite direction of the wall 𝑦𝑧 in question:
The overall pulse is therefore equal to:
𝐼𝑖 = 2𝑁𝑖,𝑘 𝑚𝑣𝑥,𝑖 = 2 (𝑁𝑖
𝐿2 𝑣𝑥,𝑖 𝑑𝑡
𝐿2 𝑣𝑥,𝑖 2 𝑑𝑡
) 𝑚𝑣𝑥,𝑖 = 𝑚𝑁𝑖
2𝑉
𝑉
(9.3)
The corresponding average force is equal to:
𝐹𝑖 = 𝐼𝑖 ⁄𝑑𝑡
(9.4)
from which the average pressure on the wall 𝑦𝑧 is obtained:
𝑝𝑖 =
𝐹𝑖
𝐼𝑖
𝑣𝑥,𝑖 2
=
=
𝑚𝑁
𝑖
𝐿2 𝐿2 𝑑𝑡
𝑉
(9.5)
Adding up all the molecules that have a velocity 𝑣𝑥,𝑖 such as to hit the wall, it holds true:
𝑝 = ∑ 𝑝𝑖 =
𝑖
𝑚
∑ 𝑁𝑖 𝑣𝑥,𝑖 2
𝑉
(9.6)
𝑖
where 𝑁 = ∑𝑖 𝑁𝑖 .
It is therefore possible to define the mean square velocity of the particles or the mean velocity that
gives the same effect of total pressure of the statistical variety of the velocities that the single particles
can take on:
̅̅̅𝑥 2 =
𝑣
1
∑ 𝑁𝑖 𝑣𝑥,𝑖 2
𝑁
𝑖
from which the pressure is:
97
(9.7)
Thermodynamic properties of fluids - Energy Conversion A – Version 6.1
𝑝 = ∑ 𝑝𝑖 =
𝑖
𝑁𝑚𝑣
̅̅̅𝑥 2
𝑉
(9.8)
Since there is however no preferential direction of motion:
̅̅̅𝑥 2 = ̅̅̅
𝑣
𝑣𝑦 2 = 𝑣̅𝑧 2
𝑣̅ 2 = ̅̅̅
𝑣𝑥 2 + ̅̅̅
𝑣𝑦 2 + 𝑣̅𝑧 2 = 3𝑣
̅̅̅𝑥 2
When defining the mean kinetic energy of the molecules as ̅̅̅
𝐸𝑘 =
be rewritten as:
̅̅̅𝑘
𝑁𝑚𝑣̅ 2 2𝑁𝐸
𝑝=
=
3𝑉
3𝑉
(9.9)
(9.10)
𝑚𝑣̅ 2
2
, the formula of the pressure can
(9.11)
That is, the pressure of the system is a function of the total weight of gas (𝑁 𝑚) contained in the
volume 𝑉 and the mean specific kinetic energy of the particles, or their mean translational square
velocity 𝑣̅ .
Now it is possible to link the pressure to the temperature in the case of an ideal gas with a final passage:
̅̅̅𝑘
2𝑁𝐸
𝑝𝑉 = 𝑛𝑅𝑢 𝑇 =
3
3𝑛
̅̅̅
𝐸𝑘 =
𝑅 𝑇
2𝑁 𝑢
(9.12)
(9.13)
Remembering that:
𝑁
• 𝑛 = 𝑁𝑎𝑣𝑜𝑔𝑎𝑑𝑟𝑜 = 6.23 ∙ 1023
•
𝑅
𝑁𝑎𝑣𝑜𝑔𝑎𝑑𝑟𝑜
J
= 𝑘 = 1.38 ∙ 1023 kg = 𝐵𝑜𝑙𝑡𝑧𝑚𝑎𝑛𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
it holds true:
3
̅̅̅
𝐸𝑘 = 𝑘𝑇
2
(9.14)
That is, the mean square velocity of the particles is known if their temperature is known.
From this relation it is also possible to obtain the specific heat at constant volume for a gas whose
particles have the only way to store energy in the increase in mean translational square velocity. The
total internal energy of the system will be given by the mean kinetic energy of the 𝑛 particles:
𝑐𝑣 =
̅̅̅𝑘
1 𝜕𝐸
3𝑘 3
(
) =
= 𝑅
𝑛 𝜕𝑇 𝑉 2 𝑛 2 𝑢
(9.15)
from which it is demonstrated that every degree of translational freedom involves a contribution equal
1
to 2 𝑅𝑢 on the 𝑐𝑣 .
98