Thermodynamic Properties of Fluids

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Thermodynamic Properties of Fluids

• Dr.Sininart Chongkhong A Dao
• ChE. PSU.

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Purpose of this Chapter

• To develop from the first and second laws the
  fundamental property relations which underlie the
  mathematical structure of thermodynamics.
• Derive equations which allow calculation of
  enthalpy and entropy values from PVT and heat
  capacity data.
• Discuss diagrams and tables by which property
  values are presented for convenient use.
• Develop generalized correlations which provide
  estimated of property values in he absence of
  complete experimental information.

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Property Relations for Homogeneous
PhasesFundamental Properties

• Although this equation is derived from the
  special case of a reversible process, it not
  restricted in application to reversible process.
• It applies to any process in a system of constant
  mass that results in a differential change form
  one equilibrium state to another.
• The system many consist of a single phase or
  several phases may be chemically inert or may
  undergo chemical reaction.

(6.1)
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• Define
• H Enthalpy
• A Helmholtz energy
• G Gibbs energy

(2.11)
(6.2)
(6.3)
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• Based on one mole (or to a unit mass) of a
  homogeneous fluid of constant composition, they
  simplified to

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• Maxwells equaitons

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Enthalpy and Entropy as Functions of T and P

• Temperature derivatives

• Pressure derivatives

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• The most useful property relations for the
  enthalpy and entropy of a homogeneous phase
  result when these properties are expressed as
  functions of T and P (how H and S vary with T and
  P).

(6.20)
(6.21)
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Property Relations for Homogeneous
PhasesInternal Energy as Function of P

• The pressure dependence of the internal energy is
  shown as

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Property Relations for Homogeneous PhasesThe
Ideal Gas State

• For ideal gas, expressions of dH and dS
  (eq.6.20-6.21) as functions of T and P can be
  simplified to as follows

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Property Relations for Homogeneous
PhasesAlternative Forms for Liquids

• Relations of liquids can be expressed in terms of
  ? and ? as follows

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Property Relations for Homogeneous
PhasesAlternative Forms for Liquids

• Enthalpy and entropy as functions of T and P as
  follows
• ? and ? are weak functions of pressure for
  liquids, they are usually assumed constant at
  appropriate average values for integration.

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Example 6.1

• Determine the enthalpy and entropy changes of
  liquid water for a change of stage from 1 bar
  25?C to 1,000 bar 50?C.

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Note that the effect of ?P of almost 1,000 bar on
H and S of liquid water is less than that of ?T
of only 25?C.
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Property Relations for Homogeneous
PhasesInternal Energy and Entropy as Function
of T and V

• Useful property relations for T and V as
  independent variables are

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• The Partial derivatives dU and dS of homogeneous
  fluids of constant composition to temperature and
  volume are
• Alternative forms of the above equations are

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Property Relations for Homogeneous PhasesThe
Gibbs Energy as a Generating Function

• An alternative form of a fundamental property
  relation as defined in dimensionless terms
• The Gibbs energy when given as a function of T
  and P therefore serves as a generating function
  for the other thermodynamic properties, and
  implicitly represents complete information.

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Residual Properties

• The definition for the generic residual property
  is
• M is the molar value of any extensive
  thermodynamics property V, U, H, S, G.
• M, Mig the actual and ideal gas properties
  which are at the same temperature and pressure.

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• Residual gibbs energy
• G, Gig the actual and ideal gas values of the
  Gibbs energy at the same temperature and
  pressure.
• Residual volume

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Fundamental property relation for residual
properties

• The fundamental property relation for residual
  preperties applies to fluids of constant
  composition.

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Enthalpy and Entropy from Residual Properties
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The true worth of the Eq. for ideal gases is now
evident. They are important because they provide
a convenient base for the calculation of real-gas
properties.
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• Example 6.3
• Calculate H and S of saturated isobutane vapor at
  630 K from the following information
• Table 6.1 gives compressibility-factor data
• The vapor pressure of isobutane at 630 K 15.46
  bar
• Set H0ig 18,115 Jmol-1 and S0ig 295.976
  Jmol-1K-1 for the ideal-gas reference state at
  300 K 1 bar
• Cpig/R 1.776533.037x10-3T (T/K)

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• Solution 6.3
• Eqs. (6.46) and (6.48) are used to calculate HR
  and SR.
• Plot (?Z/?T)P/P and (Z-1)/P vs. P
• From the compressibility-factor data at 360 K ?
  (Z-1)/P
• The slope of a plot of Z vs. T ? (?Z/?T)P/P
• Data for the required plots are shown in Table
  6.2.

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Residual Properties by Equations of State
Residual Properties from the Virial Equation of
State

• The two-term virial eq. gives Z-1 BP/RT.

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• In application ? is a more convenient variable
  than V,
• PV ZRT is written in the alternative form.

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• The three-term virial equation.

Application of these equations, useful for gases
up to moderate pressure, requires data for both
the second and third virial coefficients.
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Residual Properties by Cubic Equations of State
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The generic equation of state presents two cases.
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Ex. 6.4

• Find values for the HR and SR for n-butane gas at
  500 K
• 50 bar as given by the Redlich/Kwong Eequation.
• Solution
• Tr 500/425.1 1.176, Pr 50/37.96 1.317
• From Table 3.1

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These results are compared with those of other
calculation in Table 6.3.
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TWO-PHASE SYSTEMS
The Clapeyron eq. for pure-species vaporization
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Temperature Dependence of the Vapor Pressure of
Liquids
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Corresponding-States Correlations for Vapor
Pressure
The reduced normal boiling point
The reduced vapor pressure corresponding to 1 atm
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• Ex. 6.6
• Determine the vapor pressure for liquid n-hexane
  at 0, 30,
• 60 and 90?C (a) With constants from App. B.2.
• (b) From the Lee/Kesler correlation for Prsat
• Solution
• (a)
• (b) Eq.(6.78)
• From Table B.1,
• From Eq.(6.81) ? ? 0.298
• The average difference from the Antoine values is
  about 1.5.

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Two-Phase Liquid/Vapor System
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THERMODYNAMIC DIAGRAMS
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GENERALIZED PROPERTY CORRELATION FOR GASES
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Table E.5 - E.12
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Analytical correlation of the residual properties
at low pressure
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HR and SR with ideal-gas heat capacities

• For a change from state 1 to 2

• The enthalpy change for the process, ?H H2 H1

• Alternative form

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A three-step calculational path

• Step 1?1ig A hypothetical process that
  transforms a real gas into an ideal gas at T1 and
  P1.
• Step 1ig ?2ig Changes in the ideal-gas state
  from (T1,P1) to (T2,P2).
• Step 2ig ?2 Another hypothetical process that
  transform the ideal gas back into a real gas at
  T2 and P2.

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• Ex. 6.9
• Estimate V, U, H and S for 1-butane vapor at
  200?C, 70 bar
• if H and S are set equal to zero for saturated
  liquid at 0?C.
• Assume Tc420.0 K, Pc40.43 bar, Tn266.9 K,
  ?0.191
• Cpig/R1.96731.630x10-3T-9.837x10-6
  T2 (T/K)
• Solution

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• Step (a) Vaporization of saturated liquid
  1-butane at 0?C
• The vapor pressure curve contains both
• The latent heat of vaporization, where
  Trn266.9/4200.636

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• Step (b) Transformation of saturated vapor into
  an ideal gas at (T1, P1).
• Tr 0.650 and Pr 1.2771/40.43 0.0316

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• Step (c) Changes in the ideal gas state
• Tam 373.15 K, Tlm 364.04 K,
• A 1.967, B 31.630x10-3, C -9.837x10-6
• ?Hig 20,564 J mol-1
• ?Sig 22.18 J mol-1 K-1

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• Step (d) Transformation from the ideal gas to
  real gas state at T2 and P2.
• Tr 1.127 Pr 1.731
• At the higher P Eqs.(6.85) and (6.86) with
  interpolated values from Table E.7, E.8, E.11 and
  E.12.

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Extension to Gas Mixtures
These replace Tr and Pr for reading entries from
the table of App. E, and lead to values of Z by
Eq.(3.57), and HR/RTpc by Eq.(6.85), and SR/R by
Eq.(6.86).
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• Ex. 6.10
• Estimate V, HR, and SR for an equimolar mixture
  of
• carbon dioxide(1) and propane(2) at 450 K and 140
  bar by
• the Lee/Kesler correlations.
• Solution
• From Table B.1,

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