THERMODYNAMIC PROPERTIES OF FLUIDS CHAPTER 6 THERMODYNAMIC PROPERTIES OF FLUIDS

Chapter outline Develop property relations from 1st & 2nd Laws Derive equations to calculate DH and DS from Cp and PVT data Use ready PVT data from property tables or develop generalised correlations to estimate property values

Property relations for homogeneous phases First law for a closed system U, S and V are molar values As equation only contains properties of system (state function) so it can be applied to any process in a closed system (not necessarily reversible) and the change occur between equilibrium states (constant composition)

The primary thermodynamic properties: The enthalpy: Energy from environment during heating The Helmholtz energy: The Gibbs energy: Energy from environment and change in volume For one mol of homogeneous fluid of constant composition: These equations also apply for the entire mass of any closed system The criterion of exactness applies for these equations

General Equations for a Homogeneous Fluid at Constant Composition dU = T dS – P dV (Eq. 6.7) dH = T dS + V dP (Eq. 6.8) dA = - P dV - S dT (Eq. 6.9) dG = VdP - S dT (Eq. 6.10)

Maxwell Equation Maxwell equations gives relations which underline the mathematical structure of thermodynamics Serve as the mathematical bases for evaluation thermodynamics functions from experimental measurement of P, T, V behavior of a fluid The most useful relations are for U, H, S as a function of T, P (or specific volume)

Maxwell’s Equation

Derivation of Maxwell Equation Phase rule Where the state of homogeneous fluid is specified by two of the eight intensive thermodynamic variables, we can write the chain rule for a general function of the two independent variables, F(x,y)

Application of Chain Rule for Exact Differential Expression When F = F(x,y) Where F, x, y represents the (P, T, V, S, U. H, A and G)

The derivative of this type occur whenever one tries to relate a change in one thermodynamic functions to changes in two others (Phase Rule )

For continuous function , the order of partial differentiation can be interchanged and the following holds for continuous function with continuous first and second partial derivatives.

DH and DS as functions of T & P

Eq 6.20 Eq 6.21 Equations relate the change in S, U, H to changes in only PVT. The right sides of the equations contain only Cp, Cv, P, V, T and partial derivatives involving P, V and T

DU and DS as functions of T & V Out-class exercize: Show the derivation of equation Eqns 6.32 and 6.33

Lets derived equation Eqn. 6.20 DH as a function of T and P DH = f(T, P)

Out-class exercize: Show that Eqn 6.20 and Eq 6.21 reduce to the correct equations for ideal gases. ( See the Ideal-Gas State in page 203 Note : in other words derived the eqn 6.23 and Eqn. 6.24

Useful forms for liquids or solids: Eq 6.28 Eq 6.29

Problem Illustration 6.7 page 241 Estimate the change in enthalpy and entropy when liquid ammonia at 270 K is compressed from its saturation pressure of 381 KPa to 1200 KPa. For saturated liquid ammonia at 270 K, Vl=1.551x10-3 m3 kg-1, and β=2.095x10-3 K-1.

Example 5.1 (page 180) Source : Introductory Chemical Engineering Thermodynamics by Lira Pressure Dependence of H Derive the relation for and evaluate the derivative for water at 200C where = 2.07x10-4 cm3/g-K and = -4.9 x 10-5 cm3/g-bar , ρ = 0.998 g/cm3.

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Joke !!!!! You think of the Carnot cycle everytime you turn on your AC unit.

You try to explain entropy to strangers at your table during casual dinner conversation.

You cannot write unless the paper has both horizontal and vertical lines

You have a habit of destroying things in order to see how they work

Gibbs energy G = G (P,T) G/RT = g (P,T) Thermodynamic property of great potential utility as it can be directly measured and controlled Eq 6.10 Eq 6.37 Eq 6.38 Eq 6.39 G/RT = g (P,T) The Gibbs energy serves as a generating function for the other thermodynamic properties.

Gibbs energy as generating function

Residual properties The definition for the generic residual property: M and Mig are the actual and ideal-gas properties, respectively at the same T & P M is the molar value of any extensive thermodynamic properties, e.g., V, U, H, S, or G. The residual Gibbs energy serves as a generating function for the other residual properties:

const T Eqn 6.39 Eqn 6.46 Eqn 6.49 const T Eqn 6.3 Eqn 6.48 const T Z = PV/RT: experimental measurement .Given PVT data or an appropriate equation of state, we can evaluate HR and SR and hence all other residual properties.

These equations provide a convenient base for real gas calculations

Calculate the enthalpy and entropy of saturated isobutane vapor at 360 K from the following information: (1) compressibility-factor for isobutane vapor; (2) the vapor pressure of isobutane at 360 K is 15.41 bar; (3) at 300K and 1 bar, the ideal-gas heat capacity of isobutane vapor: Graphical integration requires plots of and (Z-1)/P vs. P.

Residual properties by equation of state: an alternative method The calculation of residual properties for gases and vapors through use of the virial equations and cubic equation of state. Z-1 = BP/RT GR/RT = BP/RT If Z = f (P,T): Eq 3-38 Eq 6-49 Eq 6-54 Eq 6-48 Eq 6-46

Equation of State Equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions. A mathematical expression yielding PVT relations. Equations of state are useful in describing the properties of fluids, mixtures of fluids and solids.

Different kinds of Equations of State Major equation of state Classical Ideal Law Cubic equations of state Van der Waals equation of state Redlich-Kwong equation of state Soave modification of Redlich-Kwong Pent-Robinson equation of state Elliot, Suresh, Donohue equation of state Non-cubic equations of state Dieterici equation of state Virial equation of state

Redlich-Kwong equation of state

Residual Properties from the Virial Equation of State two-term virial equation Eq 6.46 Eq 6.55 Eq 6.56

Three-term virial equation Application up to moderate pressure and required data for second and third virial coefficients

Using the cubic equation of state The generic cubic equation of state: Eq 3.42 in terms of density Eq 6.46 Eq 3.51 Eq 6.48 Textbook – Refer to page 218

Find values for the residual enthalpy HR and the residual entropy SR for n-butane gas at 500 K and 50 bar as given by Redlich/Kwong equation. Eq. 3.53 Eq. 3.54 Eq. 3.52 Use Table 3.1 for data Eq. 6.65 Eq.6.67 Eq. 6.68

Sample Problem 6.14 page 242. Calculate the Z, HR, and SR by the Redlich/Kwong equation for one of the following and compare results with values from suitable generalized correlations :

Two-phase systems Whenever a phase transition at constant temperature and pressure occurs, The molar or specific volume, internal energy, enthalpy, and entropy changes abruptly. The exception is the molar or specific Gibbs energy, which for a pure species does not change during a phase transition. For two phases α and β of a pure species coexisting at equilibrium: where Gα and Gβ are the molar or specific Gibbs energies of the individual phases

The latent heat of phase transition The Clapeyron equation Ideal gas, and Vl << Vv The Clausius/Clapeyron equation

The Antoine equation, for a specific temperature range The Clapeyron equation: a vital connection between the properties of different phases. Temperature dependence on vapour pressure of liquids For the entire temperature range from the triple point to the critical point The Antoine equation, for a specific temperature range The Wagner equation, over a wide temperature range.