Procedure for Determining Thermodynamic Properties
1. Procedure for Determining Thermodynamic Properties
Thermodynamic properties can be determined, once the state equation, critical con- stants, and corresponding ideal gas properties are known. Some useful formulas are listed be- low and some thermodynamic data is listed in Table 2 . The reference conditions should be specified. For example for water, the reference condition is generally specified as that of the saturated liquid at the triple point. The choice of reference conditions is arbitrary. Here,
v R ´ = v/v c ´, v c ´ = RT c /P c
v R = v/v c , Z=P R v R ´/T u
= – u Res c,R R /(RT c ) = (u o (T) –
u(T,P))/RT c ,
h c,R = (h o (T) – h(T,P))/RT c ,
s c,R = (s o (T,P) – s(T,P))/R,
c P,c R = (c P (T,P) – c P,o (T))/R,
c v,c,R = (c v (T,P) – c v,o (T))/R, µ JT,R = µ JT c p /v c ´,
g c,R = (g o (T,P) – g(T,P))/RT c ,
a c,R = (a o (T,P) – a(T,P))/RT c φ = f/P = exp((g(T,P)– g o (T,P))/RT)
f T g with g =g and T inv with µ JT =0
sat
The constants are used in the formula c po Figure 23: Schematic illustration of a method = of determining the thermodynamic properties
A o +B o T+C o T –2 +D o T 2 +E T 3 o . In the
of a material using a P–h diagram.
h(873,60
h fg )
Figure 24: Schematic illustration of the determination of enthalpy of a vapor or a real gas with respect to the values at the reference condi- tion.
range 0 K < T < 1000 K, the maximum error is less than 8%. oo. Example 41
Determine the thermodynamic properties of water at 250 bar and 600ºC using the RK equation of state. Assume that c = 28.85 + 0.01206 T + 1.002
×10 K –1 ,h fg,ref = 2501.3 kJ kg –1 ,P c = 220.9 bar, and T c = 647.3 K. The reference conditions are those for the saturated liquid at its triple point, i.e., 273.15 K and 0.006113 bar.
5 /T 2 kJ kmole –1
p,o
The molecular weight of water is 18.02 kg kmole –1 . Solution
A schematic diagram of the procedure followed is illustrated in on a P–h diagram in Figure 25 and Figure 24 . First, the reference condition is selected at which h f =s f = 0 (e.g., the saturated liquid state at the triple point of water at point A in the figure). At the reference condition h = 0 + h
(point B in Figure 25 and Figure 24 ). Therefore, s g =h fg /T = 2501.3
g fg = 2501 kJ kg
÷273 = 9.17 kJ kg –1 K ,
The entropy of the saturated vapor at 273.15 K and 0.0061 bar is 9.17 kJ kg –1 K –1 above the entropy of saturated liquid at same temperature and pressure.
For the vapor at 273 K and 0.0061 bar (i.e., at P R,ref = 0.000028 and T R,ref = 0.422) the reduced correction factor (h o – h)/RT c ≈ 0, since the pressure is low and the intermo- lecular attraction forces are weak. Therefore,
h = h = 2501 kJ kg o –1 at the triple point (point B in Figure 25 and Figure 24 ). Similarly, s = s = 9.17 kJ kg –1 K –1 o or 165.2 kJ kmole –1 K –1 . The values of h o (873 K) and s o (873 K, 250 bar) can be obtained using the specific heat relations for an ideal gas, i.e.,
Figure 25: Schematic illustration of an entropy calculation starting from a temperature of absolute zero; C: critical point.
h o –1 = 3706 kJ kg = 66782 kJ kmole –1 .
This corresponds to the point D. Similarly,
s –1 o (873 K, 250 bar) = 6.47 kJ kg K or 116.6 kJ kmole K .
The ideal gas internal energy u o at 873 K can be determined using the relation u –1
o =h o – RT = 66782 – 8.314 × 873 = 59,524 kJ kmole . The correction or residual factors at 873 K and 250 bar can be obtained. From charts
we see that at P R = 1.13, T R = 1.35, Z = 0.845. Therefore, (h o – h)/RT c = 0.735, (u o – u)/RT c = 0.526, and (s o – s)/R = 0.389. Consequently,
u = 3146 kJ kg –1 , and s = 6.29 kJ kg K . Thereafter, the enthalpy can be determined using the relation h = u + Pv = u + ZRT.
× (8.314 ÷ 18.02) × 873 = 3486 kJ kg –1 ,
which is represented by point C. Other properties can be similarly obtained.
Parts
» COMPUTATIONAL MECHANICS and APPLIED ANALYSIS
» Explicit and Implicit Functions and Total Differentiation
» Exact (Perfect) and Inexact (Imperfect) Differentials
» Intermolecular Forces and Potential Energy
» Internal Energy, Temperature, Collision Number and Mean Free Path
» Vector or Cross Product r The area A due to a vector product
» First Law for a Closed System
» First Law For an Open System
» STATEMENTS OF THE SECOND LAW
» Cyclical Integral for a Reversible Heat Engine
» Irreversibility and Entropy of an Isolated System
» Degradation and Quality of Energy
» SINGLE–COMPONENT INCOMPRESSIBLE FLUIDS
» Evaluation of Entropy for a Control Volume
» Internally Reversible Work for an Open System
» MAXIMUM ENTROPY AND MINIMUM ENERGY
» Generalized Derivation for a Single Phase
» LaGrange Multiplier Method for Equilibrium
» Absolute and Relative Availability Under Interactions with Ambient
» Irreversibility or Lost Work
» Applications of the Availability Balance Equation
» Closed System (Non–Flow Systems)
» Heat Pumps and Refrigerators
» Work Producing and Consumption Devices
» Graphical Illustration of Lost, Isentropic, and Optimum Work
» Flow Processes or Heat Exchangers
» CLASSICAL RATIONALE FOR POSTULATORY APPROACH
» Generalized Legendre Transform
» Van der Waals (VW) Equation of State
» Other Two–Parameter Equations of State
» Compressibility Charts (Principle of Corresponding States)
» Boyle Temperature and Boyle Curves
» Three Parameter Equations of State
» Empirical Equations Of State
» State Equations for Liquids/Solids
» Internal Energy (du) Relation
» EXPERIMENTS TO MEASURE (u O – u)
» Vapor Pressure and the Clapeyron Equation
» Saturation Relations with Surface Tension Effects
» Temperature Change During Throttling
» Throttling in Closed Systems
» Procedure for Determining Thermodynamic Properties
» Euler and Gibbs–Duhem Equations
» Relationship Between Molal and Pure Properties
» Relations between Partial Molal and Pure Properties
» Mixing Rules for Equations of State
» Partial Molal Properties Using Mixture State Equations
» Ideal Solution and Raoult’s Law
» Completely Miscible Mixtures
» DEVIATIONS FROM RAOULT’S LAW
» Mathematical Criterion for Stability
» APPLICATION TO BOILING AND CONDENSATION
» Physical Processes and Stability
» Constant Temperature and Volume
» Equivalence Ratio, Stoichiometric Ratio
» Entropy, Gibbs Function, and Gibbs Function of Formation
» Entropy Generated During an Adiabatic Chemical Reaction
» MASS CONSERVATION AND MOLE BALANCE EQUATIONS
» Evaluation of Properties During an Irreversible Chemical Reaction
» Criteria in Terms of Chemical Force Potential
» Generalized Relation for the Chemical Potential
» Nonideal Mixtures and Solutions
» Gas, Liquid and Solid Mixtures
» Availability Balance Equation
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