Empirical Equations Of State
11. Empirical Equations Of State
These equations accurately predict the properties of specified fluid; however, they are not suit- able for predicting the stability characteristics of a fluid (Chapter 10).
a. Benedict–Webb–Rubin Equation The Benedict Webb Rubin (BWR) equation of state which was specifically devel- oped for gaseous hydrocarbons, has the form
P = RT/v + (B 2 RT–A 2 –C /T 2 )/v 2 2 + (B 3 RT–A 3 )/v 3 +A 3 C 6 /v 6
(81) The eight constants in this relation are tabulated in the literature. This equation is not recom-
+ (D 3 /(v 3 T 2 ))(1+E 2 ) exp(–E /v 2 2 /v 2 )
mended for polar fluids. Table A-20A lists the constants.
b. Beatie – Bridgemann (BB) Equation of State This equation is capable representing P-v-T data in the regions where VW and RK
equations of state fail particularly when ρ < 0.8 ρ c . It has the form
P v 2 = R T( v +B 0 (1- ( b / v )) (1- c/( v T 3 ))- (A / v 2 0 )(1-(a/ v )). Table A-20B contains several equations and constants.
c. Modified BWR Equation The modified BWR equation is useful for halocarbon refrigerants and has the form P = 9
2 2 15 ∑ (2n –17) n = 1 A n (T)/v + exp(–v c /v ) ∑ n = 10 A n (T)/v .
P = (T / v ′ ) (1+A/ v ′ +B/ v ′ 2 5 R 2 R R R R +C/ v ′ R +(D/ v ′ R )( β+γ/ v ′ R )exp(– γ/ v ′ 2 R )), (83a) Z=P
/T = 1+A/ v +B/ v 2 5 R 2 v ′ R R ′ R R ′ +C/ v ′ R +(D/ v ′ R )( β+γ/ v ′ R )exp(– γ/ v 2 ′ R ), (83b) where A = a –a /T –a /T 2 –a /T 3 1 3 2 R 3 R 4 R ,B=b 1 –b 2 /T R +b 3 /T R ,C=c 1 +c 2 /T R , and D = d 1 /T 3 R .
The constants are usually tabulated to determine Z (0) for all simple fluids and Z (ref) for a refer- ence fluid, that is usually octane (cf. Table A-21 ). Assuming that
Z (ref) –Z (0) = ωZ (1) , (83c) Z (1) can be determined.
A general procedure for specified values of P R and T R is as follows: solve for v R ´ from Eq. (83a) with constants for simple fluids and use in Eq. (83b) to obtain Z (0) . Then repeat the procedure for the same P and T
R with different constants for the reference fluid, obtain Z , and determine Z from Eq.(83c). The procedure is then repeated for different sets of P R and T . A plot of Z (0) R is contained in the Appendix and tabulated in Table A–23A . The value of Z (1) so determined is assumed to be the same as for any other fluid. Tables A-23A and A-23B tabulate Z (0) and Z (1) as function of P R and T R .
(ref) (1)
e. Martin–Hou The Martin–Hou equation is expressed as
P = RT/(v – b) + 5 ∑
j = 2 F j (T)/(v – b) j +F 6 (T)/e Bv ,
(84) where F i (T) = A i +B i T+C i exp (–KT R ), b, B and F j are constants (typically B 4 = 0, C 4 =0
and F 6 (T) = 0). This relation is accurate within 1 % for densities up to 1.5 ρ c and temperatures up to 1.5T c .
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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