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 .