Kwong Equations

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Kwong equations are mainly used as they are simple and a temperature-dependent correction is used for EOS constant A. van der Waals Equation van der Waals proposed the first cubic EOS in 1873. The van der Waals EOS gives a simple, qualitatively accurate relation between pressure, temperature, and molar volume.
P=RT/(V-b)-a/V^2
where a is “attraction” parameter, b is “repulsion” parameter, and R is universal gas constant. Comparing this equation with the ideal gas law,P= zRT/v, we see that the van der Waals equation offers two important improvements. The prediction of liquid behavior is more accurate because volume approaches a limiting value 'b' at high pressures i.e, lim┬(p=∞)⁡〖V(p)=〗 b
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The cubic Z-factor equation can readily be solved with an analytical or a trial-and-error approach. One or three real roots may exist, where the smallest root (assuming that it is greater than b) is typically chosen for liquids and the largest root is chosen for vapors. The middle root is always discarded as a nonphysical value. For mixtures, the choice of lower or upper root is not known a priori and the correct root is chosen as the one with the lowest normalized Gibbs energy, g*. where, gy=∑_(i=1)^n▒〖(ln⁡(〗 fi(y))*yi) gx=∑_(i=1)^n▒〖(ln⁡(〗 fi(x))*xi) where yi and xi = mole fractions of vapor and liquid, respectively, and fi = multicomponent fugacity
But in RK EOS all components have a critical compressibility factor of Zc=1/3, where, in fact, Zc ranges from 0.29 for methane to 0.2 for heavy C7fractions. The Redlich-Kwong value of Zc=1/3 is reasonable for lighter hydrocarbons but is unsatisfactory for heavier components. Even the vapour pressure calculated from it were
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Here, alpha= (1+m(1-(Tr)0.5))2 and m= 0.480+ 1.574w-0.176w2
But there are some demerits of this equation too: BIP’s is recommended to be kij=0 for HC/HC pairs It grossly overestimates liquid volumes (and underestimates liquid densities) of petroleum mixtures
Even though it has some demerits it is still used as it offers an excellent predictive tool for systems requiring accurate predictions of VLE and vapor properties.
Peng and Robinson EOS
Peng robinson proposed a slightly different form of molecular attractive term so it eliminates the deficiency of liquid density prediction from SRK EOS.
For Peng and Robinson EOS: U=2, W=1 so PR EOS is P= RT/(V-b) - a/(V^2+2Vb-b^2 )
It can be written in terms of Z factor by putting (U=2, W=1) in equation 6
Z3 + Z2(B - 1)+ Z(A - 3B2 - 2B) -(AB -B3 -B2) =0 from equation 3, 4 and 5, we get

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