# Dew P Flash using PR EOS

Dew P flash calculation determine dew point pressure (P) and liquid mol fraction (Xi) for a mixture at given temperature (T) and vapor mol fraction (Yi). These calculations can be performed in excel spreadsheet using Peng Robinson Equation of State (PR EOS).

Estimate pressure P and liquid mol fraction (Xi). P can be estimated as following –

`P = 1/ Σ Yi/ Pi`

_{sat}`Pi`

_{sat}= exp[ ln(Pc) + ln(10)(7/3)(1 + ω )(1-Tc/T)]

where Pc, Tc and ω are critical constants for a component i. Liquid mol fraction is estimated as following

`Xi = Yi/Ki`

`Ki = exp[ ln(Pc/P) + ln(10)(7/3)(1 + ω )(1-Tc/T)]`

First iteration starts with estimated P and Xi. Parameters for Peng Robinson EOS are calculated for each component i.

`κi = 0.37464 + 1.54226ω - 0.26992ω²`

`αi = [ 1 + κi (1 - (T/Tc)`

^{0.5})]²`ai = 0.45724 (RTc)²α / Pc`

`bi = 0.07780 RTc / Pc`

Mixture parameters are calculated next

`aij = [(ai.aj)`

^{0.5}(1 - kij)] = aji`a = Σ`

_{i}Σ_{j}aij.Yi.Yj`b = Σ`

_{i}bi.Yi`A = aP/(RT)²`

`B = bP/RT`

where, kij’s are Binary Interaction Parameter available from literature.

Following cubic equation is solved to get Z^{V}.

`Z³ + (B-1)Z² + (A-3B² -2B)Z + (B³+B²-AB) = 0`

Above equation can be written as

`Z³ + C2.Z² + C1.Z + C0 = 0`

### Solving Cubic Equation

Cubic equation is solved using following procedure. Calculate Q1, P1 & D.

`Q1 = C2.C1/6 - C0/2 - C2³/27`

`P1 = C2²/9 - C1/3`

`D = Q1² - P1³`

If D >= 0, then equation has only one real root provided by

`Z1 = (Q1 + D`^{0.5})^{1/3} + (Q1 - D^{0.5})^{1/3} - C2/3

If D < 0, then equation has 3 real roots, following parameters are calculated

`t1 = Q1² / P1³`

`t2 = (1 - t1)`

^{0.5}/ t1^{0.5}. Q1/abs(Q1)`θ = atan(t2)`

Roots are calculated as following –

`Z0 = 2.P1`

^{0.5}.cos(θ/3) - C2/3`Z1 = 2.P1`

^{0.5}.cos((θ + 2*Π)/3) - C2/3`Z2 = 2.P1`

^{0.5}.cos((θ + 4*Π)/3) - C2/3

Roots thus calculated are arranged in descending order, highest root gives Z^{V} and lowest root gives Z^{L}.

### Fugacity

Based on Z^{V}, vapor fugacity φ_{i}^{V} is calculated for each component.

As a next step, Liquid phase fugacity is calculated. Mixture properties are estimated as following –

`a = Σ`

_{i}Σ_{j}aij.Xi.Xj`b = Σ`

_{i}bi.Xi`A = aP/(RT)²`

`B = bP/RT`

Cubic equation is solved as per method shown above to get Z^{L}.

`Z³ + (B-1)Z² + (A-3B² -2B)Z + (B³+B²-AB) = 0`

Based on Z^{L}, liquid fugacity φ_{i}^{L} is calculated for each component.

Liquid phase mol fraction is calculated as

`Xi = Yi.φ`_{i}^{V}/φ_{i}^{L}

New values of Xi thus calculated are again used to estimate φ_{i}^{L} and thereafter Xi. This iteration is repeated till there is no further change in Xi values. Typically, in 25 iterations change in Xi values become negligible.

At the end of iteration ΣXi is calculated, if it is close to 1, results are obtained. If not, new value of P is estimated such that ΣXi is close to 1. In excel it can be achieved by using GOAL SEEK function, in which P value is changed to make summation equal to 1.

### Note

For some initial values of Pressure, Xi become equal to Yi and summation ΣXi becomes 1, it happens when initial guess for P falls in critical region. For such cases use lower value of pressure such that summation is not equal to 1 and then use Excel GOAL SEEK function to estimate Dew Point Pressure and liquid mol fractions Xi.

### Spreadsheet

All above calculations along with iterative procedure for flash calculation have been modeled in below spreadsheet.