Bubble P Flash using PR EOS

# Bubble P Flash using PR EOS

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

Estimate pressure P and vapor mol fraction (Yi). P can be estimated as following –

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

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

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

First iteration starts with estimated P and Yi. 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 = ΣiΣj aij.Xi.Xj`
`b = Σi bi.Xi`
`A = aP/(RT)²`
`B = bP/RT````

where, kij’s are Binary Interaction Parameter available from literature.
Following cubic equation is solved to get ZL.

``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 + D0.5)1/3 + (Q1 - D0.5)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 / t10.5. Q1/abs(Q1)`
`θ = atan(t2)````

Roots are calculated as following –

````Z0 = 2.P10.5.cos(θ/3) - C2/3`
`Z1 = 2.P10.5.cos((θ + 2*Π)/3) - C2/3`
`Z2 = 2.P10.5.cos((θ + 4*Π)/3) - C2/3````

Roots thus calculated are arranged in descending order, highest root gives ZV and lowest root gives ZL.

### Fugacity

Based on ZL, liquid fugacity φiL is calculated for each component.

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

````a = ΣiΣj aij.Yi.Yj`
`b = Σi bi.Yi`
`A = aP/(RT)²`
`B = bP/RT````

Cubic equation is solved as per method shown above to get ZV.

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

Based on ZV, vapor fugacity φiV is calculated for each component.

Vapor phase mol fraction is calculated as

``Yi = Xi.φiL/φiV``

New values of Yi thus calculated are again used to estimate φiV and thereafter Yi. This iteration is repeated till there is no further change in Yi values. Typically, in 25 iterations change in Yi values become negligible.

At the end of iteration ΣYi is calculated, if it is close to 1, results are obtained. If not, new value of P is estimated such that ΣYi 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, Yi become equal to Xi and summation ΣYi 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 Bubble Point Pressure and vapor mol fractions Yi.

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

Spreadsheet for Bubble P Flash using PR EOS

## 3 Replies to “Bubble P Flash using PR EOS”

1. Uwe Raabe says:

The formula for estimating the initial pressure
Pisat = exp[ ln(Pc) + ln(10)(7/3)(1 + ω )(1-T/Tc)]
doesn’t match the formula used in the spreadsheet. Instead of T/Tc the spreadsheet uses the inverse Tc/T. The same happens with the calculation of Ki.