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Bubble T Flash using PR EOS

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

Estimate temperature T and vapor mol fraction (Yi). T can be estimated as following –

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

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 T 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.

Liquid fugacity phi using Peng Robinson EOS

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 Fugacity using Peng Robinson EOS

Vapor phase mol fraction is calculated as

Yi = Xi.φiLiV

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 T is estimated such that ΣYi is close to 1. In excel it can be achieved by using GOAL SEEK function, in which T value is changed to make summation equal to 1.

Note

For some initial values of Temperature, Yi become equal to Xi and summation ΣYi becomes 1, it happens when initial guess for T falls in critical region. For such cases use different value of temperature, such that summation is not equal to 1 and then use Excel GOAL SEEK function to estimate Bubble Point Temperature and vapor mol fractions Yi.

Spreadsheet

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

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Bubble P Flash using PR EOS

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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.

Liquid fugacity phi using Peng Robinson EOS

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 Fugacity using Peng Robinson EOS

Vapor phase mol fraction is calculated as

Yi = Xi.φiLiV

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.

Spreadsheet

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

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Solving Cubic Equation of State

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Equation of State are used to predict pure component and mixture properties such as compressibility, fugacity and mixture equilibrium.

Soave – Redlich – Kwong (SRK) EOS

Equation is defined as

 P = RT / (V - b)  - a α / V(V + b)

where

 a = 0.42748 R²Tc²/Pc
 b = 0.08664 RTc/Pc
 α = [1 + (0.48 + 1.574ω - 0.176ω² )(1 - Tr0.5)]²

Above equation is translated into polynomial form.

 Z³ - Z² + Z (A - B - B²) - AB = 0
 Z = PV/RT
 A = 0.42748 α Pr/ Tr²
 B = 0.08664 Pr/ Tr

where, Pr is Reduced Pressure (= P / Pc), Tr is Reduced Temperature (= T / Tc).

Newton-Raphson Method

Newton Raphson is an iterative procedure for finding roots of a function f(Z). Function f(Z) and its derivative f ‘(Z) is calculated. An initial guess is made for the root Z, successive vales for Z’ are estimated using below relation till there is negligible difference between successive Z values.

 f(Z)  = Z³ - Z² + Z (A - B - B²) - AB
 f'(Z) = 3Z² - 2Z + (A - B - B²)
 Z'    = Z - f(Z)/f'(Z)

Example
Calculate compressibility factor for Methane based on SRK EOS at 30 bar, 285 °K. Critical parameters are Tc : 190.6 °K, Pc : 46 Bar, ω : 0.008.

Based on above equations f(Z) & f'(Z) is calculated as following.

 f(Z)  = Z³ - Z² + 0.0596 Z - 0.0037
 f'(Z) = 3Z² - 2Z + 0.0596

It is solved iteratively using Newton-Raphson method.

Cubic Equation of State Solution

There is negligible error in successive values of Z after 6th iteration.

 Z = 0.9409

Peng – Robinson (PR) EOS

Equation is defined as

 P = RT / (V - b)  - a α / [V(V + b) + b(V - b)]

where

 a = 0.45724 R²Tc²/Pc
 b = 0.07780 RTc/Pc
 α = [1 + (0.37464 + 1.54226ω - 0.26992ω² )(1 - Tr0.5)]²

Above equation is translated into polynomial form.

 Z³ - (1 - B)Z² + Z (A - 2B - 3B²) - (AB - B² - B³) = 0
 Z = PV/RT
 A = 0.45724 α Pr/ Tr²
 B = 0.07780 Pr/ Tr

where, Pr is Reduced Pressure (= P / Pc), Tr is Reduced Temperature (= T / Tc). Above relation is then solved for values of Z using Newton-Raphson method.

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