## Flash Calculation (Raoult’s Law)

This article shows step by step procedure to do Bubble Point, Dew Point and Flash Calculation based on Raoult’s Law.

### Bubble Point Calculation

Bubble point of a system is the temperature at which liquid mixture begins to vaporize.

Obtain process parameters
Get liquid mixture molar composition ( Xi ) and Pressure (P) of the system. Obtain equilibrium ratios ( Ki ) for the components. Ki can be calculated from Antoine equation.

````Yi = Ki.Xi`
`Ki(T) = (e A - B / (T + C) )/ P````

where Yi is vapor phase molar composition in equilibrium with liquid and A,B,C are Antoine equation constants.

Calculate Bubble Point
At Bubble Point temperature summation of vapor phase molar fraction should be 1.

``Σ Ki(T).Xi = 1``

Above equation can be solved iteratively using Newton Raphson method. An initial temperature T is assumed. Function F(T) is calculated as following.

``F(T) = Σ Ki(T).Xi - 1``

Derivative of F(T) is calculated as following.

``F'(T) = Σ (B.Ki(T)/(T+C)² ).Xi``

New estimate of temperature is calculated as following.

``TNew = T - F(T)/F'(T)``

Function F(T) and F'(T) are calculated based on new temperature and this process is repeated till there is negligible difference in between T and TNew. Bubble point temperature thus obtained is then used to calculate vapor phase composition based on above equations.

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### Dew Point Calculation

Dew point is the temperature at which liquid begins to condense out of the vapor.

Obtain process parameters
Get vapor mixture molar composition ( Yi ) and Pressure (P) of the system. Obtain equilibrium ratios ( Ki ) for the components.

````Yi = Ki.Xi`
`Ki(T) = (e A - B / (T + C) )/ P````

Calculate Dew Point
At Dew Point temperature summation of liquid phase molar fraction should be 1.

``Σ Yi / Ki(T) = 1``

Above equation can be solved iteratively using Newton Raphson method. An initial temperature T is assumed. Function F(T) is calculated as following.

``F(T) = Σ Yi / Ki(T) - 1``

Derivative of F(T) is calculated as following.

``F'(T) = Σ -Yi.( B/(Ki.(T+C)² ))``

New estimate of temperature is calculated as following.

``TNew = T - F(T)/F'(T)``

Function F(T) and F'(T) are calculated based on new temperature and this process is repeated till there is negligible difference in between T and TNew. Dew point temperature thus obtained is then used to calculate liquid phase composition based on above equations.

### Flash Calculation

A mixture when flashed to conditions between bubble and dew point separates in vapor and liquid phases. Flash calculation is done to determine vapor fraction and composition of liquid, vapor formed when a mixture is flashed at a given pressure and temperature.

Obtain process parameters
Get molar composition ( Zi )of the mixture and flash conditions mainly pressure (P) and temperature (T) of the system. Obtain equilibrium ratios ( Ki ) for the components.

``Yi = Ki.Xi``

Solve Flash Equations
Based on material balance on the system

````1 = V + L`
`Zi = V.Yi + L.Xi````

where V & L are vapor and liquid fractions. Solving above equations for Xi gives

``Xi = Zi / ( V.( Ki - 1) + 1 )``

At Flash conditions

``0 = Σ Yi - Σ Xi``

Above equation can be solved iteratively using Newton Raphson method. An initial vapor fraction V is assumed. Function F(V) is calculated as following.

````F(V) = Σ Yi  - Σ Xi`
`     = Σ [Zi.(Ki - 1)/(V.(Ki - 1) + 1)]````

Derivative of F(V) is calculated as following.

``F'(V) = Σ -[Zi.(Ki - 1)² /( V.(Ki - 1) + 1)²]``

New estimate of vapor fraction is calculated as following.

``VNew = V - F(V)/F'(V)``

Function F(V) and F'(V) are calculated based on new vapor fraction and this process is repeated till there is negligible difference in between V and VNew. Vapor fraction thus obtained is then used to estimate vapor and liquid molar composition based on above equations.

Spreadsheet for Flash Calculation based on Raoult’s Law

## Bubble T Flash using PR EOS

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-T/Tc)]````

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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.

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### 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 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.