## 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

## PT Flash Calculation using PR EOS

PT Flash calculation determines split of feed mixture F with a molar composition Zi, into Vapor V and Liquid L at pressure P and temperature T. These calculations can be done in a excel spreadsheet using Peng Robinson Equation of State (PR EOS). To start with bubble point pressure (PBubble) and dew point pressure (PDew) are determined for feed mixture.

• P < PDew, Mixture exists as super-heated vapor.
• P > PBubble, Mixture exists as sub-cooled liquid.
• PDew < P < PBubble, mixture exist in vapor and liquid phase.

Initial guess of vapor fraction V and Ki is made as following.

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

Based on initial Ki values, iteration is done to get value of V which satisfies material balance on system.

Yi = Ki.Xi
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 )

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At Flash conditions

Σ Yi - Σ Xi = 0

Above equation can be solved by iteration using Newton Raphson method. Function F(V) is defined as:

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

Derivative of F(V) is calculated as:

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

New estimate of vapor fraction is calculated as:

V New = 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 (Yi & Xi).

### Iteration for Ki

Vapor (Yi) and Liquid (Xi) mol fractions estimated above are used to generate values for Ki. 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

#### φiL Calculation

Mixture parameters are calculated.

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

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

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

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#### φiV Calculation

Mixture parameters are calculated.

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

Cubic equation is solved 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.

Ki is calculated as:

Ki = φiLiV

New values of Ki thus calculated are again used to estimate V and thereafter Xi & Yi. Iteration is repeated till there is no further change in Ki values. Typically, in 10 iterations change in Ki values become negligible.