# 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 = φiL/φiV``

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.