# 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-Tc/T)]`

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 )`

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 V_{New}. 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`

#### φ_{i}^{L} Calculation

Mixture parameters are calculated.

`aij = [(ai.aj)`

^{0.5}(1 - kij)] = aji`a = Σ`

_{i}Σ_{j}aij.Xi.Xj`b = Σ`

_{i}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 Z^{L}.

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

Roots calculated are arranged in descending order, highest root gives Z^{V} and lowest root gives Z^{L}.

Based on Z^{L}, liquid fugacity φ_{i}^{L} is calculated for each component.

#### φ_{i}^{V} Calculation

Mixture parameters are calculated.

`a = Σ`

_{i}Σ_{j}aij.Yi.Yj`b = Σ`

_{i}bi.Yi`A = aP/(RT)²`

`B = bP/RT`

Cubic equation is solved to get Z^{V}.

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

Based on Z^{V}, vapor fugacity φ_{i}^{V} is calculated for each component.

Ki is calculated as:

`Ki = φ`_{i}^{L}/φ_{i}^{V}

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.

### Spreadsheet

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