1. Lecture 5: Isothermal Flash Calculations 1 In the last lecture we: Described energy and entropy balances in flowing systems Defined availability and lost work Derive a Gibbs Phase Rule for flowing systems Described system specification Showed how DePriester charts are used to tabulate K-value data for hydrocarbon systems In this lecture we’ll: Describe an isothermal flash separation Derive the Rachford-Rice equation Show how to use Newton’s method to find the roots of the RR equation Use the Rachford-Rice procedure, including Newton’s method and DePriester equilibrium data to solve a hydrocarbon isothermal flash problem.

2. Lecture 5: Isothermal Flash Calculations 2 Isothermal FlashConfiguration Liquid Feed Consider the following operation which produces a liquid-vapor equilibrium from a liquid feed: Vapor out Flash Drum F, z i, T F, P F, h F For each stream: n: molar flow rate: F, L, V z i : composition variables: x,y,z T: temperature P: pressure h: enthalpy Q: heat transfer L, x i, T L, P L, h L Liquid out V, y i, T V, P V, h v Q

3. Lecture 5: Isothermal Flash Calculations 3 Isothermal Flash Variables Liquid Feed We showed in the last lecture that there are C+5 degrees of freedom. If we specify F, z i, T F, P F we have specified (C+3) variables and we can specify two additional variables. Vapor out Flash Drum F, z i, T F, P F, h F L, x i, T L, P L, h L Liquid out V, y i, T V, P V, h v Q Common Specifications: T V,P V Isothermal Flash V/F=0, P L Bubble-Point Temperature V/F=1, P V Dew-Point Temperature V/F=0, T L Bubble-Point Pressure V/F=1, T V Dew-Point Pressure Q=0, P V Adiabatic Flash Q, P V Non adiabatic flash V/F, P V Percent Vaporization Flash For this system there are 3C+10 variables: F, V, L, T F, P F, T V, P V, T L, P L, Q, {x i, y i,z i } C

4. Lecture 5: Isothermal Flash Calculations 4 Isothermal Flash Equations variables F, T v, P v, T F, P F, Z i If we specify the Then remaining variables must be found from: Equations A) mole fraction summations B) K-Value relationships C) Mass balances D) Energy balance E) Thermal and mechanical equilibrium Total Note that if T and P of each product stream are not considered as variables, then we wouldn’t have equations for thermal and mechanical equilibrium in the drum. If less than C+5 variables are specified, then the system is undetermined (underspecified). If more than C+5 variables are specificed, then the system is overdetermined (overspecified).

5. Lecture 5: Isothermal Flash Calculations 5 Isothermal Flash Equations We have 2C+5 variables to determine from 2C+5 equations. The expression gives the total number of equations for this system of 2C+3 (the system creates a two- phase equilibrium). The two additional equationswe need come from our assumption of thermal and mechanical equilibrium in the drum. We have total material balance: We have component material balances, one for each component: We have the mole fraction summations for each phase (or stream): In equilibrium, we have a K-value relationship for each component:

6. Lecture 5: Isothermal Flash Calculations 6 Rachford Rice Derivation It is convenient to define the Vapor Fraction as follows: Substituting into our total material balance: For the component material balances: Using the K-Value and solving for the liquid phase mole fraction: We use the K-Value to get:

7. Lecture 5: Isothermal Flash Calculations 7 Rachford Rice Equations We use the mole fraction summations: Substituting in our expressions for the mole fractions: Gives us the Rachford-Rice Equation: The roots of this equation give us the compositions, and vapor fraction of the Isothermal Flash operation. To solve this equation, we need to use some procedure for finding the roots: Iterative Graphical

8. Lecture 5: Isothermal Flash Calculations 8 Newton’s Iterative Method To solve the Rachford-Rice equation we can use Newton’s method to find  : Newton’s method estimates a better root using the last guess and the ratio of the function to its derivative at that guess: For the Rachford-Rice Equation this becomes:

9. Lecture 5: Isothermal Flash Calculations 9 Rachford-Rice Procedure The Rachford-Rice procedure using Newton’s method is then: Step 1: Thermal equilibrium Step 2: Step 3:Solve Rachford-Rice for V/F where the K-values are determined by T L, and P L. Mechanical equilibrium Step 4: Steps 5 and 6: Step 7: Step 8: Can use Newton’s method here. Determine V Determine L Determine Q

10. Lecture 5: Isothermal Flash Calculations 10 Example: Rachford-Rice A flash chamber operating at 50ºC and 200kPa is separating 1000 kg moles/hr of a feed that is 30 mole %propane, 10 % n-butane, 15 % n-pentane, and 45 % n-hexane. What are the product compositions and flow rates? 1) Using the Depriester Chart we determine that: K 1 (propane) = 7.0 K 2 (n-butane) = 2.4 K 3 (n-pentane) = 0.80 K 4 (n-hexane) = 0.30 2) We first write the Rachford-Rice Equation and substitute in the composition and K-values:

11. Lecture 5: Isothermal Flash Calculations 11 Depriester Determination of K-Values

12. Lecture 5: Isothermal Flash Calculations 12 Example: Rachford-Rice We can either plot the Rachford-Rice Equation as a function of V/F or use Newton’s method: Guess V/F=0.1 To obtain a new guess we need the derivative of the RR equation:

13. Lecture 5: Isothermal Flash Calculations 13 Example: Rachford-Rice So our next guess is To obtain a new guess we need the derivative of the RR equation:

14. Lecture 5: Isothermal Flash Calculations 14 Example: Rachford-Rice So: Using: X 1 (propane) = 0.0739 X 2 (n-butane) = 0.0583 X 3 (n-pentane) = 0.1670 X 4 (n-hexane) = 0.6998 Y 1 (propane) = 0.5172 Y 2 (n-butane) = 0.1400 Y 3 (n-pentane) = 0.1336 Y 4 (n-hexane) = 0.2099

15. Lecture 5: Isothermal Flash Calculations 15 Summary In this lecture we discussed: Variables, Equations and degrees of freedom for an isothermal flash separation An isothermal flash configuration The derivation and solution of the Rachford Rice equation Newton’s iterative procedure to solve for the roots of the RR equation A numerical example to demonstrate this approach. Next Lecture will cover: Bubble point pressure and Dew Point temperature calculations