Rachford Ride Equation

Lecture 5: Isothermal Flash Calculations 1

Lecture 5: Isothermal Flash Calculations

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 andDePriester equilibrium data to solve a hydrocarbon isothermal flash problem.


Isothermal FlashConfiguration

Liquid Feed

Consider the following operation which produces a liquid-vapor equilibrium from a liquid feed:

Vapor out

Flash Drum

F, zi, TF, PF, hF

For each stream:n: molar flow rate: F, L, Vzi: composition variables: x,y,zT: temperatureP: pressureh: enthalpyQ: heat transfer

L, xi, TL, PL, hL

Liquid out

V, yi, TV, PV, hv

Q


Isothermal Flash Variables

Liquid Feed

We showed in the last lecture that there are C+5 degrees of freedom.If we specify F, zi, TF, PF we have specified (C+3) variables andwe can specify two additional variables.

Vapor out

Flash Drum

F, zi, TF, PF, hF

L, xi, TL, PL, hL

Liquid out

V, yi, TV, PV, hv

Q

Common Specifications:TV,PV Isothermal FlashV/F=0, PL Bubble-Point TemperatureV/F=1, PV Dew-Point TemperatureV/F=0, TL Bubble-Point PressureV/F=1, TV Dew-Point PressureQ=0, PV Adiabatic FlashQ, PV Non adiabatic flashV/F, PV Percent Vaporization Flash

For this system there are 3C+10 variables: F, V, L, TF, PF, TV, PV, TL, PL, Q, {xi , yi ,zi}C


Isothermal Flash Equations

variables F, Tv, Pv, TF , PF , Zi F C 5If we specify the

Then remaining variables must be found from:

EquationsA) mole fraction summations B) K-Value relationships

C) Mass balances

D) Energy balance

E) Thermal and mechanical equilibrium

Total

1 C

C

1

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.

C 1 32C 5

2C 5

2 1

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).


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.

C 1

We have total material balance:

FZi VYi LXi

F V L

We have component material balances, one for each component:

We have the mole fraction summations for each phase (or stream):

Xii 1 Yi

i 1 Zi

i 1

In equilibrium, we have a K-value relationship for each component:

kiL,V

YiV

XiL

TL TV PL PV


Rachford Rice Derivation

It is convenient to define the Vapor Fraction as follows:

VF

Substituting into our total material balance:

FZi VYi LXi

L F F

For the component material balances:

Using the K-Value and solving for the liquid phase mole fraction:

We use the K-Value to get:

Zi V

FYi

F FF

Xi Zi Yi Xi Xi

Zi Yi Xi XiXi

ZiKi 1

Yi KiZi

Ki 1

Yi KiXi

Zi KiXi Xi Xi


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

Xi Zi

Ki 1 Yi KiZi

Ki 1

Xii 1 Yi

i 1 Yi Xi

i 0

KiZiKi 1

Zi

Ki 1

i 0 f ( )

Zi Ki 1 Ki 1 i

0


Newton’s Iterative Method

To solve the Rachford-Rice equation we can use Newton’s method to find :

k1 k f k f ' k

k1 k

Zi Ki 1 k Ki 1 1i

Zi Ki 1 2

k Ki 1 1 2i

Newton’s method estimates a betterroot using the last guess and the ratioof the function to its derivative at thatguess:

For the Rachford-Rice Equationthis becomes:


Rachford-Rice Procedure

The Rachford-Rice procedure using Newton’s method is then:

Step 1: TL TV Thermal equilibrium

Step 2:

Step 3: Solve Rachford-Rice for V/F where the K-values are determined byTL, and PL.

PL PV Mechanical equilibrium

Zi Ki 1 Ki 1 i

0

Step 4: V F

Steps 5 and 6:

Step 7:

Step 8:

Xi Zi

Ki 1 Yi

KiZiKi 1

L F F

QVhv Lhl Fh f

Can use Newton’s method here.

Determine V

Determine L

Determine Q


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:

K1 (propane) = 7.0K2 (n-butane) = 2.4K3 (n-pentane) = 0.80K4 (n-hexane) = 0.30

2) We first write the Rachford-Rice Equation and substitute in the composition and K-values:

fV

F

Zi Ki 1 Ki 1 1i

fV

F

0.3 7.0 1 7.0 1 1

0.1 2.4 1

2.4 1 10.15 0.8 1 0.8 1 1

0.45 0.3 1 0.3 1 1


Depriester Determination of K-Values



We can either plot the Rachford-Rice Equation as a function of V/F or use Newton’s method:

fV

F

Zi Ki 1 Ki 1 1i

fV

F

0.3 7.0 1 7.0 1 1

0.1 2.4 1

2.4 1 10.15 0.8 1 0.8 1 1

0.45 0.3 1 0.3 1 1

Guess V/F=0.1

f 0.1 0.3 7.0 1 0.1 7 1 1

0.1 2.4 1 0.1 2.4 1 1

0.15 0.8 1 0.1 0.8 1 1

0.45 0.3 1 0.1 0.3 1 1

0.8785

f 'V

F

0.3 7.0 1 2

7 1 1 20.1 2.4 1 2

2.4 1 1 20.15 0.8 1 2

0.8 1 1 20.45 0.3 1 2

0.3 1 1 2

To obtain a new guess we need the derivative of the RR equation:

f ' 0.1 0.3 7.0 1 2

0.1 7 1 1 2

0.1 2.4 1 2

0.1 2.4 1 1 20.15 0.8 1 2

0.1 0.8 1 1 20.45 0.3 1 2

0.1 0.3 1 1 24.631



So our next guess is

2 0.10.8794.631

0.29

f 0.29 0.3 7.0 1 0.29 7 1 1

0.1 2.4 1

0.29 2.4 1 10.15 0.8 1 0.29 0.8 1 1

0.45 0.3 1 0.29 0.3 1 1

0.329

To obtain a new guess we need the derivative of the RR equation:

f ' 0.29 0.3 7.0 1 2

0.29 7 1 1 2

0.1 2.4 1 2

0.29 2.4 1 1 2

0.15 0.8 1 2

0.29 0.8 1 1 2

0.45 0.3 1 2

0.29 0.3 1 1 21.891

3 0.29 0.3291.891

0.46

f 0.46 0.066 f ' 0.46 1.32

4 0.46 0.0661.32

0.51

f 0.51 0.00173



So: V /F 0.51

Xi Zi

Ki 1 Yi

KiZiKi 1 Using:

X1 (propane) = 0.0739X2 (n-butane) = 0.0583X3 (n-pentane) = 0.1670X4 (n-hexane) = 0.6998

Y1 (propane) = 0.5172Y2 (n-butane) = 0.1400Y3 (n-pentane) = 0.1336Y4 (n-hexane) = 0.2099

V 510kg / hr


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

Rachford Ride Equation

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