Computer Program (subroutine) for Molecular Weight and Density

! The outputs (average density and molecular weight) of the subroutine ! are in FPS unit

! because they will be used in the tray hydraulic Equation (9.18).

! Input: X1 (mole fraction)

! Outputs: MWA (lbm/lbmol), DENSA (lb/ft3) SUBROUTINE MWDENS(X1,MWA,DENSA)

IMPLICIT NONE

REAL*8,INTENT(IN)::X1

REAL*8,INTENT(OUT)::MWA,DENSA REAL*8::X2,MW1,MW2,DENS1,DENS2

!——Initialization——!...‘1’ used for ethanol and ‘2’ for Water MW1=46.0634 D0

MW2=18.0152 D0

DENS1=0.789*62.42587...!1 g/cm3 = 62.42587 lb/ft3 DENS2=1.00*62.42587

X2=1-X1

MWA=(X1*MW1)+(X2*MW2)

DENSA=(X1*DENS1)+(X2*DENS2) END SUBROUTINE MWDENS

9.4.6 Vapour–Liquid Equilibrium (VLE)

An essential ingredient in the design of distillation column is a knowledge of the vapour–liquid equilibria of the system to be separated. Because the cost and time involved in obtaining experimental equilibrium data increase rapidly with the number of components, the distillation practitioner has turned to thermodynamics in search of effective predictive methods. Many such methods are discussed in Chapter 8.

The ideal equilibrium stage, also known as the theoretical stage or theoretical plate or ideal stage, is one which has the exit phases/streams in thermodynamic equilibrium, each phase/stream being removed from the stage without entraining any of the other phase/stream. If the outgoing vapour and liquid streams are in thermal equilibrium (same temperature) but not in phase equilibrium, the tray efficiency is used to correlate the composition in the ideal phase with that in the actual phase as stated for vapour-phase in Subsection 9.4.4. Phase equilibrium exists when the chemical potential (function of temperature, pressure and phase composition) of each component in the liquid phase is the same with that in the vapour phase.

Vapour–liquid equilibrium refers to the relationship between the liquid and vapour compositions in each equilibrium stage. In general, the vapour composition as well as the stage temperature are determined by solving the bubble point correlations with known liquid composition and pressure. The bubble point temperature is the temperature at which a bubble of vapour is formed in a liquid solution.

The composition of the bubble is different from the liquid and is richer in the more volatile component.

Usually, the bubble point temperature is considered as the stage temperature of a distillation column.

Vapour composition and temperature in an equilibrium stage

Ideal mixtures: If the vapour and liquid both are in ideal state, the calculations of the vapour composition and stage temperature are very simple and straightforward. Now for the ideal case, we will study the useful equations that are involved in VLE predictions for a binary system.

Dalton’s law (for ideal vapour-phase):

...(9.22)

where Pt is the total pressure. For component i, Pi is the partial pressure and is the equilibrium vapour composition.

Raoult’s law (for ideal liquid-phase):

...(9.23)

where is the vapour pressure of pure component i. Liquids that obey Raoult’s law are called ideal.

Combining Equations (9.22) and (9.23), we have ...(9.24)

Next, we will know several possible ways by which the VLE can be predicted at ideal situation.

Mainly based on the available information, we can choose a suitable approach. In the subsequent discussions, subscript ‘1’ will be used for one constituent element and ‘2’ for another constituent element of a binary mixture.

A. Calculation of vapour composition

Initially, we will consider two cases (Case 1 and Case 2) in which no calculation of temperature is involved; only vapour composition is computed. The equilibrium vapour composition of the constituent elements 1 and 2 ( and ) can be calculated by the following ways as per the availability of the coefficient.

Case 1

Problem statement:

Given: liquid composition (x1), equilibrium coefficient (k1) Unknowns: vapour compositions ( and )

Solution technique: can be computed from the following correlation:

...(9.25) For a binary mixture,

...(9.26)

Case 2

Problem statement:

Given: liquid composition (x1), relative volatility (12) Unknowns: vapour compositions ( and )

Solution technique: can directly be obtained using:

...(9.27)

Now, can be calculated from Equation (9.26). Note that this approach (Case 2) is used in the dynamic simulation of a compartmental distillation model (Chapter 6) and of an ideal binary distillation model (Chapter 7).

B. Calculation of vapour composition and temperature Problem statement:

Given: liquid composition (x1), pressure (Pt)

Unknowns: vapour compositions ( and ), temperature (T)

Solution technique: In order to compute the unknowns, any iterative convergence technique (details in Chapter 2) is required to use. Here, the Newton–Raphson algorithm is outlined to determine the vapour compositions and temperature for the present problem.

The following computational steps are involved.

Step 1: Guess temperature T (say, at time step t).

Step 2: Calculate vapour pressures . The vapour pressure solely depends on the stage temperature according to the following form of Antoine equation:

...(9.28)

where Ai, Bi, and Ci are the Antoine constants for pure component i. The constants are reported for various components in Table 9.4.

Table 9.4 Antoine constants (vapour pressure in mm Hg and temperature in K) Component (formula) Temp. range

Ethyl amine (C2H7N) McGraw-Hill Book Company, New York) McGraw-Hill Companies.

Step 3: Compute equilibrium vapour compositions ( and ) using a form of Equation (9.24) as:

...(9.29)

Step 4: Check whether the absolute value of  tolerance limit (the range of 10–5 to 10–10 is adequate to achieve reasonable accuracy)? If yes, note down the temperature and vapour compositions and go to Step 7, otherwise go to next step.

Step 5: Let

...(9.30)

that means,

...(9.31)

and then calculate the derivative of Ft with respect to temperature based on:

...(9.32)

Step 6: Compute T for the next time step t + 1 using:

...(9.33) Next go to Step 2 to continue the iteration.

Step 7: Stop.

Nonideal mixtures: For a nonideal mixture, Equation (9.24) is modified to:

...(9.34)

This is sometimes called modified or extended Raoult’s law. Here, i is the liquid-phase activity coefficient of component i. This coefficient is incorporated within the k-value formulation to account for the nonideality, which is generally severe in the liquid phase even at low pressures (close to 1 atm). In general, it is assumed that the vapour phase is remained at ideal state in low pressure operations.

Different useful models for activity coefficient predictions are given in details in Chapter 8.

We can notice one point that is obvious in Equation (9.34). Two main factors, which make the vapour and liquid compositions different at equilibrium, are: the pure component vapour pressure and the nonidealities in the liquid phase.

Problem statement:

Given: liquid composition (x1), pressure (Pt)

Unknowns: vapour compositions ( and ), temperature (T)

Solution technique: The solution approach is very similar to the approach as presented previously for ideal mixtures; only the calculation of activity coefficient is included in the present technique. The following computational steps can be followed to solve this problem.

Step 1: Guess temperature T (say, at time step t).

Step 2: Calculate activity coefficients for both the components (1 and 2) using an activity coefficient model. Different coefficient models are: Margules, Van Laar, Wilson, NRTL, UNIQUAC, UNIFAC, Hildebrand equations, etc.

Step 3: Compute vapour pressures employing Antoine Equation (9.28).

Step 4: Calculate equilibrium vapour compositions ( and ) from:

...(9.35)

This equation, which is obtained with the rearrangement of Equation (9.34), is the same with Equation (8.6); only difference is in the notations used.

Step 5: Check whether the absolute value of  tolerance limit? If yes, note down the temperature and vapour compositions and go to Step 8, otherwise go to next step.

Step 6: Suppose

that means,

...(9.36)

and then determine the temperature derivative as

...[from Equation (9.32)]

Step 7: Compute T for the next time step t + 1 using Equation (9.33) and then go to Step 2 to repeat computations for the next time step.

Step 8: Stop.

For the prescribed batch distillation column, the above algorithm has been used to compute the equilibrium vapour compositions of the binary components and tray temperature. The liquid-phase activity coefficients have been predicted by the use of the Wilson thermodynamic model. In the example of batch rectifier, the Wilson parameters ( ij) are assumed to be temperature independent and constant.

The Fortran (90) code for the computations of vapour compositions and tray temperature in the batch distillation simulator is given in Program 9.5.

PROGRAM 9.5 Computer Program (subroutine) for Equilibrium Vapour Compositions and Tray