Patel MODELING There are three basic categories of VLE

Patel and Teja (1982) proposed a third constant c
into the van der Waals equation’s attraction term, resulting in Eq. (2.7),
and to find a cubic EOS suitable for representing low and high pressure
behavior and vapor liquid equilibria. When c=b,
equation reduced to the Peng-Robinson equation and when c=0, it reduced to Redlich-Kwong equation. The equation of state
allows for adjustment of the critical compressibility factor Zc instead
of predicting a fixed value. This added flexibility improves saturation
property predictions for polar fluids while maintaining a simple form that
doesn’t require a large amount of pure component or mixture experimental data
for accuracyCHAPTER 3

MODELING

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There are three basic categories of VLE
calculations which are extensively used in design:

Estimation of equilibrium
temperature, T and vapor or liquid phase compositions, y or x, at constant
pressure (bubble point or dew point temperature calculations).
The prediction of the system
pressure, P, and vapor or liquid compositions, y or x under isothermal
conditions (bubble and dew point pressure calculations).
The evaluation of the phase
compositions, x, y or K-values at a given temperature and pressure (flash
calculations).

 

Figure 3.1 Basic categories of VLE calculations

In this course project work I used bubble
P calculation to find the Vapor liquid equilibria using ? – ? approach. Steps
to perform Bubble P calculation using three parameter Tsai & Jan EOS to
find vapor phase non-ideality and Wilson equation for liquid phase
non-ideality.

 

3.1    
Bubble
P calculation

 

During the calculation of the bubble P,
data required are:

       
Liquid
phase composition at a particular temperature,

       
Temperature,

       
Critical
Properties (TC, PC, ZC) of all component that
are present in the system,

       
Values
of acentric factors of all component that are present in the system,

       
Antoine
equation constants,

       
Value
of Wilson parameters for the given system.

 

 

 

 

Following are the steps for performing
bubble P calculation

1           
List
the data required.

2           
Find
Vapor pressure of the compounds at the given Temperature using Antoine Equation.
Antoine equation is given as:

                                                                    
                 ……… (3.1)

Where, A, B and C are Antoine constants And Psat
is saturated vapor pressure in mmHg
and T is temperature in Kelvin.

3           
Find
activity coefficient by using Wilson equation. Wilson Equation can be given as,

                                  ……… (3.2)

                                  ……… (3.3)

Where, ? is activity coefficient, A12 and A21 are Wilson equation constants.

4           
Use
modified Roult’s to find Bubble P, Modified Roult’s low can be give as,

                                                                                   ………
(3.4)

Where, P is total pressure, y is vapor composition, x is liquid composition, ? is fugacity coefficient and ? is activity coefficient and Psat is saturated vapor
pressure.

5           
First
assume ? = 1, that reduces equation (3.4) as follow,

                                                                                 
……… (3.5)

6           
Find
valve of fugacity coefficients by use of Tsai and Jan EOS.  The maximum value of fugacity coefficient is
related to vapor phase non-ideality. In this calculation value of P is required that is not available at
start so we have to make an estimate by performing steps 3, 4 and 5.

7           
Convert
equation (3.4) in terms of y to find
vapor composition,

                                                 
      ……… (3.6)

8           
Use
y values found by step 7 in equation (3.4) with ? value that is
found by step 6 to find new value of P.

9           
Recalculate
steps 6, 7 and 8 till following
condition is satisfied,

                                                                  
……… (3.7)

10        
To
obtain P-x-y data perfume all steps
for different values of x.

 

 

3.2    
Bubble
T calculation

During the calculation of the bubble T,
data required are:

       
Liquid
phase composition at a particular temperature,

       
Pressure,

       
Critical
Properties (TC, PC, ZC) of all component that
are present in the system,

       
Values
of acentric factors of all component that are present in the system,

       
Antoine
equation constants,

       
Value
of Wilson parameters for the given system.

 

Following are the steps for performing
bubble T calculation

1           
List
the data required.

2           
Estimate
value of temperature for the compounds at the given pressure using Antoine
Equation. From equation (3.1) find value of T1sat
and T2sat and
estimated T becomes,

                                                                                   
……… (3.8)

3           
Find
P1sat and P2sat using Test in equation (3.1),

4           
Find
activity coefficient by using Wilson equation. Eq. (3.2) and (3.3),

5           
Find
valve of fugacity coefficients by use of Tsai and Jan EOS.  The maximum value of fugacity coefficient is
related to vapor phase non-ideality. In this calculation value of T is required that is not available at
start so we have to make an estimate by performing steps 2.

6           
Find
Value of y1 and y2 by using Eq. (3.6).

7           
Use
y values found by step 6 in equation
(3.4) with ? value that is found by step 5
to find new value of P.

8           
Recalculate
steps 6, 7 and 8 till following
condition is satisfied,

                                                                  
……… (3.9)

9           
When
above condition satisfied value of temperature obtain is Bubble T. To obtain T-x-y data perfume all steps for different values of x.

 

 

 

 

 

 

 

3.3    
Dew
T calculation

During the calculation of the dew T, data
required are:

       
Vapor
phase composition at a particular temperature,

       
Pressure,

       
Critical
Properties (TC, PC, ZC) of all component that
are present in the system,

       
Values
of acentric factors of all component that are present in the system,

       
Antoine
equation constants,

       
Value
of Wilson parameters for the given system.

 

Following are the steps for performing Dew
T calculation

1           
List
the data required.

2           
Estimate
value of temperature for the compounds at the given pressure using Antoine
Equation. From equation (3.1) find value of T1sat
and T2sat and
estimated T becomes,

                                                                                   ……… (3.10)

3           
Find
P1sat and P2sat using Test in equation (3.1),

4           
Take
? = 1 and ? = 1 and use Roult’s law to find x1 and x2,

5           
Find
valve of fugacity coefficients by use of Tsai and Jan EOS.  (The maximum value of fugacity coefficient is
related to vapor phase non-ideality.)

6           
Find
the value of ? by using (3.2) and (3.3).

7           
Recalculate
value of P using equation (3.4).

8           
Recalculate
steps 2 to 7 till following condition is satisfied,

                                                                  ……… (3.11)

9           
When
above condition satisfied value of temperature obtain is Bubble T. To obtain T-x-y data perfume all steps for different values of x.

 

3.4    
Dew
P calculation

During the calculation of the dew P, data
required are:

       
Vapor
phase composition at a particular pressure,

       
Temperature,

       
Critical
Properties (TC, PC, ZC) of all component that
are present in the system,

       
Values
of acentric factors of all component that are present in the system,

       
Antoine
equation constants,

       
Value
of Wilson parameters for the given system.

 

Following are the steps for performing Dew
P calculation

1           
List
the data required.

2           
Find
P1sat and P2sat using T in equation (3.1),

3           
Take
? = 1 and ? = 1 and use Roult’s law to find x1 and x2,

4           
Find
valve of fugacity coefficients by use of Tsai and Jan EOS.  (The maximum value of fugacity coefficient is
related to vapor phase non-ideality).

5      
Find
the value of ? by using (3.2) and (3.3).

6      
Recalculate
value of P using equation (3.4).

7      
Recalculate
steps 2 to 7 till following condition is satisfied,

                                                                 
……… (3.12)

8      
When
above condition satisfied value of temperature obtain is Bubble T. To obtain T-x-y data perfume all steps for different values of x.

 

3.5    
Flash
calculation

Flash calculation is used when any of the
two values (From P, T, x,
y) are given. In flash calculation K-
value are used for calculation. It does serve as a measure of the “Lightness”
of a constituent species, i.e., its tendency to favor the vapor phase. When its
value is greater that one means species i
exhibits a higher concentration in vapor phase and if less than higher
concentration in liquid phase. K-values can be found from the nomographs.

                                                                                                                     
……… (3.13)

                                                                                                                
……… (3.14)

                                                                                                             
      ……… (3.15)

 

Above equations are use to for VLE
calculation. Equation (3.13) used when T
& P or x & y values are
given. Equation (3.14) use for bubble point calculation and (3.15) use for dew
point calculation.

 

Nomographs and Algorithms for all the calculation
are provided in the Appendix.

 

 

CHAPTER 4

DESIGN IN EXCEL

In this chapter, the procedure for
performing calculations for bubble P for generating P-x-y data in Excel is
shown. The work in this course project has been restricted to only bubble P
calculation. Following are the steps to perform Bubble P calculation.

 

Step 1.      
Generation
of Proper Data Base System.

Proper Data Base will
make work easy when we need to call values for particular systems. In this work,
I include three different types of the Data Base systems for my calculation.

1.    
For
critical properties of individual compounds,

2.    
For
binary systems and their Wilson parameters,

3.    
For
comparison with experimental data.

First Data Base contains
critical properties, Antoine constants, acentric factor etc. for the
calculation purpose (Fig. 4.1). There are two columns containing no, first for calling the compounds and
second for calling the values.

 

Figure 4.1 Data
Base for critical properties of components

 

 

Second Database (Fig 4.2)contain
Wilson parameter constants and slope and intersects values to generate parity
plot. First column is for giving unique identity for particular system. And
second and third columns are unique identities of first and second compound in
systems respectively. Last two columns are slope and intercepts to generate
parity plot. Ex no is unique identity for the experimental data and use to call
experimental data from Third Data Base. Third Data Base contains P-x-y data of
experiments.

 

Figure 4.2 Data
Base for Wilson parameter constants

 

Step 2.      
Listing
properties and parameters,

List all the Parameters
and properties that are remain constant during the calculation. For this
calculation following are the parameters and properties that are constant
during the calculation of particular system.

 

Table 4.1 Constant
Parameters and Properties.

 

To call the values of different properties
and parameters vlookup function is
use. Syntax for vlookup can be given
as follow,

= vlookup (lookup_value, table_array, column_index_no,
range_lookup)

 

 

 

Step 3.      
Design
Tools,

In this work, the button
and combo box from the developer tab were used. To use that one first need to
make visible the developer tab which is hide default in any MS word. To make it
visible follow step below for MS Office 2016 or Office 2013

1.    
Go
to File ? Options ? Customize Ribbon,
check the box before the developer.

Figure 4.3 Excel Option for developer tab

2.    
From
developer select insert ? combo box,

Figure 4.4 Combo box

3.    
After
selecting combo box draw box of appropriate size, and then follow window will
open,

Figure 4.5 Format control for combo box

In input range
select the cells whose value one want to show in box, in cell link select any blank cell, it will show the number of a
system that is selected from the list i.e. if u select fifth system from the
list (combo box) then it will show 5
in the cell and that is helpful to call the values using vlookup.

 

To create button, go to developer ? insert ? button,

 

Figure 4.6 Button tool

 

After selecting button draw a button of an appropriate
size and after that on window will open to ask for macro but we don’t make it
one so just close the window.

Step 4.      
Use
the equations mention in chapter 3 for Tsai and Jan to calculate the ? values and Wilson equation to find ?. To calculate follow the step given
bellow,

                             
1.         
List
the values of x at uniform intervals,

                             
2.         
Find
the values of the parameters required in the calculation of Tsai & Jan EOS
(Calculate till A, B, C).

                             
3.         
From
A, B, C generate the equation for Z using equation (2.31). and compare it
with following equation,

                                                                   
………. (4.1)

By doing so,

   ……… (4.2)

To solve this 3rd
degree polynomial equation following equations are use,

f = ((3*c/a) – (b^2/a^2))/3

g = ((2*b^3/a^3) – (9*b*c/a^2)
+ (27*d/a))/27

h = (g^2/4) + (f^3/27)

i = SQRT ((g^2/4)-h)

j = I ^ (1/3)

k = ACOS(-(g/(2*i)))

L = j*(-1)

M = COS(k/3)

N = SQRT (3) * SIN(k/3)

P = (b/(3*a)) * (-1)

R = -(g/2) + SQRT(h)

S = R ^ (1/3)

T = – (g/2) – SQRT(h)

U = T ^ (1/3)

For h > 0, one real
root and two imaginary

Z1 = (S+U) –
(b/3a),

Z2 = (-
(S+U)/2-(b/3a)) + i (S-U) *SQRT
(3)/2,

Z3 = (-
(S+U)/2-(b/3a)) – i (S-U) *SQRT (3)/2,

For h ? 0, three unique
and real roots

Z1 = 2*j*COS(k/3)
– (b/(3*a)),

Z2 = L*(M+N) +P,

Z3 = L*(M-N) +P;

For f, g, h = 0, only one
real root

Z = (d/a) ^ (1/3) * (-1);

For the all possible
values of Z obtain highest real root
is consider to calculate ? for both
the component from the system and same method is used to find ?b for equation
(2.26). 

                             
4.         
Use
equation (3.2) and (3.3) to find ?.

                             
5.         
Calculate
P1sat and P2sat from Antoine
equation

Step 5.      
Bubble
P calculation,

To calculate bubble P in
excel follow the bellow steps,

1.    
Make
a column for estimate P, which will
be found by equation (3.5),

2.    
Calculate
yi using equation (3.6),

3.    
Make
a column for calculated P, which is
calculated by equation (3.4),

4.    
Copy
calculated value on estimate as paste as
value only,

5.    
Make
column for ?P and ?y. ?P
is difference between estimate and calculated P and ?y is summation of
y values.

6.    
Use
solver to make ?y = 1 and ?P = 0, 

 

 

 

Step 6.      
Make
use of solver,

Solve is not installed by
default in Excel. So, to install and use solver use steps given bellow for
office 2013 or above,

1.    
Go
to File ? options ? add-ins ? solver add-in, click on GO.

2.    
To
use solver, go to data ? solver,
following window will open,

Figure 4.7 Solver window

 

3.    
Make
set cell as ?P, to value 0 and by
changing cell as copied P cell which
is paste as value only. Add constrain that ?y
cell becomes 1. Run the solver.

Step 7.      
Macro
for solver,

Repetitive use of solver
is very tedious and time consuming work. So, to avoid that we have to create macro.
Before that one has to make sure that its macro is enable. For that go to developer ? macro security ? enable all
macro. After that follow steps given bellow,

1.    
Go
to developer and click on record macro,

2.    
Perform
on single solver process,

3.    
Go
to developer and click on stop recording,

4.    
Go
to developer and click on Visual Basic,

5.    
Copy
the code and paste it in the sheet where calculation is performed,

6.    
Sheet
can be found from the list of all active sheet shown in top-left penal,

7.    
Go
to tools ? reference and tick mark solver,
press OK,

8.    
Edit
the code as follow,

9.    
After
creating macro assign it to button mention in step 3.

 

Sub Macr()

‘ Macr Macro

 

Range(“BK51:BK71”).Select

Selection.Copy

Range(“BD51″).Select

Selection.PasteSpecial
Paste:=xlPasteValues, Operation:=xlNone, SkipBlanks _

:=False,
Transpose:=False

 

solverReset

SolverOk
SetCell:=”$BN$51″, MaxMinVal:=3, ValueOf:=0,
ByChange:=”$BD$51″, _

Engine:=1,
EngineDesc:=”GRG Nonlinear”

SolverAdd
CellRef:=”$BO$51″, Relation:=2, FormulaText:=”1″

SolverSolve True    

 

In above coding first
paragraph is add for copy the calculated value and paste as value only so that
solver can be run. In second paragraph first and last line are added fist line
is for reset solver after every titration and last line is to avoid clicking OK every
time solver run. Coding between that is generated by recording macro. Copy
second paragraph and paste it for the number of time one would run it for
different rows or cells. Change the value of cell references with appropriate
values (i.e. if one want to use for next row then $BN$51 becomes $BN$52 and so
on…). Save coding and save excel file as macro enable file.     

Step 8.      
Graph
plot.

After generating P-x-y data draw a scatter graphs of

1.    
P vs x,

2.    
P vs y,

3.    
y vs x.

Step 9.      
Comparison
plots,

To create comparison plot
we have to generate P-x-y data for the
same value of x that is given in experimental data and compare the calculated y and P values with experimental one. After that, draw various comparison
graphs. (All graphs are mentions in appendix.) 

CHAPTER 5

RESULTS AND DISCUSSION

In this excel file
around 25 random systems are included and the result obtained are compared with
the actual experimentally 4 generated data. For Ethanol – Benzene
system at 298.15 K temperature following data are generated and comparison with
actual data generate graphs shown in figure 5.1.

Table 5.1 P-x-y
data for Ethanol – Benzene system

x

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

y

0.00

0.41

0.53

0.59

0.63

0.65

0.67

0.69

0.70

0.74

1.00

P (mmHg)

59.1

90.9

107.5

116.4

121.1

123.6

124.7

125.0

124.4

121.3

95.2

 

Figure 5.1
Comparison plots for Ethanol – Benzene system

 

Just like this
more systems can be calculated in Excel. For around 25 azeotropic and non
–  azeotropic system calculation has been
done and results are compiled in table 5.2 and table 5.3 with error in pressure,
y values, parity plot error etc.

 

Figure 5.2 P-x-y diagram and x-y diagram for ethanol –
benzene system

 

 

Table 5.2 Result
for non azeotropic systems

 

Table 5.3 Result
for azeotropic systems

In above tables values of
?Y and ?P average error are found by using equation given bellow,

 

The number in the final
column are show how good the fitting of the curve is, and 1. Excellent, 2.
Good, 3. Moderate, 4. Poor, 5. Very Poor. Rating is done on the bases of how
many points are deviating from the experimental values and how much is the
deviation. i.e., for benzene – toluene systems all the points are perfectly
match except one so, there is possibility that the point is due to experimental
error, thus it is considered as perfect match.         

 

 

CONCLUSION

 

From the above result I
can conclude that,

v
some
of the non azeotropic systems with few of the azeotropic systems shows the very
good match with experimental data, but most of the azeotropic systems are
showing deviation from the experimental values.

v More over system with combination of
aliphatic and aromatic compounds are showing more deviation compare to same
systems.

 

More details about this
EOS can be obtain by including more compounds.