• Multicomponent Rectification: A New Method of Calculation


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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
MULTICOMPONENT RECTIFICATION: A NEW METHOD OF
CALCULATION
J. Reyes, A. Gómez & A. Marcilla
Dpto. Ingeniería Química, Universidad de Alicante, Apdo. de Correos nº 99,
Alicante 03080, España.
This poster presents a series of calculation procedures for computer design
of ternary distillation columns overcoming the iterative equilibrium calculations
necessary in these kind of problems and, thus, reducing the calculation time. The
proposed procedures include interpolation and intersection methods to solve the
equilibrium equations and the mass and energy balances. The calculation
programs proposed also include the possibility of rigorous solution of mass and
energy balances and equilibrium relations.
In 1995 Marcilla et al. proposed an extension of The Ponchon-Savarit method
to solve the problem of the separation of a binary mixture in a complex column as
those of the Fig. 1. The equations are the following:
( j) ( j) ( j)
∑ Pk ⋅ z p,k + D ⋅ x D − ∑ A k ⋅ z A ,k
( j)
δk = k k (1)
∑ Pk + D − ∑ A k
k k
∑ Pk ⋅ H p, k + D ⋅ H D + Q D − ∑ A k ⋅ H A , k − ∑ Q E , k + ∑ Q A , k
Mk = k k k k
∑ Pk + D − ∑ A k
k k
(2)
L k ,i +1
=
(y
(1) (1)
k ,i − x k ) (
=
y ( c,) − x ( c)
ki k
=
)
H k,i − M k ( )
( ) ( )
(3)
Vk , i (1)
x k ,i +1 − x k(1) ( c)
x k ,i +1 − x k( c) (
h k ,i +1 − M k )
Equations (1) and (2) show the composition (δk) and enthalpy (Mk), respectively, of
the net flow (∆k) in the section k of the column of Figure 1 in the Ponchon Savarit
diagram and Eqn. (3) represents a operative line in the enthalpy-composition
diagram.
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
Figure 1. Generalized distillation column.
Figure 2 shows a sketch of the location of the net flow points (∆i) and the
graphical solution of a simple column for a ternary system, as in the classical
Ponchon-Savarit method for binary mixtures. Obviously, systems with more than
three components (and ternary systems with great difficulties) cannot be graphically
represented and the problem must be solved by the use of computational methods.
The extension of this procedure to multicomponent mixtures proposed
by Marcilla et al. (1995) can be easily made if explicit functions for the saturated
vapor enthalpy, saturated liquid enthalpy and saturated liquid composition of the type
H = H(yi, T, P), h = h(xi, T, P) and xi = x(yi, T, P) are available. In the other case,
approximate methods are necessary. In this work, we have generated these enthalpy
functions from fittings of the equilibrium data in all the composition range to
polynomial functions. These functions have been combined with different suggested
interpolation methods in order to obtain the equilibrium compositions at each stage.
To find the analytical solution to the problem, two poblems must be solved:
a) Establish a procedure to calculate, the vapor-liquid equilibrium and to
obtain the liquid phase in equilibrium with a given vapor phase (or
viceversa) in each tray,
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
b)To set a procedure to obtain the intersection point between the
operative line and the enthalpy-composition surface (in the
enthalpy/composition n-dimensional space).
Figure 2. Graphical representation of the Ponchon-Savarit
method for a ternary mixture.
PROCEDURES TO CALCULATE THE VAPOR-LIQUID EQUILIBRIUM
The methods proposed in this work fall into two categories:
a) the rigorous method, where the equilibrium data have been obtained using the
NRTL model and the procedure proposed by Renon et al. (1971) (this method
obviously involves an iterative calculation method wherever an equilibrium
composition is required).
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
b) the approximate methods suggested, where the equilibrium data in all the
composition range are generated at once by the previous rigorous procedure forming
two ordered lattices of equilibrium points. The data obtained in such a way, are used
in combination with any of the interpolation methods suggested, allowing (without
any iterative calculation) the determination of the composition in equilibrium with
any given phase in equilibrium.
The interpolation procedures proposed in this work are based in the
following steps:
1) In the network corresponding to the phase containing the point whose
equilibrium composition is to be calculated (Po) to find the three nearest
points (Pi, i ≠ 0) and to calculate their distances, di, from point Po in the
composition space.
2) To find the points P’i in equibrium with Pi in the network
corresponding to the conjugated phase.
3) To stablish an appropiate relation between Po and Pi.
4) Translate the previous relation to the other phase for pass from Pi’ to
Po’.
Figures 3 and 4 shows one of this procedures proposed with the existing
relation between the corresponding points.
Figure 3. Intercepting lines for interpolation
with four points. Localization of point of interpolation.
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
The interpolated point, P’o, is obtained from the intersection between the straight
lines P5’P6’ and P7’P8’ (Fig. 4).
d5 ' = d5 ⋅
[ P1 ' P2 '] d6' = d6 ⋅
[ P3 ' P4 ']
[ P1P2 ] [ P3P4 ]
(4)
[ P ' P ']
d7 ' = d7 ⋅ 1 3
[ P ' P ']
d 6 ' = d5 ⋅ 2 4
[ P1P3 ] [ P2 P4 ]
Figure 4. Intercepting lines for interpolation
with four points. Localization of interpolated point.
An Interpolation from linear fitting of compositions is also proposed, based
on expressing the composition of point Po as a linear combination of the
compositions of the three nearest points, P1, P2, P3:
P0 = a ⋅ P1 + b ⋅ P2 + c ⋅ P3 (5)
When these coordinates are defined:
(
Pk = x (1) , x ( 2) ,..., x ( c)
k k k ) (6)
the solution of the system of equations shown in Eqn. (7) allows the calculation of the
coefficients a, b and c:
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
X0 = X ⋅ A
( )
T
X 0 = x (1) , x ( 2) ,..., x ( c)
0 0 0
T
A = ( a , b, c)
(7)
 x (1) x12)
(
... x1c) 
(
 1 
X =  x (1)
2 x ( 2)
2
( c) 
... x 2
 (1) ( 
x
 3 x 32)
(
... x 3c) 
a, b and c can be used to find the point P’0, from points P’1, P’2 and P’3, in
equilibrium with P1, P2 and P3.
PROCEDURES TO CALCULATE THE MASS AND ENERGY BALANCES
i) Approximate methods
The approximate procedure proposed in this work require a fitting of the equilibrium
points in each phase to the corresponding enthalpy-composition polynomial
functions, as shown in Eqn. (8):
a ⋅ ( z(1) ) 2 + b ⋅ ( z( 2) ) 2 + c ⋅ z(1) + d ⋅ z( 2) + e ⋅ H = 1 (8)
where z can be x or y and H can be h or H depending on the phase. This equation
represents parabolic surfaces, and their intersection with the operative lines (Eqn. 3)
allows the mass and energy balances to be solved using a simple method.
This method involves very short calculation time, yields very good results and
can be improved by updatting the fitting of the enthalpies using the nearest points to
the intersection point versus their compositions.
ii) Rigorous method
The problem has been solved by the procedure shown in Figure 5: calculation
begins at an equilibrium point P1, near to the intersection point; its enthalpy h1
allows the calculation of the composition of a point P2 (by Eqn. 3), with the same
enthalpy as P1, but not in the equilibrium surface. A rigourous calculation with the
composition of P2, allows P3 to be obtained, on the equilibrium surface with the
same composition but different enthalpy. This point is used for the next iteration.
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
This process continues until the difference between the two consecutive values of
enthalpy is less than a small quantity previously fixed.
Figure 6 shows the general flowsheet corresponding to the algorithm of
calculation used in this work and Eqn (9) shows the criterion has been used to test if
the condition for the change of sector (and of ∆k) has been reached: - the last tray in
a sector is that which verifies Eqn. (9), where the subindex A refers to the side stream
separating sectors k and k+1.
x (1,)i
k
x ( 3)
k ,i
≤1 (9)
(1
x A)i
,
x A ,)i
(3
Figure 5. Sketch of the iterative calculation
to determine the intersection point betwee n
an operative line and the enthalpy-composition
surface.
RESULTS
As a example, Figure 6 shows the comparison between the results obtained for
a column with the characteristics of the table 1 by the approximate methods, using a
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
network with 231 points and varying the interpolation procedure to solve the
equilibrium and compared to the results obtained by the rigorous method. Figure 7
compares the results obtained from the rigorous method presented in this work and
the results from Renon et al. (1971).
Table 1. Characteristics of distillation column
Feed Flow rate (mol/h) 100
Feed Composition (molar fraction) Benzene: 0.600
Cyclohexane:0.006
Toluene:0.394
Distillate flow rate (mol/h) 60
Reflux ratio (LD/D) 2
% Separation of key component 1 97.3072
% Separation of key component 3 97.39858
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
Figure 6. Flowsheet of proposed program for calculation of a rectification
column.
Figure 6. Comparison between approximate (∆) and rigorous methods ( ), for
column of table 1 using a lattice with 231 points (increment in the composition in
mole fraction of 0.05). Interpolation 1, 2, 3 and 4 refers to interpolation methods
showed in figures 3, 4 an 6 and to interpolation method based in linera fitting of
conpositions, respectively.
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7th Mediterranean Congress of Chemical Engineering (EXPOQUIMIA’96) P37, PAG.77 Separation Process
Figure 7. Comparison between the rigorous method proposed in this work ( )
and those from Renon et al. (1971) (∆), for column 1
CONCLUSIONS
• The results obtained by the methods for multicomponent distillation complex
column calculations proposed are very close to those obtained by the rigorous
methods, using highly reduced calculation times.
• Approximate methods yield good results, even for lattices with a relatively small
number of points. Results greatly depend on the type of interpolation used. A
simple interpolation method is the one yielding the worse results and the
interpolation with intercepting straight lines with four points shown in figures 3
and 4, is the one yielding the better results. In all cases studied, a remarkable
improvement of the results can be noted when increasing the number of points in
the lattice.
• Semirigorous methos, using a rigorous calculation of vapor-liquid equilibrium and
the intersection procedure proposed to solve the balances in the column lead to
very good results.
• Rigorous method using the rigorous calculation of the vapor-liquid equilibrium and
the procedure showed in figure 5 to solve the balances in the column allow the
same results as the rigorous methods found in the bibliography (Renon et al.,
1971).
References
Marcilla, A., Ruiz, F. and Gómez, A., ”Graphically find trays and minimum reflux for
complex binary distillation for real systems”, Latin American Applied Research,
25, 87 (1995).
Renon, H., Asselineau, L., Cohen, G. et Raimbault, C., "Calcul sur ordinateur des
équilibres liquide-vapeur et liquide-liquide", Publications de l'Institut Français
du Pétrole, Collection "Science et Technique du Pétrole", nº 17 (1971).
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