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Multicomponent Distillation Column Design A Semi-rigorous Approach.

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MuIticomponent Dist iIIation Column Design:
A Semi-rigorous Approach
S.N. Maiti, S. Ganguly‘, A.K. Das’ and D.N. Saraf
Department of Chemical Engineering
Indian lnstitufe of Technolorn Kanpur 208 076, INDIA
A new semi-rigorous approach for the design and simulation of multicomponent
distillation columns is presented which combines a rigorous Naphtali-Sandholm
algorithm with Edmister’s shortcut method. The column is assumed to comprise of single
and multi-tray units with the number of trays inside the multi-tray unit treated as an
iterative variable. The assumption of linear variation of flows inside a multi-tray unit
made by earlier workers (which often lead to convergence d.iBculties or erroneous
results) has been replaced with a more rigorous calculation procedure. Examples are
presented to demonstrate the usefilness of the new method for both design and simulation
calculations. The semi-rigorous model equations can be solved directly to obtain a
reavonubly accurate and practical design, whereas most other available methodr require
repeated calculations. Computer memory requirement is drastically reduced in this semirigorousformulation as compared to rigorous methods.
Introduction
Multistage distillation methods are readily available in the journal literature and will
not be considered here. The two basic approaches are a rigorous design method and a
column rating procedure. Naphtali and Sandholm [ 11 used stagewise grouping of the
distillation equations with subsequent linearization to give the resulting set of
equations a bIock mdiagonal structure, permitting solution by a simple technique.
Ohmura and Kasahara [2] reported a semi-tray-by-trayapproach to solve simulation
problems. It combined the tray-by-tray procedure of Tomich [3] with the shortcut
method of Edmister 141, using an effective stripping factor for a group of trays which
were considered as a single unit. The method of Ohmura and Kasahara [2] had
several shortcomings. There are a large number of dependent variables which lead to
the difficulty of modulation inside the two loops, especially for large problems. A
number of simplifying approximations were made, which may not always be
justified. Linearity of flow profiles inside a multi-tray unit is often objectionable. In
this algorithm, there is no provision for inclusion of nonideal thermodynamics and
plate efficiencies. Most of the shortcomings of Ohmura and Kasahara‘s work have
subsequently been removed by Ganguly [51; Ganguly et al., [61 and Maiti [7l.
Based on the approach of Ohmura and Kasahara, a semi-rigorous algorithm has
been developed in this study which includes the my-by-my model of Naphtali and
Sandholm and the shortcut model of Edmister. A more rigorous calculation
I. Presently at Engineers India Ltd. (R & D), Gurgaon, India.
2. Presently at Indian Oil Corporation (R & D). Faridabad, India.
*Authorfor correspondence.
37
SN.Maiti, S. Ganguly, A.K. Das and DN. Sarrrf
p e d u r e replaces the assumed linearity offlow profiles inside the multi-tray units.
Provision has been made for use of nonideal thermodynamics and stage efficiencies.
The concept of incorporating the number of trays inside a multi-tray unit as an
independent variable was exploited to convert the original simulation method into an
effective design tool.
Development of the Model
The general procedure of solving multicomponent, multistage separation problems is
based on the solution of the MESH (Mass,Equilibrium, Summation and Enthalpy)
equations, and using appropriate convergence techniques. The use of a NewtonRaphson type convergence method requires selection and ordering of the unknown
variables and the corresponding functions (MESH equations). Figure 1 shows a
general equilibrium stage. The corresponding MESH equations in the NaphtaliSandholm formulation are given in Table 1 for a column with N equilibrium stages
and C components.
*
i
V.i+1, j
'i, j
Figure 1. Schematic representation of a general single tray unit.
Table 1. The MESH Equations
1. Component materid balmcc:
Mi,=
(l+S,L)lij+(l+S~)~;j-li-,j-ui+~j-f;j=~;
2. Fqiilibrium relationship:
38
i = I ,....N ; ~ =,...,
I c (I)
Midticomponent Distillation Colwnn Design: A Semi-rigorousApproach
Semi-rigorous Method
The column model incorporates two types of units - single tray and multi-tray. In
Figure 2, the single tray units are the simple equilibrium trays, whereas the multi-tray
units are a group of single trays. In the single tray units, the iterated variables are lb
YO and Ti.in the multi-tray unit, instead of the stage temperature, the number of
equilibrium stages inside the unit (Mi) is taken as an independent variable in addition
to li. and vij The MESH equations for single my units are the same as in the
Napf&ili-Sandholrnmethod. The M-, H- and S- equations for multi-tray units are
also the same as for the single trays but the E- equations are different.
Condenser>
El
H
product
Top
p
G
product
ream
Side stream
Feed
Single tray units
b
Re boikr Bottom
Bottom
product
product
(b) Semi rigorous model with
(a) Conventional traysingle and multitray units
by-tray model
Figure 2, Schematic represeluation of a complex multistage distillation column.
39
SN. Maiti, S . Ganguly. A.K. Das and DN. Sard
Effective Stripping Factor and Equilibrium Relation
for Multi-tray Units
In the multi-tray unit i, the vapor rising from the unit is not in simple equilibrium
with the descending liquid as in single tray units (see Figures 2 and 3). Ohmura and
Kasahara obtained a relationship between the vapor flowrate and the liquid flowrate
leaving the unit by using effective stripping factor of Edmister [4], namely:
where
mister's effective slripping factor (Se$ is defined by:
'i1,{1
+ S'M~) + 0.25 - 0.5
(9)
with
S'ilj = K'il,{l
+ SiV)Vi /L'ii
(10)
and
(11)
The K'i1. and K > M ~values are the equilibrium constants of component j at the first
)
inside the ith multi-tray
and the &t plate temperatures (pi1 and T ' ~ Mrespectively,
unit (see Figure 3). Primes refer to variables/parameters inside the multi-tray units,
and an additional subscript is used to specify the tray number within these units.
For the Naphtali-Sandholm formulation, the equilibrium relation for the ith
multi-tray unit was derived as follows:
40
MulticomponewDistillation Column Design: A Semi-rigorousApproach
Hence, the E-equation is
Equations (9) to (14) show that the effective stripping factors depend on the flow rate
of liquid from the first plate (L'il) and flow rate of vapor from the last plate (V'M)
inside the ith multi-tray unit These values are calculated as described below.
vi+l,j
\i,j
Figure 3 . Schematic representation of a general multi-tray unit.
Calculation of L;I and V$M
Ohmura and Kasahara [2] incorporated L'i1 as an independent variable. Calculation
of v ' i ~assumed that the load changed linearly inside the multi-tray unit, as given
by:
v '=~
Vi+l - ( [(1+ SiL, Li -L'i1] / (Mi - 1 ) )
(16)
41
SN.Maiti, S.Ganguly. A.K.Das and D . N.Saraf
In the proposed method these two profiles are calculated by solving component
material balance, enthalpy balance and equilibrium relations inside the multi-tray
unit as discussed below.
The vapor leaving the first plate of the multi-tray unit is at its dew point
temperature (T;1) as given by:
The above equation is solved for T'i1. Hence, the component flow rates of liquid
from the first plate in the unit are given by the equilibrium relation:
(vii I Vi)- K'ilj (l'ilJ/L'il)= 0
(18)
The enthalpy and total material balance equations for this plate are given by
Equations (19) and (20) respectively:
Since H'i2 is a function of T'i2, and is given by the dew point relation at the second
plate in the unit as:
then Equations (18) to (21) can be solved simultaneouslyto obtain L i l .
Similarly, the vapor rising from the last plate in the multi-tray unit ( v ' i ~is)
calculated using the material balance, equilibrium relation and enthalpy balance for
the bottom plate in the unit The bubble point relation at the last plate in the unit is
given by:
and T;.M is obtained by solving Equation (22). The equilibrium relation at this plate
is:
42
Multicomponent Distillation Column Design: A S m X g o r o u s Approach
The total material balance equation is:
L'iM-1=
(I+sf)Li + v ' M - v i + l
=f(V'iM)
(25)
From Equations (24) and (Z)
the
,only unknowns are V'N and h ' i ~ - l=f(TiM-1)1.
[
The latter is obtained by considering the bubble point relation at the last but one plate
in the unit, which is given by:
Hence, V ' ~ Mcan be obtained by solving Equations (23) to (26) iteratively.
Computation Technique
The model equations are solved simultaneously by a modified Newton-Raphson
method making use of a damping factor whiIe correcting the solution vector. The
convergence criterion used is described by Henley and Seader [8], namely:
where
s=N(2C+1)
("C F .
2 ) 10-10
i=l
1
fv,
Specifying N ,
Tf Pf,Pi, Sf, S/ and Qi, the remaining N(2C + 1 ) unknown
variables ( l b Vij and Ti or Mi) can be obtained by solving the N(2C + 1) model
equations simultaneously.
Design Specifications
For the design of a two-product column, only two specifications can be accepted.
These are usually given by specifying one composition each in the top and the
bottom products. In the Naphtali-Sandholm method, such specification is included by
dropping out one MESH equation and including a dummy equation in its place. For
example, in order to specify the top product composition o)oi>, replace the enthalpy
balance for the condenser by an equation of the form:
v1j -
&
(29)
v1j) YDj = 0
In the semi-rigorous formulation, consideration of the variables and the equations for
each unit revealed that an additional specification in each multi-tray unit was
possible. Since there are one or two single trays between the multi-tray unit and top
product, either the same specification can be demanded at the multi-tray unit
resulting in a conservative design, or the composition at the multi-tray unit can be
43
SN.Mairi. S.Ganguly.A.K. Dar and D . N.Swaf
estimated from the given product specification. Therefore, unlike the NaphtaliSandholm formulation, the enthalpy balance at the condenser can be retained and an
additional product composition specification can be included in the multi-tray unit.
In the case of a column with side-stream product draw-off, only the fraction of the
liquid and/or the vapor flow can be specified in the Naphtali-Sandholm method.
Replacement of equations with other specifications is not possible. However, the
semi-rigorous method allows the side-seeam product composition as well as flow
rate to be specified. The semi-rigorous method for rating of existing columns allows
the number of trays inside the multi-tray unit as an additional specification.
Computer Program
A computer program called CADPRO has been developed by the authors (contact
DN. Saraf for details) which includes distillation calculations in both the rigorous
design and rating approaches. In the design mode, one design specification at each
multi-tray unit is provided and the number of trays, the feed location and product
distribution are calculated. In the ratings approach, the number of trays grouped in
each multi-tray unit is specified, and the temperature of each tray and product
distribution are calculated. A thermodynamic package is also provided to account for
nonideal systems. Figure 4 shows the flow chart of the design algorithm in
CADPRO. This program was executed for some test problems using an HP 9000/850 computer.
Results and Discussion
The following three problems have been solved to demonstrate the application of the
semi-rigorous algorithm.
Problem I :
A 16-plate column including a partial reboiler and a total condenser. The saturated
liquid feed is introduced at the 81h plate from the top at a rate of 100 kmol/h
containing propane 23% n-butane 37% and the rest of n-pentane at 280 K and 17
am. The distillate rate and reflux ratio are 22.6 kmoVh and 5 respectively.
Problem 2:
A 20-plate column including a partial reboiler and a total condenser. The feed flow
rate, composition, temperature and pressure are same as in Problem 1. The distillate
rate and reflux ratio are also the same, but the saturated liquid feed was introduced at
the 10th plate.
Problem 3:
The column consisted of 17 trays with a partial condenser and a partial reboiler. The
feed is a mixture of ethane 3%,propane 20% n-butane 37%. n-pentane 35% and the
rest of n-hexane, temperature 374 K at 17 am, introduced at the 91h tray at 100
kmol/h, reflux = 150kmolh, and distillate = 23 kmol/h (Henley and Seader, 1981).
Problems 1 and 2 have been solved both in the design and ratings modes,
whereas Problem 3 has been solved for design only.
44
Multicomponent Distillation Column Design: A Semi-rigorous Approach
Specify one liquid or
vapor flow in each
multitray unit
1
Specify No. of trays
in each multitray unit
Mi
.
Compute initial
, gueasesof
vij and l i j
r
I
Compute internal flows:
L:, and V& in
multitray units
Compute sum of the squares of
the discrepency functions
Is s u m
L
5e?
1 No
I
Compute correction vector
using Newton-Raphson
Compute optimal step
length and new values
of v ; j , I i j ,
or Mi
from summation Equation
Figure 4. Flowchartfor semi-rigorourdesign method.
45
SN.Maiti, S.Ganguly. A.K. Das and D. N.Saraf
Design
In the design approach, one specificationequation in each multi-tray unit is required.
In principle, it should be possible to specify any combination of liquid and vapor
flow rates for the top and bottom sections, although a solution may not always be
achieved. It was observed for the present case, that if the liquid flow rates in both the
units were specified,the problem did not converge.
Problem 1 was configured as an 11 unit, 10 unit and 9 unit column with two
multi-tray units - one each in the rectifying and stripping sections. These were solved
by the semi-rigorous method for three different design specifications. The design
results obtained are given in Table 2. For all the column configurations and product
specifications,the calculated number of stages in each multi-tray unit (when rounded
to the nearest integer) correspond to the actual number of trays (given in the sixth
column for comparison).
units
unit
multi-tray units
MI
M2
11
6
042, 082
3.1284
3.2803
3.2125
4.4249
4.1824
4.3279
4.2619
4.2302
3.9391
3.8202
3.7581
3.7010
3,9007
3.8155
3.7517
5.1019
4.6449
4.9282
v42, vK3
v41, %2
10
5
v3lr v72
v32s v72
v32, 1173
9
5
1131, 172
w3ar
+a
v33, h 2
multi-tray units
3, 4
iterations
4 4
495
5
3
2
3
4
5
4
3
3
* M,and M2are the number of trays in the top and bottom multi-tray unit respectively.
Figure 5 shows the progress of convergence of the number of trays inside the
multi-tray units as iterations proceed. Figures 6, 7 and 8 show the convergence of
liquid and vapor flows and temperature profiles respectively for the same case.The
convergence can be seen to be quite fast and reasonably smooth.
To investigate the effect of different design specifications and initial guesses,
Problem 1 was solved considering a total of 9 units (case3 in Table 2). The results
of the computations are shown in Table 3. It was observed that both the design
specification and the initial guess had a profound effect on the final design, the latter
having most influence. For the same design specifications, different initial guesses
resulted in markedly different designs. Clearly, the design problem has a multiplicity
of solutions. This is to be expected because Edmister's relation (Equation 15) is
highly nonlinear in the number of stages (Mi)inside the multi-tray unit. To check if
the various solutions thus obtained would indeed meet the design requirements, the
column equations were solved in rating mode using the values for A41 and M2 as
obtained from the design calculations (without rounding off). Both the designs gave
essentially the same product distribution. It is well known that different
combinations of total number of stages and feed location can lead to the same
product specifications. The mathematical multiplicity of solutions observed in the
present study is indeed a manifestation of the above physical reality.
46
Multkomponent Distillation Column Design: A Semi-rigorousApproach
6
5Unit-9
1-
0;
1
I
I
1
2
3
4
No. of Iterations
I
5
5. Convergence inside a multi-tray unit for problem I .
47
SN.Maiti. S. Ganguly. A.K. Das and D . N.Saraf
Iturotion no.1
Final solution
(iteration no.' 5)
I
I
I
3
1
-
I
I
I
I
I
5
7
U nit number
I
I
9
11
Figure 6. Convergence of liquidflow profilefor problem 1.
Table 3: Resultsfor various design specifications and dferent initial guesses.
Case Specifications
Second initial gums
First initial gums
No.
in multi-tray
units
Mi
1
2
mi.
~ .I
.n.
4.26
Ma
5.10
%l, 17,
3.93
4.93
3
~
4.76
5.62
4.42
4
4
3
4
48
i 17s,
VJI, c*3
4.48
Iterations
A&
4
2.76
2.49
2.27
. -%
7.77
8.26
14.7
diverged
Ikrdions
4
4
7
Multicomponent Distillation Column Design: A Semi-rigorousApproach
160
140
-
4
r
:
120 0
E
Y
-
:loo
c
0
L
-z 80-
.c
-Initial guess
L
60>
40
Iteration no.1
Fino1 solution
(iteration no. 5 )
o
0
6
-
20
3
1
5
7
9
11
Unit number
Figure 7 . Convergence of vaporflow profile for problem I .
Although all designs obtained for a given product specification are valid, an
optimum is required. Numerous procedures for calculating optimum feed stage
location are available in the literature [9,10]. The simplest and most commonly used
is an empirical rule for feed-plate location where the ratio of key component liquid
mole fraction on the feed stage should be as close as possible to that in the liquid
portion of the feed s m m . For design Problem 1, the key component ratio in the feed
is 0.62. In cases 1 and 2 (Table 3) it was found that this ratio is 0.62 and 0.61
respectively for the first initial guess, and 0.89 and 0.93 respectively for the second
initial guess. Therefore, the designs obtained by the first initial guess are the optimal
designs. However, as mentioned by Hanson and Newman [lo], empirical rules do
not always work well, and hence a more general procedure must be evolved.
For a particular reflux and product specification, the design which yields the
minimum number of stages is the optimum. Therefore, an optimization problem can
be formulated as:
NmR
Nms
minf= C Mi+ C Mi
i= 1
i= 1
subject to all model equations as constraints. Here N d and Nms are the number of
multi-tray units in the rectifying and stripping sections respectively.
49
SN.Maiti, S.Ganguly, AX. Das and D.N.Swaf
Computations were performed to compare the designs obtained by using a linear
vapor profile inside the multi-tray unit [2] and the present method. In all the 11 cases
studied (not presented here), the present method converged to desired solutions,
whereas a linear profile failed to converge in five cases.
Problems 2 and 3 were solved in the design mode and the results were found to
be similar to those of Problem 1, and thus vindicated the conclusions already drawn.
Further details on these and other computationsare available from the authors.
410
3 90
*u.370
L
a
c
2
u
E
r-" 35c
3 30
310
1
I
3
I
I
I
I
I
5
7
Unit number
Figure 8. Convergence of temperature profilefor problem 2.
1
I
9
I
11
Rating
The number of plates inside each multi-tray unit was varied to reduce the total
number of units. The rating results for Problem 1 are shown in Table 4. It was
observed that the product compositions for different configurations remained almost
the Same with the separation between components slightly decreasing as the total
number of units decreased. Similar results were obtained for Problem 2.
50
MulticomponeruDistillation Column Design: A Semi-rigorousApproach
Table 4. Rating results by semi-rigorous method compared with Naphtali-Sandholm
methodfor Problem 1 .
S1.
NO. of Feed
NO.
units
16'
Trays in
Top product . Bottom product
multi-tray units cornposition
composition
1
21.5363
1.4637
-9
8
1.0614
35.9385
39.9978
0.0022
2
11
6
3, 4
21.4573
1.5427
35.8604
1.1396
0.0030
39.9970
1.8406
3
7
4
5, 6
21.1594
35.587
1.4130
39.973
0.0269
* Calculation by Naphtali-Sandholm method.
unit
NO. of
iterations
2
3
4
Since the number of units is smaller compared to the total number of trays in the
column, the computer memory requirement is significantly reduced for the semirigorous method.
Conclusions
A computer-aided design package, CADPRO, has been developed which
incorporates a semi-rigorous design procedure. It enables calculation of the total
number of stages and the feed location for a given set of design specifications
without repeated simulations. However, different initial guesses may lead to a
multiplicity of solutions because of the nonlinear nature of the design equations.
Selection of an optimal design using an empirical relationship works well in the
examples cited. Further investigations are required to establish the usefulness of the
suggested optimization approach.
The semi-rigorousmethod when used for rating problems gives results which are
comparable to those obtained from the rigorous Naphtali-Sandholm procedure, but
requiring much less computer memory. When the number of trays inside a multi-tray
unit exceeds 5 or 6, the validity of the effectivestripping factor approximation begins
to break down. Hence for very large columns it may become necessary to provide
more than one multi-tray unit in each section.
Acknowledgement
Financial support from the Department of Science and Technology, New Delhi,
under Project No. In - 5 (67) /87 - ET, is gratefully acknowledged.
References
D.P.1971. Multicomponent separation calculation by
Linearization.AIChE J.. 17(1), 148-153.
2. Ohmura, S. and Kasahara. S. 1978. New distillation calculation method utilizing salient
features of both short-cut and tray-by-tray method. J . Chem.Eng. Japan. ll(3). 185-193.
3. Tomich, J.F. 1970. A new simulation method for equilibrium stage processes. AIChE J..
16(2). 229-232.
4. Edmister, W.C.1943. Design for hydrocarbon absorption and stripping. Id.Eng. Chem.,
35(8), 837-839
1. Naphtali, L.M. and Sandholm,
SH. Mairi. S. Ganguly, A.K. Das and D. N.Saraf
5. Ganguly, S. 1985. M.Tech. Thesis, Indian Inrtitute of Technology, Kanpw.
6. Ganguly, S., Das, A.K. and Saraf,DN. 1985. Paper presented at National Symposium on
Modeling and Simulation in Chemical Engineering, Indian Institute of Science,
Bangalore, India, August 22-24
7. Maiti. S.N. 1989. M.Tech. Thesis, Indian Institute of Technology, Kanpw.
8. Henley, E.J. and Seader, J.D. 1981. Equilibrium-Stage Separation Operations in Chemical
Engineering, John Wiley & Sons,New York.
9. King, C.J. 1980. Seperation Processes,McGraw-Hill., New York.
10. Hanson, D N . and Newman, J.S. 1977. Calculation of distillation columns at the optimum
feed plate location. I d . Eng. Chem. Process. Des. D o . , 16(2), 223-227.
Received: 18 Janurary 1993; Accepted gfter revision: 2 August 1993.
52
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