42
Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 42-48
A Simple Algorithm for Sensitivity and Operability Analysis of Separation Processes Esteban A. Brignole, Raflqul Gani,‘ and Jos6 A. Romagnoli P h t a Piloto de Ing. Quimica, UNS-CONICET,
Bahia Blenca, Argentina
A methodology is described in which simple analytical tools are given to guide the design engineer in selection of distillation column control structures and/or predicting column operability and control during the design stage. Quantltative and easy to implement criteria are given to Identify and monitor the sensitive dominant regions in a column, based on the idea of maximum enrichment factor. These criteria are applled to study the effect of changes in the enrichment factor and column topoloay. From these results, guidellnes are obtained wlth regard to distiktion operation and control. Results are given for several practical examples.
Introduction The conventional approach to the design of control systems, particularly for distillation processes, often avoids some of the primary questions that a designer must consider, that is, determining the structure of the control system rather than a detailed analysis of the best adjustment of control system parameters. This stands in sharp contrast to the general approach to problems in control theory. The designer of any feedback control sptem must consider the process characteristics (static and dynamic), the operability of the process under different operating conditions, and the sensitivity with respect to the operating and design parameters. Based on that information, he defines what is called the measurement and control structure, i.e., the kind of measurement, the points where measurements must be made, the variables to manipulate, and the controller to be used. The development and application of optimal control structures to chemical processes is presently a subject of active research (Alvarez et al., 1981; Morari and Stephanopoulos, 1980; Jensen et al., 1982; Arkun and Ramakisnan, 1982). Gani and Romagnoli (1982) have already laid the theoretical groundwork for the implementation of optimal structures with distillation processes which present sharp temperature gradients or complex behavior. In this work, based on an inferential point of view, a methodology is developed for the solution of a minimization problem to select the optimal arrangement of the sensors. Although the usefulness of the proposed methodology was demonstrated for what it was intended, it is not ideally suited for other purposes such as operability analysis and to study how this operability is affected by changes in the different operating and design variables of the distillation process. This article describes a methodology in which simple analytical tools are given to guide the design engineer in the selection of distillation control structures and/or for predicting column operability and control during the preliminary design steps of a separation unit. A quantitative and easy to implement criterion is given (based on physical considerations) for the synthesis of an optimal measurement structure of the control system. This criterion is applied to study the effect of changes in the feed and column topology on the column measurement structure. From these results, guidelines are obtained with regard to distillation column control policies. The results of the simplified procedure are found in agreement with 0196-4305/85/ 1124-0042$0 1.50/0
a rigorous optimal location of ‘measurement (OLM) algorithm developed by Gani and Romagnoli (1982). In consequence, the methodology provides the process and control engineers with computer-generated indicators of interactions between control variables and aids in location of sensitive control points and the design engineers with a procedure to approach selection of a control strategy in a structural manner. The final objectives are to improve the consistency of control applied to separation units and thereby improve performance in the plants. Location of the Sensitive Zones Enrichment Factor. There are zones in the distillation column where the component flow rates (compositions) and temperatures undergo significant changes for minor changes (in absolute terms) in the quality or temperature of the overhead or bottom products. The high ratio between changes in temperature or composition in this region to the corresponding change in the specification of the column products makes this region a suitable location for the insertion of sensors for the control of the column. For a sharp separation with operating conditions close to the minimum reflux, there are “pinch” zones in both ends of the column, corresponding to the fractionation of an almost pure component. In these zones the change in conditions in each plate is very small, there is also a zone of potential “pinch” condition in the region of the feed plate. Therefore, for a normal equilibrium curve there are two zones between the “pinch” regions that depict a large separation power. With the purpose of locating these zones, the “enrichment factor” in a multicomponent mixture for a given plate is defined as NC-1
u = ic= l
IY*i,n-
Yi,n+ll
for the rectifying section, and
u’ =
NC-1
C
IY*i,n
-
Y’i,n+lt
1=1
for the stripping section, where y*i,n is the composition of the vapor leaving an ideal stage and corresponds to the equilibrium composition with the liquid and yi,n+lor y:,n+l are the average composition of vapor entering plate n (in the rectifying or stripping section, respectively) and can be obtained as a function of through a material balance. For a particular specification of the column, given feed compositions and flow rate, distillate flow rate and com0 1984 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 1, 1985 43
1
x,
a
1 - 91’
where XS
positions, reflux rate, feed plate location, and column pressure, the equilibrium curve and operating lines are fixed. Under these conditions it is expected from the McCabe-Thiele diagram of Figure 1 that U depicts a maximum value in the rectifying section and U‘ in the stripping section. The values of U and U’are directly the difference in ordinate of the equilibrium and operating curves at a given value of liquid composition. The condition for the points of maximum are consequently defined as dU/dx = 0
(3)
dU’/dx = 0
(4)
Therefore, the plate composition of maximum enrichment factor are obtained from dyi,n+l
(5)
dxi dxi for the rectifying section and dY*i,n --
--dY’i,n+l
(6) dxi dxi for the stripping section. Analytical Solution for Binary Mixtures. Analytical expressions are readily obtained for a binary mixture for the compositions x , and x , that maximize U and U‘ 1)
- 1]/b
and 2,
1
1-9;
(1 -
a1and a2are the roots of
NR is numbered counting from the top plate and Ns is numbered counting from the reboiler. If we are dealing with a high-purity separation of a binary mixture, the composition will tend to unity in the distillate for the light component and in the bottom product for the heavy component. Under these conditions and using the Taylor expansion to approximate the equilibrium relations, it is possible to compute NRand Ns with very simple relationships
Ns = In [(l - p ’ ) / y a / ( a - 1)2((1/p’)ll2- l)]/ln p’ (14)
where
r’+ 1 (15) ra r ’a Extention to Multicomponent Mixtures. For the location of U,, and U’,, in a multicomponent distillation column, it should be recognized that for a given set of specifications, the values of U and U’ are only a function of position (plate) in the column. Therefore using a steady-state model of a column, an exhaustive search over N will render the location of the zones of maximum separation power. A rigorous location of these zones could be done very easily with the help of Underwood’s equations. In the specification of a distillation operation for multicomponent mixtures, the compositions of the top and bottom products are estimated on the basis of the distillate or bottoms flow rates and the recovery of the heavy and light keys. If it is assumed that the non-keys do not distribute, the distillate consists of a mixture of light components plus the keys. On the other hand, the bottom product is then a mixture of the heavier components plus the keys. If we further assume that the composition of the heavy non-key components are negligible a few plates above the feed, the computation of U for the plates of the rectifying section can be made starting from the distillate composition and going down the column using Underwood’s relations
p = -*r + l
p’=-
+ [ (at)iz r
x, =
--
XD
XF
Figure 1. Typical McCabe-Thiele diagram for a binary system.
-dY*i,n =-
. - a i xs
-
=
[ (&y -
1]/[.
- 13
- 11
(7)
(8)
where r is the reflux ratio and r’ is the reboil ratio. The location of the plates that render U and U’maximum for the rectifying and stripping sections are readily obtained using Underwood’s equation. For the rectifying section
which can be derived from where CP1 and r
Cpz
are the roots of a
+ 1 = -xD a-0
For the stripping section
1 + -(I 1-9
- xD)
\
