process model based control of distillation columns …
TRANSCRIPT
PROCESS MODEL BASED CONTROL OF DISTILLATION COLUMNS
by
RUPAK SINHA, B.S. in Ch.E.
A THESIS
IN
CHEMICAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
IN
CHEMICAL ENGINEERING
Approved
Accepted
December, 1988
—o
-•o
T3
^f^ ACKNOWLEDGEMENTS Ci>pv
I would like to express my sincere appreciation to my graduate
advisor, Dr. James B. Riggs, for his guidance and support throughout
this work. I wish to express my thanks to Dr. R. Russell Rhinehart for
his constructive suggestions.
I would like to dedicate this work to my family, especially my
parents, for their unbending support and love throughout this project
and my academic career.
Last, but not least, I would like to thank my friends who always
took time to listen and make suggestions. I am thankful to Kamal M.
Mchta for his friendship, and Lisa M. Trueba for proofreading this
document.
Appreciation is also extended to Dow Chemicals U.S.A. for
providing the financial support that made this research possible.
11
b J
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
LIST OF TABLES v
LIST OF FIGURES vi
NOMENCLATURE vii
CHAPTER
1. INTRODUCTION 1
2 . LITERATURE REVIEW 4
2.1 Dynamic Simulation of a Distillation Column 4
2 . 2 Distillation Control 7
2 . 3 Sidestream Control 17
2.4 Control of High-Purity Columns 19
3 . DYNAMIC MODELING OF A DISTILLATION COLUMN 22
3.1 Tray-to-Tray Model 22
3 . 2 Modelling of High-Purity Columns 26
4. PROCESS MODEL-BASED CONTROL OF A HIGH-PURITY COLUMNS 31
4.1 Process Model-Based Control 31
4.2 Parameterization of Process Model-Based Controller 32
4.3 Implementation of Process Model-Based Controller 36
5 . RESULTS AND DISCUSSION 39
6. DEVELOPMENT AND VERIFICATION OF A STEADY-STATE APPROXIMATE MODEL OF A DISTILLATION COLUMN WITH A SIDESTREAM DRAWOFF 66
6 .1 Model Derivation 66
111
i
6 . 2 Implementation of the Approximate Model 71
6 . 3 Approximate Model Verification 74
7 . CONCLUSIONS AND RECOMMENDATION 85
7 .1 Conclusions 85
7 . 2 Recommendations 86
LIST OF REFERENCES 87
APPENDICES
A. LISTING OF THE COMPUTER CODE FOR PROCESS MODEL-BASED CONTROL OF HIGH-PURITY COLUMN 89
B. DETAILED DERIVATION OF THE APPROXIMATE MODEL FOR A DISTILLATION COLUMN WITH A SIDESTREAM DRAWOFF 106
C. LISTING OF THE COMPUTER CODE FOR THE APPROXIMATE MODEL 115
iv
LIST OF TABLES
PAGE
TABLE 5 .1 Data for PMBC base case ^3
TABLE 6.1 Base case results for approximate model 77
TABLE 6. 2 Data used in the comparison tests 78
TABLE 6.3 Recoveries and Process gains for a change in reflux rate 79
TABLE 6.4 Recoveries and Process gains for a change in sidestream drawoff rate 81
TABLE 6.5 Recoveries and Process gains for a change in boilup rate 83
LIST OF FIGURES
PAGE
FIGURE 2.1 Schematic of the P-DELTA algorithm (Process Design by limiting thermodynamic approximation) 19
FIGURE 2.2 General configuration of the Adaptive Predictive Control System 20
FIGURE 2.3 Desired temperature profile in face of feed
composition change in propane (Dl; colximn 1) 21
FIGURE 3.1 Typical seive tray N 28
FIGURE 3 .2 Flowsheet of the main driver program 29
FIGURE 3 . 3 Flowsheets of subroutines 30
FIGURE 4.1 Flowsheets 37
FIGURE 4.2 Flowsheet of subroutine CONT 38
FIGURE 5.1 Effect of an increase in XSP on bottom tray 44
FIGURE 5.2 Effect of an increase in XSP 45
FIGURE 5.3 Effect of a decrease on XSP on Tray 1 46
FIGURE 5.4 Effect of a decrease in YSP 47
FIGURE 5. 5 Effect of an increase in YSP 48
FIGURE 5.6 Increase in XSP and 5X increase in feed rate; Tray 1....49
FIGURE 5 - 7 Increase in XSP and 102 increase in feed rate; Tray 1... 50
FIGURE 5.8 Increase in XSP and 152 increase in feed rate; Tray 1...50
FIGURE 5.9 Increase in XSP and 102 decrease in feed rate; Tray 1...51
VI
FIGURE 5.10 Increase in XSP and 152 decrease in feed rate; Tray 1...51
FIGURE 5.11 Increase in YSP and 52 increase in feed rate; Tray 1 52
FIGURE 5.12 Increase in YSP and 102 increase in feed rate; Tray 1...52
FIGURE 5.13 Increase in YSP and 152 increase in feed rate; Tray 1...53
FIGURE 5.14 Increase in YSP and 52 decrease in feed rate; Tray 1...54
FIGURE 5.15 Increase in YSP and 102 decrease in feed rate; Tray 1...55
FIGURE 5.16 Increase in YSP and 152 decrease in feed rate; Tray 1...56
FIGURE 5.17 +52 Upset in feed composition; Tray 1 57
FIGURE 5.18 +52 Upset in feed composition; Top Tray 57
FIGURE 5.19 +102 Upset in feed composition; Tray 1 58
FIGURE 5.20 +102 Upset in feed composition; Top Tray 58
FIGURE 5.21 Effect of a 102 upset in feed composition 59
FIGURE 5.22Effect of a 102 upset in feed composition using NLDMC...60
FIGURE 5.23 Effect of a 52 upset in feed composition 61
FIGURE 5.24 Effect of a 52 upset in feed composition using NLDMC....62
FIGURE 5.25 Effect of an increase in XSP 63
FIGURE 5.26 Effect of a decrease in XSP 64
FIGURE 5.27 Effect of a change in gain 65
FIGURE 6.1 Split of the distillation column with a
sidestream drawoff 75
FIGURE 6 .2 Flowsheets for approximate model 76
FIGURE B.l Split of the distillation column with a sidestream drawoff 114
vii
NOMENCLATURE
M Molar tray holdup (moles)
L Liquid flow rate (moles/sec)
V Vapor rates (moles/sec)
h Liquid enthalpy (Btu/lbn,-°F)
H Vapor enthalpy (Btu/lbm-^F)
X(i,j) Liquid composition of component j on tray i
Y(i,j) Vapor composition of component j on tray i
hf Enthalpy of the feed (Btu/lbn,-OF)
F Feed rate (moles/sec)
z Liquid feed composition of the light key
K Equilibrium constant
T Temperature on all trays (°F)
T- Temperature at the top of the column (°F)
T j Temperature at the bottom of the column (°F)
Tf Temperature of the feed coming into the column (°F)
P Pressure in the column (Psia)
S Separation factor
B Bottoms rate (moles/sec)
D Distillate rate (moles/sec)
G Sidestream drawoff rate (moles/sec)
viii
f
f Fractional recovery of comoponet i in the bottoms
g Fractional recovery of component i in the sidestream
dd Fractional recovery of component i in the distillate
a Constant relative volatility.
IX
CHAPTER 1
INTRODUCTION
Distillation is a process widely used in the petroleum and chemical
process industry to separate a mixture into its components. The
separation is based on the fact that the vaporized portion of a liquid
mixture has a composition richer in the more volatile components from
that of the liquid. Binary columns have been widely studied and linear
controllers such as the Proportional-integral (PI) and Proportional-
integral-derivative (PID) usually perform satisfactorily for low purity
columns that also have consistent feed quality and feed flowrate.
However, at moderate purities the binary column exhibits nonlinear
behavior and coupling effects become important. Today, many industrial
areas, such as pharmaceuticals, plastics, and polymer production,
require very high purity products. High purity columns are both
extremely nonlinear and highly coupled thus limiting the applications of
linear model-based multivariable controllers and classical PID
controllers.
Process Model-Based Control (PMBC) is a new controller that has the
potential to overcome the limitations of linear model-based controllers.
PMBC is a multivariable model-based controller which uses an approximate
model directly for control purposes. The approximate model does not
have to be a rigorous simulator, but does need to contain the major
characteristics of the process. The approximate model is adaptively
updated on-line in order to keep it "true" to the process as changes in
the process and process operating conditions occur. Since the PMBC
controller has a relatively accurate description of the process, it
provides nonlinear feedback control along with nonlinear feedforward
control. Therefore, the PMBC controller is able to "anticipate" the
required control action to absorb feed composition and feed flowrate
changes through its feedforward capabilities, as well as absorb the
results of an unmeasured disturbance (heavy or light feed composition
change, cooling water temperature change, head losses, etc.) using its
feedback features. As a result, PMBC can offer significant advantages
for distillation columns that produce high purity products. The
objectives of the first phase of this research are to develop a rigorous
dynamic simulation model for a high purity binary distillation column,
and then test the implementation of the PMBC algorithm to control the
column.
The second phase of this research involves the study of
distillation columns with sidestream drawoffs. A distillation column
that uses a sidestream drawoff can provide substantial economic savings.
Sidestream drawoff can reduce the number of columns required for certain
multicomponent separations. For instance, if we had to separate a
mixture of ABC, it would require two columns, one to separate A from BC
and the other to separate B and C whereas, a distillation column with a
sidestream drawoff could perform the required separation in a single
column. They also can be used for binary separations to obtain
different purities of the two components. Despite these process
advantages of columns with sidestream drawoff columns, control is
inherently more difficult than with conventional columns since there are
more degrees of freedom. Product quality control loops and material
balance control loops are more complex, less direct, often more
sensitive and more interacting. Sometimes unexpected dynamic and steady
state behavior can be observed due to transient and steady state changes
in internal liquid and vapor rates. In order to use all the economic
advantages of a distillation column with a sidestream drawoff, better
control techniques will have to be developed. In order to apply PMBC to
the sidestream column we need an approximate model of a column with a
sidestream drawoff. The last phase of the research includes the (1)
development of an extension of the Smith and Brinkley approximate model
to incorporate the sidestream drawoff and (2) the verification of the
approximate sidestream drawoff model.
CHAPTER 2
LITERATURE REVIEW
2.1 Djmamic Simulation of a Distillation Column
The advent of analog computers in the early 50's allowed attempts
to be made, to model distillation dynamics in a reasonable manner, but
simplifications were imposed by the limitation of the analog equipment
However, the widespread availability of digital computers in the 60's
promted new attack on the d3mamics problem, but a number of earlier
simplifications remained. For instance, Huckaba et al. (1963) limited
their attention to binary distillation at constant pressure, with
constant liquid holdups and negligible vapor holdups. Waggoner and
Holland (1965) required independent specifications of the transient
behavior of the liquid holdups, and vapor holdup was once again
neglected. Levy, Foss, and Grens (1969) effectively treated the
varying liquid holdups but made the following assumptions:
(1) Constant vapor holdup that is small compared to the liquid holdup.
(2) Perfect liquid mixing at every stage.
(3) Negligible holdup in condenser.
(4) Adiabatic column.
(5) Constant liquid holdup in reflux drum and reboiler.
(6) Stationary process, described by linear differential equations for
small perturbations from steady state.
However, in the 70's, linearized, dynamic models were developed.
In their book, Rademaker et al., (1975) extensively cover linearized
dynamic modeling of distillation columns, and they use these models for
stability analysis and control systems design.
Luyben (1973) extensively covers the dynamics of multicomponent
distillation columns, and presents an algorithm using Euler's method to
solve the differential equations. In his approach almost all possible
nonlinarities are eliminated by local linearization, using the following
assumptions:
(1) There is one feed plate onto which vapor and liquid feed are
introduced.
(2) Pressure is constant on each tray but varies linearly up the
column.
(3) Coolant and steam dynamics are negligible in condenser and
reboiler.
(4) Vapor and liquid products are taken off the reflux drum and in
equilibrium. Dynamics of vapor space in reflux drum are
negligible.
(5) Liquid hydraulics are calculated from the Francis weir formula.
(6) Volumetric holdups in the reflux drum and column base are held
constant by changing the bottoms and distillate rates.
(7) Dynamic changes in internal energy on trays are negligible compared
with latent-heat effects, so the energy equation on each tray is
just algebraic.
Luyben also presents a code for the algorithm in his text. However,
with the availability of some differential equation solving packages,
Euler's method is inefficient by contrast.
Sourisseau and Doherty (1982) studied various different dynamic
models and classified them according to the state variables employed.
Following their definitions, a model in which the state vector consists
of only liquid compositions was called the C-model. If both
compositions and enthalpies are included, the CE-model results. The
most complex model is the CHE-model and has a differential equation for
each state variable on each tray (composition, holdup and enthalpy).
The constant molar-overflow model (CMO-model), assumes fast holdup and
energy changes as well as fixed liquid and vapor rates at all times.
Sourisseau and Doherty studied all five dynamic models for various
distillation problems involving relatively ideal mixtures. They
concluded that the transient response results for all of the models were
in good agreement. Furthermore, they concluded that the CE and CHE
models were too time consuming considering the relatively little
additional information obtained; they preferred the use of the C or CMO
models. These conclusions are in agreement with earlier work by Levy et
al. (1969), which indicate that the significant dynamics in distillation
processes are retained in the differential equation modeling liquid
phase compositions.
Chimowitz, Anderson and Macchietto (1985) used the C and CMO models
of Sourisseau and Doherty to present an algorithm for the dynamics of a
multicomponent distillation column using local thermodynamic and
physical property models. They split the system into two tiers, as
shown in Figure 2.1, the inside tier uses an approximate process model
and local thermodynamic models, the outside tier contains a rigorous
thermodynamic model which is used to update the validity of the local
model based upon rigorous thermodynamic evaluations. The system they
investigated was a non-ideal ternary system. One of the characteristics
of this system is that the composition trajectories for adjacent trays
during a transition period can be distinctly different. In their
conclusions they acknowledge the fact that this approach requires larger
storage space, but it significantly improves the execution time, often
by a factor of 5 - 10 when compared to algorithms that rigorous
thermodynamic evaluations. This is a significant improvement especially
when we are considering on-line control of multicomponent distillation
columns.
2.2 Distillation Control
Feedback, feedforward, material balance control, decoupling and
cascade control are some of traditional approaches to distillation
control. All these generally involve PI or PID controllers, however,
distillation processes are non-stationary and nonlinear in nature, and
their operating conditions change frequently. For this reason
computerized distillation control has become an active research topic
for the past decade.
This was the main motivating factor for the development of adaptive
control. Model reference adaptive systems (MRAS), self-tuning
regulators or controllers (STR or STC) and adaptive predictive control
systems (APCS) have been developed from different perspectives.
Martin-Sanchez (1976) developed the adaptive predictive control
system which is related to the traditional dead-beat control idea of
bringing a system to its final state or set point in minimum time. It is
characterized by the following principles:
(1) At each step a future desired process output is generated, and the
control input is computed in order to make the predicted process
output equal to the desired process output.
(2) The predicted output is based on an adaptive predictive (AP)
model, whose parameters are estimated by a recursive estimation law
with the objective of minimizing the prediction error.
(3) The previously mentioned desired process output belongs to a
desired output trajectory, that satisfies a certain performance
criterion, e.g., this trajectory can start from the current 'state'
of the plant and evolve according to some chosen dynamics to the
final desired setpoint.
However, the dead-beat control idea seems impractical because (i) it
requires an exact knowledge of the process model and (ii) its implicit
objective function is closed in nature and generally requires an
excessive control effort. Predictive control as defined by APCS is
unlike dead-beat control. It is in fact a very practical and powerful
strategy because it includes the concept of a desired output trajectory
based on a finite-time horizon objective.
Martin-Sanchez and Shah (1984) introduced the adaptive predictive
control methodology with special emphasis on the key issues involved in
the practical applications of APCS to real processes, using SISO and
MIMO control of a binary distillation column. Figure 2.2 shows a
general confguration of APCS, the specific functions at each control
instant are explained as follows:
The driver block generates a future desired process output value,
that belongs to a desired output trajectory.
The adaptive predictive model is used to generate a control signal
that makes the predictive process output equal to the desired output
generated by the driver block.
The adaptive mechanism: (i) adjusts the adaptive predictive model
to minimize the prediction error and (ii) allows the driver block to
redesign the desired output trajectory for the optimization of the
control system performance.
Martin-Sanchez and Shah also provide the mathematical formulation
and implementation of APCS. Their experimental results easily out
10
perform all classical techniques, and have put APCS beyond the
theoretical stage.
Yu and Luyben (1984) used multiple temperatures for the control of
distillation columns. They studied two columns, one had three
components and thirty-two trays and the other had five components and
twenty trays. Using a steady-state model they proposed three different
control systems as follows:
(1) The Single Temperature Control involves selecting the 'optimum'
tray which gave minimum steady-state error in distillate
composition for the 'worst' disturbance.
(2) The Temperature/Differential Temperature Control follows these
design procedures:
a) find the 'optimum' single temperature control tray;
b) generate the desired temperature profiles for the worst
disturbance case (e.g.. Figure 2.3);
c) locate the section of the column where there is the most
change in temperature differential.
This control works well for lighter than light key (LLK) feed
composition changes, but it may not work as well for light key (LK)
and heavy key (HK) changes in feed composition.
(3) The Temperature/Dual Differential Temperature Control (TD2T) is
based on adjusting the temperature controller using two temperature
differentials. It is almost like the Temperature Differential
Controller except that it works with the temperature that least
11
disturbs the LLK composition. LK and HK feed composition changes
are handled better using this type of controller.
Yu and Luyben concluded that the TD2T controller gave the best overall
performance on both columns. It also has several advantages over the
more complex and conventional 'inferential controls.'
Bryan (1985) studied the heat-integration technique to control
distillation columns. In this method a sequence of distillation columns
has to be chosen and then integrated to provide a joint effect. Heat
integration has some economic advantages in that it can reduce energy
consumption nearly 502 (Roffel and Fontein, 1979) when compared to a
conventional system using steam reboilers and water-cooled condensers.
However, heat integration does have some drawbacks which present many
control problems. Product qualities are difficult to maintain, because
the common reboiler - condenser affects operation in both columns. Also,
changes in the vapor rate in the high-pressure column effect the
performance of the low-pressure column. But the main objective of
Bryan's research was to design a control strategy that could overcome
these problems and still maintain the economic incentives of heat
integration.
There have been some attempts at developing a control strategy for
nonlinear systems, most notable by Morari and Economu (1986). They
extended the Internal Model Control approach to the nonlinear system.
Their paper describes the differences in treating the linear and
nonlinear systems both from a mathematical as well as control point of
12
view. The article shows that even when dealing with processes that have
relatively mild nonlinearities, no linear controller can match the
performance and robustness of a rationally designed nonlinear
controller.
Process Model Based Control (PMBC) is a technique based on the
dynamic simulation of the real process. However, selection of a process
model can be a key factor in the design of the control system; Cott,
Reilly, and Sullivan (1986) present a procedure for the selection of a
process which can be used for PMBC. The ideal model would have the
following qualities:
(1) It would exactly predict the operation of the real process over the
entire operating region with only one set of parameters.
(2) It would require very little computational effort.
However, it is impossible to find a perfect model for a real process.
The two main criteria for selecting a model should be :
(1) Model Accuracy: preference would go to the model that most closely
reflects the real process over the operating region.
(2) Computational Effort: the model requiring the fewest calculations
would be preferred.
They also suggest an algorithm for the integration of the model
selection techniques, accuracy and computational effort as follows:
13
(1) Select candidate model on rough computational effort criterion,
based on control computer capacity.
(2) Determine the accuracy of the candidate models by validating with
respect to the process data.
(3) Determine the computational effort required for each model.
(4) Select the 'best' model, based mainly on accuracy criterion.
This algorithm provides a basis for a model selection procedure for
model-based control.
They also present a detailed application of these techniques to
distillation control. They use the model presented by Luyben (1973) as
the rigorous simulation model and then evaluate four shortcut methods
using the model selection procedure. Process Model Based Control is
then applied using the selected model, which was the Smith-Brinkley
model. The implementation of model-based controllers involves two
steps: a model parameter update and control action calculations. The
controller follows the following pattern:
(1) Model Parameter Update
measure D,B,y,x,L,P from the column;
' back-calculate a pseudo feed stream based on products;
solve for the Smith-Brinkley parameters.
(2) Control Action Calculation
determine the product set points;
measure F,z,P from the column and put through digital filters;
using the filtered data, solve model for L and VP;
implement L and VP on the column.
They also compared PMBC with two other strategies (1) Internal Material
Balance and (2) Dynamic Matrix Control and concluded that PMBC out
performs both techniques.
2.3 Sidestream Control
Despite the process advantages of a sidestream distillation column,
control is inherently more difficult than with conventional columns,
since there are more control variables, interactions and degrees of
freedom.
Luyben (1966), presents a qualitative discussion and comparison of
ten different schemes to control distillation columns with sidestream
drawoffs. The configurations he presents range from simple temperature
and composition control loops to internal reflux or vapor and feed
forward control, using a ternary system. If the feed contains a small
amount of light component, then the light component is taken off the
top. the sidestream is a liquid in the rectifying section. However, if
the feed contains a small amount of the heavy component then the system
is reversed, and the sidestream will be vapor in the stripping section.
In his discussion he has assumed pressure dependence of temperature, and
states that "variations in pressure due to barometric or column pressure
drop changes, may have more effect on temperature than changes in
composition." However, pressure dependence can be reduced by placing
15
the sidestream tray farther away from the end of the column to a spot
where the temperature gradient is steeper and more composition
dependent. The choice of scheme is governed by the economic and process
considerations of the application.
Tyreus and Luyben (1975) applied the sidestream drawoff to study
the control of a binary distillation column. Various schemes were
simulated on a digital computer, but due to the limited range of steady-
state operability, none of the schemes proved satisfactory. But.
controlling the sidestream composition by varying the drawoff tray
location proved to be very successful. The overhead composition was
approximately controlled by holding the temperature of a tray near the
top of the column with reflux flow. Bottoms composition was similarly
controlled by the temperature in the lower section of the column.
Doukas and Luyben (1981) present the 'L' and the 'D' --schemes to
control a two-column configuration consisting of a prefractionator
column and a sidestream column. The 'L'-scheme used the manipulation of
the sidestream drawoff tray location to control one of the sidestream
compositions. While the 'D'-scheme utilized the overhead distillate
product rate from the prefractionator to control one of the sidestream
compositions. The 'D'-scheme can handle the lightest component in the
feed better, while the 'L'-scheme the larger changes in the heaviest
component better. However, they concluded that the 'D'-scheme is much
easier to implement and hence the more favorable scheme to use.
Alatiqi and Luyben (1986) compare the controllability of two
sidestream drawoff configurations. The systems they studied are the
16
sidestream column/stripper configuration (SSS) and the two column
sequential 'light-out-first' (LOF) confguration. They tested a ternary
system of benzene/toluene/o-xylene. In the LOF system the heat inputs
and reflux flow rates to each column can be manipulated, the heat input
to feed ratio (QB/F) provided good control of the column. For the SSS
system they maintained the temperatures of the trays above and below the
sidestream drawoff tray constant by manipulating the sidedraw rate. The
SSS was controlled by four PI controllers. They conclude that the load
response of the SSS was as good as, if not better than the LOF system.
2.4 Control of High-Purity Columns
Fuentes and Luyben (1983) studied the dynamics and controllability
of high-purity columns. They first studied the dynamic responses of the
open-loop system for changes in various manipulated and disturbance
variables in order to gain some insight into the dynamic difficulties
associated with the control of these columns. Then several types of
closed-loop systems were investigated. The system they used had
purities ranging from 5 mol 2 to 10 ppm (molar) impurity in both
distillate and bottoms product for two values of relatively volatility
(a=2 and a-4). They used the following assumptions: constant relative
volatility, equimolal overflow, theoretical trays, total condenser,
partial reboiler, and saturated liquid feed and reflux. To study the
17
open-dynamics they linearized the nonlinear ordinary differential
equations using the Lamb and Rippin technique. From their results they
concluded that the responses are highly nonlinear. The response is
completely different for a positive change than for a negative change.
There is little difference in the dynamic behavior of systems with
different relative volatilities when purity levels are low. However, as
the purity increases, the dynamic response begins to differ greatly for
different relative volatilities. For systems with high relative
volatility the response is quite fast and highly nonlinear. Disturbance
in feed composition is felt rapidly in the bottom of the column.
Fuentes and Luyben then applied a closed-loop control on the
column. They basically used feedback controllers for each end of the
column: reflux was controlling distillate composition and vapor boil-up
controlling bottoms composition. These controllers worked well for low
relative volatility, however, for higher relative volatility the results
were very poor and large errors occurred in the product purities. So it
was concluded that simple product composition controllers cannot be used
for high-purity columns with high relative volatilities. In order to
overcome this problem they studied another controller; the Temperature/
Composition Cascade Controller. This gave a lot better results for the
high relativity columns. In conclusion they state that high purity
columns can be effectively controlled despite their highly nonlinear
behavior. They also conclude that high purity columns respond much
.re quickly than predicted by linear analysis. This fact must be mo]
18
recognized when specifying analyzer cycle times and in designing control
systems.
Georgiou, Georgakis and Luyben (1988) compared the conventional
diagonal control with the Dynamic Matrix Control (DMC) design for
moderate and high purity columns and showed that the performance of DMC
can be significantly improved by the use of nonlinear transformations of
the composition measurements. They studied three systems: one had a
product composition of 992 and 1.02 light component at the top and
bottom, respectively, the other had product purities of 99.92 and 0.12.
The third column was the same as that studied by Fuentes and Luyben
(1983). The first two columns worked well with standard DMC and the
third column was rejected because it had an unstable closed-loop DMC
response. However, using the nonlinear transformations they were able
to control the third column. In conclusion they state that the DMC
performed better that conventional controllers; however, simple
nonlinear output transformations improve significantly the performance
of DMC for high purity columns.
19
• • • 1 i « v • I c
t(i|a«ie>0*a*«iC
O A t * • • • t l
FIGURE 2.1 Schematic of the P-DELTA algorithm (Process Design by limiting thermodynamic approximation)
20
SET POINT ^ onivER BLOCK
i
DESIRED OUTPUT ADAPTIVE
PREOICTIVC MODEL
.
CONTROL SIGNAL ^
•
'
P H W ^ C O O
ADAPTIVE MECHANISM
i
PROCESS OUTPUT ^
FIGURE 2.2 General configuration of the Adaptive Predictive Control System
21
-• i2
i —r 74.
TE/r. a>€s cj
FIGURE 2.3 Desired temperature profile in face of feed composition change in propane (Dl; column 1)
CHAPTER 3
DYNAMIC MODELING OF A
DISTILLATION COLUMN
3.1 Tray-to-Tray Model
Performing dynamic mass and energy balances around each tray are
the first steps in developing a dynamic model of a distillation column
For a typical sieve tray shown in Figure 3.1, the equations are
Total Mass Balance:
dM(N)/dt - L(N+1) + V(N-l) - L(N)- V(N) (3.1)
Component Mass Balance:
dM(N)X(N,J)/dt - L(N+1)X(N+1,J) + V(N-l)Y(N.l,J)
+L(N)X(N,J) - V(N)Y(N,J) (3.2)
Overall Energy Balance:
dM(N)h(N)/dt - L(N+l)h(N+l) + V(N-1)H(N-1) -
L(N)h(N) - V(N)H(N) (3.3)
where there are N-tray and J-components. For every tray there are one
each of equations (3.1) and (3.3) and J-1 of equation (3.2). All the
above equations assume that vapor holdup on each tray is negligible
compared to liquid holdup. For the feed tray the equations are
22
23
Overall Mass Balance:
dM(I)/dt - L(I+1) + V(I-l) - L(I) - V(I) + F (3.1a)
Component Mass Balance:
dM(I)X(I,J) - L(I+1)X(I+1) + V(I-1)Y(I-1) -
L(I)X(I,J) - V(I)Y(I,J) + FXF(J) (3.2a)
Overall Energy Balance:
dM(I)h(I) - L(I+l)h(I+l) + V(I-1)H(I-1) -
L(I)h(I) - V(I)H(I) + Fhf (3.3a)
where I is the feed plate, F is the feed rate. XF(J) is feed composition
of the Jth component and hf is liquid feed enthalpy. The equations for
the condenser and reboiler are, respectively.
Overall Mass Balance:
dM(l)/dt - L(2) - V(l) - L(l) (3.1b)
dM(N)/dt - V(N-l) - L(N) - D (3.1c)
Component Mass Balance:
dM(l)X(l,J)/dt-L(2)X(2,J) - V(1)Y(1,J) - L(1)X(1,J) (3.2b)
dM(N)X(N,J)/dt - V(N-1)Y(N-1,J) - (L(N)+D)X(N,J) (3.2c)
Overall Energy Balance:
dM(l)h(l)/dt - L(2)h(2) - V(1)H(1) - L(l)h(l) + QR (3.3b)
dM(N)h(N)/dt - V(N-1)H(N-1) - (L(N)+D)h(N) - QC (3.3c)
where tray N is the condenser, QR is the reboiler duty, QC is the
condenser duty and L(l) is the bottoms flow rate.
24
To complete the model we need algebraic equations for phase
equilibria, thermal properties, and equations for the flow rates
(Luyben, 1973):
Phase Equilibrium:
Y - f(X,T,P) (3.4)
Equations of Motion:
L - f(M,V,X,T,P) (3.5)
V - f(P,Y.T) (3.6)
Thermal Properties:
h - f(X,T) (3.7)
H - f(Y,T.P) (3.8)
hf - f(Xf,Tf). (3.9)
Equations (3.1)-(3.3) can be solved simultaneously and then
equations (3.4)-(3.9) can be used to calculate column profiles of L, V,
T, X, Y and P to any step input. Equations (3.1)-(3.3) form a 'stiff
set of differential equations. This means that the system is made up of
a system of equations that represents both slow and fast dynamics. The
overall mass balance being the fastest to reach steady-state and the
component mass balance being the slowest. Numerous numerical techniques
have been proposed to solve this system of equations. Lamb et al.
(1961), used the Runge-Kutta method with the Adams predictor-corrector.
Luyben (1973), used Euler's method. Gear's method which was designed to
solve stiff systems was used by Holland and Liapus (1983).
25
3.2 Modeling of High-Purity Columns
Modeling of a high purity column poses a very difficult problem,
because of the composition split. One of the objectives of this study
was to simulate a high purity distillation column. The system chosen
was ethane-butane splitter, with light component compositions between
0.012 and 99.992 in the bottom and the top. respectively. Equations for
this system are the same as those discussed in section 3.1. In the
course of this study a couple of different numerical techniques were
tried, but the Runge-Kutta method was implemented and proved to be quite
efficient. The assumptions used in modeling this column are:
(1) Constant pressure throughout each tray, but varies linear up the
column.
(2) Coolant and steam dynamics are negligible in condenser and
reboiler.
(3) Constant liquid and vapor holdups on the trays, reflux drum and
condenser.
(4) Perfect liquid mixing at every tray.
(5) Molar holdup on trays were calculated from the Francis weir
formula.
(6) There is only one feed tray on which either liquid or vapor feed is
introduced.
Most of these assumptions are standard for a column. However,
assumption (3) assumes that the vapor and liquid holdup on any tray is
constant. The liquid and vapor dynamics for a system such as ours are
26
so fast that they hardly provide any extra information but consume a lot
of extra computer time. While the component balance alone can provide
all accuracy and information required, previous work done by Sourisseau
and Doherty (1982) support this assumption. The feed can be either a
liquid or vapor or a mixture; the only changes that have to be made are
in equation (3.2a). Assumption (3) also eliminates the need for any
thermal properties or the use of equations for flow. The phase
equilibrium can be calculated using one of two options; one is by doing
a bubble point calculation on each tray and the other is only valid for
a binary system and that is to use constant relative volatility (a) that
varies linearly. In this work constant relative volatility was used,
because it provides the required accuracy in less computational time and
is easier to implement. Constant relative volatility is defined as:
a - K1/K2 - Y1X2A2X1 (3.10)
and hence, the equilibrium curve can be calculated using:
Yi - Xia/(l+(a+l)Xi). (3.11)
The molar holdups on the trays are calculated using the Francis weir
formula:
L - 3.33whl-5 (3.12)
where L is the liquid rate (ft3/sec)
w is the weir length (ft)
h is the weir height (ft).
Figure 3.2 is the flowsheet for the code used in this project. The
main program is the driver program; all the data needed to run the
column is fed directly into the first few lines of the program and the
composition data is read in from the data section of the program,
27
subroutine DATAIN contains all the initial data for the controllers,
comment cards throughout the program indicate exactly where all the data
is to be fed in. The bottom section of the main program contains the
ordinary differential equation solver, in this case, the Runge-Kutta
method. Subroutine FX is called from within the equation solver and it
contains the call statements for the subprograms that perform the phase
equilibrium and molar holdup calculations, it also has the ordinary
differential equation required to do dynamic calculations; Figure 3.3
illustrates the flowsheet for the subroutine FX and the subprograms
called from it. A fully documented listing of this program appears in
appendix A,
28
' N
Ln+1 Vn
Ln Vn-1
FIGURE 3 .1 Typica l s e i v e t r a y N
29
sun
feed in alldau
initialize
set tareet tune
ves
no
controller
call subrouone func to setup ODES
equation soKrr
ves
parameterize
no
pnnt icsults
FIGURE 3.2 Flowsheet of the main driver program
w
30
CaU LDHUP
Calculate Y
Set up the ODE'S
END
(a)
(b)
FIGURE 3.3 Flowsheets of subroutines a) Subroutine FX b) Subroutine LDHUP
CHAPTER 4
PROCESS MODEL-BASED CONTROL
OF A HIGH-PURITY COLUMN
4.1 Process Model-Based Control
Process model-based control (PMBC) is a nonlinear control scheme,
which, as its name indicates, is based directly on an approximate model
of the process in study. The advantage of using mechanistic process
models in control is that they reflect the nonlinearities of the real
process and they account for interaction among process variables. Using
this type of a model to predict the control action required to meet the
control objectives can be expected to improve the control performance
over the simple traditional cause-and-effect controllers. Therefore,
PMBC can predict control actions to move all controlled variables to
their setpoints while single loop may interact with each other.
Cott et al. (1986) presented selection techniques for approximate
models of process model based controllers for distillation columns. In
their article they state what the qualities of an ideal model ought to
be and then go on to develop a selection procedure using the following
guidelines:
(1) Model Accuracy
(2) Model Selection Procedure
(3) Model Parameter Update.
31
32
They considered the following short-cut models:
(1) the Douglas-Jafaery-McAvoy model
(2) the Edmister Group method
(3) the Fenske-Underwood-Gilliland model
(4) the Smith-Brinkley (SB) model
and concluded that the SB model was the best-suited based on its
accuracy and computational ease for controlling a distillation column.
Based on their conclusion we used the SB model to design our process
model-based controller.
4.2 Parameterization of Process Model-Based Controller
In the implementation of PMBC, the model parameter and control
action calculations are the two major steps. The model parameters,
which are the theoretical stages in the stripping section (M) and the
number of theoretical stages in the entire column, are updated only once
and the control action is performed periodically at each control
interval.
Steps involved in updating the model parameters are as follows:
(1) Measure the distillate rate (D), bottoms rate (B), the liquid and
vapor compositions (x,y), liquid rate (L) and the pressure (P)
from the column.
33
(2) Calculate pseudo-feed qualities from products.
FP - D + B (4.1)
zP - (Dy + Bx)/FP. (4.2)
(3) Calculate reflux ratio (R) and internal flowrates (L) and (V).
R - L/D (4.3)
LP - F + L (4.4)
VP - LP - B ( .5)
V - VP. (4.6)
(4) Calculate feed, top (Tt) and bottom (Tb) stage temperatures.
zPKi(Tf,P)+(l-zP)K2(Tf,P8)-l (4.7)
y/Ki(Tt.P)+(l-y)/K2(Tt.P) - 1 ( -8)
xKi(Tb.P)+(l-x)K2(Tb.P) - 1 ( -9)
where Tf is the feed temperature.
(5) Calculate the average temperatures in the rectifying and stripping
sections.
Tn - (Tt + Tf)/2 ( -10)
Tm - (Tf + Tb)/2 ( -11)
where n and m are tray numbers in the rectifying and stripping
sections, respectively.
(6) Calculate the separation factors (S) for each section.
Sni - Ki(Tn.P)V/L i - 1.2 ( - 2)
Smi - Ki(Tn„P)V/L i-1.2. (^13)
(7) Calculate h-factors.
hi-[Ki(Tn,.P)L(l-Sni))/lKi(T„.P)LP(l-Sn,i)] i-1.2. (4.14)
(8) Calculate current recoveries for each component.
34
fl ' Bx/(FP2P) (4.15)
f2 - B(l-x)/(FP(l-zP)). (4.16)
(9) Solve for model parameters using the recoveries. The two equations
are then solved simultaneously to determine the number of
theoretical stages in each section.
fi-l(l-Sni^"")+R(l-Sni)]/lalpha] i-1,2 (4.17)
alpha-(l-SniN-M)+R(l-Sni)+hiSniN-M(l.Snii"+^).
Figure 4.1 is a flowsheet of the subroutines involved in parameterizing
the system. The subroutine PARM is the parameterizing subroutine called
by the main program when initializing a system. Subroutine PARM then
calls subroutine DERP, which generates the Jacobian from the recoveries,
that are generated in the subroutine FUNC which is called by DERP.
Subroutine FUNC solves equations (4.1) through (4.17) to evaluate the
required recoveries. Subroutine DERP then generates the Jacobian. Then
Newton's method is used to solve n nonlinear equations containing n in
our case n-2, unknowns as follows:
n
(6fk/6xi)l d(J)i- -fk(x(J)) k - 1,2 n (4.18)
i-l IxU)
where Xi(J+l) - Xi(J) + d^W
where fy is the function and Xi is the independent variable. This
Jacobian is a 2 X 2 matrix. The linear forms of the two nonlinear
equations is given by:
35
A(1,1)AI + A(1,2)AN - -B(l) (4.19)
A(2,1)AI + A(2,2)AN - -B(2) (4.20)
where Al and AN are deviations in the number of theoretical
stages. The solutions to these equations are then used to update the
guess values for the number of trays in each section of the column.
Convergence is assumed when the difference between the deviation in the
number of theoretical stages is less than the set error limit.
4.3 Implementation of the Process Model-Based Controller
Once the system has been parameterized, the control actions can be
put to work. Control action is taken after a certain time step which is
set by the main program. The calculations involved are as follows:
(1) Determine the setpoints (y* and x*), and the number of trays in
each section(N and M) from the model parameter update.
(2) Measure the feed rate, composition and pressure (F, z, and P) from
the column.
(3) Calculate external flowrates required to obtain the desired
products.
D - F(z-x*)/(y*-x*) (4.21)
B - F - D. (4.22)
(4) Calculate recoveries for the desired products.
fl - Bx*/(Fz) (4.23)
f2 - B(l-x*)/(F(l-z)). (4.24)
(5) Calculate the new top and bottom temperatures.
36
yVKi(Tt,P) + (l-y*)A2(Tt.P) - 1 (4.25)
x*Ki(Tb,P)+(l-x*)K2(Tb,P) - 1. (4.26)
(6) Solve equations (4.3) and (4.7) to solve for reflux ratio and
feed stage temperatures.
(7) Calculate the new liquid and vapor rates.
L* - RD (4.27)
LP* - F + L* (4.28)
VP* - LP*-B. (4.29)
(8) Implement control changes L* and VP* on column.
Figure 4.2 illustrates how the control subroutine (CONT) is structured.
CONT calls the subroutine DERC to obtain the Jacobian for the change,
DERC works similarly to DERP and it also calls FUNC to calculate the
recoveries and that generates the Jacobian. CONT uses the Jacobian in a
manner similar to PARM and obtains the required control actions which
are then implemented to the liquid and vapor rates.
37
intiol data
call DERP for Jacobian
solve equation for deviation in parameter
(a)
Stan
call func for Jacobian 1 caJcD(l)«t B(2)
update N
caJl tunc
caJc A ( I . l ) & A( l . : )
updJic M
caJc AC.DcS;: AC.Z)
return
(b)
FIGURE 4.1 Flowsheets
a) Subroutine PARAM b) Subroutine DKRl'
38
Stan
feed ill all data
call DERC
caJc control action for
update V & L
return
FIGURE 4.2 Flowsheet of subroutine CONT
CHAPTER 5
RESULTS AND DISCUSSION
The PMBC controller was tested on a simulation of a high purity
binary column with several setpoint changes and a variety of
disturbances. Table 5.1 shows the base case conditions with all the
values. The tests performed were: changes in the top and bottom
setpoints, disturbances in the feed rate and composition, variation in
the liquid and vapor with an increase or decrease in the bottom
setpoint, and we also compared how the controller behaves for a SOX
increase or decrease in the gain. In order to fully understand the
effect of these tests, the top and bottom product compositions were
observed during each test. Results from PMBC were qualitatively
compared with results from a similar high-purity column studied by
Luyben et al. (1988) using nonlinear djmamic matrix control (NLDMC).
In order to study the effects of changes in the setpoint for the
bottom light component composition (XSP), the value of XSP was increased
by 15X and the decreased by lOZ (it could not be decreased more than 10%
due to total reflux restrictions). Figure 5.1 shows the effects of an
increase in XSP on the LK composition in the bottoms product. Whereas,
it has no effect on the LK composition on the top product. From Figure
5.1 we can see that around the 10- and 15-minute mark it shows a
disturbance. This happens throughout the whole column. Figures 5.2 (a)
and (b) show the same effects on trays 5 and 8, and this was found to
occur throughout the whole column for every upset
39
40
or disturbance tested. The effects of a decrease in XSP can be seen in
Figure 5.3 and again there is a disturbance at the 10- to 15-minute mark
and then the system reaches 95X of the steady state changes in about 5
time constants (process time constant is about 5 min). Once again there
is no significant change in the LK composition on the top product.
Effects of an increase in setpoint of the LK composition (YSP) can be
seen in Figures 5.4 (a) and (b). Figure 5.4 (a) shows the effect that
this change had on the LK composition in the bottoms and Figure 5.4 (b)
shows the effects on the top composition. Disturbance rejection caused
by a setpoint change in the top (YSP) causes a sharp drop in the bottoms
composition. This shows that the bottoms composition control is very
sensitive. The bottoms composition behaves in a similar way to a
decrease in the value of YSP. Figure 5.5 (b) shows the effect of a
decrease of YSP on the LK composition in the top of the column.
The next set of tests was done using a small upset in either XSP or
YSP along with a step change in the feed rate of 5, 10 or 15X. Figures
5.6 through 5.8 show how the LK component compositions in the bottoms
product for a small increase in XSP and for a 5, 10 or 15X increase in
feed rate, respectively. LK compositions for the top product are not
shown because there is less than a O.OOOIX change in that composition
during these tests. Similarly, Figures 5.9 and 5.10 show the behavior
of the LK composition on the bottom tray for a decrease in the feed rate
by 5. 10 or 15X. In all these cases the LK composition in the bottoms
behaves in an almost identical manner except that it reaches steady-
state faster in cases with higher feed rates. These tests were repeated
for a small increase in the value of YSP, Figures 5.11 through 5.13
41
show the trend of the LK compositions in the bottoms product for an
increase in feed rate while Figures 5.14 through 5.16 show the trends
for a decrease in the feed rates. There was a strong similarity in the
results from each of these test for both the top and bottoms products.
The next set of disturbances studied were in the feed composition
of the LK. Disturbance of +5X and +10X was made in the feed composition
of the LK and the trends of the LK composition for the top and bottoms
product were studied. Figure 5.17 shows how the LK composition changes
in the bottoms and Figure 5.18 shows how it is affected in the top of
the column for a +5X step change in the feed composition. Similarly
Figures 5.19 and 5.20 show the trends for the bottoms and top,
respectively, for +10X step change.
Luyben et al. (1988) extensively compared linear and nonlinear DMC
for three sets of columns, the moderate purity column which is 99X and
IX LK in the top and bottom, respectively, the high purity column with
product purities of 99.9X and O.IX of the LK in the top and bottom,
respectively, and then the very high-purity column with impurities
down to 10 ppm. The column in study for this project falls in the high-
purity category, and so some of the results from this study were
quantitatively compared to their results, even though the system may not
be exactly the same. Figures 5.21 (a) and (b) and 5.22 (a) and (b) are
a comparison of the behavior of the HK on the top product for a +10X and
+5X step change in the feed composition. While Figures 5.23 (a) and (b)
and 5.24 (a) and (b) show the trend of LK on the bottoms tray for the
same changes, respectively. From both these figures it seems that in
42
their column the controller had more of an effect on the top product
than this study. Their column also seems to show an initial disturbance
in the bottom tray like the one used in this study. Based on these
observations it seems that the PMBC is faster since it reaches 95X
steady state in 5 time constants while NLDMC takes 10 time constants,
also observed was that the PMBC gave a lot less upset than the NLDMC.
Effects of an increase or decrease in the bottom setpoint on the
liquid and vapor flowrate in the column are shown in Figures 5.25 (a)
and (b) and 5.26 (a) and (b), respectively. In both cases we can see
that the vapor rate is effected a lot more than the liquid rate, that
could be explained by the fact that both graphs are for an upset in the
bottoms setpoint. For a decrease in the setpoint both values increase
to compensate and then level off, however, for an increase they decrease
and then level off.
Figures 5.27 (a) and (b) prove that the controller has to be
carefully tuned to obtain the right results. In both the increase and
decrease the system becomes unstable and can no longer be controlled.
This goes to show that the PMBC is very sensitive to the control
parameters.
TABLE 5.1 Data for PMBC base case
Xf
F
R
V
XSP
YSP
N
M
NT
K1
K2
0.3
4000
6000
7000
0.47864 E-3
0.99888
21.3
8.872
18
2.05
2.05
43
SYSTEM : C2 C4 MIXTURE
44
>^ (fl u iJ
e o 4J U O
JO
a o cu CO
X
o VI
0) u o C C
o
O (U
tM
i n
UJ
:^ o (—1
(% 3noH) Mcaisodrioo iNBNOdrDO iHon
1#
2.24.
2 . 2 2 -l{
y o JiS.
^ ' z Q. • -en O OL
5 O
»--UJ
z o a •-• >
^
5 U -J
2 .2
2 . 2 a
2.2B ,
2 . 2 2
2 .2
2 .16
2 . I d
2 . i a
2 .12
•J
r Q
O
a
UJ
O a u •— I c
45
—r-20 'lO
— 1 —
CO — I —
BO 100 120
TIMC (MIN)
a
2 1.3 -f
TrMC ( M I N )
b
FIGURE 5.2 Increase in XSP a) Effect on Tray 5 b) Effect on Tinv 8
46
o
o -*
o r< »-
o o r-
o 03
O u
o -i
O n
^^ z 5 UJ
1-
r-l
n T
ray
o
n X
SP
o 0) V) (Q 0)
U 0)
CO
(4-1
o u u
3 E
ffe
•
ICU
RE
5
Uu
<% JlCiN) NCIilSOdltCO iN3NOdr*:X) iHOH
I
47
C.Ci
li 0.0J.7B -
0.0-176 -
Q0.04.7A -
^O.GA72 -
B O.OA7 -1 CO
po.OAca -
§ o.OAea -•- 0 . 0< id4 -
z Z O.O.ifi2 H o
^ O.OAC -4 ^o.OAsa -I
•^O.OiSfl - I o
0.0-t?4. -J
0 . 0 4 J 2 -I r —T— 3 0
t
0 0 I I I
flO 100
T1»/.C (MIN)
a
1 2 0 l A O lao — I leo
9 9 . 9
SS 09 -
^ 99 .00 W
- 9 9 . 0 7 -J g ^. 9 9 . 2 0 -
^ 9 9 . 0 * -] UJ
^ 99 02 ->
O 99 .02 -
9 99 .01
99.S -r-A, a
T >
> I O 1
TlMC ( M I N )
b
1 a 18
FIGURE 5.4 E f f e c t of d e c r e a s e in YSP a) Tray 1 b) Top t r a y
0.04S
in
go.0450 -§ 0 . 0 4 6 6 -O ^ 0 . 0 4 6 4 H UJ z O . 0 4 6 2 H O
% 0 .046 H
O0.045Q
g 0 . 0 4 5 6 -
"^0 .0454-
0.0452 4
99.9
A 99.09 H
UJ
2
99.00 -
99.07
-00 99.06 O Q.
^ 99.05 -
^ 9 9 . 0 4 -UJ
2 99.03 -
O gg.02 -. 99.01
99.0
48
20 40 I
60 —r~ 00
—I r 100 120
I 140 160 100
TIMC ( M I N )
1^
2 4 6 0 10 I
12 14 16 -J—
10
TIME ( M I N )
FIGURE 5.5 Effect of increase in YSP a) Tray 1 b) Top tray
49
0 . 0 5
0.04.9S -
0 .04-96 -
0.04.94. -Ul
2 0.04-92 \ - / Z O 0.04.9 H t
p 0 .04 .88 H
Q 0 . 0 4 5 6 - I
1 0 . 0 4 8 4 . - I O
0 . 0 4 8 2
0 . 0 4 8 -
0 . 0 4 7 8 - — I — 20
—T" 4 0
—1— BO
T r "I 1 1 r T r BO 1C0
TIME (MIN)
1 2 0 140 160
FIGURE 5.6 I n c r e a s e i n XSP and 5X inc rease in feed r a t e ; Tray 1
50
r\ r.n
0.04.39 -
0 .049B -
_ O.OAga -O ^O.OA92 H S ^ C.04.9 -8 ^..,o.OAee -
o *° 0.0- ie J.
p .04f i2 -
0 .04a -
O.OJ.78
TIMC (MIN)
FIGURE 5.7 Increase in XSP and lOX increase in feed rate; Tray 1
r. f\*^
Q •^O.OJ-SA -
t:0.0A92 -
o & o.oag -p ^o.ooc- -
^0.0-tse -o
8 •-O.O.i£2 4 I o Ij o.o-ie -
o.o«t~s — I —
20 A.O eo — I —
eo I I —
100 120 — I 1
I A O ICO
TIME (MIN)
FIGURE 5.8 Increase in XSP and 15X increase in feed rate; Tray 1
51
y 0.04-98 -Q ^ 0 . 0 4 . 9 4 . H
t : 0 . 0 4 . 9 2 H en O 9: 0 .049 -
o . 0 4 e e -
^ 0 - 0 4 . B B O a 5 O.o-teA -
» -0 .04e2 g - I 0 . 0 4 S
0 . 0 4 . 7 8
2 0 I
4 0 CO OO 1 0 0
TIME ( M I M )
— I 1 2 0 1 4 0 ICO
FIGURE 5.9 Increase in XSP and lOZ decrease in feed rate; Tray 1
. ^ •0 .04 -9 3 -
• y 0 . 0 4 . 9 6 -O
^ 0 . 0 4 9 4 z Q t 0 . 0 4 9 2 -01 O 9: 0 .049 -
-I Z 0 . 0 4 Q 6
O , II °> O.U4-e4.
t - 0 . 0 4 e 2 -X -1 0 . 0 4 8 -
O . O 4 7 0
/ y
-I r :o
— I —
e o 8 0 1 0 0
Tlf^C ( M I N )
— I r i r o 1 4 0 i c ;
FIGURE 5.10 Increase in XSP and 15X decrease in feed rate; Tray 1
52
0 . 0 4 7 9
C O . 0 4 7 8 -
•^ 0 . 0 4 7 7
r 0 . 0 4 7 0 -
^ 0 . 0 4 . 7 5 H
a t> 0 . 0 4 . 7 4 -a 5 0.04.73 A
o a 0 . 0 4 7 2 -
~1
0 .04 .71 -X o D 0 . 0 4 7 -
0 . 0 4 f i 9 — I — 2 0
— I — 4 0
~T r CO
— I —
eo 1O0 I Z O
T I M E ( M I N )
FIGURE 5.11 Increase in YSP and 5X increase in feed rate; Tray 1
0 . 0 4 7 9
^. /•• /-, ^r» o _ \i
y 0 . 0 4 7 7 -o
• ^ 0 . 0 4 7 0 -
C, t 0 . 0 4 7 S H
& 0 . 0 4 7 4 -
^. 0 . 0 4 7 2 -
2 0 . 0 4 7 2 H
d 5 0 . 0 4 7 1 -O
•- 0 . 0 4 7 -I c - I 0 . 0 4 f i 9 -
o.04£e 2 0 4 0
— I —
eo — I —
OO — I 1 1 r 1 0 0 i r o 1 AC
T I M E ( M I N )
FIGURE 5.12 Increase in YSP and lOZ increase in feed latc; Tr.TV 1
53
o ti T "
o o * -
o CI
^^ u to
o ^
o r
4 ^ \
^ t:
v ^
u 2 1 -
Tray 1
• « 0)
feed rat
c
ease i
u u c
•*4
»4 i n r-l
P and
CO
rease in Y
o c
UJ oi
o • -
r -*
o o
r -^
o d
o> 03 -4 o
t o
in ID
-i O
•
o (% 31CV^) NOmSOdkVOO irONOdl-'OO iHOn
54
\
v..
^—I—r on
o
ID Iv
O o I
o
o t
o
O *
o
O I
o
^ d j
'i O
IN (i3 -f O
a) m
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C'
o
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o o
o OO
o u
o ^
o rt
- r
5 V w *
u 2 h-
.-(
e; Tray
i j (B
feed r
c . f t
0) w (tJ 0) k l
o
and
5X
e in YSP
V)
ncrea
lU
q d o
TlCi'-O NC1il50dk*X) irONOd^'iDO iHOH
i n
UJ
55
g o d
o I
o
Ii 5 ' s ^ ? 5 5 ?! ' "J o q (.-) o 9 9 9 9 d 9 9 o o o o o o o o o :, TicvV NCiiisod^voa iN3NOd ' *oo man
-i
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n !3 O O
u H
(U iJ cfl
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u o (U
T3
o
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in
0)
(T3 0)
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i n
i n
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o
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o o
I T 1 1 1 f' r4 fv nJ ?? •«*
=f=r ft to
5 5 Ii 3 g "-/ 3 O O O o' ^ ^ * 6 6 6 c o o
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o fl
o o
-r
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T
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"
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—
o - o ^
o ' tt)
o 03
O
o ' r
1
^
/ ^ i .
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Ui "«;
F
ray
H
0) 4J Cd V4
•d <u
M-l
c ••-1
0) en Cd
u 0) •0 »4 »n t - i
•0 C Cd
5«
C • f l
0)
crea
s
r - (
•
9 ^ o • o
• •
o o
i n
o (—1 U4
56
57
C O 4.73
^ 0 . 0 4 . 7 8 -
y 0 . 0 4 7 7 H
§ 0 . 0 4 . 7 6 -
2 0 . 0 4 7 5 -
Q S 0.04.74. -O % 0 .0473 -
0 .0472 -»-S 0.04.71 HI Z
2 0 — I — ' t o
I BO
— I — BO 1 0 0 1 2 0 1 4 0
T IMC ( M I N )
FIGURE 5.17 +5% Upset in feed composition; Tray 1
9 9 . 8 9
- 9 9 . e s -
V 9 9 . 8 7 -\
c/i 2 99.BB -
99.85 -
UJ
2 "• ' '^• '
2 99.83 -
O 9 9 . 8 2 -
9 9 . 8 1 -
9 9 . 8 2
-T -e
I
a — I — I O
— I — 1 1 — I 1 — I — I — 1 2 1 4 1 e 1 8
TIMC ( M I N )
FIGURE 5.18 +5X Upset in feed composition; Top Tray
58
0.04 .79
0.04.7B
"«'0.04.77
^ 0 . 0 4 . 7 B
3^0.04.75
0.04 .74 -
^ 0 . 0 4 . 7 i
2o.04.72
0 0.04.71
•_ 0 . 0 4 7
y 0 . 0 4 e 9
0 . 0 4 e B -
i 0 . 0 4 e 7
^_o.04aB X g o . 0 4 e 5 - I
O.OAfiit
0 . 04 f l3 1 1 1 1 1 1 1 1 1 1 1 1 1 r r —
O 2 0 4 0 BO BO TOO 1 2 0 1 4 0 1 0 0
TIMC ( M I N )
FIGURE 5.19 +10Z Upset in feed composition; Tray 1
9 9 . 9
^ 9 9 . 6 6 -a C 9 9 . a s H
2 9 9 . 8 4 -\ z O a 9 9 . 8 3 H "5
8 o
9 9 . 8 2 -
9 9 . 8 1
9 9 . 8 -+ — I —
I O — I —
2 0
T i v r ( M I N )
— I —
.30 — 1 J O
FIGURE 5.20 +103: Upset in feed composition; Top Trav
0 . 0 0 2
o.ooig -
2 0 . 0 0 1 B -
g
JC0.0017 -\
CO.OOIB -
59
Z 0 . 0 0 1 S -
0 .0014 . -
^ O . O O l 3 -»-X
3 0 . 0 0 1 2 -
0 . 0 0 1 1 -
0.001 2 0 4 0 6 0 BO 100
TIMC (MIN)
a
1 2 0 — 1 1 1 1 — 1 4 0 160 180
0.04.79
0.0470 -
^ 0 . 0 * 7 7 H Ui r^0.0*76 o ^ 0 . 0 4 7 3 -
gO.04.74 -
S0.0473H
2o.0472 H
0 0 . 0 4 - 7 1 O
K- 0 . 0 4 7 -
^ 0 . 0 4 6 9 -
g 0 . 0 4 G 0 -d o . 0 4 6 7 -O ^ 0 . 0 4 6 6 -X g 0 . 0 4 G 5 --J
0 . 0 4 6 4 -
0 . 0 4 6 3 1 6 0
FIGURE 5.21 Effect a of lOX upset in feed composition a) (1-XD) on top tray b) Tray 1
60
- - 14 .0 O
T r T > r 1 r-
^ 12.0 - ; V
. /
<. 10.0 cc
§ 8 .0
^^.
*^NLDMC
a X I
6 .0 -L
eO.O 160.0 240 .0 320.0 400.0 480.0
TIME (MIN)
4 4 . 0
2 36.0 X
-T 2B.0 »-o -<• IZ u.
1 — r T — ' — r
2 0 . 0 -J o I
C3 1 2 . 0 X
4 . 0 I L
BO.O 1 6 0 . 0 2 4 0 . 0 3 2 0 . 0 4 0 0 . 0 4 8 0 . 0
TIME tKJN)
FIGURE 5.22 Effect a of lOZ upset in feed composition using rJLDMC a) (1-XD) on top tray b) Tray 1
Ii
61
O.O02
O.OO J 19 -
o.ooie -
0 . 0 0 1 7 -
Vo.ooie -X o'o.oois -I I .'O.O014 -
0 . 0 0 1 3 -
O.0012 -
0 . 0 0 1 1 4
0 .001 t—r
0.0479
^ 0 . 0 4 7 0 -H ujO.0477 -_ j
§0.0476 H
= 0 . 0 4 7 5 -Q ^ 0 . 0 4 7 4 -O ^ 0 . 0 4 7 3 H O <->0.0472 -
i j jO.0471 -
a 0 .047 H
o 0 . 0 4 6 9 H
O 0 . 0 4 6 0 -
0 . 0 4 6 7 -
0.0466 4
—I r 2 0 eo
I
BO 1 0 0 1 2 0 1 4 0 leo 1 8 0
TWkCC ( M I f . ' )
a
60 00
TIMC ( M I N )
1 40
FIGURE 5.23 Effect a of 5Z upset in feed composition a) (1-XD) on top tray b) Tray 1
62
Q X
I
20 .
0 1
LV
^ . LOMC.
CD X
80 ,
40 ..
0 1
NLDMC
0 30 60 90 120 TIME (MINUTES)
150 180
FIGURE 5.24 Effect a of 5X upset in feed con,position using NLDIU a) (1-XD) on top tray b) Tray 1
63
TIMC (MIM)
a
1.90
u
\ C".
o • UI
< a' Q
X O c t >
I . t c - r -
O 2 0 4 0 eo Tl?/.C ( ' . ' I* . ! )
?
eo i c ;
FIGURE 5.25 Effect of an increase in XSP a) Liquid rate b) Vapor rate
1 . 4 S
o Ui \ (A
UI
a:
9
O 3
64
O BO
TIJ/.C (MtM)
eo 1 0 0
1.9
' J UJ VI
t/1 UJ
d
or
O
O
5
I.OS A
1.B
1.75 H
1.7
i.es
i.e -
1.55 -
o 2 0 —r— 4 0 eo
Tll/.C (»XIN>
eo I C O
FIGURE 5.26 Effect of a decrease in XSP a) Liquid rate b) Vapor rale
0.075
UI
0.11
0.1 -
0.09 -
o c
S' c a « 8 -
Ul
c
o.oc
0.07
O.OB
0.C5
y.
c O.C.3 -
O
TWAZ, (MIN)
a
65
T'M^ (*'"N)
FIGURE 5.27 Effect of a change in gain a) Increase b) Decrease
CHAPTER 6
DEVELOPMENT AND VERIFICATION OF A STEADY-STATE
APPROXIMATE MODEL OF A DISTILLATION COLUMN
WITH A SIDESTREAM DRAWOFF
6.1 Model Derivation
This material presents the general development of an approximate
model of a distillation column with a sidestream drawoff. A
relationship between the concentration of any given component and the
stage number can be developed using the calculus of finite differences
as follows:
write a component balance around stage n+1 for any component to obtain
Lr,+2Xn+2 + V^Yn - Ln+lX^+i + Vn+i^n^l (6.1)
since Y^-K^Xn and Yn+i-K^+iXn+i
then equation (6.1) becomes
Ln+2Xn+2 + V^K^Xn - l^+lXn+l + "^u-^l^n-^l^-^l- (6-2)
Rearranging gives
Xn+2 - [(Kn+lVn+l/Ln+2) + (Ln+l/Ln+2)]Xn-l + (^'^n/^^2)^n " 0.(6.3)
To simplify things the distribution coefficient and phase rates must be
assumed constant within the column section under consideration. Making
these assumptions and writing the equation in terms of a linear operator
form gives
66
67
[E2 . (KV/L + 1)E + KV/L]Xn - 0 (6.4)
or
(E - KV/L)(E - l)Xn - 0 (55)
where E defined as the linear operator.
Let S and SA represent the two roots
S - KV/L and SA - 1.0.
The solution is then
Xn - C (S)n + CA. (6.6)
To implement this on our column the column is divided into three
sections as shown in Figure 6.1. Based on these assumptions and Figure
6.1 the following equations can be written for the different sections:
section I
Xi - CiSi^ + C2 (6.7)
section II
XN - C3S2N + C4 (6.8)
section III
XM - C5S3M + C6 (6.9)
where I, N, and M are the number of trays in sections I, II and III,
respectively, and S is the separation factor and C are integration
constants. The equation for the separation factor for each section is:
68
SI - KiVi/Li I - 1,2.3. ^6.10)
The constants in equation (6.7) can be eliminated by simultaneously
solving component balances around the condenser and the top tray.
Similarly constants in equation (6.9) can be determined using a
component balance around the reboiler and tray 1 or bottom most tray.
Then C3 and C4 can be determined by doing component balances on the tray
above the feed tray and below the sidestream drawoff tray.
The final forms of these equations are as follows:
section I
Xi-(l-f-g)A{Li(l/Ki.l)/V){SiI-i-l)/KiD(l/Si-l)
(l-f-g)A/KiD (6.11)
section III
XM-[fA/(B(l-S3)){l-(K3V+B)/L3){S3M-l-l))]+fA/B (6.12)
section II
XN-(l/(l-S2")(A/F.gA/G)(S2N.l) + A/F. (6.13)
All the nomenclature is defined in appendix A. The product rates f and
g for the bottoms and sidestream drawoff respectively can be obtained by
doing component balances on the feed tray and sidestream drawoff tray
and solving them simultaneously. The two product rates are as
follows:
69
(aOi+a+VK2di/G+L2/G+VK2/G+l] [ -L2dKf/K2-L2Kf/K2+VKf+L3+F] -
[-L2/G](aOi+a+(di+l)VK2/F]
^" (6 .14)
(VK3( -1)/B][aOi+a+VK2di/G+L2/G+VKg/G+l]-[aOi+a]
(-L2d/G]
and
(aOi+a+(di+l)VK2/F]-[aOi+a)f
S" . (6.15)
(aOi+a+(dx+l)VK2/G+L2/G+l]
and from a material balance
dd- 1 - f - g (6.16)
where d - (S2-l)/(l-S2") (6.17)
dl - (S2"-l)/(l-S2") (6.18)
0 - (S3»-2-l)/(l.S3)(l.(VK3+B)/L3) (6.19)
01 - Li(Sil-i-l)/(l/Si-l)(l/Ki-l)/V (6.20)
a - Li/(Kx D). (6.21)
The solutions of equations (6.14), (6.15) and (6.16) will result in
the required recoveries of component I in each of the exit stream.
Section 6.2 discusses how the program that solves these equations is
implemented along with parameterization of the system.
70
6.2 Implementation of the Approximate Model
6.2.1 Data Entry
The first step in implementing the approximate model is to
parameterize the model using initial guesses for the number of
theoretical stages in each of the sections. The temperatures and
pressures for the sections are set by the distillate, sidestream, feed
and bottoms temperatures and pressure, respectively. Values for the
feed rate, bottoms rate, boilup rate, sidestream drawoff rate and the
reflux rate are read into the program. All other values can be obtained
from material balances. The program is also given the values of the
recoveries in the bottoms, sidestream and the distillate.
6.2.2 Parameterization
To parameterize the column the main program calls the subroutine
PARAM that calculates the number of theoretical stages in each section
of the column. This subroutine uses the same approach as described for
the parameterization of the SB model, used in the high purity column
control in chapter 4, section 4.2. Only for this system there are three
parameters instead of two. Figure 4.1 is a flowsheet of the subroutines
involved in parameterizing the system. Subroutine EVAL calculates the
function values for the Jacobian. The function values are the errors in
the recovery equations (6.14) through (6.16). That is, since the number
of theoretical stages are unknown equations (6.14) through (6.16) are
not exact, even though all the other values are known. Subroutine
71
DERP numerically generates the full Jacobian using equation (4.18). The
Jacobian forms a 3 X 3 matrix which is converted into the following set
of equations by Subroutine PARAM
A(1.1)DI + A(1.2)DN + A(1,3)DM - -B(l) (6.22)
A(2.1)DI + A(2,2)DN + A(2.3)DM - -B(2) (6.23)
A(3.1)DI + A(3.2)DN + A(3.3)DM - -B(3) (6.24)
where Dl. DN, and DM are the changes in the number of theoretical
stages in each section of the column.
These equations are solved by Cramer's rule to give the deviation
in the number of theoretical stages in each section. The entire process
is repeated until the deviations become acceptably small.
Parameterization can be done using either the LK or the HK. To be
consistent the recoveries for each component are used to parameterize
the model. That is, number of theoretical stages of each section
obtained from each component is averaged to give the number of
theoretical stages for each section of the column.
6.2.3 Implementation
Once the system has been parameterized, then the approximate model
can be used to estimate fractional recoveries for a variety of
conditions. Figures 6.2 (a) and (b) are the flowsheets of the program
that evaluate the product recoveries f. g, and d. The main part of the
program is used to call the evaluating subroutine EVAL, read in the data
as stated in the first paragraph of this section, and print out the
results. Subroutine EVAL calls subroutine CONST and KVAL for each
component. Subroutine KVAL calculates the K-values using equilibrium
72
constants for each component. Whereas subrotine CONST calculate the
constant 0, d. a, etc., asing equations (6.17) through (6.21). Finally,
subroutine EVAL solves equations (6.14) through (6.16) to obtain the
required recovery estimates. These values are then compared in the next
section for a specified system with values obtained from a commercial
package.
6.3 Approximate Model Verification
A ^1 Model
The system that was studied is a binary with ethane as the light
key (LK) component and propane as the heavy key component (HK). A
commercial program was used to calculate the initial recoveries by which
the approximate model was parameterized. Table 6.1 shows the
comparison between the recoveries of the approximate model and the
commercial program.
6.3.2 InJM'al Parameters
Table 6.2 shows the parameters used in the model and in the
commercial program. A comparison between the model and the commercial
design package was performed by making changes in the reflux rates, the
sidestream drawoff rates, and the boilup rates.
6.3.3 Description of the Data Tables
The three values predicted by the approximate model were the
recoveries in the distillate, sidestream drawoff and bottoms. These
values and the process gains (K) for each of the changes predicted by
73
the approximate model and the commercial package were compared. Process
gain which is referred to as the gain in the rest of this section is
evaluated as follows:
Kc-Change in recovery (f. g, or d)/Change in (L, G. or V) (6.25)
Tables 6.3 (a) and (b) show a comparison of the recoveries when the
reflux rate is increased or decreased by 3Z and Tables 6.3 (c) and (d)
show a comparison of the gain caused by these changes in both the model
and the commercial program. Tables 6.4 (a) and (b) are the comparisons
for a 3Z increase and decrease in the sidestream drawoff rate, while
Tables 6.4 (c) and (d) are gains for the increase and decrease in the
sidestream drawoff rates. Similarly. Tables 6.5 (a) and (b) are the
comparisons for a 3Z change in the boilup rate and Tables 6.5 (c) and
(d) show the slopes for the changes.
6,3.4 Case Study for Process Gain
Tho process gairt p1- icts the direction of change of the measured
value in comparison to the change in a manipulated variable, the sign of
the gains is extremely important because it also predicts the future
values of the liquid and vapor compositions on each tray. For instance,
if changes caused by a change in the reflux rate in both systems are
studied; then logically for an increase in the reflux rate, it would be
expected that the recovery in the distillate and sidestream drawoff to
increase for the LK and decrease for the HK, indicatingP°^^t^"^® gains
for the LK and negative for the HK. Similarly the sign of the gains is
expected to be the same for a decrease in the reflux rate, however, from
Tables 6.3 (c) and (d) we can see that while the model
74
seems to follow this expected result, whereas, the commercial design
package differs in some places. One of the reasons for this difference
in results could be due to the equilibrium package used by the
commercial program, since there is no technique to duplicate the exact
package and it may not be well suited for this system. Also the values
in certain cases are so small that a minor roundoff error in the
commercial program could have an effect on the sign of the gain.
6.3.5 Conclusions
Based on these comparisons we can conclude that the extension of
the Smith and Brinkley approximate model to include the sidestream
drawoff can be used as an independent approximate model for a
distillation column with sidestream drawoff, and also for further
control studies.
II
III
75
D
^
B
FIGURE 6.1 Split of the distillation column with a drawoff sidestream
76
Read in all data
Yes Parameterize
no
Evaluate
rccoveries
Print results
End
Do # of comp.
Calc. K-Values
Calc. constants
Other calcs.
Calc. f,g. & d
FIGURE 6.2 Flowsheets for approximate model
a) Main program b) Subroutine EVAL
77
0) •o o E
0) (fl
E
X o Wi a a CO u o
LU
> > < LLt
O O
o o
T—
r m (D ^ CO o
CO o IO r T -
o
O) C\J V —
CO CO ^
o o
UJ
o o
00 CD
CO
C\J
d o d
w
3 V) 0) i-t
a; en CO u a> (0 CO
CQ UJ
o cc CL
o o
CO CO o> (O CD O O
-^ CO o y—
CO y—
o
r CM o CNJ CO h-o
U -J OQ <
I
o LU a o
r CJ o o 00 o
T —
o CO 00 y—
CD a> CO CO r
O)
TABLE 6.2 Data used in the comparison tests
CASE STUDIED C2 - C3 MIXTURE
78
1.556 mole/sec
0.2778 mole/sec
0.70056 moles/sec
0.7223 moles/sec
L(1)
•R
Tf
Tb
R
Pf
Pb
0.32478 moles/sec
18.3 DEGF
53.6 DEG F
108.6 DEGF
254.10
254.3 Psia
254.6 Psia
79
TABLES 6.3 Recoveries and Process gains for change in reflux rate a) Recovery for 3Z increase b) Recovery for 3Z decrease
f
g
d
LIGHT KEY
MODEL
0.077337
0.18636
0.73630
COMM PROG.
0.084423
0.18175
0.733827
HEAVY KEY
MODEL
0.76862
0.21815
0.013229
COMM PROG.
0.808164
0.179725
0.012111
f
g
d
MODEL
0.082776
0.18562
0.73160
a
COMM PROG
0.089460
0.175913
0.734627
MODEL
0.78113
0.20856
0.010307
COMM PROG
0.81915
0.170305
0.010545
80
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81
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u u > > o o o o (U (U
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84
TABLE 6.5 Recoveries and Process gains for change in boilup rate c) Process gain for 3X increase d) Process gain for 3X decrease
df
dg
dd
LIGHT KEY
MODEL
-0.0001396
0.00001102
-0.00001286
COMM PROG.
-0.000333
-0.000000615
0.0003558
HEAVY KEY
MODEL
-0.00009821
0.00007782
0.00002036
COMM PROG.
-0.000114
0.0000138
-0.00003757
df
dg
dd
MODEL
-0.000159
0.00001244
-0.0001462
COMM PROG
-0.00007587
-0.000005196
-0.009314
C
MODEL
-0.00009821
0.00007962
0.00001871
COMM PROG
-0.00000495
-0.00001839
0.00008633
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
7.1 Conclusions
7.1.1 PMBC
Based on the following conclusions PMBC is an excellent strategy
for controlling high purity columns.
(1) In a qualitative comparison to NLDMC used by Luyben et.al. (1988),
PMBC performs better.
(2) PMBC is also almost twice as fast as the NLDMC and also the upset
due to the disturbance is a lot less in the PMBC column.
(3) Similar to Luyben's observation, we also observed that the bottoms
product response time is much more sensitive than the top to the
various upsets and changes.
(4) A very strong similarity was found in the response graphs for all
the tests that were done. That is, for any disturbance or step
change in which ever direction, the path changes slightly, but on
the overall the response path still remains the same.
7.1.2 Sidestream Approximate Model
This approximate model is an extension of the Smith and Brinkley
approximate model for a distillation column to include sidestream
drawoffs. It is computational easy to implement and compares very well
with commercial packages currently in use.
85
86
7.2 Recommendations
Recommendation for continued research in PMBC for high purity
columns should include:
(1) Further exact comparison with other nonlinear control techniques
such as djmamic matrix control (DMC) and internal model control
(IMC).
(2) Extension of this simulator to include multicomponent columns.
Further research on the approximate model should include:
(a) More detailed validation using other commercial packages.
(b) Update the current approximate model to include a vapor
sidestream.
(c) Adding parameters to include pumparounds on the drawoff streams.
(d) Implement this approximate model along with a distillation column
simulator that has a sidestream drawoff.
LIST OF REFERENCES
1. Alatiqi, I.M., and Luyben, W.L., "Alternative Distillation Configurations for Separating Ternary Mixtures with Small Concentrations of Intermidiate in the Feed," Ind. Eng. Cliem. Proc. Pes. Dev.. v24 (1986).
2. Bryan, K.E., Design and Control of a Heat Integrated Distillation Train, M.S. Thesis, TTU, Lubbock, TX (1985).
3. Chimowitz, E.H., Anderson, T.F., and Macchietto, S., "Dynamic Multicomponent Distillation Using Thermodynamic Models," Chem. Engng. Sci.. v40, no 10 (1985).
4. Cott, B.J., Reilly, P.M., and Sullivan, G.R., "Selection Techniques for Process Model Based Controllers," Presentation at AIChE Meeting. July 1986.
5. Doukas, P.N., and Luyben, W.L., "Control of an Energy-Conserving Prefractionators/Sidestream Column Distillation System," Ind. Eng. Chem. Proc. Pes. Dev.. v20, (1981).
6. Funentes. C., and Luyben, W.L., "Control of High-Purity Distillation Columns," Ind. Eng. Chem. Proc. Pes. Dev.. v22 (1983).
7. Georgiou, A., Georgakis. C., and Luyben, W.L., "Nonlinear Pynamic Matrix Control for High-Purity Columns," AIChE Journal. (1988).
8. Holland, C.P., and Liapis, A.I., Computer Methods for Solving Dynamic Separations Problems. McGraw-Hill, New York, (1983).
9. Huckaba, C.E., May, F.P., and Franke. F.R., "An Analysis of Transient Conditions in Continous Pistillation Operation," AIChE Symposium Series. v46, n59 (1963).
10. Lamb, P.E., Pigford, R.L., and Rippin, P.W.T., "Pynamic Responses and Analouge Simulation of Pistillation Columns," Chem. Eng. Prog. Sym. Ser. . 57, (1961).
11. Levy, R.E., Foss, A.S., and Gren, E.A., "Response Modes of a Binary Pistillation Column," I & E.G. Fundamentals. vl8, n4, (1969).
12. Luyben, W.L., Process Modelling. Simulation, and Control for Chemical Engineers. McGraw Hill, New York (1973).
13. Luyben, W.L., "10 Schemes to Control Pistillation Columns with Sidestream Prawoffs," ISA Journal. vl3, n7 (1966).
87
88 14 Martin-Sanchez, J.M., "Adaptive Predictive Control Systems,"
U.S.A. patent no.4, 197,576 (1976).
15. Martin-Sanchez, J.M., and Shah, S.L., "Multivariable Adaptive Predictive Control of Binary Distillation Column," Automatica. v20, n5 (1984).
16 Morari, M., and Economu, C.G., "Internal Model Control. 5. Extension to Nonlinear Systems," Ind. Eng. Chem. Proc. Des. Dev.. v25 (1986).
17 Rademaker, 0., Rijnsdrop, J.E., and Maarlveld, A., Dynamics and Control of Continous Distillation Units. Elsevier, Amsterdam (1975).
18. Reid, R.C., Prausnitz, J.M., and Polling, B.E., The Properties of Gases and Liquids. Mc-Graw Hill, New York (1987).
19. Roffel, B., and Fontein, H.J., "Constraint Control of Distillation Processes, "Chem. Eng. Science, v34 (1979).
20. Smith, B.D., and Brinkley, W.K., "General Short-Cut Equation for Equilibrium Stage Process," AIChE Journal. Sept. (1960).
21. Sourisseau, J., and Doherty, M.F., "On Dynamics of Distillation Process-IV. Uniqueness and Stability of the Steady State in Homogenous Continous Distillation," Chem. Engng. Sci.. v37, (1982)
22. Tyreus, B., and Luyben, W.L., "Control of a Binary Distillation Column with a Sidestream Drawoff," Ind. Eng. Proc. Des. Dev.. vl4 (1970).
23. Yu, C C , and Luyben, W.L. , "Use of Multivariable Temperatures for the Control of Multicomponent Distillation Columns," Ind. Eng. Chem. Proc Des. Dev.. v23 (1984).
APPENDIX A
LISTING OF THE COMPUTER CODE FOR PROCESS MODEL-BASED
CONTROL OF HIGH PURITY COLUMN
89
90
c********************************************************************* C*** PROGRAM FOR THE SIMULATION *•* C*** AND PROCESS MODEL BASED CONTROL *** C*** OF HIGH PURITY COLUMNS *** C*** *** C*** PROGRAMMER: RUPAK SINHA (B.S. CHE) **• C*********************************** **•*****••*•*••*******•******•****
c THIS PROGRAM FIRST SIMULATES A BINARY HIGH PURITY COLUMN AND THEN
PARAMETRIZES THE SMITH AND BRINKLEY APPROXIMATE MODEL TO OBTAIN THE NUMBER OF THEORETICAL STAGES IN THE WHOLE COLUMN AND THIS APPROXIMATE MODEL IS USED TO CONTROL THE COLUMN.
C C C C C C C C C C c**********************************************************************
THE SYSTFJ1 STUDIED WAS A BINARY MIXTURE OF 02 - Ot WITH 30X C2 AND 702 a*. RANGE-KUTTA METHOD WAS USED TO INTEGRATE THE ODE'S. THE EQUILIBRIUM CALULATIONS ARE DONE USING A LINEARLY VARYING CONSTANT RELATIVE VOLATILITY.
C C C C C C C c c C C C C c c C c c c c c c c c C c c c c c c c c c c
NOMENCLATURE X - THE LIQUID COMPOSITION (MOLES) Y - VAPOR COMPOSITION (MOLES) DYDX - IS THE ODE FOR THE COMPONENT BALANCE ON EACH TRAY TAU - ARE THE TIME CONSTANTS FOR THE CONTROL LAW K - ARE THE EQUILIBRIUM CONSTANTS TFC - TEMPERATURE OF THE FEED IN THE COLUMN (DEG F) TBC - TEMPERATURE IN THE BOTTOM OF THE COLUMN (DEG F) TTC - TEMPERATURE IN THE TOP OF THE COLUMN (DEG F) PC - PRESSURE IN THE COLUMN (PSIA) XF - COMPOSITION OF THE FEED (MOLES) F - FEED RATE (MOLES/SEC) R - REFLUX RATE (MOLES/SEC) V - BOILUP RATE (MOLES/SEC) XB - LIGHT COMPOSITION IN THE BOTTOM (MOLES) YD - LIGHT COMPOSITION IN THE TOP OF THE COLUMN (MOLES) NP - NUMBER OF THEORETICAL TRAYS IN THE WHOLE COLUMN MP - NUMBER OF THEORETICAL TRAYS IN THE STRIPPING SECTION NT - TOTAL NUMBER OF TRAYS IN THE COLUMN ALPHA - CONSTANT RELATIVE VOLATILITY KI & K2 - GAINS FOR THE CONTROL LAW F.RLIM - ERROR LIMIT FOR CONVERGENCE NITR - TOTAL NUMBER OF ITERATIONS HWS - WEIR HEIGHT IN THE STRIPPING SECTION (FT) MUR - WEIR HEIGHT IN THE RECTIRTING SECTION (FT) XLWS - WEIR LENGTH IN THE STRIPPING SECTION (FT) XIWR - WEIR LENGTH IN THE RECTIFYING SECTION (FT) DTS - DIAMETER OF THE COLUMN IN THE STRIPPING SECTION (FT) DTR - DIAMETER OF THE COLUMN IN THE RECTIFYING SECTION (FT)
DEN - DENSITY OF THE COMPONENTS VR - VOLUME OF THE REBOILER (FT3) CAY1.CAY2.CAY3. & CAY^ ARE THE INTERMIDIATE CALULATIONS FOR
THE RUNGE-KUTTA METHOD. T - TIME (SEC) DT - STEP CHANCE
91
C XSP - SETPOINT FOR THE BOTTOM COMPOSITION C YSP - SETPOINT FOR THE TOP COMPOSITION C ALL OTHER VARIABLES ARE DEFINED AS THEY APPEAR IN THE PROGRAM.
C MAIN PROGRAM C
C
C c c
c c c
c c
c c c
IMPLICIT REAL*8(A-H.0-Z) DIMENSION XM(50).X(50).Y(IOO),DYDX(100),TAU(2) DIMENSION CAY1(100).CAY2(100).CAY3(100).CAY4(100).Y1(100) COMMON /ONE/DC. BC, XC. YC,TFC. TTC, TBC. PC. FC. ZC, KT. KC COMMON /TWO/ ERLIM.NITR,K1.K2.TAU.XINT.YINT.DI.ITYPE COMMON /ON9/ALFA(50),NT.R.V,F.XF.NF.DEN,HWS,HWR,XLUS,XLWR,DTS.DTR COMMON /TW9/ XLL(50).XB.YD.VR.VA.YC1(50) REAL*8 L1.NP.MP.NA.M1.K1(2),K2(2).KT(2).KC(2)
READ IN SOME OF THE CONTROLLER DATA
CALL DATAIN
INITIAL DATA FOR THE COLUMN
XB-.A786AE-3 Y D - . 9 9 8 8 8 NTRAYS-16 NT-NTRAYS+2
NF IS THE FEED TRAY
NF-10 THE LARGEST AND SMALLEST VALUE FOR THE CONSTANT RELATIVE
VOLATILITY
AT-A.56 A B - 4 . 0 5 BETA - 0 . GAM - l . O D - 0 1 DA-(AT-AB) /17 . DO 77 I - l . N T
77 ALFA(1)-AB+DA*FL0AT(I-1) R - 6 0 0 0 . 0 / 3 6 0 0 . V - 7 0 0 0 . 0 0 / 3 6 0 0 . FO - 3 4 0 0 . / 3 6 0 0 . FN - A 0 8 0 . / 3 6 0 0 . XO - 0 . 3 XN - 0 . 3 5 F-BETA*FN + (1-BETA)*F0 XF-GAM*XN+(1-GAM)*X0 DEN-0.9 VR-.120. VA-VR H U S - 1 . / 6 . HWR-HWS XLWS-8.6 XLWR-6.8 D T S - 1 0 . 5
92
DTR-8.5
DO 821 I-l.NT
821 READ 822.Y(I)
822 FORMAT( E12.5)
PRINT 838.(Y(I).I-l.NT) 838 FORMAT( U\i YI-.5E12.5)
T-O.O
NE-NT
C TMAXO IS THE STEP SIZE FOR THE CONTROLLER
C AND INTEGRATOR.
TMAXO-60.
DI-120.
C P IS THE INTEGRATION FACTOR FOR RUNGA-KUTTA C
P-.OOOl VL-V
DO 1000 II-1.180
TMAX-TMAXO*FLOAT(II)
FC-F
ZC-XF
BC-FC+R-V
DC-V-R
XC-Y(l)
YC-Y(NT)
TFC-40.
TTC-8.
TBC-118.
PC-254.
,\.SI'-0.6 786/.D.03
YSP-.99890
Z1-(YC*DC+XC*BC)*30./((YC*DC+XC*BC)*30.+44.*((1.-YC)*DC+(1.-XC)*BC
D) ZE-Zl/30./(Zl/30.+(l.-Zl)/4<..)
FE-(YC*DC+XC*BC)/ZE
VI-V
Ll-R
NP-21.3
MP-8.872
C
C CALL THE PARAMETERIZATION PROGRAM TO PARAMETERIZE THE SIMITH AND
C BRINKLEY APPROXIMATE MODEL
C
CALL PAKAM(NP.MP.Vl.Ll)
C IF(I.NE.55)STOP
N6-NP
Ml-MP
C
C CALL TWE CONTROLLER FOR UPDATED VALUES OF THE VAPOR AND
C LIQUID FLOWRATE OF THE COLUMN
C
CALL CONT(NA.Ml.XSP.YSP.VI.Ll)
R-Ll
V-Vl
C CALL SUBROUTINE TO CALCULATE THE INTIAL DYDX FOR THE SYSTEM
C
93
1 CALL F'X(T.Y.DYDX) C
C START THE STEP SIZE CALCULATOR FOR THE RUNGE-KUTTA METHOD C
DDD-DYDX(l) IM-1 IF(DABS(DyDX(l)).LT.1.E-10)DDD-1.E-10 DXM-DABS^*Y(1)/DDD) DO 2 1-2.NE IF(DABS(DYDX(I)).LT.1.E-10)TEST-1.E20 IF(DABS(DYDX(l)).LT.l.E-10)GO TO 2 TEST-DABS(P*Y(I)/DYDX(I)) IF (DABS(TEST).LT.DXM)IM-I
2 IF(DABS(TEST).LT.DXM)DXM-TEST DT-DXM IF(DT.GT.600.)DT-600. XT-DT+T IF(XT.GT.TMAX)DT-TMAX-T
C C START THE RUNGE-KUTTA INTEGRATION TECHNIQUE C
DO 33 I-l.NE CAY1(I)-DYDX(I)
33 Yl(I)-Y(I)+DYDX(I)*DT/2. Tl-T+DT/2. CALL FX(Tl.Yl.DYDX) DO 34 I-l.NE CAY2(I)-DYDX(I)
34 Yl(l)-Y(I)+DYDX(I)*DT/2. CALL FX(Tl.Yl.DYDX) DO 35 I-l.NE CAY3(I)-DYDX(I)
35 Y1(I)-Y(I)+DYDX(I)*DT T1-T4DT CALL FX(Tl.Yl.DYDX) DO 36 I-l.NE CAY4(I)-DYDX(I)
36 Y(I)-Y(I)+DT*(CAYl(I)+2.*CAY2(I)+2.*CAY3(I)4CAY4(I))/6.
T-T+DT
IF(T.LT.TMAX)GO TO 1
C PRINT RESULTS
C 272 PRINT 224.1.Y(1).YC1(1) 224 FORMAT( IX.'TRAYM2 . 2X. 3H X-. E12 . 5 . 2X. 3H Y-. E12 5) 1000 CONTINUE
STOP END
C******** END OF MAIN PROGRAM
C C ****** START SUBROUTINE TO LINEARIZE EQUAATIONS
C
C THIS SUBROUTINE USES THE THOMAS METHOD TO LINEARIZE THE
C NONLINEAR EQUATIONS.
C SUBROUTINE TM(N.C.D.E.B.X)
94
IMPLICIT REAL*8(A-H.0-Z) DIMENSION C(50).D(50).E(50).B(50).X(IOO).BETA(50).GAM(50) BETA(1)-D(1) GAM(1)-B(1)/BETA(1) DO 10 1-2,N BETA(I)-D(I)-C(I)*E(I-1)/BETA(1-1)
10 GAM(I-1)-(B(I)-C(I)*GAM(I-1))/BETA(I) X(N)-GAM(N) DO 20 1-2.N
J-N-I+1 20 X(J)-GAM(J)-E(J)*X(J+1)/BETA(J)
RETURN END END SUBROUTINE TM C****
C
c ***** c c c c c
c c c
c c c
START SUBROUTINE FX
C C C
THIS SUBROUTINE CALCULATES THE ODES USING THE COMPONENT BALANCE AT THE TRAY. IT ALSO CALULATES THE VAPOR COMPOSITION USING CONSTANT RELATIVE VOLATILITY.
SUBROUTINE FX(T,Y,DYDX) IMPLICIT REAL*8(A-H,0-Z) DIMENSION Y(IOO).DYDX(IOO).XL(50).XM(50).X(50).YC(50),YX(50) COMMON /0N9/ALFA(50),NT.R.V,F.XF.NF.DEN.HWS.HWR.XLWS.XLWR,DTS.DTR COMMON /TW9/XLL(50).XB.YD,VR.VA.YC1(50) XLS-R+F XLR-R DO 88 I-1,NF
88 XLL(I)-XLS NFP-NF+1 DO 89 I-NFP.NT
89 XLL(I)-XLR DO 84 I-l.NT
84 XL(I)-XLL(I) CALL SUBROUTINE TO CALCULATE LIQUID HOLDUP
CALL LHDUP(XM.XLS,XLR)
EF IS THE EFFEICIENCY OF THE TRAY
EF-.75
START CALCUI.ATING THE VAPOR COMPOSITION
DO 2 I-l.NT YX(I)-ALFA(I)*Y(I)/(1.+(ALFA(I)-1.)*Y(I))
2 IF(YX(I).GT.1.)YX(I)-1.
YC(1)-YX(1) YC(NT)-YX(NT) NTM-NT-1 DO 9 1-2.NTM
9 YC(I)-YC(I 1)+EF*(YX(I)-YC(I-1))
SETUP THE ODE ODE FOR THE REBOILER
95
DYDX(1)-(Y(2)*XLS-YC(1)*V.(XLS-V)*Y(1))/XM(1) NTM-NT-1 DO 3 1-2,NTM
C
C ODE FOR ALL TRAYS C
3 DYDX(I)-(XL(I+1)*Y(I+1)-Y(I)*XL(I)+V*(YC(I-1)-YC(I)))/XM(I) C C ODE FOR FEED TRAY C
DYDX(NF)-DYDX(NF)+XF*F/XM(NF) D-V-R
C C ODE FOR CONDENSER C
DYDX(NT)-(V*YC(NT.1)-R*Y(NT)-D*Y(NT))/XM(NT) DO 666 I-1,NT
666 YCl(I) - YC(I) RETURN END
C****** END SUBROUTINE FX C C***** START SUBROUTINE LHDUP C
SUBROUTINE LHDUP(XM.XL) C THIS SUBROUTINE CALCULATES THE MOLAR HOLDUP ON EACH TRAY C USING THE LIQUID FLOWRATE AND THE FRANCIS WEIR FORMULA C
IMPLICIT REAL*8(A-H.0-Z) DIMENSION XL(50).XM(50).H(50) COMMON /0N9/ALFA(50).NT.R.V.F,XF.NF.DEN.HWS.HWR.XLWS.XLWR.DTS.DTR COMMON /TW9/XLL(50).XB.YD.VR.VA.YCl(50) NFP-NF+1 DO 1 I-l.NF H(I)-((R+F)/(3.33*XLWS*DEN))**.6667+HWS
1 XM(I)-H(I)*3.14*DTS*DTS*DEN/4.
DO 2 I-NFP.NT H(I)-(R/(3.33*XLWR*DEN))**.6667+HWR
2 XM(I)-H(I)*3.14*DTR*DTR*DEN/4. C C HOLDUP ON IN THE REBOILER AND CONDENSER ARE CONSTANT
C
XM(l)-440. XM(NT)-440. RETURN END
C***** END SUBROUTINE HDLUP C C***** START SUBROUTINE DATAIN
C SUBROUTINE DATAIN
C FEED IN ALL THE DATA FOR THE CONTROLLER
C IMPLICIT REAL*8(A-H.0-Z) COMMON /TWO/ ERLIM.NITR.Kl.K2.TAU(2).XINT.YlNT.DT.ir.'PE
96
COMMON /THREE/XSP.YSP.N.M
COMMON /EDATA/TCl.TC2.PCI.PC2.W1.W2
COMMON /FOUR/ BHV.THV.STMHV.FHV.HWMAX.HWMIN
REAL*8 N.M.KT(2).KB(2).L.K1(2).K2(2)
ITYPE-1
ERLIM-l.E-6
NITR-75.
TAU(1)-1.0
TAU(2)-1.2
Kl(l)-2.05
Kl(2)-2.05
K2(l)-1.0E-03
K2(2)-1.0E-03
XINT-0.0
YINT-0.0
TCl-305.4
TC2-425.2
PC1-48.8*14.7
PC2-38.0*14.7
W1-.099
W2-.199
STMHV-928.
THV-4552.
BHV-5808.
FHV-150.
HWMAX-4.E6
HWMIN-.5E6
RETURN
END
C ***** END SUBROUTINE DATAIN
C
C****** START SUBROUTINE CONT
C
C************************ ABSRACT *••*****•**•••**••*************•****• C
C
C
C
C
C
C
C
C
C
THIS SUBROUTINE CALCULATES THE VAPOR AND LIQUID FLOW RATES (V & L)
BASED UPON THE PROCESS MODEL BASED CONTROL LAW. THE CONTROL LAW
DETERMINES MODIFIED COMPOSITION FOR THE LIGHT COMPONENT IN THE BOTTOMS
AND THE OVERHEAD (XSPP AND YSPP. RESPECTIVELY). THEN A NETOWN'S
SEARCH IS USED TO FIND V AND L THAT SATISFY THE SMITH-BRINKLEY MODEL
WITH XSPP AND YSPP- THIS CONTROLLER CAN ALSO BE USE TO CONTROL
THE BOTTOMS COMPOSITION (X) USING ONLY V ( USE ITYPE-2) OR TO
CONTROL THE OVERHEAD CONPOSITION USING ONLY L (USE ITYPE-3).
NOMENClj\TURE ******************************** C***^i*******************
c THE PARTIAL OF THE ITH EQUATION WITH RESPECT TO THE JTH
UNKNOWN (J-1 IS V; J-2 IS L)
THE FUNCTION VALUE OF THE ITH EQUATION
THE RELATIVE ERROR CRITERIA FOR CONVERGENCE OF THE NEWTON'S SEARCH
THE CHANGE IN THE UNKNOWNS CALCUUVTED BY THE NEWTON'S SEARCH
-1 BOTH X .& Y CONTROLLED; -2 X ONLY; -3 Y ONLY
THE K VALUE OF THE ITH COMPONENT FOR THE AREA OF COLl'M^ BELOf
THE FEED TRAY(1-1 LIGHT COMPONENT; 1-2 HEAVY COMPONENT)
A(I.J)
C(I)-ERLIM-
C. (I) -
ITVPE-
KB(I)-
97
C KT(1)- THE K VALUE OF THE ITH COMPONENT FOR THE ARFJK OF COLUMN ABOVE C THE FEED TRAY (I-l LIGHT COMPONENT; 1-2 HEAVY COMPONENT) C K1(I)- TUNING PARAMETER FOR THE PROPORTIONAL TERM IN THE CONTROL LAW C K2(I)- TUNING PARAMETER FOR THE INTEGRAL TERM IN THE CONTROL LAW C L.Ll- THE REFLUX FLOW RATE (#MOLES/HR) C M.Ml- THE NUMBER OF THEORITICAL STAGES BELOW THE FEED TRAY C N.NI- THE TOTAL NUMBER OF THEORITICAL STAGE IN THE COLUMN C NERS- THE NUMBER OF VARIABLE THAT DO NOT MEET CONVERGENCE CRITERIA C TB- THE BOTTOMS TEMPERATURE (DEG F) C TEST- THE RELATIVE ERROR IN A VARIABLE C TF- THE TEMPERATURE OF THE FEED (DEG F) C TM- THE AVERAGE TEMPERATURE IN THE STRIPPING SECTION (DEG F) C TN- THE AVERAGE TEMPERATURE IN THE RECTIFYING SECTION (DEG F) C TT THE TEMPERATURE AT THE TOP OF THE COLUMN (DEG F) C V.Vl- THE VAPOR BOIL-UP RATE (/!IM0LES/HR) C XINT- THE INTEGRAL TERM VALUE FOR THE BOTTOMS COMPOSITION C XSP- THE SET POINT FOR THE LIGHT COMPONENT IN THE BOTTOMS (MOLE FRAG) C XSPP- THE MODIFIED COMPOSITION FOR THE LIGHT COMP IN THE BOTTOMS C YINT- THE INTEGRAL TERM VALUE FOR THE OVERHEAD COMPOSITION C YSP- THE SETPOINT FOR THE HEAVY COMPONENT IN THE OVERHEAD (MOLE FRAC) C YSPP- THE MODIFIED COMPOSITION FOR THE HEAVY COMP IN THE OVERHEAD C C************************************************************************ c
SUBROUTINE C0NT(N1.Ml.XSP.YSP.VI.LI) IMPLICIT REAL*8(A-H.0-Z) DIMENSION A(2.2).C(2) COMMON /ONE/ D.B.X.Y.TF.TT.TB.P.F.Z.KT.KB COMMON /TWO/ ERLIM.NITR.K1.K2,TAU(2).XINT.YINT.DT.ITYPE REAL*8 N,M.KT(2).KB(2).K1(2).K2(2).L1.L.G(2).N1.M1
L-Ll V-Vl N-Nl M-Ml
C CALCULATE AVERAGE TEMPERATURES IN STRIPPING AND RECTIFYING SECTIONS
TN-.5*(TT+TF) TM-.5*(TF+TB)
C CALCULATE K VALUES FOR BOTH SECTIONS OF THE COLUMN CALL EQL(TN.P,KT) CALL EQL(TM.P.KB)
C APPLY CMC CONTROL LAW I.E.. CALCULATE MODIFIED COMPOSITIONS XSPP-X-TAU(1)*K1(1)*(X-XSP)-TAU(1)*K2(1)*X1NT YSPP-Y-TAU(2)*K1(2)*(Y-YSP)-TAU(2)*K2(2)*Y1NT
IF(XSPP.LT.l.E-6)XSPP-l.E-6 IF(YSPP.CT.l.)YSPP-.9999
C C BEGIN THE ITERATIVE NEWTON'S SEARCH FOR L AND V THAT SATISFY THE C SMITH-BRINKLEY MODEL WITH XSPP AND YSPP
C ICT-O
1000 ICT-ICT+1 C CALCULATE THE JACOBIAN AND THE FUNCTION VALUES
CALL DERC(N.M.V.L.XSPP.YSPP.A.C)
C CALCULATE THE CHANGE IN V AND L C ( 2 ) - ( - A ( 2 . 1 ) * C ( l ) + A ( l . I ) * C ( 2 ) ) / ( A ( 2 . 1 ) * A ( 1 . 2 ) - A ( l . l ) * A ( 2 . 2 ) )
98
C(l)-(-C(l)-A(1.2)*G(2))/A(l.l) C CHECK FOR BOTTOMS CONTROL ONLY
IF(ITYPE.EQ.2)C(1)—C(l)/A(l.l) lF(ITYPE.EQ.2)G(2)-0.0
C CHECK FOR OVERHEAD CONTROL ONLY
IF(ITYPE.EQ.3)G(2)—C(2)/A(2.2) IF(ITYPE.EQ.3)G(l)-0.0
C CHANGE VAPOR AND REFLUX RATES V-V+G(l) L-L+C(2) NERS-0
C TEST CONVERGENCE FOR V TEST-DABS(G(l)/V) IF(TEST.GT.ERLIM)NERS-NERS+1
C TEST CONVERGENCE FOR L TEST-DABS(G(2)/L) IF(TEST.GT.ERLIM)NERS-NERS+1 IF(ICT.GT.NITR)GO TO 2000 IF(NERS.NE.0)GO TO 1000
C
C CONVERGENCE OBTAINED C
Vl-V Ll-L
C
C CALCUUVTE CONTRIBUTION TO INTEGRAL TERMS XINT-XINT+(X-XSP)*DT YINT-YINT+(Y-YSP)*DT RETURN
C NEWTON'S METHOD DID NOT CONVERGE; PRINT THAT RESULT AND RETURN 2000 PRINT 25
25 FORMAT( 'NEWTON METHOD DID NOT CONVERGE FOR CONT') RETURN END
C
C**************************** ABSTRACT ********************************
C C THIS SUBROUTINE CALCUUVTES THE DERIVATIVE OF THE TWO EQUATIONS THAT C RESULT FORM THE SMITH-BRINKLEY MODEL WITH RESPECT TO V AND L. THE C DERIVATIVES ARE CALCULATED BY FINITE DIFFERENCE APPROXIMATIONS. THE C ANSWERS ARE STORED IN A(I.J). A(I.J) IS THE JACOBIAN OF THE TWO C NONLINEAR EQUATIONS. C C*************************** NOMENCALTURE ***************************** C C A(I.J)- THE JACOBIAN OF THE ITH EQUATION WITH RESPECT TO THE JTH C VARIABLE ( J-1 V; J-2 L) C B(I)- THE VALUE OF THE ITH EQUATION C DELTA- THE FRACTIONAL CHANGE IN V AND L USED TO CALCULATE THE C NUMERICAL DERIVATIVES C F(I)- THE EQUATION VALUE OF THE ITH EQUATION C L.L1.L2- THE REFLUX LIQUID FLOW RATE (#MOLES/HR) C M.Ml- THE NUMBER OF THEORITICAL STAGES IN THE RECTIFYING SECTION C N.NI- THE NUMBER OF THEORITICAL STAGES FOR THE COLUMN C V .V1.V2- THE VAPOR BOIL-UP RATE FOR THE REBOILER (ilfMOLES/HR)
99
C X.Xl C Y.Yl C
THE COMPOSITION OF THE LIGHT COMP IN THE BOTTOMS THE COMPOSITION OF THE HEAVY COMP IN THE OVERHEAD
^*****1t**1t******ir***-k**********
SUBROUTINE DERC(N1.Ml.VI.LI.XI.Yl.A.B) IMPLICIT REAL*8(A-H.0-Z) DIMENSION A(2.2).B(2).F(2) REAL*8 N.N1,M.M1.L.L1.L2 N-Nl M-Ml V-Vl l^Ll X-Xl Y-Yl DELTA-.03
DETERMINE BASE CASE VALUE OF F(I) CALL FUN(N,M.V,L.X.Y,F) B(l)-F(l) B(2)-F(2) L2-L*(1.+DELTA)
INCREMENT L CALL FUN(N.M.V.L2.X.Y.F)
CALCULATE NUMERICAL DERIVATIVE WITH RESPECT TO L A(1.2)-(F(1)-B(1))/L/DELTA A(2,2)-(F(2)-B(2))/L/DELTA
INCREMENT V V2-V*(1.+DELTA) CALL FUN(N,M.V2.L.X.Y.F)
CALCULATE NUMERICAL DERIVATIVE WITH RESPECT TO V A(1.1)-(F(1)-B(1))/V/DELTA A(2.1)-(F(2)-B(2))/V/DELTA RETURN END
C************************** ABSTRACT *********************************
C THIS SUBROUTINE CALCULATES THE K VALUES FOR EACH COMPONENT FROM
THE TEMPERATURE AND PRESSURE
NOMENCALTURE C*************************
c K(I)- THE K VALUE OF THE ITH COMPONENT ( 1-1 LIGHT; P- PRESSURE (PSIA)
******************************
C c c c c c c c c c c c c
1-2 HEAVY )
PCI- THE CRITICAL PRESSURE OF THE LIGHT COMP (PSIA) PC2- THE CRITICAL PRESSURE OF THE HEAVY COMP (PSIA) PH- THE FUGACITY COEFICIENT FOR THE VAPOR PHASE PHI- THE FUGACTIY COEFICIENT FOR THE HEAVY COMP PHX- THE FUGACITY COEFICIENT FOR THE LIGHT COMP PL- A-B/(T+C) IN THE ANTOINE EQUATION PR- THE REDUCED PRESSURE T- TFMPERATURE (DEG F) TCI- THE CRITICAL TEMPERATURE OF THE LIGHT COMP (DEG K) TC2- THE CRITICAL TEMPERATURE OF THE HEAVY COMP (DEG K) TK- TEMPERATURE (DEG K)
100
C TR- THE REDUCED TEMPERATURE C VP- THE VAPOR PRESSURE (PSIA) C Wl- THE ACENTRIC FACTOR FOR THE LIGHT COMP C W2- THE ACENTRIC FACTOR FOR THE HEAVY COMP C C******************************************^k.********^****^t^*^^^^^^*^^^^***
C SUBROUTINE EQL(T.P.K) IMPLICIT REAL*8(A-H.0-Z) COMMON /EDATA/TCl.TC2.PCI.PC2.Wl.W2 REAL*8 K(2)
C CALCULATE TEMPERATURE IN DEGREES KELVIN TK-(T+460.)*5./9.
C CALCULATE REDUCED TEMPERATURE AND PRESSURE FOR LIGHT COMPONENT TR-TK/TCl PR-P/PCl
C CALCULATE VAPOR PRESSURE FOR LIGHT COMPONENT PL-IO.072-1976.1/(TK+12.894) VP-14.69*DEXP(PL)
C CALCULATE FUGACITY COEFICIENT FOR VAPOR PHASE PHX-PH(TR.PR.Wl)
C CALCULATE K VALUE FOR LIGHT COMPONENT
K(1)-VP/(PHX*P) C CALCULATE REDUCED PRESSURE AND TEMPERATURE FOR HEAVY COMPONENT
TR-TK/TC2 PR-P/PC2
C CALCULATE VAPOR PRESSURE FOR HEAVY COMPONENT PL-9.4928-2067.3/(TK-13.437) VP-14.69*DEXP(PL)
C CALCULATE FUGACITY COEFICIENT FOR VAPOR PHASE PHI-PH(TR.PR.W2)
C CALCULATE K VALUE FOR HEAVY COMPONENT K(2)-VP/(PHI*P) RETURN END
C C**************************** ABSTRACT *******************************
C C THIS FUNCTION SUBROUTINE CALCULATES THE FUGACITY COEFICIENT FOR C A COMPONENT FROM THE REDUCED TEMPERATURE. REDUCED PRESSURE, AND THE
C ACENTRIC FACTOR C C************************* NOMENCLATURE **•*••***•******•*********•***
C C PH- FUGACITY COEFICIENT C PR- THE REDUCED PRESSURE C TR- THE REDUCED TEMPERATURE C W- THE ACENTRIC FACTOR
C C***********************************************************************
C FUNCTION PH(TR.PR.W) IMPLICIT REAL*8(A-H.0-Z)
P-( 1445+.073*W)/TR-(.33-.46*W)*TR**(-2)-( 1385+.5*W)*TR**( 3)
l-(.0121*.097*W)*TR**(-4)-.0073*W*TR**(-9)
101
PH-10.**(PR*P/2.303) RETURN END
C C*************************** ABSTRACT *********•***•***•**•*•***••***• C
C THIS SUBROUTINE CALCULATES THE ERROR IN EACH OF THE TWO NONLINEAR C EQUATIONS RESULTING FROM THE APPLICATION OF THE SMITH-BRINKLEY MODEL C C************************* NOMENCLATURE **•******•*****•••••***••**•** C C B- BOTTOMS PRODUCT FLOW RATE (JJIMOLES/HR) C D- OVERHEAD PRODUCT FLOW RATE (#MOLES/HR) C F- FEED FLOW RATE (#MOLES/HR) C FX(I)- THE ERROR IN THE ITH EQUATION C FD(I)- THE RECOVERY OF THE ITH COMP IN THE BOTTOMS PRODUCT C (I-l LIGHT; 1-2 HEAVY) C HD(I)-C KB(I)- THE K VALUE IN THE STRIPPING SECTION (I-l LIGHT; 1-2 HEAVY) C KT(I)- THE K VALUE IN THE RECTIFYING SECTION (I-l LIGHT; 1-2 HEAV ') C L- THE REFLUX LIQUID FLOW RATE (IMOLES/HR) C R- THE REFLUX RATIO (I.E.. R-L/D) C SM(I)- THE SEPARATION FACTOR FOR THE ITH COMP IN THE STRIPPING SECTIO C SN(I)- THE SEPARATION FACTOR FOR THE ITH COMP IN THE RECTIFTING SECT C V- THE VAPOR BOIL-UP RATE FROM THE REBOILER (idMOLES/HR) C X- THE MOLE FRACTION OF THE LIGHT IN THE BOTTOMS PRODUCT C Y- THE MOLE FRACTION OF THE LIGHT IN THE OVERHEAD PRODUCT C Z- THE MOLE FRACTION OF THE LIGHT IN THE FEED C C***********************************************************************
c SUBROUTINE FUN(N.M.V.L.X.Y.FX) IMPLICIT REAL*8(A-H.0-Z) COMMON /ONE/ D.B.Q.E.TF.TT.TB.P.F.Z.KT.KB REAL*8 N.M.L.KT(2).KB(2).SN(2).SM(2).FX(2).FD(2).HD(2) D-F*(Z-X)/(Y-X) B-F-D R-L/D VB-V/B
C CALCULATE THE SEPARATION FACTORS DO 1 1-1.2 SN(I)-KT(I)*((R+1.)/R)
1 SM(I)-KB(I)*VB/(VB+1.) DO 99 1-1.2
99 IF((SN(I).LT.O.).OR.(SM(I).LT..O))PR1NT 55.SM(I).SN( I) .V.L. F
55 FORMAT( 3H S-.5E12.5) C CALCULATE THE H FACTORS
DO 2 1-1.2
2 HD(I)-SM(I)*(1-SN(I))/(SN(I)*(I-SM(I)))
C CALCULATE THE RECOVERIES IN THE BOTTOMS
FD(1)-B*X/F/Z FD(2)-B*(1.-X)/(F*(1 -Z))
C CALCUUTE THE ERROR IN THE SMITH-BRINKLEY MODEL EQUATIONS DO 3 1-1.2 X1-(1.-SN(I)**(N-M))*R*(1.-SN(I))
102
XX-(1.-SN(I)**(N-M))+R*(1.-SN(I)) X9-HD(I)*SN(I)**(N-M)*(1.-SM(I)**(M+1)) X2- X9/X1 FX(I)-1./((1.+X2)*FD(I))-1.0
C PRINT 88.X1.X2.FX(I).X9 3 CONTINUE RETURN END
C C************************ ABSRACT *************************************
C C THIS SUBROUTINE CALCULATES THE NUMBER OF THEORITICAL STAGES FOR C THE COLUMN AND FOR THE STRIPPING SECTION. FILTERED STEADY-STATE DATA C FOR THE REFLUX RATE. THE VAPOR BOIL-UP RATE. FEED RATE. AND THE C COMPOSITON OF THE LIGHT COMPONENT IN THE BOTTOMS AND OVERHEAD ARE C PROVIDED TO THIS SUBROUTINE. THE VALUES OF N AND M ARE FOUND USING A C A NEWTON'S SEARCH TO SATISFY THE SMITH-BRINKLEY MODEL. C C************************ NOMENCLATURE ********************************
C C A(I,J)- THE PARTIAL OF THE ITH EQUATION WITH RESPECT TO THE JTH C UNKNOWN'. (J-1 IS N; J-2 IS M) C C ( I ) - THE FUNCTION VALUE OF THE ITH EQUATION C ERLIM- THE RELATIVE ERROR CRITERIA FOR CONVERGENCE OF THE NEWTON'S C SEARCH C G(I)- THE CHANGE IN THE UNKNOWNS CALCULATED BY THE NEWTON'S SEARCH
C ICT- AN ITERATION COUNTER C KB(I)- THE K VALUE OF THE ITH COMPONENT FOR THE AREA OF COLUMN BELOW C THE FEED TRAY(I-1 LIGHT COMPONENT; 1-2 HEAV ' COMPONENT) C KT(I)- THE K VALUE OF THE ITH COMPONENT FOR THE AREA OF COLUMN ABOVE C THE FEED TRAY (I-l LIGHT COMPONENT; 1-2 HEAVY COMPONENT) C L.Ll- THE REFLUX FLOW RATE (l!fMOLES/HR) C M.Ml- THE NUMBER OF THEORITICAL STAGES BELOW THE FEED TRAY C N Nl- THE TOTAL NUMBER OF THEORITICAL STAGE IN THE COLUMN C NERS- THE NUMBER OF VARIABLE THAT DO NOT MEET THE CONVERGENCE CRITERI C TB- THE BOTTOMS TEMPERATURE (DEG F) C TEST- THE RELATIVE ERROR IN A VARIABLE C TF- THE TEMPERATURE OF THE FEED (DEG F) C TM- THE AVERAGE TEMPERATURE IN THE STRIPPING SECTION (DEG F) C TN- THE AVERAGE TEMPERATURE IN THE RECTIFYING SECTION (DEG F) C TT- THE TEMPERATURE AT THE TOP OF THE COLUMN (DEG F) C X- MOLE FRACTION OF THE LIGHT COMP IN THE BOTTOMS PRODUCT C Y- MOLE FRACTION OF THE LIGHT COMP IN THE OVERHEAD PRODUCT
^ J . J . . I . . I * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C******************************** C
SIIHROUTINF rARM(Nl .Ml . V I . LI)
IMPLICIT REAL*8(A-H.0-Z) DIMENSION A ( 2 . 2 ) . C ( 2 ) COMMON /ONE/ D . B . X . Y . T F . T T . T B . P . F. Z. KT. KB COMMON /TWO/ ERLIM.NITR.K1.K2.TAU(2).XINT.YINT D T m P E REAL*8 N . M . K T ( 2 ) . K B ( 2 ) . K 1 ( 2 ) . K 2 ( 2 ) . L 1 . L . G ( 2 ) . N 1 . M 1
L-Ll V-Vl N-Nl
103
M-Ml
TN-.5•(TT-^TF) TM-.5*(TF+TB)
CALL EQL(TM.P.KB)
PRINT A4.KT(1).KT(2).KB(1),KB(2) 44 FORMATC 4H KT-.4E12.5)
Xl-X Yl-Y
C
C ITERATIVELY SOLVE FOR N AND M C
ICT-O 1000 ICT-ICT+1
C CALCULATE THE JACOBIAN OF THE TWO NONLINEAR EQUATIONS CALL DERP(N.M.V.L.X1,Y1.A.C)
C CALCULATE THE CHANCE IN N AND M C G(2 ) - ( .A(2 ,1 )*C(1 )+A(1 .1 )*C(2 ) ) / (A(2 .1 )*A(1 .2 ) -A(1 .1 )*A(2 2) ) C G ( 1 ) . ( . C ( 2 ) - A ( 2 . 2 ) * G ( 2 ) ) / A ( 1 . 2 ) ^^^
G ( l ) - ( - A ( 2 . 2 ) * C ( l ) + A ( 1 . 2 ) * C ( 2 ) ) / ( A ( 2 . 2 ) * A ( l . l ) - A ( 1 . 2 ) * A ( 2 1)) G ( 2 ) - ( - C ( 2 ) - A ( 2 . 1 ) * G ( l ) ) / A ( 2 . 2 )
IF(ABS(G(1)).GT..3*N)G(1)-.3*N*ABS(C(1))/G(1) IF(ABS(G(2)).CT..1*M)G(1)-.1*M*ABS(G(2))/G(2) PRINT 22.V.L.N.M.G(1).G(2) N-N+G(l) M-M+G(2) NERS-O
PRINT 2 2 , A ( 1 . 1 ) . A ( 1 . 2 ) . A ( 2 . 1 ) . A ( 2 . 2 ) . C ( 1 ) . C ( 2 ) PRINT 22.V.L.N.M.G(1).G(2) PRINT 23
22 FORMAT( 5H C0N-,6E11.4) 23 FORMAT( / )
C CHECK FOR CONVERGENCE TEST-DABS(G(l)/N) IF(TEST.GT.ERLIM)NERS-NERS+1 TEST-DABS(G(2)/M) IF(TEST.GT.ERLIM)NERS-NERS+1 IF(ICT.CT.NITR)GO TO 2000 IF(NERS.NE.O)GO TO 1000 Nl-N Ml-M RETURN
C IF NEWTON'S METHOD DID NOT CONVERGE. INDICATE THRU A PRINT AND RETURN 2000 PRINT 25
25 FORMAT( 'NEWTON METHOD DID NOT CONVERGE IN PARM') RETURN END
C C>i» ************************** ABSTRACT ******************************** C C THIS SUBROUTINE CALCUU TES THE DERIVATIVE OF THE TWO EQUATIONS THAT C RESULT FORM THE SMITH-BRINKLEY MODEL WITH RESPECT TO N AND M THE
104
C DERIVATIVES ARE CALCULATED BY FINITE DIFFERENCE APPROXIMATIONS THE C ANSWERS ARE STORED IN A(I.J). A(I.J) IS THE JACOBIAN 0 ^ ™ ° TWO C NONLINEAR EQUATIONS. C C*************************** NOMENCALTURE ************************^^* C
C A(I.J)- THE JACOBIAN OF THE ITH EQUATION WITH RESPECT TO THE JTH C VARIABLE ( J-1 V; J-2 L) C B(I)- THE VALUE OF THE ITH EQUATION C DELTA- THE FRACTIONAL CHANGE IN V AND L USE TO CALCULATE THE C NUMERICAL DERIVATIVES C F(I)- THE EQUATION VALUE OF THE ITH EQUATION C L.Ll- THE REFLUX LIQUID FLOW RATE (#MOLES/HR) C M.M1,M2- THE NUMBER OF THEORITICAL STAGES IN THE RECTIFYING SECTION C N.N1.N2- THE NUMBER OF THEORITICAL STAGES FOR THE COLUMN C V.Vl- THE VAPOR BOIL-UP RATE FOR THE REBOILER (#MOLES/HR) C X.Xl- THE COMPOSITION OF THE LIGHT COMP IN THE BOTTOMS C Y.Yl- THE COMPOSITION OF THE HEAVY COMP IN THE OVERHEAD C c*--C
SUBROUTINE DERP(N1.Ml.VI.LI.XI.Yl.A.B) IMPLICIT REAL*8(A-H.0-Z) DIMENSION A(2,2),B(2).F(2) REAL*8 N.N1.M,M1,L,L1,L2.M2.N2 N-Nl M-Ml V-Vl L-Ll X-Xl Y-Yl DELTA-.01
C DETERMINE F(I) FOR THE BASE CASE CALL FUN(N.M,V.L.X,Y.F) B(l)-F(l) B(2)-F(2)
C INCREMENT M M2-M*(1.+DELTA) CALL FUN(N.M2,V.L.X,Y.F)
C DETERMINE PARTIAL DERIVITATIVES NUMERICALLY A(1,2)-(F(1)-B(1))/M/DELTA A(2.2)-(F(2)-B(2))/M/DELTA
C INCREMENT N N2-N*(1.+DELTA) CALL FUN(N2.M.V.L.X.Y.F)
C CALCULATE PARTIAL DERIVITATIVES NUMERICALLY A(1.1)-(F(1)-B(1))/N/DELTA A(2.1)-(F(2)-B(2))/N/DELTA RETURN END
//GO.SYSIN DD * 0.47864E-03 0 15271E-02 0,38336E-02 0.94629E-02
105
0.23015E-01 0.54153E.01 0.11832E+00 0.22S22E+OO 0.35515E+00 0.46831E+00 0.70799E+O0 0.84922E+00 0.92741E+O0 0.96633E+00 0.98474E+00 0.99325E+00 0.99712E+00 0.99888E+00
/* //
APPENDIX B
DETAILED DERIVATION OF THE APPROXIMATE MODEL
FOR A DISTILLATION COLUMN WITH A
SIDESTREAM DRAWOFF
106
.
107
Based on Figure B.l and \ising the general form of the equation for
the composition in any section derived in Chapter 6, section 6.1, we
shall derive the equations for the recoveries in the bottoms and
sidestream.
For section I:
Consider the condenser
dA - Y^D - XiKiD
Xi - dA/KiD - CiS4 + C2 ( -D
doing a material balance on the top tray we have
LlYi| I VYi
LlXt • T VYi-l
LiYi + VYi.i - VYi - LlXi - 0
using Yi - K^Xi
we have
Yi.i - dA/D + LidA/VD(l/Ki -1)
which implies that
Xi.i - dA/KiD{l+Li(l/Ki-l)/V) - CiSi-h -»- C2 (B.2)
now solving equations (B.l) and (B.2) for Ci and C2 we get the equation
for Xi as follows
108
XI - (l-f-g)A/KiD(l/Si-l)[Li(l/Ki-l)/V][sI-ii-l]+(l-f.g)A/KiD. (B.3)
Now for the bottom section the following procedure is used.
Considering the reboiler
XiB - fA
Xi - fA/B - C3S3 + C4 (B.4)
now consider the tray above the reboiler,
UX2^ f VYI
U BXl doing a material balance we have
L3X2 - VYi - BXl - 0
and using Yi - K^Xi
109
we have
X2 - fA/BL3(VK3 + B) - C3S23 + C4 (B.5)
solving equations (B.4) and (B.S) we have
XM - fA/B(l-S3)[l-(VK3+B)/L3][sM-l3-l) + fA/B. (B.6)
For the middle section we shall perform the material balance on the
tray immidiately above the feed tray and the tray right below the
sidestream drawoff tray.
Now considering the tray above the feed tray (remember this tray 1
for this section)
L2X: I f VYI
CZZIl L2X4 f VYf
The material balance is
L2X2 + VYF - L2X1 - VYi - 0 (B.7)
110
where
X2 - C5S22 + C5; Xi - C5S2 + Ce; Yl - K2X1 Yp - KpXp - AK2/F
substituting these in equation (B.7) we have the following equation
containing C5 and C5
C5 + Ce - A/F. (B.S)
Now considering the tray below the feed tray (this tray will be tray n
for this section)
VYn-l • ^ L2Xn
VYn.2 T • LlXn-l
the material balance yields:
L2Xn + VYn.2 " L2Xn-l ' VY^.i - 0 (B.9)
1-2.
where
Xn - gA/G Xn-l - C5Sn-l2 + Cg and Xn.2 - C5Sn-^2 + ^6
and Yn-i - Kp.iXn-i; "^n-2 ' Xn.2Kn.2
substituting these in equation (B.9) we get the following equation.
C5S' 2 + C6 - gA/G. <»' 0>
Ill
Solving equations (B.8) and (B.IO) for C5 and C6 we get and
substituting in the general form for the liquid composition in the
middle section we have:
XN - {l/(l-S^2))[VF-gA/G][sN2 -1] + A/F. (B.ll)
Now that we have all the liquid compositions required, we can perform
material balances on the feed tray and the sidestream drawoff tray.
now consider the feed tray;
L2X1^ • VYf
A-
(XIF) , . 1 f VYM
L3Xf^ I
the material balance is:
L2X1 + VYm-1 - VYm - L3Xm + A - 0 (B.12)
where XF - Xm - A/F; XF-1 - Xm-1 - C3Sm-13 + C4 and XI - C5S2 + C6
and Y - KX
112
substituting these in equation (B.12) yields the following equation in
terms of f and g:
(VK30/B + VK3/B)f + (-L2b/G)g - (VK3 +L3-L2-L2b)/F -1 (B.13)
where
b - (S2-l)/(l.Sn2)
and
0 - (Sm-23-l)/(l-S3)[l-(VK3+B)/L3].
Now consider the material balance on the sidestream drawoff tray:
LlXl ^ f VYn
GXn
L2Xn ^ f VYn-l
the material balance is:
LlXl + VYn-l - L2Xn - VYn - GXn - 0 (B.14)
where Xn - gA/G; XI - ClSl + C2 and Xn-l - C5Sn-12 + C6
now substituting in equation (B.14) we get the following in terms of f
and g:
113
(-a01-a)f-h(-a01-a-VK2bl/G-L2/G-VK2/G-l)g-a01-a-(VK2bl-VK2)/F (B.15)
where
01 - Ll(Sl-il -1)A(1/S1-1){1/K1-1))
bl - (Sn-12 -1)/(1-Sn2)
and
a - Ll/KID.
Now equations (B.15) and (B.13) can be simultaneously solved to give
the values for f and g as follows:
[aOi+a+VK2di/G+L2/G+VK2/G+l][-L2dKf/K2-L2Kf/K2+VKf+L3+F]-
(-L2/G] [aOi+a+(di-i-l)VK2/F]
f-
[VK3( -1)/B][aOi+a+VK2di/G+L2/G+VKg/G+l]-[aOi+a]
[-L2d/G]
and
[aOi+a+(di+l)VK2/Fl-(aOi+a]f
g- _
(aOi+a+(di+l)VK2/G+L2/G+l)
and from a material balance
dd- 1 - f - g.
^
T ^ f - . . . I
F
*'
LI I 1 I I
L ( i ) i !
T
II
III
B
- - D
"L> G
^
FIGURE B.l Splic^of the disciUation column „ich a sidescrea.
APPENDIX C
LISTING OF THE COMPUTER CODE FOR THE APPROXIMATE MODEL
115
116
C*** COMPUTER CODE FOR THE APPROXIMATE •*•• C*** MODEL OF A DISTILLATION COLUMN WITH **** C*** A SIDESTREAM DRAWOFF **•* C*** **** C*** PROGRAMMER RUPAK SINHA **•* C***********************************************************
C******* MAIN PROGRAM ******* C C THIS PROGRAM IS THE APPROXIMATE MODEL FOR A DISTILLATION COLUMN C UlTH A SIDESTREAM DRAWOFF TRAY IN THE STRIPPING SECTION OF THE COLUMN C IT ALSO CALLS THE PROGRAM TO PARAMETERIZE THE COLUMN. C C NOMENCLATURE C F - FEED RATE (MOLES/SEC) C B - BOTTOMS RATE (MOLES/SEC) C G - SIDESTREAM DRAWOFF RATE (MOLES/SEC) C V - BOILUP RATE (MOLES/SEC) C D - DISTILLATE RATE (MOLES/SEC) C R - REFLUX RATIO C L(l) - REFLUX RATE (MOLES/SEC) C L(2) - LIQUID FLOWRATE IN MID SECTION OF COLUMN (MOLES/SEC) C L(3) - LIQUID FLOWRATE IN BOTTOM SECTION OF COLUMN (MOLES/SEC) C TT - ARE TEMPERATURES IN THE COLUMN IN THE DIFFERENT SECTIONS C PP - ARE PRESSURES IN THE COLUMN C FMB - REQUIRED RECOVERY OF THE COMPONENT IN THE BOTTOMS C GMB - REQUIRED RECOVERY OF THE COMPONRNT IN THE SIDESTREAM C DMB - REQUIRED RECOVERY IN THE DISTILLATE C II - NUMBER OF TRAYS IN THE TOP SECTION OF THE COLUMN C Nl - NUMBER OF TRAYS IN THE MIDDLE OF THE COLUMN C Ml - NUMBER OF TRAYS IN THE LOWER SECTION OF THE COLUMN C X - LIQUID COMPOSITION (MOLES) C Y - VAPOR COMPOSITION (MOLES) C C THIS PROCESS USES C2 AND C3 AS THE TWO COMPONENTS
IMPLICIT REAL*8(A-H.0-Z) COMMON /VAR/ F.B.D.G.R.NT.NJ COMMON /VAR1/AA(3).BB1(3).CC(3).DD(3).X(3.3).P(3).T(3).TC(3).PC(3) COMMON /VAR2/FMB.GMB.DMB DIMENSION XX(3).Y(3).Z(3).FX(3).L(3).TT(5).PP(5) REAL *8 K.L.Il.Nl.Ml.IL.IM.NL.NM.ML.MX DO 1 NN - 1.3
C C READ IN ALL INITIAL DATA C
READ(5.14)AA(NN).BBl(NN).CC(NN).DD(NN).TC(NN).PC(NN) 14 F0RMAT(6D12.5) 1 CONTINUE F - 5600./3600. B - 2600.0/3600. V - 2522.0/3600. C - 1000.0/3600 L(l) - 1169 2/3600. L(2) - L(l)-G L(3) - L(2)+F
117
c c c c
c c c c c c
CALCULATE ALL AVARAGE TEMPERATURE AND PRESSURES FOR EACH OF THE THREE SECTKWS
TT(1) - 1 8 . 3 TT(2) - 5 3 . 1 TT(3) - 53.60 TT(4) - 108.6 PP(1) - 254 .10 PP(2) - 254 .20 PP(3) - 254 .3 PP(4) - 254.60
P ( l ) - ( P P ( l ) + P P ( 2 ) ) / 2 . P(2) - (PP(2 )+PP(3 ) ) / 2 . P(3) - ( P P ( 3 ) + P P ( 4 ) ) / 2 . T ( l ) - ( T T ( l ) * T T ( 2 ) ) / 2 . T(2) - (TT(2) -HT(3) ) /2 . T(3) - (TT(3) -HT(4) ) /2 . D - F-G-B R - L ( l ) / D FMB - 0 .8416 GMB - 0 .0827 DMB - 1-FMB-GHB IL - 4 . 9 9 1 8 NL - 4 .986795 ML - 9.992715 II - IL Nl - NL Ml - ML XX(1) - 0 . Y d ) - 0.
CALL SUBROUTINE TO PARAMETERIZE THE PROGRAM INITIALLY
CALL PARM(I1.NI,M1,V,L) CALL SUBROUTINE TO EVALUATE THE RECOVERIES FOR THE SYSTEH
100 CALL E V A L d l . N l . M l . V . L . X X . Y . F F ) CONTINUE
C C C C C C
PRINT RESULTS
DO 3 J - l . N J WRITE(6.20)F1.GI
20 FORMAT(6X,2D12.5) STOP END
C * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C THIS FUNCTION SUBROUTINE CALCULATES THE FUGACITY COEFICIENT FOR
A COMPONENT FROM THE REDUCED TEMPERATURE. REDUCED PRESSURE. AND THE ACENTRIC FACTOR
ABSTRACT *******************************
C C C C C * * * * * * * * * * * * * * * * * * * * * * * * *
c NOMENCLATURE
118
C I'll- FUGACITY COEFICIENT C PR- THE REDUCED PRESSURE C TR- THE REDUCED TEMPERATURE C W- THE ACENTRIC FACTOR C C-J
c •••***•**••*•••*••••*•*•
FUNCTION PH(TR.PR.W) IMPLICIT REAL*8(A-H.0-Z)
P-(.1445+.073*W)/TR-(.33-.46*W)*TR**(-2)-(.1385+.5*W)*TR**(-3) l-(.0121+.097*W)*TR**(-4)-.0073*W*TR**(-9) PH-10.**(PR*P/2.303) RETURN END
C C************************ ABSRACT ************************************* C
C THIS SUBROUTINE CALCULATES THE NUMBER OF THEORITICAL STAGES FOR C THE COLUMN AND FOR THE STRIPPING SECTION. FILTERED STEADY-STATE DATA C FOR THE REFLUX RATE. THE VAPOR BOIL-UP RATE. FEED RATE. AND THE C COMPOSITON OF THE LIGHT COMPONENT IN THE BOTTOMS AND OVERHEAD ARE C PROVIDED TO THIS SUBROUTINE. THE VALUES OF N AND M ARE FOUND USING A C A NEWTON'S SEARCH TO SATISFY THE SMITH-BRINKLEY MODEL. C C************************ NOMENCLATURE **•*•********•***********••*•**• C
C A(I,J)- THE PARTIAL OF THE ITH EQUATION WITH RESPECT TO THE JTH C UNKNOWN (J-1 IS N; J-2 IS M) C C(I)- THE FUNCTION VALUE OF THE ITH EQUATION C ERLIM- THE RELATIVE ERROR CRITERIA FOR CONVERGENCE OF THE NEWTON'S C SEARCH C G(l)- THE CHANGE IN THE UNKNOWNS CALCULATED BY THE NEWTON'S SEARCH C ICT- AN ITERATION COUNTER C KB(I)- THE K VALUE OF THE ITH COMPONENT FOR THE AREA OF COLUMN BELOW C THE FEED TRAY(I-1 LIGHT COMPONENT: 1-2 HEAVY COMPONENT) C KT(I)- THE K VALUE OF THE ITH COMPONENT FOR THE AREA OF COLUMN ABOVE C THE FEED TRAY (I-l LIGHT COMPONENT; 1-2 HEAVY COMPONENT) C L.Ll- THE REFLUX FLOW RATE (iJIMOLES/HR) C M.Ml- THE NUMBER OF THEORITICAL STAGES BELOW THE FEED TRAY C N.NI- THE TOTAL NUMBER OF THEORITICAL STAGE IN THE COLUMN C NERS- THE NUMBER OF VARIABLE THAT DO NOT MEET THE CONVERGENCE CRITERI C Til- THE BOTTOMS TEMPERATURE (DEC F) C TEST- THE RELATIVE ERROR IN A VARIABLE C TF- THE TEMPERATURE OF THE FEED (DEG F)
C TM- THE AVERAGE TEMPERATURE IN THE STRIPPING SECTION (DEG F) C TN- THE AVERAGE TEMPERATURE IN THE RECTIFYING SECTION (DEG F) C TT- THE TEMPERATURE AT THE TOP OF THE COLUMN (DEG F) C X- MOLE FRACTION OF THE LIGHT COMP IN THE BOTTOMS PRODUCT C Y- MOLE FRACTION OF THE LIGHT COMP IN THE OVERHEAD PRODUCT
C C************************************************************************
C Sl'BROUTINE PARMdl .Nl .Ml .V .L) IMPLICIT REAL*8(A-H.0-Z) DIMENSION A ( 3 . 3 ) . C ( 3 )
119
COMMON /ONE/ D.B.X.Y.TF.TT.TB.P.F.Z.KT.KB COMMON /TWO/ ERLIM.NITR.K1.K2.TAU(2).XINT.YINT.DT.ITYPE REAL*8 I.N,M.KT(2),KB(2).K1(2).K2(2).L(3).GG(3).N1.M1.I1 I-Il N-Nl M-Ml WRITE(6.666)1.N.M
666 FORMAT(6X,'TRAY IN PARM'.3D12.5) C Xl-X C Yl-Y C C ITERATIVELY SOLVE FOR N AND M C
ERLIM - 1.0D-03 NITR - 100 ICT-O
1000 ICT-ICT+1 C CALCULATE THE JACOBIAN OF THE TWO NONLINEAR EQUATIONS
CALL DERPd.N.M.V.L.Xl.Yl.A.C) C CALCULATE THE CHANGE IN N AND M
ALPHA-A(3.2)-(A(3.1)*A(1.2)/Ad.l)) RHO-A(3.3)-(A(3.1)*Ad.3)/A(l.l)) GAMMA-C(2)-(A(2.1)*Cd)/A(l.l)) PHI - C(3).(A(3.1)*C(1)/A(1.1)) OMEGA-A(2,3)-(Ad.3)*A(2.1)/A(l,l)) BETA - A(2.2)-(A(1.2)*A(2.1)/A(1.1)) GG(3) - (PHI-(ALPHA*GAMMA/BETA))/(RHO-(ALPHA*OMEGA/BETA)) GG(2) - (GAMMA/BETA)-(0MEGA/BETA)*GG(3) GG(1) - (C(1)/A(1.1))-(A(1.2)/A(1.1))*GG(2)-(A(1.3)/A(1.1))*GG(3) IF(ABS(GG(l)).GT..l*I)GG(l)-.l*I*ABS(GG(l))/GGd) IF(ABS(GG(2)).GT..1*N)GG(2)-.1*N*ABS(GG(2))/GG(2) IF(ABS(GG(3)).CT..1*M)GG(3)-.1*M*ABS(GG(3))/GG(3) PRINT 22.I,N.M I-I+GG(1) N-N+GG(2) M-M+GG(3) NERS-0 PRINT 24.A(1.1).A(1.2).A(1.3).C(1).GG(1) PRINT 24.A(2.1).A(2.2).A(2.3).C(2).GG(2) PRINT 24.A(3.1).A(3.2).A(3.3).C(3).GG(3)
PRINT 22.I.N.M PRINT 23
24 FORMAT( 5H ACG-.5E12.5) 22 FORMAT( 5H INM-.3E11.4)
23 FORMAT( /) C CHECK FOR CONVERGENCE
TEST-DABS(GG(1)/I) IF(TEST.GT.ERLIM)NERS-NERS+1 TEST-DABS(GG(2)/N) IF(TEST.GT.ERLIM)NERS-NERS*1 TEST-DABS(GG(3)/M) IF(TEST.GT.ERLIM)NERS-NERS+1 IF(1CT.GT.NITR)G0 TO 2000 IF(NERS.NE.0)GO TO 1000 Il-I
120
Nl-N MI-M RETURN
C IF NEWTON'S METHOD DID NOT CONVERGE. INDICATE THRU A PRINT AND RETURN 2000 PRINT 25
25 FORMAT( 'NEWTON METHOD DID NOT CONVERGE IN PARM') RETURN END
C C********************;******** ABSTRACT ********************************
C THIS SUBROUTINE CALCULATES THE DERIVATIVE OF THE TWO EQUATIONS THAT C RESULT FORM THE SMITH-BRINKLEY MODEL WITH RESPECT TO N AND M. THE C DERIVATIVES ARE CALCULATED BY FINITE DIFFERENCE APPROXIMATIONS. THE C ANSWERS ARE STORED IN A(I.J). A(I.J) IS THE JACOBIAN OF THE TWO C NONLINEAR EQUATIONS. C C*************************** NOMENCALTURE ***************************** C C A ( I . J ) - THE JACOBIAN OF THE ITH EQUATION WITH RESPECT TO THE JTH C VARIABLE ( J - 1 V; J - 2 L) C B(I)- THE VALUE OF THE ITH EQUATION C DELTA- THE FRACTIONAL CHANGE IN V AND L USE TO CALCULATE THE C NUMERICAL DERIVATIVES C F(I)- THE EQUATION VALUE OF THE ITH EQUATION C L.Ll- THE REFLUX LIQUID FLOW RATE (^MOLES/HR) C H.M1,M2- THE NUMBER OF THEORITICAL STAGES IN THE RECTIFYING SECTION C N.N1.N2- THE NUMBER OF THEORITICAL STAGES FOR THE COLUMN C V.Vl- THE VAPOR BOIL-UP RATE FOR THE REBOILER (<>MOLES/HR) C X.Xl- THE COMPOSITION OF THE LIGHT COMP IN THE BOTTOMS C Y.Yl- THE COMPOSITION OF THE HEAVY COMP IN THE OVERHEAD C (;•*•>•;••>*******•***************•*•*•****•*•*•*****•*****•***•**••****••**•**
C SUBROUTINE DERP(II.Nl.Ml.V.L.Xl.Yl.A.B) IMPLICIT REAL*8(A-H.0-Z) DIMENSION A(3.3).B(3).F(3) REAL*8 I.11.N.N1.M.MI.L(3).L2.M2.N2.I2 l-Il N-Nl M-Ml X-Xl Y-Yl DELTA-.01
C DETERMINE F(I) FOR THE BASE CASE CALL EVALd.N.M.V.L.X.Y.F) B(l)-F(l) B(2) —F(2) B(3)-F(3)
C INCREMENT I I2-I*(1.+DELTA) CALL EVAL(I2.N.M.V.L.X.Y.F)
C DETERMINE PARTIAL DERIVITATIVES NUMERICALLY A(1.1)-(F(1)-B(1))/1/DELTA A(2.1)-(F(2)-B(2))/1/DELTA
121
A(3.1)-(F(3)-B(3))/1/DELTA C INCREMENT N
N2-N*(1.+DELTA) CALL EVAL(I.N2.M.V.L.X.Y.F)
C CALCULATE PARTIAL DERIVITATIVES NUMERICALLY Ad.2)-(F(1).B(1))/N/DELTA A(2.2)-(F(2)-B(2))/N/DELTA A(3.2)-(F(3)-B(3))/N/DELTA
C INCREMENT M M2-M*(1.+DELTA) CALL EVAL(I,N.M2.V.L.X.Y.F) A(1.3)-(F(1)-B(1))/M/DELTA A(2.3)-(F(2)-B(2))/M/DELTA A(3.3)-(F(3)-B(3))/M/DELTA RETURN END SUBROUTINE CONST(K.L.V,I.N.M)
C C THIS SUBROUTINE CALCULATES THE VALUES FOR THE CONSTANTS REQUIERD C IN THE INTERMIDIATE CALCULATIONS FOR THE RECOVERIES. C
IMPLICIT REAL*8(A-H.0-Z) COMMON /VAR/F.B,D.G.R.NT.NJ COMMON /VAR3/PHI.PHIl.ALPHA,EGA.EGAl.GAMMA.GAMMAl,S(3).BETA DIMENSION K(3).L(3) REAL*8 K.L.I.N.M DO 200 II-l.3
C SEPARATION FACTORS CALCULATION C
Sdl) - K(II)*(V/L(II)) 200 CONTINUE
GAMMA - (S(2)-1.)/(1.-S(2)**N) XM - (S(3)**(M-2)-l.)/(l-S(3)) XM2 - (V*K(3)+B)/L(3) EGA - (XM/B)*(1-XM2) ALPHA - L(1)/(K(1)*D) GAMMAl - (S(2)**(N-1)-1.)/(1-S(2)**N) XXMl - L(l)*(l/K(l)-1.)/V XXM - (S(l)**(l-I)-l.)/(l/(S(l))-l.) EGAl - XXM1*XXM RETURN END
C************** CALCULATE K VALUES •*•*••***•**•***** C THIS SUBROUTINE CALCUUVTES THE K-VALUES FOR EACH OF THE COMPONENTS C USING ANTONINES CONSTANTS. C
SUBROUTINE KVAL(K.NJ.J) IMPLICIT REAL*8(A-H.0-Z) COMMON /VAR1/AA(3).BB1(3).CC(3).DD(3).X(3.3).P(3).T(3).TC(3).PC(3) DIMENSION K(3).P0(3).L(3).T1(3).TR1(3).PPC(3).TTC(3).W(3)
REAL*8 K.L C PPC ARE CRITICAL PRESSURES (PSIA) C. TTC - CRITICAL TEMPERATURE (DEC F) C W - ACTIVITY COEFFICIENT C
122
PPC(l) - 709.8 PPC(2) - 617.4 PPC(3) - 550.7 TTC(l)-550.0 W d ) - .1064 TTC(2)- 665.9 W(2)-0.1538
C AA. BB. 6, CC ARE THE ANTOINES CONSTANTS C
AA(1) - 5.38394 BBld) - 2847.921 CCd) - 434.898 AA(2) - 5.353418 BB1(2) - 3371.048 CC(2) - 414.488 DO 1 N - 1,3 TR - TTC(J)/T(N) PR - PPC(J)/P(N)
POP - AA(J) - (BB1(J)/(T(N)+CC(J))) C CALCULATES PARTIAL PRESSURE
PO(N) - DEXP(POP)*PPC(J) PHX - PH(TR,PR,W(J))
C PHX - 1. C CALCULATES THE K-VALUE
K(N) - PO(N)/(P(N)*PHX) 1 CONTINUE RETURN END
C************* EVALUATE FOR RECOVERY RATES ************* C
C THIS SUBROUTINE CALCULATES THE RECOVERIES. AS WELL AS THE C THE DEVATION FROM THE REQUIRED RECOVERIES C
SUBROUTINE EVAL(I.N.M.V.L.XX.Y.FF) IMPLICIT REAL*8(A-H.0-Z) COMMON /VAR/F.B.D.G.R.NT.NJ COMMON /VAR2/FMB.GMB.DMB COMMON /VAR3/PHI .PHIl. ALPHA. EGA. EGAl .GAMMA.GAMMAl .S( 3) . BETA DIMENSION XX(3).Y(3).Z(3).A(3.3).BB(3).K(3).FF(3).FX(3).L(3) REAL *8 K.L.I.N.M J -1 DO 1 J - 1.2 CALL KVAL(K.NJ.J) CALL CONST(K.L.V.I.N.M) A(l.l) - (V*K(3)*ECA)+(V*K(3)/B) Ad.2) - -L(2)*GAMMA/G A(2.1) - -ALPHA*(EGA1+1.) A(2.2) - A(2.1)-V*K(2)*GAMMA1/G-L(2)/C-(V*K(2)/G)-1. BB(i) - (V*K(3)+L(3)-L(2)-L(2)*GAMMA-F)/F BB(2) - A(2.1)-V*K(2)/F*(1+GAMMA1)
Fl -(A(2.2)*BB(1)-A(1.2)*BB(2))/(A(I.1)*A(2.2)-A(1.2)*A(2.1))
CI - (BB(l)-A(l.I)*Fl)/A(l.2)
Dl - 1-Fl-Gl WRITE(6.102)F1.G1.D1
123
c c C c c c
FF ARE THE DEVIATIONS OF THE CALCULATED RECOVER FROM THE REQUIRED RECOVERY
FFd)-Fl/FMB-l. FF(2)-G1/CMB-1. FF(3)-D1/DMB-1.
102 F0RMAT(6X.'F1-'.D12.5.'G1-',D12.5.'D1-'.D12.5) 1 CONTINUE RETURN END
//CO.SYSIN DD * 5.38389D+00 2.84792D-t-03 5.35342D+00 3.37711D+03 5.74162D+00 4.12639D+03
//
4.34898D+02-2.03539D+00 3.05400IH02 4.14488D+02-1.38551D+00 3.69800D+02 4.09518D+02-3.13O03D+00 5.63100D+02
4.88000D+01 4.25000D4 01 4.42000D+01
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