AN ABSTRACT OF THE THESIS OF
Marcio Matandos for the degree of Master of Science in Chemical
Engineering presented on December 12, 1991
Title:
.
Use of Orthogonal Collocation in the Dynamic Simulation of Staged
Separation Processes
Abstract approved:
Redacted for Privacy
Keith L. Levien
Two basic approaches to reduce computational requirements for solving
distillation problems have been studied: simplifications of the model based on
physical approximations and order reduction techniques based on numerical
approximations.
Several problems have been studied using full and reduced-order
techniques along with the following distillation models: Constant Molar
Overflow, Constant Molar Holdup and Time-Dependent Molar Holdup.
Steady-state results show excellent agreement in the profiles obtained using
orthogonal collocation and demonstrate that with an order reduction of up to
54%, reduced-order models yield better results than physically simpler models.
Step responses demonstrate that with a reduction in computing time of the
order of 60% the method still provides better dynamic simulations than those
obtained using physical simplifications. Frequency response data obtained
from pulse tests has been used to verify that reduced-order solutions preserve
the dynamic characteristics of the original full-order system while physical
simplifications do not.
The orthogonal collocation technique is also applied to a coupled columns
scheme with good results.
Use of Orthogonal Collocation in the
Dynamic Simulation of Staged Separation Processes
by
Marcio Matandos
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Completed December 12, 1991
Commencement Tune 1992
APPROVED:
Redacted for Privacy
Assistant Professor of Chemical Engineering in charge of major
rm
i
n
Redacted for Privacy
Head of Depa4t#nent of Chemical Engineering
Redacted for Privacy
Dean of Gr....6.r
Date thesis is presented December 12, 1991
Typed by Marcio Matandos for Marcio Matandos
ACKNOWLEDGEMENTS
It is hard to imagine that in a personal project such as this one, so many
inevitably become involved and contribute in so many ways.
Among those to whom I feel particularly indebted are Dr. Keith Levien,
whose trust, enthusiasm and guidance throughout the entire course of this
work have been decisive in accomplishing all the goals that were initially set. I
gratefully acknowledge the Teaching Assistantship provided by our Chemical
Engineering Department on Fall 1990 (it was quite an experience!) and thank
very much all faculty members for their dedication and friendship.
To all our friends in Corvallis, thanks a whole lot for making all those
days something to be missed forever.
To my parents, Nicolas and Maria Matandos, and parents-in-law,
Valentino and Oddete Chies, for their love and support throughout the entire
course of this work, my deepest appreciation.
My very special thanks to my son, Bruno, for bringing a new meaning
to even the most trivial things in life and to his mommy, Maria, for her love,
care and patience. Thank you VERY MUCH for being so nice to be with. The
cooperation I received from you guys was simply out of this world!!! I'm
afraid words would never be enough to express all my gratitude to both of
you.
TABLE OF CONTENTS
1-INTRODUCTION
1
1.1-Terminology
3
1.2-Approach to the Problem
1.2.1-Equilibrium-Staged Separation Operations
1.2.2-Systems of Differential and Algebraic Equations
1.2.3-Orthogonal Collocation
4
4
8
9
1.3-Literature Survey
13
2-DEVELOPMENT OF THE MODELS
18
2.1-General Assumptions
18
2.2-Full-Order Models
2.2.1-The Constant Molar Overflow (CMO) Model
2.2.2-The Constant Molar Holdup (CMH) Model
2.2.3-The Time-Dependent Molar Holdup (TDMH) Model
19
25
2.3-Reduced-Order Techniques
2.3.1-Constant Molar Overflow
2.3.2-Constant Molar Holdup
2.3.4-Time-Dependent Molar Holdup
36
41
41
41
3-NUMERICAL IMPLEMENTATION
43
3.1-Thermodynamic Properties
3.1.1-Enthalpies
3.1.2-Equilibrium Constants
3.1.2.1-Bubble Point Temperatures
3.1.2.2-Fraction of Feed Vaporized
3.1.3-Liquid Densities
43
43
44
45
46
47
3.2-Collocation Points
48
3.3-Software for Initial Value Problems
49
28
32
4-EXAMPLE PROBLEMS
51
4.1-Steady-State Results
4.1.1-Temperature Profiles
4.1.2-Composition Profiles
4..3-Liquid and Vapor Flowrate Profiles
51
4.2-Step Tests
66
4.3-Pulse Tests
76
4.4-Coupled Columns
78
4.5-Conclusions
93
NOTATION
94
BIBLIOGRAPHY
97
57
57
64
APPENDICES
A.
Sequence of Computations
B.
The Polynomial Interpolation Technique
99
103
LIST OF FIGURES
Figure
Page
1.1 - Schematic diagram and notation for an equilibrium stage.
5
1.2 - Schematic diagram illustrating the
continuous position variable s within
a separation module.
10
2.1 - Schematic diagram and notation for a distillation column.
20
2.2 - Schematic diagram and notation for equilibrium stage
21
2.3 - Schematic diagram and notation for condenser.
22
2.4 - Schematic diagram and notation for reboiler.
23
2.5 - Mass and energy balance envelopes utilized in the
derivation of the equations for calculating liquid and
vapor flowrates.
31
2.6 - Geometry utilized for calculating liquid flowrates
leaving an equilibrium stage using the Francis weir
formula.
33
2.7 - Schematic diagram and notation utilized when the
order-reduction technique is applied to a distillation
column.
37
4.1
Steady-state temperature profiles for example
problem in Sec.4.1
58
4.2 - Steady-state C1 composition profiles for example
problem in Sec.4.1.
59
Steady-state C2 composition profiles for example
problem in Sec.4.1.
60
4.4 - Steady-state C3 composition profiles for example
problem in Sec.4.1.
61
4.5 - Steady-state C4 composition profiles for example
problem in Sec.4.1.
62
4.3
Page
Figure
4.6 - Steady-state C5 composition profiles for example
problem in Sec.4.1.
63
4.7 Steady-state liquid and vapor flowrate profiles for example
problem in Sec.4.1.
65
4.8 - Initial and final temperature profiles for example problem
in Sec.4.2.
68
4.9 - Step responses for distillate temperature for example
problem in Sec.4 2
69
4.10 - Step responses for C2 distillate composition for example
problem in Sec.4.2
70
4.11 - Step responses for C3 distillate composition for example
problem in Sec.4.2
71
4.12 Step responses for C4 distillate composition for example
problem in Sec.4 2
72
4.13 - Initial and final molar holdup profiles for example
problem in Sec.4.2
75
4.14 - Pulse responses for C2 distillate composition for example
problem in Sec 4 3
77
4.15 - Bode diagram for full-order pulse test responses in Sec.4.3.
79
4.16 - Bode diagram for the TDMH model pulse test responses
in Sec.4 3
80
4.17 - Bode diagram for the CMH model pulse test responses
in Sec.4.3
81
4.18 - Bode diagram for the CMO model pulse test responses
in Sec.4.3
82
4.19 - Specifications for the coupled columns scheme utilized
in Sec.4.4
4.20 - Schematic diagram utilized when the order reduction technique
is applied to the coupled columns problem in Sec.4.4.
84
85
Figure
Page
4.21 - Steady-state temperature profiles for the
example problem discussed in Sec 4 4
87
4.22 - Steady-state C2 liquid composition profiles
for the example problem discussed in Sec.4.4.
88
4.23 - Steady-state C3 liquid composition profiles
for the example problem discussed in Sec.4.4.
89
4.24 - Steady-state C4 liquid composition profiles
for the example problem discussed in Sec.4.4.
90
4.25 - Step responses for C2 distillate composition
for the example problem discussed in Sec.4.4.
91
4.26 - Step responses for C3 distillate composition
for the example problem discussed in Sec.4.4.
92
LIST OF APPENDIX FIGURES
Figure
Page
A.1 - Flowsheet diagram demonstrating the sequence
of computations utilized for solving a distillation
problem using the CMO model.
100
A.2 - Flowsheet diagram demonstrating the sequence
of computations utilized for solving a distillation
problem using the CMH model.
101
A.3 - Flowsheet diagram demonstrating the sequence
of computations utilized for solving a distillation
problem using the TDMH model.
102
B.1 - C2 steady-state composition profile for the rectifying
module in the column treated in Section 4.1 obtained
using the lx1 reduced-order TDMH model.
104
LIST OF TABLES
Table
2-1
Page
Design variables defining the distillation models on Sec 2 2
27
3-1 - Coefficients for liquid and vapor enthalpies
44
Coefficients for equilibrium constants.
45
3-2
3-3 - Critical constants for liquid densities.
47
4-1 - Specifications for the example problems discussed in Chapter 4.
52
4-2 - Collocation points utilized for determining steady-state
conditions for the example problem of Sec.4.1 using
the orthogonal collocation technique.
53
4-3 - Steady-state conditions for distillate and bottoms
products for example problem discussed in Sec.4.1.
55
4-4 - Mean squared errors for steady-state
profiles of example problem in Sec.4.1.
56
4-5 - Collocation points utilized in the solution
of the example problem discussed in Sec.4.2.
67
4-6 - Relative computing times and integral absolute errors
for distillate product in example problem of Sec 4 2
73
4-7 - Collocation points utilized in the solution of
the example problem discussed in Sec.4.4.
86
USE OF ORTHOGONAL COLLOCATION IN THE
DYNAMIC SIMULATION OF STAGED SEPARATION PROCESSES
1- INTRODUCTION
Rigorous analytical representation of equilibrium-staged separation
operations includes a large number of nonlinear mass, energy and equilibrium
relationships which must be simultaneously satisfied at each stage. Therefore, a
significant computational effort is usually required in order to solve
multistage/multicomponent separation problems. One approach to reduce
computational requirements is to somehow reduce the number of equations
involved. Two alternatives have been used: either simplify the model based on
assumptions about physical properties and material and energy relations or
simplify the solution method by solving the stage equations only at certain
locations (collocation points).
The first approach has been extensively applied to graphical and shortcut
multistage calculations, where pressure, molar overflow, relative volatility, and
other parameters are assumed constant within sections of the column. Treybal
(1985) and McCabe and Smith (1985) present detailed treatments on graphical
multistage calculations. Henley and Seader (1981) and Wankat (1988) discuss
approximate analytical methods applied to steady-state calculations. It will be
demonstrated that even though the computing effort can be significantly
reduced, such physical simplifications can lead to substantial errors and
2
therefore are suitable only for preliminary analysis and design of distillation
columns.
The second approach, however, will be shown to yield surprisingly
accurate yet computationally cheap results. The models based on this approach
are called reduced-order models. This work attempts to demonstrate the
advantages of one method of reduced-order modeling, orthogonal collocation
(Stewart et al., 1985), for the dynamic simulation of distillation columns based
on more detailed physical models than previously used. To accomplish this,
the following physical models have been treated:
Model # 1 - Constant Molar Overflow (CMO)
Model # 2 - Constant Molar Holdup (CMH)
Model # 3 - Time-Dependent Molar Holdup (TDMH)
The approach utilized to achieve the goals proposed in this thesis consists
of assuming that the full-order TDMH model above yields "true" dynamic
responses and steady-state results. Several distillation problems have been
studied using full and reduced-order techniques for each of the three physical
models. The following aspects have been compared against the full-order
TDMH model:
Steady-state profiles for temperature and composition within a column
3
have been used to verify the accuracy of each method in reproducing
steady-state profiles obtained by the "true" model.
Dynamic responses of column products to step changes of input variables
have been used to verify the ability of each method to predict column
dynamics and to compare the computing time required to solve the
problem.
Dynamic pulse tests have been simulated and analyzed to generate the
frequency response data necessary for determining whether the dynamic
characteristics of the "true" model are well represented.
In addition, in order to demonstrate the suitability of the method for
treating more complex separation systems, orthogonal collocation is applied to
the dynamic simulation of a coupled-columns scheme.
1.1-Terminology
The following definitions are used in this thesis:
Full-order model refers to a physical model where all pertinent
equations are applied at each stage.
4
Reduced-order model refers to a simplification of the physical model
where the number of locations at which the equations are applied has
been reduced by use of numerical techniques.
Section refers to a physical compartment within a column where a
specific separation operation is taking place.
Module represents a set of (not necessarily physical) equilibrium
stages placed one atop the other where no sidestreams are introduced
nor drawn.
Computing time represents the time (expressed in CPU seconds)
required to solve a simulation and depends on the accuracy criteria and
machine being used.
Computing effort refers to the overall amount of mathematical
operations required to solve a simulation problem.
1.2-Approach to the Problem
1.2.1-Equilibrium-Staged Separation Operations
Rigorous models describing equilibrium-staged separation
operations are widely accepted and used in the analysis and design of
staged separation processes. The basic physical element utilized in these
models is the equilibrium stage shown schematically in Fig. 1.1.
5
Li-1
14
hi -1
Hi
yij
xi-ij
1
1
Mi, hi, xii
Li
hi
xii
Vi i-1
Hi+1
yi+i,i
Figure 1.1 - Schematic diagram and notation
for an equilibrium stage.
6
Assuming negligible mass and energy holdup in the vapor phase,
application of balance relations to equilibrium stage i (where i indicates
the stage number on a module counting down from the top) containing
nccomponents yields the following set of equations:
Overall material balance
dM.
dt
z =V. +L.
,-1 -V.-L.
Balance for species j (j=1,...,nd
(1-2)
dt
Energy balance
dM.h.
=VI. +1H1 . +1
. -1
dt
-ViHi -L i hi
(1-3)
Equilibrium relationships
y.. = K; x..
(1-4)
Ex 1.4=1 ; E y = 1
(1-5)
Summation relations
n,
j=i
By combining Eqs (1-2) and (1-5), Eq.(1-1) can be obtained and
consequently, one of the j- species balance equations could be disposed
7
of. It is advisable however, not to do so since accumulated round -offs in
the computations could eventually cause one of the compositions to go
negative. Therefore, in order to completely represent the dynamic
behavior of a M-stage, nc-component separation module, a set of
M*(nc+2) differential equations, Eqs (1-1), (1-2) and (1-3), plus a number
of usually nonlinear algebraic relations dependent upon the modeling
assumptions are required.
Gani et al. (1986a) present a generalized dynamic model for
distillation columns. Some algebraic relations included in that model are
listed below to demonstrate the degree of complexity involved in the
rigorous approach.
- Multicomponent non-ideal thermodynamic properties.
- Non-adiabatic columns.
- Pressure dependent vapor flowrates.
- Liquid flowrates dependent on plate geometry/type.
- Entrainment, weeping and flooding effects.
- Bubble, froth and wave formation.
Since these relations must be satisfied at each stage, dynamic
simulation of staged separation problems using such detailed models is
computationally demanding. The models treated in this work involve
8
somewhat simpler algebraic relations but are adequate to accomplish the
goals proposed.
1.2.2-Systems of Differential and Algebraic Equations
Several numerical techniques are available for integrating systems of
coupled nonlinear differential and algebraic equations such as those
encountered in the dynamic simulation of distillation columns. Holland
and Liapis (1983) present a detailed discussion on many of these.
In order to illustrate the computational effort required, the following
system of equations is utilized
du
=f(u,v,t)
dt
(1-6)
0 =g(u,v,t)
(1-7)
where u is the solution vector of length k, v is the vector of k dependent
variables and g represents a set of k algebraic relations. Depending on
the nature of this system, several different methods can be used to carry
its integration. Gani et al. (1986b) suggested that backwarddifferentiation (BDF) and diagonally implicit Runge-Kutta (DIRK)
methods are efficient means for solving the equations arising from
distillation modeling. In order to compute the solution vector u, these
9
methods require evaluation of the Jacobian (matrix of partial
derivatives) given by
Sf
Sf .8v
(1-8)
Su Sv Su
and a number of operations involving matrices of dimension k *k,
demonstrating that computational requirements are influenced in a
quadratic fashion by the number of equations in the system.
1.2.3-Orthogonal Collocation
Orthogonal collocation is a class of methods of weighted residuals,
which are numerical techniques utilized to solve differential equations.
An extensive treatment on these methods is available in Finlayson
(1980).
The fundamental idea behind the application of orthogonal
collocation to staged separation processes lies in the assumption that
variables defined within a separation module can be treated as
continuous functions of the position variable s. Figure 1.2 shows the
schematic diagram of a separation module consisting of M equilibrium
stages where Eqs (1-1) to (1-5) are to be applied. Let l(s,t) and v(s,t)
respectively represent any variable related to the liquid and vapor
phases. As shown by Stewart et al. (1985), this problem can be
10
'MI..
s=0
i=0
11
to
VI
i=1
s =1
iv
t
11
V2
i=2
s=2
s =M-1
VM
i=M
s=M
A1
i =M +1
Mk
s=M+ I
Figure 1.2 - Schematic diagram illustrating the
the continuous position variable s within a
separation module.
11
approximated by one of lower order. To do so, l(s,t) and v(s,t) are
approximated by polynomials using tr/1v1 interior grid points,
sn (n=1,...,m), plus the entry points s0=0 for the liquid and sni+1=M+1 for
the vapor. This gives the following expressions
m
Rs,t)=E w,,(s)Rs,,,t)
..o
0<s
M
m.+1
ti(s,,t)=E w,(s)z-(s,t)
1 5_ s 5.. M+1
(1-9)
(1-10)
ne,1
where a tilde (-) represents an approximation to the variable of interest
and the W-functions are Lagrange interpolating polynomials calculated
using the formulas
m ss
W =11
1c=0 S n
hen
m+1
W,(s) =
k
Sk
ss
k=i sn-sk
Ti
ir..
n =0,...,m
(1-11)
n =1,...,m+1
(1-12)
As a result, variables l(s,t) and v(s,t) have been approximated using
m-th order polynomials in s, given by Eqs (1-11) and (1-12), and time
varying coefficients, l(sn,t) and v(s,t) in Eqs (1-9) and (1-10). Insertion of
Eqs (1-9) and (1-10) into Eqs (1-1), (1-2) and (1-3) for a separation
module yields the following expressions which can be considered mass,
species and energy "stage" balances around locations s1,...,s, which are
12
the collocation points.
dia(si,t)
= V(s i+1,t) + 1:(s i-1,t) -1-7(si,t)
dt
[(set)
(1-13)
d A-4(s i,t)I.(s i,t)
= V(si+1,t)g i(s i+1,t) + f(si-1,01i(s i-1,t)
dt
d
17(s i,t)g
i(si,t)
L(s i,t)ii(si,t)
i,t)Ii(s i,t)
dt
(1-14)
= V(s i+1,014(s i+1,t) +
17(s i,t)14(s i,t)
i-1,t)
(1-15)
1.:(s i,t)li(s i,t)
Inlet conditions for liquid at s0=0 and vapor at sm+1=M+1 are boundary
conditions for the separation module.
The number and position of the collocation points within the
separation module play an important role in the accuracy that can be
obtained with this method. Based on the requirement that the
summation of squared residuals for variables l(s,t) and v(s,t) be
minimized at locations s=1,2,...,M (corresponding to the location of the
actual physical stages), Stewart et al.(1985) recommended that
collocation points be selected as roots of Hahn orthogonal polynomials
Q(x;cc,(3,N), a,I3>-1, which satisfy the following orthogonality criteria
E w(x;aA3 ,N) Q,(x;a,P,N) Qn(x;a,f3,N) = 0
x=0
x=s-1 ; N=M-1
m*n
(1-16)
13
where the weighing function w(x;a,(3,N) is given by
(a+1) ((3 +1)N-x
W(X;a,(3,N) =
(1-17)
x!(N-x)!
The choice a=13 yields symmetrical collocation points within the
module and the selection a=f3=0, gives uniform weighing in Eq.(1-16).
When the number of collocation points is made equal to the number
of stages in the module, Hahn orthogonal polynomials yield collocation
points which fall exactly at the physical stages and consequently, the
full stagewise solution is recovered.
Recursive relationships for obtaining Hahn orthogonal polynomials,
the appropriate choice of parameters a and 13, its effects and other
properties of these polynomials are discussed by Stewart et al.(1985). All
example problems treated in this work using the order reduction
technique use Hahn orthogonal polynomials with parameters a=f3=0.
1.3 - Literature Survey
Early attempts to solve staged separation problems using the assumption
of continuous variables within a separation module have been presented by
Wilkinson and Armstrong (1957) who applied this idea to the dynamic
simulation of binary distillation columns. The model was approximated by
linear partial differential equations which were then analytically solved using
14
Laplace transforms.
Osborne (1971) numerically solved a PDE-approximated model for
multicomponent systems using finite differencing. In order to assure numerical
stability of the integration when large steps were taken in time, step sizes for
the spatial variable were taken to be smaller than the distance between stages.
Wong and Luus (1980) applied orthogonal collocation to the control
problem of a staged gas absorber using a state-space representation of the
differential equations. In their approach the partial derivatives of state
variables with respect to space were approximated by Lagrangian interpolation
using collocation points selected as the roots of shifted orthogonal Legendre
polynomials which obey the orthogonality relationship
fP .(x)P.(x)dx =0
in#n
(1-18)
Joseph and co-workers presented a series of papers exploring various
aspects of orthogonal collocation applied to the dynamic simulation of staged
separation operations. The approach utilized was similar to that of Wong and
Luus but differed in that collocation points were selected as the roots of
orthogonal Jacobi polynomials, Pn"(z), defined by the relationship
P(1 z)azi (aP(z)dz=0
0
j= 0,1,...,n -1
(1-19)
15
In parts I and II of the series, Cho and Joseph (1983a,b) developed the
order reduction technique and used it to simulate the dynamic response of
single module gas absorbers.
Stewart et al. (1985) extended the general approach proposed by Cho and
Joseph (1983a) to obtain steady-state results and step responses for distillation
columns, the major difference being the choice of polynomials used to select
the collocation points. They observed that both Legendre and Jacobi orthogonal
polynomials are orthogonal for continuous variables and therefore, unsuited to
staged systems. They demonstrated that optimal collocation points for staged
systems, selected as the roots of Hahn orthogonal polynomials, gave better
results than those obtained using Jacobi polynomials. Only models using these
Hahn polynomials converge exactly to the full-order solution when the number
of collocation points is made equal to the actual number of stages present in
the column. Methods that use other orthogonal polynomials introduce errors
even if the number of collocation points is made equal to the number of stages.
In Cho and Joseph (1984) the method was applied to columns with
sidestreams. Since one single approximating polynomial was used for each
variable throughout the entire column, discontinuities in the profiles
introduced by the sidestreams could only be partially accounted for and the
location of feed and withdrawal stages had to be approximated to the
collocation point closest to their actual location. Even though outlet conditions
could be reasonably predicted, this approach resulted in oscillatory profiles
16
within the column, yielding poor accuracy at internal points.
Srivastava and Joseph (1987a) addressed the sidestream problem using
different approximating polynomials for each section of the column and an
additional collocation point at the feed location similarly to the treatment
proposed by Stewart et al. (1985). The complete profile was then approximated
by splines which resulted in improved accuracy for predicting conditions
within the entire column.
Srivastava and Joseph (1985) performed an error analysis to determine the
appropriate choice of parameters a and 13 for Hahn and Jacobi polynomials
and compared the solutions obtained using each of these polynomials. They
conduded that Hahn polynomials are preferable when columns with small
number of trays are being treated. In addition, they introduced the concept of
an order reduction parameter, defined as ORP=N In A (where A =L /KV is the
absorption factor), used as an indicator of the compromise between order
reduction and accuracy attainable for a separation module. The term L /KV
represents the ratio of the slope of the operating line to that of the equilibrium
line of the full-order system at steady-state conditions. For a linear, binary
system this absorption factor can be graphically obtained but for non-linear,
multicomponent systems, the availability of steady-state profiles is necessary
for calculating an effective absorption factor based on a geometric average of
the A's for each stage in the module. Large absolute values of the order
reduction parameter indicate the presence of steep composition profiles within
17
the column which, for reasonable accuracy, require higher order approximating
polynomials.
Srivastava and Joseph (1987b) introduced the idea of global and local
collocation points as an approach for treating flat and steep composition
profiles. Global collocation points are used for approximating non-steep
profiles (such as those observed for key components, flowrates and
temperatures) and local collocation points are used to approximate steep
profiles resulting from the presence of non-key components.
Swartz and Stewart (1986) also used Hahn polynomials in orthogonal
collocation for optimal distillation column design. The approach consisted of
defining an objective function based on economic aspects, constraints involving
the recovery of key components and using the number of stages as the
decision variables. Results obtained for a simple binary system were in good
agreement with those obtained by the McCabe-Thiele graphical construction.
18
2-DEVELOPMENT OF THE MODELS
2.1-General Assumptions
All models treated in this work have been developed based on the
following general assumptions.
Adiabatic stages Columns are well insulated and therefore heat
losses to the surroundings can be neglected.
Good mixing of phases Each phase on each stage is well mixed and
homogeneous.
Physical equilibrium at stages Liquid and vapor streams leaving a
stage are in thermal equilibrium. In addition, there is a well defined
relationship between the compositions of these streams.
Negligible vapor holdup The density of the vapor phase is negligible
when compared to that of the liquid. Consequently, all the mass present
in the system is assumed to be that of the liquid and no balances are
made around the vapor phase.
19
Constant pressure within the column Pressure variations within the
column and the resulting effects on its dynamics are negligible.
Ideal thermodynamic properties All fluid mixtures involved have
been treated as ideal. In practice this is equivalent to ignoring mixing
effects and considering all thermodynamic properties to be additive on a
mole fraction basis.
In addition to these, other assumptions relative to each particular model
have been made and will be discussed prior to the model's development.
2.2-Full-Order Models
A schematic representation of a distillation column is presented in Fig.2.1.
The standard elements present in this column are the internal equilibrium
stages, condenser and reboiler, shown in more detail in Figs. 2.2, 2.3 and 2.4
respectively. In the interest of simplicity, condensers and reboilers used in the
example problems have been assumed to be partial and therefore, the distillate
product is withdrawn as saturated vapor and the bottoms product as saturated
liquid. Application of Eqs (1-1), (1-2) and (1-3) to each of these elements yields
the following expressions, used in the development of the different distillation
models, CMO, CMH, and TDMH treated in this thesis.
20
D
''''...._
i=f-1
Il
i=f
Figure 2.1 - Schematic diagram and notation
for a distillation column.
21
Li -1
j
F
XF j
XW
HF
Hw
Mi, hi, xii
IP]
ILi
hi
xii
Hi+i
yi+li
Figure 2.2 - Schematic diagram and notation
for equilibrium stage.
22
D,HD, YD,j
i=1
Figure 2.3 - Schematic diagram and notation
for condenser.
23
i=N
Figure 2.4 - Schematic diagram and notation
for reboiler.
24
Condenser
dM0
dt
dMo x
= ViYij
dt
dMoho
=V1 -L0 -D
(2-1.a)
Loxoi DYDJ
(2-1.b)
=Villi-Loho-DHD-Qc
(2-1.c)
=Vi+1+Li_1 +F-Vi-Li-W
(2-2.a)
dt
Internal Stages
dM.
dt
dM.x..
dt
Vi+iyi+14 +Li_ixi_i +FXFi- V iyij-
WXwj
dM.h.
(2 -2.b)
(2-2.c)
d;
Reboiler
dMN+1
B
(2-3.a)
- VN ÷iyN+1,;-BXB4
(2-3.b)
dt
dA4N+1xN+1,1.
dt
dMN+ihN+1
dt
= LNhN Vs+1HN+1 BhB + QR
(2-3.c)
25
Liquid and vapor enthalpies are given by
hi=h({xid,Ti) ;
(2-4)
and vapor-liquid equilibria are represented by the following functional
relations
yii = E. Ki4(
)x.4
nc
E yi4 =1
(2-5)
(2-6)
J=1
where i=0,1,...,f,...,N+1, is the stage number.
2.2.1-The Constant Molar Overflow (CMO) Model
This is the dynamic/multicomponent version of the simple binary
model used in the McCabe-Thiele graphical method. The following
assumptions are made in addition to those presented in Sec. 2.1.
Constant molar holdup The total number of moles present on any
stage is constant and consequently, the left-hand sides of Eqs (2-1.a),
(2-2.a) and (2-3.a) are set to zero.
Constant molar overflow Occurs when molar latent heats for each
26
species are equal and sensible heat changes for liquid and vapor are
neglected. Thus, each mole of vapor condensed on a stage will vaporize
exactly one mole of liquid and therefore, the left-hand sides of
Eqs (2-1.c), (2-2.c) and (2-3.c) are set to zero.
This model is thus defined by nc*(N+2) ordinary differential
equations, Eqs (2-1.b), (2-2.b) and (2-3.b) and a set of (tic +5)*(N+2)
algebraic relations given by Eqs (2-1.a), (2-2.a) and (2-3.a), Eqs (2-1.c), (22.c) and (2-3.c), Eqs (2-4) and Eqs (2-5) and (2-6).
Design variables required by this model are presented in Table 2-1
which, when combined with the overall mass and energy balance
relations, result in constant liquid and vapor flowrates for stages within
the stripping and rectifying sections of the column as follows:
Stripping section
B=F-D
Vs=RB*B
(2-7)
L =V +B
S
S
Rectifying section
LT=Ls-(1-11)*F
(2-8)
VF =VS 4-11*F
where W=VIF is the fraction of feed vaporized
The main reason for exploring this model lies in the fact that when
flowrates are constant, the j-species material balance (given by
27
Design Variables
CMO
CMH
TDMH
Total number of stages, N
R
R
R
Location of feed stage, f
R
R
R
Number of components, nc
R
R
R
Feed flowrate, F
R*
R*
R*
Feed temperature, TF
R
R
R
Feed composition, 4
R
R
R
Species vaporization efficiency, Ei
R
R
R
R*
R*
R*
NR
NR
Reboiler duty, QR
NR
R*
R*
Plate geometry, cd, lw, hw
NR
NR
R
Stage molar holdup, M.
R
R
NR
Distillate rate, D
Boil-up ratio, RB
Table 2-1 - Design variables defining the distillation models
presented on Sec.2.2. (R-required, NR-not required, (*)-used as a
manipulated variable in the example problems).
28
Eq.(2-2.b)) for any stage i depends on the composition of the remaining
nc-1 components on that stage, as well as on that of the stages directly
above and below it. Consequently, the differential system of equations
defining this model using liquid compositions xii as state variables, can
be represented by a banded matrix with 2*nc-1 elements on each side of
the band which involves computationally cheap Jacobian evaluations
and provides a good opportunity to compare the efficiency of reducedorder models with physical models that are numerically easier to solve.
2.2.2-The Constant Molar Holdup (CMH) Model
The following assumptions are made in addition to those in Sec. 2.1.
Constant molar holdup Same as in Sec. 2.2.1.
Algebraic expression for energy balances Assumption of constant
molar holdup allows the left hand side of Eqs (2-1.c), (2-2.c) and (2-3.c)
to be written as
dM =mdhi
dt
'
(2-9)
dt
As demonstrated by Howard (1970) and under the simplifying
assumption of negligible mixing effects by Cho and Joseph (1983b), the
29
differential term dhildt in equation (2-9) can be written as an algebraic
expression in terms of dxij/dt as follows:
At constant pressure, the term dhildt can be expanded into
dhi
ahidTi
dt
aT dt
ahi
(2-10)
dt
Differentiating the bubble point relation given by Eqs (2-5) and (2-6)
d
dK..
c
+K
(EE.K..x..)=EE.(x1.
dt j=1
14 14
4 dt
dt
j1
(2-11)
dK.. dT.
n
=E
dT dt
dt
Which can be solved for dTildt
nc
E E K14
dTi
j=i
nc
dt
dx14..
dt
(2-12)
dK..
EEx
14
11 4
dT
Substituting back into Eq.(2-10) yields
nc
dx..
dhi
EK
14
ahii71 1 74 dt
dt
aT
n
c
ah1
dx..14
dK.. j.i
E Ei x.
14
j.1
4 dT
axis
dt
nc
(i =0,1,...,N +1)
.
(2-13)
30
Eqs (2-1.c), (2-2.c) and (2-3.c) are not independent but subject to
equilibrium (bubble point temperature) requirements.
As in the CMO model, this model is defined by the same nc*(N+2)
differential equations and (nc +5)*(N+2) algebraic relations but, due to
the addition of Eq.(2-13) result in a more involved system of equations.
Design variables required by this model are given in Table 2-1 and
the following equations, obtained by applying mass and energy balances
around the bottom of the column and stage i as demonstrated in Fig.2.5,
are used to calculate liquid and vapor flowrates along the column
i+1
B(hB-Hi+1)+F* (Hi+1-HF*
L=
"2
E
R
k.N +1
dhk
M k dt
(2-14)
hi-Hi+1
(2-15)
Vi+1=Li+F* -B
i=N,N-1,...,1,0
where B=F-D, dhk/dt in Eq.(2-14) is calculated using Eq.(2-13) and F* and
H; are given by
=0
= (1-W)F
F* = F
;
;
;
HF. =0
i>f
11; =
i=f
hF
H; = HF+hF
Due to the summation term in Eq.(2-14), liquid flowrates leaving
stage i and vapor flowrates entering it will be dependent upon the
31
i=0
Figure 2.5 - Mass and energy balance envelopes utilized in the
derivation of the equations for calculating liquid and vapor flowrates.
32
liquid composition on all stages below i. As a result, the system's matrix
(using xi4 as state variables) has an upper triangular structure with a
lower bandwidth of 2*nc-1 elements.
2.2.3-The Time-Dependent Molar Holdup (TDMH) Model
The following assumptions are made in addition to those in Sec.2.1.
Algebraic expression for energy balances Same as in Sec.2.2.2.
Hydraulic relations for liquid flowrates Liquid flowrates leaving an
internal stage, as shown in Fig.2.6, depend upon the height of liquid on
that stage and can be calculated using the following expression, which is
a modified version of the Francis weir formula
Q. =1,(
h°'
0.48Fw
)3"
(2-15)
(i=1,...,N)
where Q. is the volumetric liquid flowrate in gallons per minute, /w is
the length of the weir, Fw is a weir constant and the height of vapor-free
liquid over the weir, ho, is given by
M.
how,.
pi
(2-16)
33
hOW
hw
Figure 2.6 - Geometry utilized for calculating liquid flowrates
leaving an equilibrium stage using the Francis weir formula.
34
Introduction of independent equations for liquid flowrates as
determined by Eq.(2-15) results in requiring the molar holdups for
internal stages to be additional state variables and consequently, an
extra set of differential equations, the overall material balance, given by
Eq.(2-2.a) has to be solved.
Perfect level control for condenser and reboiler drums Liquid level
in the drums is maintained constant using a perfect level control policy.
Assuming that the drums are vertical cylindrical tanks, this corresponds
to maintaining the liquid volume constant. The level control equations
can be obtained by expressing the derivatives in Eqs(2-1.a) and (2-3.a) as
dMi
d(V.p.)
dt
dt
dpi
1 dt
7
(2-17)
The derivative dpi/dt above can be further expanded into
dpi
dt
ap dTi
nc
+
aT dt
j.1
ap d
axii dt
(2-18)
Substituting the term dT1/dt from Eq.(2-12) yields
nc
dpi
dt
dx..
E
EK
app
I 14 dt
P j:c1
E Ex.
J .1
I
DK..
14
14 aT
i=0,N+1
n
vpi
j=i ax
A
(2-19)
dt
35
This model is thus defined by a system of (nc+1)*(N+2)-2 differential
equations given by Eqs (2-2.a), (2-1.b), (2 -2.b) and (2-3.b), and
nc*(N+2)+6N+12 algebraic relations given by Eqs (2-1.c), (2-2.c) and
(2-3.c) along with Eq.(2-13), Eqs (2-1.a) and (2-3.a) along with Eq.(2-19),
Eq.(2-15) and Eqs (2-4), (2-5) and (2-6).
Design variables required by this model are given in Table 2-1.
Liquid and vapor flowrates leaving the reboiler are determined using
the following expressions
dhN
QR + LN(hN /113 )+MN+1
VN+1
(2-20)
dt
HN+1-hB
B =LN 17N+1
dMN+1
(2-21)
dt
Vapor flowrates leaving internal stages are calculated using
+F HP. -Bh B R
L1.-1h2.-1 +Q
V.=
2
E
I
k-N
"
dM h
dt
(2-22)
H.
(i=N,N-1,...,2)
Application of a mass balance around stages N+1 to 1 and energy
balance around the condenser yield the following expressions for
calculating liquid and vapor flowrates leaving and entering the
condenser
36
F HF + Q
v1=
dM
+h (
R
°
dt°
N+1 dM
-D)+BhB+E
.
k1
=
dt
k
(2-23)
H1 -h0
L0=V1-D
dM0
(2-24)
dt
F" and liFt follow the same definition presented in Sec.2.2.2 and the
term dMkhK/dt is calculated using the following formula
dMkhk
dt
dhk
k dt
dM
k dt
(2-25)
where dhk/dt is given by Eq.(2-13) and Dmk/dt is given by Eqs (2-19) and
(2-17).
Similarly to the previous model, this system's matrix (with state
variables x14 and Mt (i=1,..N)) also presents an upper triangular structure
but a lower bandwidth with 2*nc elements due to the addition of one
extra differential equation per internal stage given by Eq.(2-2.a).
2.3-Reduced-Order Techniques
Figure 2.7 shows the schematic diagram utilized when the order-reduction
technique of Sec.1.2.2 is applied to a distillation column. As opposed to
Sec.1.2.2, in this section a tilde (-) has been used only to designate streams and
variables entering a collocation point, which must be determined by
37
i=0
Figure 2.7 - Schematic diagram and notation utilized when
the order-reduction technique is applied to a distillation column.
38
interpolation. Recalling the definition from Sec.1.1, side feeds and withdrawals
are not allowed in a module and therefore, in order to provide the boundary
conditions necessary for applying the order-reduction technique to the
rectifying and stripping modules, the stages above and below the feed have
been treated separately. This results in a matching of interpolating polynomials
for the two modules which satisfy mass and energy balances within the entire
column. The following expressions are obtained by combining Eqs (1-6) to (1-9)
with the overall material, species and energy balance equations from Sec.2.2.
Condenser
dMo
dt
(2-26.a)
=Vo-Lo-D
dMoxoi
DYD,i
(2-26.b)
=V0H0-Loho-DHD-Qc
(2-26.c)
dt
dMoho
dt
Stages within a module
dM.
dt
dMixij
=Vi+Li-Vi-Li
Viyii-Lixii
(2-27.a)
(2 -27.b)
dt
dMihi
dt
=ViHi+Lihi-ViHi-Lihi
(2-27.c)
39
Reboiler
dMm +m +3
2
1
, V
=L
dt
ml +M2 +a
ni l
(2-28.a)
B
+m2+3
dMmi +m2+3 Xm, +m2+34
,-,, +m2+3 m, +m2
dt
dMmi
BXBi
,j
2
1
hml+m2
+m2+3
+3
dt
=L m l+m2 +3 rm
1
+M2 +3
(2-28.b)
3.A7
Vm 1 +m2 +3 Hml +m2 +3
(2-28.c)
BhB 4- QR
Stage Above Feed Location
dMm
dt
+1
1
1
d
dt
1
= Vnii+2Ymi +24. +(m1+1imi+1,j
+ qi F Y.F.4
dM _ohm
dt
1
+1.
1
m1
(2-29.a)
+WF-Vm +i-Lm
2 + 1:
1
dM +ixm
mi
=V.
Vm, +1 Y M, +,i,j
(2-29.b)
L M, +1 X M, + Li
+1
1
=1/m1+2Hrni +2 + 1,- m + 1 h. m 1 + 1
+TFHF-V
m1+1
H
m1+1
(2-29.c)
-L
h
m1+1 m1+1
Stage Below Feed Location
dMm 1
dt
V-m1+2 +
Lm1+1+ (1 -W )F V mi + 2 -L m, +2
(2-30.a)
40
dMrn +2Xrn +2;
= I ni,+2gm1+2,j + Lmi+1Xm,+14
dt
+ (1 -1" )FXF4 -V m,.2ymi+24 L
(2-30.b)
Xmi
+2
dt
+Lnyihrni+1
(2-30.c)
+ (1-110F h F V mi+2H mi+2 - L m1+2 h m1+2
where m1 represents the number of collocation points in the rectifying module.
As demonstrated by Srivastava and Joseph (1987a), depending on the
distillation model being used - CMO, CMH or TDMH the stages above and
below the feed location do not necessarily need to be treated separately. In this
work however, these stages have been treated individually to guarantee
closure of mass and energy balances, independent of the quality of the feed.
By using this approach, any specified discontinuities in column profiles at the
stages above or below the feed location are exactly accounted for in the
collocation solution.
The incorporation of the order reduction technique to the distillation
models of Sec.2.2 can be accomplished simply by replacing the full-order
material, species and energy balance equations with their corresponding
reduced-order counterparts. The assumptions, algebraic relations and design
variables (except for the addition of the desired number of collocation points
for each module) remain unchanged.
41
2.3.1-Constant Molar Overflow
For this model, liquid and vapor compositions for streams entering a
stage are determined by interpolation and flowrates are calculated using
the same equations as in the full-order model, Eqs (2-7) and (2-8). The
resulting model is defined by a system of ne(mi+m2+4) differential
equations (where m2 represents the number of stages in the stripping
section) and the same algebraic relations as given in Sec.2.2.1.
2.3.2-Constant Molar Holdup
In this model, compositions and enthalpies for liquid and vapor
streams have to be interpolated. The model is defined by
nc *(mi +m2 +4)
differential equations and algebraic relations as in Sec.2.2.2. Liquid and
vapor flowrates are calculated using endaround material and energy
balances as previously done for obtaining Eqs (2-14) and (2-15).
2.3.3-Time-Dependent Molar Holdup
Variables calculated by interpolation are composition and enthalpy
for liquid and vapor streams and also, liquid flowrates. The resulting
system contains (nc+1)*(mi+m2+4)-2 differential equations plus the
42
algebraic relations of Sec.2.2.3. Vapor and liquid flowrates leaving the
condenser and reboiler are calculated using endaround material and
energy balances as for Eqs (2-20) to (2-24).
It should be observed that in order to calculate unknowns within a
module, the polynomial interpolation technique utilizes variables from each
collocation point in that module and consequently, the system of equations
arising from reduced-order modeling always present a dense matrix structure.
43
3-NUMERICAL IMPLEMENTATION
This chapter describes the means by which the distillation models of
Chapter 2 have been implemented.
3.1-Thermodynamic Properties
Since thermodynamic data has been extensively correlated for
hydrocarbons and because hydrocarbon mixtures are frequently encountered in
separation operations, the example problems treated in this thesis involve
separations of methane, ethane, propane, n-butane and n-pentane, denoted by
C1 through C5 respectively.
3.1.1-Enthalpies
Liquid and vapor enthalpies are given by
nc
(3-1.a)
hi= E
J.1
nc
Hi =
yiiHi*(Ti)
(3-1.b)
Species molar enthalpies, hi' for the liquid and Hi* for the vapor, are
44
calculated using
hi*(T) = a
(3-2.a)
2
1+b 1T +c 1T.
(3-2.b)
H.*(T)=A.1 +B.T+CT2
where hj* and Hj*, are given in Btu/lbmol, T in deg. F and coefficients aj,
bj, cj and Aj, Bj, Cj are presented in Table 3-1.
Species
aj
bi
ci
Aj
C.
Bi
1
C1
0.0
14.17
-1.782E-3
1604
9.357
1.782E-3
C2
0.0
16.54
3.341E-3
4661
15.54
3.341E-3
C3
0.0
22.78
4.899E-3
5070
26.45
0.0
C4
0.0
31.97
5.812E-3
5231
33.90
5.812E-3
C5
0.0
39.68
8.017E-3
5411
42.09
8.017E-3
Table 3-1 - Coefficients for liquid and vapor enthalpies conditions:400 psia
and 0 to 300 deg.F (Reproduced from Henley and Seader (1981)).
3.1.2-Equilibrium Constants
Equilibrium constants are calculated using the expression
Ki(T)=aj + jT +yiT2+ 873
where T is in deg. F and coefficients aj, 13j, yj and Si are given in
Table 3-2.
(3-3)
45
Species
cc i
i3;
Yi
Si
C1
4.35
2.542E-2
-2.0E-5
8.333E-9
C2
0.65
8.183E-3
2.25E-5
-2.333E-8
C3
0.15
2.383E-3
2.35E-5
-2.333E-8
C4
0.0375
5.725E-4
1.075E-5
-2.5E-10
C5
0.0105
2.692E-4
2.55E-6
1.108E-8
Table 3-2 - Coefficients for equilibrium constants - conditions: 400 psia and
0 to 300 deg.F (Reproduced from Henley and Seader (1981)).
3.1.2.1-Bubble Point Temperatures
Bubble point temperatures for saturated liquid mixtures of known
composition are calculated by combining Eqs (2-5) and (2-6) to obtain
n,
F(T)= E x.EKi(T)-1= 0
(3-4)
Newton's method for finding a root for such functions is given by
F(Tk)
(3-5)
Tk., = Tk-
F (Tk)
where subscript k indicates the trial number and the derivative F'(T) is
calculated by
nc
Fi(T)=Ex.E
dK.(T)
dT
(3-6)
46
From an initial estimate of Tk, Tk+i can be calculated using
Eq.(3-6). The method proceeds by successively updating Tk with its new
estimate Tk+1, until their difference is less than a desired convergence
parameter e.
3.1.2.2-Fraction of Feed Vaporized
In order to calculate the fraction of feed vaporized, W=V/F, resulting
from the flash vaporization of a feed stream of known composition z
and temperature T, the '11 function is utilized
nc
FOP) =E
z (1 -K.(T ))
I
F
(3-7)
=0
1 -411(Ki(T F) -1)
The fraction of feed vaporized can be calculated by applying
Newton's method of Sec.3.1.2.1 to Eq.(3-7) using 'If as the independent
variable. The composition of the resulting liquid/vapor feed splits is
calculated as follows:
IP=0
XI .=Z
1.
x.=
1
z,
1 +41(K-1)
I
y.=z.
;
y.=K.x.
I I
0 <'F <1 1
'
=1
(3-8)
47
3.1.3-Liquid Densities
Liquid molar densities are given by
nc
(3-9)
pi =E
i =i
and species molar densities, pi, for saturated liquids are calculated
using the following formula
pis(T)=
Pcd
(3-10)
(1-T/T .)217
where pct, Zci and Tai are the species critical density (mol/cm3),
compressibility factor and temperature (K), given in Table 3-3.
Species
pct (mol/cm3)
;4
Tci, K
C1
1.010E-2
.288
190.6
C2
6.757E-3
.285
305.4
C3
4.926E-3
.281
369.8
n-C4
3.922E-3
.274
425.2
n -05
3.289E-3
.262
469.6
Table 3-3 - Critical constants for liquid densities (Extracted from Smith and
Van Ness (1987)).
48
3.2-Collocation Points
The roots of Hahn polynomials are calculated using the following recursive
relationship
AnQn+1(x;a43,N) =( -x +An +BN)Qn(x;a43,N) -BnQn_1(x;a43,N)
(3-11)
with initial members given by
(20(x;c0,N)=1
(3-12)
(21(x;a,(3,N).1_ (a+i3+2)x
N(a+1)
(3-13)
and coefficients
A=
(N-n)(n+cc+1)(n+a+(+1)
(3-14)
(2n+a+(3+1)2
B= n(n+13)(N+n+a+13+1)
(3-15)
(2n +a +(3)2
where: (a)k=0
(4=(a)(a+1)...(a+k+1)
k=0
k>0
Computation of the zeros of Qn+1(x;a,f3,N) is accomplished by combination
of Newton's method with supression of previously determined roots using a
scheme similar to that proposed by Villadsen and Michelsen (1978) for
determining the zeros of Jacobi polynomials.
49
3.3-Software for Initial Value Problems
Gani et al. (1986b) discussed numerical and computational aspects related
to the solution of systems of differential and algebraic equations arising from
dynamic distillation models. They concluded that successful numerical
methods must satisfy the following requirements:
Robustness The method should be able to solve a wide variety of
problems.
Appropriateness The method should suit the mathematical problem
being solved.
Efficiency The solution should evolve in a reasonable manner
considering problem size and CPU time.
Based on these criteria they suggested that when high accuracy is required
for the solution, backward-differentiation methods should be used.
The example problems presented in this thesis have all been simulated
using a general purpose integrating package, subroutine DASSL
(Differential/Algebraic System SoLver, Petzold, 1983), which during the course
of this work has been found to satisfy the criteria above. Subroutine DASSL
uses the backward-differentiation formulas of orders one through five to solve
differential/algebraic systems of the form g (t,u,u1=0 where u is the solution
50
vector and u' is the vector of derivatives of the solution with respect to time.
For the distillation problems treated in this thesis, the solution vector
corresponds to the state variables xi4 (and Mi for the TDMH model) and the
vector of derivatives corresponds to the right-hand side of the overall and
species mass balance differential equations for the state variables.
51
4-EXAMPLE PROBLEMS
In order to compare the accuracy and efficiency of the models discussed in
Chapter 2, several problems using full and reduced-order techniques have
been solved. The results obtained demonstrate that the orthogonal collocation
solution of a more detailed model can yield better results than the full-order
solution of a simplified physical model.
4.1-Steady-state results
In order to compare the accuracies resulting from the use of each
distillation model discussed in Chapter 2, example problem 1 (treated by
Henley and Seader (pp.575-580, 1981)) specified in Table 4-1 is utilized.
Collocation points selected for solving this problem are shown in Table 4-2
where the notation mRxms designates the number of collocation points utilized
for the rectifying and stripping sections of the column and provide an order
reduction of roughly 54 and 38%, corresponding to a reduction in the number
of differential equations from 65 (76 for the TDMH model) to 30 (34) and 40
(46) respectively. The convergence criteria adopted for testing steady-state
conditions requires that all derivatives of state variables with respect to time
be less than 1.E-6 Oil for xii's and mol/h for /v1( s). In order to test the greatest
reduction of order, the simple lx1 reduced-order CMO model was used.
Design Variables
Total number of stages
Location of feed stage
Number of components
Feed temperature, deg. F
Feed flowrate, lbmol/hr
Species feed molar flowrate, lbmol/hr, and vaporization
efficiency
C1
C2
C3
C4
C5
Distillate rate, lbmol/hr
Boil-up ratio
Reboiler duty, kBTU/hr
Column diameter, m
Weir length, m
Weir height, m
Liquid level on condenser and reboiler drums, m
Problem 1
Problem 2
11
18
6
14
5
3
105.0
170.0
800.0
900.0
160.01.0
-
370.0// 1.0
300.0/1.0
300.0/1.0
300.0/1.0
530.0
300.0
2.7037
0.4073
3,199
6,000
0.5
0.5
0.1
0.1
0.05
0.05
0.15
0.15
240.0/1.0
25.0/1.0
5.0/1.0
Table 4-1 - Specifications for the example problems discussed in Chapter 4.
53
Steady-state profiles obtained were then interpolated and used as initial
conditions for other runs. By proceeding in this fashion, substantial savings in
computing time have been achieved since initial conditions for subsequent
runs consisted of numerical approximations of steady-state profiles from
previous runs.
Rectifying Module
0=condenser
5=vapor feed stage
Stripping Module
6=liquid feed stage
12=reboiler
1x1
2x2
0.0000
2.5000
5.0000
0.0000
1.3820
3.6180
5.0000
6.0000
9.0000
12.0000
6.0000
7.5858
10.4142
12.0000
Table 4-2 - Collocation points utilized for
determining steady-state conditions for the
example problem of Sec.4.1 using the
orthogonal collocation technique.
Using a dynamic model to find steady-state solutions is inefficient and so,
the computing time spent in obtaining steady-state results for this problem is
of little significance and will not be discussed. Pugkhem (1990) addresses the
appropriate methods for steady-state simulation of distillation columns using
orthogonal collocation.
Outlet conditions for distillate and bottoms product obtained by the
authors and also using the models and techniques discussed in this thesis are
54
presented in Table 4-3. Comparison of results obtained by Henley and Seader
with those obtained using the full-order TDMH model shows that distillate
and bottoms mole fractions for non-key components C1, C4 and C5, agree to
four figures. Outlet compositions for key components C2 and C3 however, are
slightly different probably due to the convergence criteria used by Henley and
Seader. Since they did not report numerical values for the profiles, further
analyses assume that full-order TDMH model yields "true" steady-state
profiles.
Steady-state profiles for temperature, composition and liquid and vapor
flowrates are used to illustrate the accuracy of each method in reproducing the
"true" solution. Table 4-4 shows the errors for these profiles at the actual
stages, calculated using a mean squared error norm, MSE (since this was the
criterion used by Stewart et al.(1985) for selecting collocation points), given by
N+1
E (Ai-Ai)2
MSE = i=°
N +2
where the generic variable Ai represents the "true" solution at the actual stages
and Ai are approximations to the solution. Errors for full-order models have
been calculated using the full-order TDMH model as a reference and errors for
reduced-order models have been determined by interpolating the
approximated solution to the location of the actual stages and calculating the
errors using the full-order model they originate from as a reference.
TDMH
CMH
CMO
H&S
Full
2x2
lx1
Full
2x2
lx1
Full
2x2
lx1
TD (deg. F)
14.4
14.40
14.40
11.35
14.40
14.40
11.75
13.22
13.42
9.49
YD,1
.3019
.3019
.3018
.3022
.3019
.3018
.3020
.3018
.3018
.3020
YD,2
.6894
.6901
.6904
.6971
.6901
.6904
.6962
.6934
.6931
.7023
YD,3
.0087
.0081
.0078
.0009
.0081
.0078
.0020
.0048
.0051
-.0041
YD,4
.0000
.0000
.0001
.0002
.0000
.0001
-.0002
.0000
.0000
-.0002
YD,5
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
TB (deg. F)
161.6
161.99
161.99
164.61
161.99
161.99
164.71
163.18
162.97
166.69
XB,l
.0000
.0000
.0001
-.0004
.0000
.0001
-.0004
.0000
.0001
-.0003
XB,2
.0171
.0158
.0152
.0035
.0158
.0152
.0030
.0093
.0099
-.0082
X13,3
.8718
.8731
.8737
.8855
.8731
.8737
.8858
.8796
.8789
.8970
XB,4
.0926
.0926
.0925
.0929
.0926
.0925
.0931
.0926
.0925
.0930
XBrs
.0185
.0185
.0185
.0185
.0185
.0185
.0185
.0185
.0185
.0185
Table 4-3 - Steady-state conditions for distillate and bottoms products for example problem
discussed in Sec.4.1(T - deg.F, H&S - results obtained by Henley and Seader).
TDMH
CMH
CMO
2x2
lx1
Full
2x2
lx1
Full
2x2
lx1
T (deg. F)2
2.2046
13.6116
0.0
2.2108
15.4667
17.1679
2.4954
59.1864
C,
1.0244E-5
7.9348E-5
0.0
1.0245E-5
7.8835E-5
4.2191E-7
1.0137E-5
8.0747E-5
C2
1.5661E-4
8.4852E-4
0.0
1.5703E-4
1.0442E-3
1.6912E-3
7.7287E-5
4.6632E-3
C3
1.3021E-4
7.7218E-4
0.0
1.3063E-4
9.7452E-4
1.7562E-3
5.2612E-5
4.8081E-3
C4
1.7379E-6
1.5783E-5
0.0
1.7389E-6
1.5958E-5
1.6579E-5
1.9228E-6
1.7826E-5
C5
3.1974E-8
1.8699E-7
0.0
3.1958E-8
1.8778E-7
6.8813E-7
3.7575E-8
2.1291E-7
L (mol/h)2
22.9425
121.6697
1.6692E-7
22.9886
147.0180
22229.
0.0
0.0
V (mol/h)2
22.7371
124.6475
3.5685E-6
22.7668
140.3544
22229.
0.0
0.0
Table 4-4 - Mean squared errors for steady-state profiles of example problem in Sec.4.1.
Note: Errors computed for full-order models are relative to the TDMH model and errors computed
for reduced-order models are relative to the base model.
57
4.1.1-Temperature Profiles
Inspection of profiles in Fig.4.1(a), and respective errors from Table
4-4 indicate that TDMH and CMH models yield essentially the same
steady-state profiles since at steady-state, mass and energy requirements
are satisfied by both these models. However, energy effects are not
accounted for in the CMO model, and the temperature profile obtained
is noticeably different (MSE=17.1679). Errors in Table 4-4 indicate that
even lx1 collocation approximation applied to the CMH model yields
smaller errors (MSE=15.4667) than the full-order CMO model. Errors
obtained by reduced-order methods applied to either TDMH, CMH or
CMO models have the same order of magnitude demonstrating that the
method yields acceptable results regardless of the physical model being
used.
4.1.2-Composition Profiles
Composition profiles for components C1 through C5 are shown in
Figs.4.2, 4.3, 4.4, 4.5 and 4.6 respectively. Inspection of Fig.4.2 and Table
4-4 shows that when a steep profile exists, such as for component C1, 1
and 2 point collocation techniques are unable to reasonably predict the
composition in the rectifying section and negative mole fractions may
180
150
150
Ei
cit
0
120
.8
120
90
90
A
tq]
60
a.
3
0
0
TDMH
0000
00000 CMH
00000 cmo
8
30
60
I
I
I
I
1
2
3
4
1
5
1
I
1
1
1
1
6
7
8
9
10
11
4
12
180
150
150
.8 120
120
90
90
60
60
30
30
oa
6 7
8
STAGE NUMBER
(c)
7
8
9
10
9
10
11
12
(b)
(a)
180
5
6
STAGE NUMBER
STAGE NUMBER
4
5
9
10
11
12
0
1
2
3
4
7
B
6
5
STAGE NUMBER
11
12
(d)
Figure 4.1 - Steady-state temperature profiles for example problem in Sec.4.1. (a) - Full-order models,
(bJ - TDMH full and reduced-order, (c) - CMH full and reduced-order, (d) CMO full and reduced-order.
ui
oo
0.10
0.10
- - 0 0 0 TDMH
0.09
0.09
00000 CMH
[moon CMO
0.08
0.07
0.08
0.07
0.06
0.06
0.05
0.05
0.04
0.04
0.03
U
0.02
0.03
0
a
it
0.02
0.01
0.00
0
'
I
1
2
'
3
l
4
l
1
T
?
5
6
7
8
STAGE NUMBER
I I?
9
10
11
0.01
0.00
4
12
5
6
7
8
9
10
11
12
9
10
11
12
STAGE NUMBER
(a)
(b)
0.10
0.10
0.09
0.09
0.08
zo
0.07
y 0.07
0.06
IA-
0.05
Ili 0.05
0.08
0.06
0.02
0
2 0.04
o
5 0.03
a
=, 0.02
0.01
0.01
0.04
0.03
0.00
0.00
4
5
6
7
8
STAGE NUMBER
(c)
9
10
11
12
0
1
2
3
4
5
6
7
8
STAGE NUMBER
(d)
Figure 4.2 - Steady-state C1-composition profiles for example problem in Sec.4.1. (a) Full-order models,
(13)-- TDMH full and reduced-order, (c) - CMH full and reduced-order, (d) - CMO full and reduced-order.
ul
.0
1.0
1.0
o 0.8
5
00
0
0.9
00000
TDMH
CMH
0.9
0.8
Docioo CMO
O
0.7
0.7
0.6
0.6
0.5
0.5
x 0.4
0.4
n
u..,/
0.3
E. 0.2
0.2
0.1
0.1
0I
0.0
0.0
4
5
6
7
8
9
10
11
4
12
STAGE NUMBER
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
8
6 7
STAGE NUMBER
5
(c)
6
7
8
9
10
11
12
(b)
(a)
4
5
STAGE NUMBER
9
10
11
12
0
1
2
3
4
7
8
5
6
STAGE NUMBER
9
10
(d)
Figure 4.3 Steady-state C2-composition profiles for example problem in Sec.4.1. (a) - Full-order models,
(1:4 - TDMH full and reduced-order, (c) - CMH full and reduced-order, (d) - CMO full and reduced-order.
11
12
1.0
0.9
0.8
1.0
-
o
li
il
ki
a
@
0.9
0.8
W
0.7
0.7
0.6 ---,
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-
19
0
o
0.5
@
0.4
0
0.3
.. 0 0 0 TDMH
a
0000 CMO
111111[1
o
1
IIII
00000 CMH
@
2
3
4
5
6
7
8
STAGE NUMBER
9
10
11
0.2
0.1
0.0
4
12
5
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
7
8
5
6
STAGE NUMBER
(c)
7
8
9
10
9
10
11
12
(b)
(a)
1.0
4
6
STAGE NUMBER
9
10
11
12
0
1
2
3
4
5
6
7
8
11
12
STAGE NUMBER
(d)
Figure 4.4 Steady-state C3-composition profiles for example problem in Sec.4.1. (a) Full-order models,
(1:9-- TDMH full and reduced-order, (c) - CMH full and reduced-order, (d) - CMO full and reduced-order.
cn
--+
0.10
0.10
0000
0.09
0.08
0.07
00000
0000
0.09
0.08
0.07
TDMH
CMH
CM()
0.06
0.05
0.04
0.03
0.02
0.06
0.05
0.04
0.03
0.02
0.01
0.01
0.00
0
1
2
3
11111
4
5
6
7
8
0.00
I
I
I
9
10
11
4
12
STAGE NUMBER
0.10
0.09
0.10
0.09
o 0.08
o 0.08
g
0.07
0.06
7
8
9
10
11
12
0.07
E 0.06
0.02
t2, 0.05
0
0.04
0.03
0.02
0.01
0.01
0.00
0.00
ti 0.05
0
2
6
(b)
(a)
Li-
5
STAGE NUMBER
0.04
5 0.03
4
5
6
7
8
STAGE NUMBER
(c)
9
10
11
12
0
1
2
3
4
8
7
6
5
STAGE NUMBER
9
10
11
12
(d)
Figure 4.5 - Steady-state C4-composition profiles for example problem in Sec.4.1. (a) - Full-order models,
(b) - TDMH full and reduced-order, (c) CMH full and reduced-order, (d) - CMO full and reduced-order.
rs.)
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0 0 0 0 0 TOMH
00000
CMH
Emoo CMO
0.01
0.00
0
1
2
3
4
I
I
I
5
6
7
lit
8
9
10
0.01
I
11
0.00
4
12
5
6
7
8
9
10
11
12
STAGE NUMBER
STAGE NUMBER
(b)
(a)
0 0.05
m 0.04
o
5 0.03
o
=, 0.02
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.01
0.10
0.09
zo 0.08
0.07
6- 0.06
0.00
0.00
4
5
6
7
8
STAGE NUMBER
(c)
9
10
11
12
0
1
2
3
4
5
6
7
8
9
10
11
12
STAGE NUMBER
(d)
Figure 4.6 - Steady-state C5-composition profiles for example problem in Sec.4.1. (a) - Full-order models,
(133- - TDMH full and reduced-order, (c) - CMH full and reduced-order, (d) CMO full and reduced-order.
a.
w
64
arise (as seen in Figs.4.2(b), (c) and (d)). This agrees with Srivastava and
Joseph's (1985a) observations and consequently for this case, the full-
order CMO model yields smaller errors than those obtained by either
reduced-order method. Nevertheless, these reduced-order profiles are
useful in illustrating the polynomial interpolation where lx1 and 2x2
collocation profiles are represented by straight lines and parabolas (first
and second-order approximations) respectively.
Profiles and resulting errors obtained for remaining components, C2
through C5, using the order reduction technique applied to either
TDMH or CMH models once again indicate that even the lower order
1x1 approximation yields better profiles than the full-order CMO model.
However, since the compositions of non-key component C4 in the
rectifying section are low and form a relatively steep profile, the
solution obtained using orthogonal collocation yields negative values for
stages close to the top of the column. This could be avoided using a
higher order approximation, e.g. 3x2.
4.1.3-Liquid and Vapor Flowrate Profiles
Figure 4.7 presents liquid and vapor flowrate profiles obtained by
each method of solution utilized. Comparison of full-order profiles for
the three physical models show that constant flowrates from the CMO
2000
2000
.--.
..
.E
1500
Zo
E
Liquid
E
F 1000
.iii
iL
1000
&
3
&
3
0
..,
1500
-6
0
_t
1,_
500
500
0
4
5
6
7
8
9
10
11
0
12
11111111111
1
2
3
4
5
6
7
8
9
10
11
12
STAGE NUMBER
STAGE NUMBER
(b)
(a)
2000
oo 000 CMO
Vapor
00000 2*2
o*
2 1500
1*1
O
Liquid
1.7.W
0
1000
0
500
0 11111IIIIII
0
1
2
3
4
8
5 6 7
STAGE NUMBER
(C)
9
10
11
12
1
I
I
1
2
3
I
4
I
I
I
I
I
1
5
6
7
8
9
10
I
11
12
STAGE NUMBER
(d)
Figure 4.7 - Steady-state liquid and vapor flowrate_profiles for example problem in Sec.4.1. (a) - Full-order
models, (b) TDMH full and reduced-order, (c) - U.4H full and reduced, (d) - CMO full and reduced.
cr.
66
model are not very accurate (MSE=22229). However, profiles obtained
by the reduced-order technique for TDMH and CMH models are
noticeably better.
4.2-Step Tests
Example problem 2 from Table 4-1 (devised for illustration purposes in this
work) is utilized to demonstrate the reduction in computing time and accuracy
associated with the use of reduced-order techniques. This problem deals with
the transient response of a column due to the introduction of a step decrease
in the distillate flowrate (from 300 to 250 mol/h) and consists of determining
distillate conditions and computing time spent in simulating 60 process hours.
Since the reboiler duty is maintained constant, the decrease in distillate
flowrate is equivalent to an increase in the reflux ratio.
Collocation points utilized for solving the problem using the order
reduction technique are presented in Table 4-5 and provide a reduction in the
number of equations from 60 (78 for the TDMH model) to 24 (30), 27 (35) and
30 (38) corresponding to an order reduction of approximately 60, 55 and 50%
respectively. Since CMH and CMO models use constant stage molar holdups,
these values were calculated as the average initial steady-state stage holdups
from the corresponding TDMH model.
Figure 4.8 shows the change of the column temperature profile caused by
67
the introduction of the step in the distillate flowrate, where a decrease in
distillate temperature results from the increase in purity of the lighter
component C2. This is as expected since a decrease in distillate flowrate
corresponds to an increase in reflux ratio. Since the effect of the step on
bottoms product temperature is less, only the distillate product response will
be discussed.
Rectifying Module
0=condenser
13=vapor feed stage
Stripping Module
14=liquid feed stage
19=reboiler
2x2
3x2
4x2
0.0000
3.0479
9.9521
13.0000
0.0000
1.8902
6.5000
11.1098
13.0000
0.0000
1.4067
4.5034
8.4966
11.5933
13.0000
14.0000
15.3820
17.6180
19.0000
14.0000
15.3820
17.6180
19.0000
14.0000
15.3820
17.6180
19.0000
Table 4-5 - Collocation points utilized in the
solution of the example problem discussed in Sec.4.2
Dynamic responses obtained for distillate temperature and composition are
shown in Figures 4.9, 4.10, 4.11 and 4.12. Table 4-6 presents the computing
times (relative to the full-order TDMH model) required to determine these
responses (on a 386DX/33MHz machine using FORTRAN coding compiled
using the Microsoft FORTRAN 4.1 compiler) and respective errors, calculated
68
200
..6. 160
CS
w
0
W
eg 120
ig
o.
z
W
1
80
40
10
15
20
STAGE NUMBER
Figure 4.8 - Initial and final temperature profiles for
example problem in Sec.4.2. (Solid lines represent
initial steady-state conditions and dashed lines
represent conditions after 60 process hours).
c:
C 90
90
TDMH
CMH
0
0
oI
V
CM0
----
65 ;
:011
o.
...
65
2
TDMH
111i1j11111
40
20
10
50
30
40
TIME (hours)
4+2
3+2
1
60
(a)
C 90
2.2
40
0
20
10
I
30
40
TIME (hours)
6
CM0
4«2
3+2
2+2
-o
La
et
D
ii..,
a.
W
.
...
.....
65
2
ik 65
a.
2
I--
CMH
- -- 4+2
5-.1
8 40
2.2
1
O
1
1
10
'
I
20
'
1
I
I
30
40
TIME (hours)
(c)
1
1
50
....
5
3+2
f
60
(b)
C 90
0
1
50
8 40
1
60
1
10
I
I
20
I
1
30
40
TIME (hours)
1
50
(d)
Fligure 4.9 - Step responses for distillate temperature for example problem in Sec.4.2. (a) Full-order models, (b) -
WMH full and reduced-order, (c) - CMH full and reduced-order, (d) - CM0 full and reduced-order.
60
1.00
1.00
........
0.90
......................
.."-
'
-.../.
0.80
'''' .............
-
0.90
..
.
.
0.80
,
0.70
T
II II
1
20
10
4*2
3*2
2*2
CM0
I
I
40
30
TIME (hours)
0.70
I
50
0
60
1.00
1.00
1
0.90
0.80
CMH
i
I
1
10
20
II
3*2
2*2
1
40
30
TIME (hours)
(c)
1
I
20
r
I
I
I
50
30
40
TIME (hours)
60
(b)
....................................
CM0
4*2
3.2
2*2
0.70
T
50
I
10
0.80 --,
4.2
0.70
TDMH
TDMH
CMH
(a)
0.90
...................................
OP
60
I
0
10
1
I
20
i
I
1
I
40
30
TIME (hours)
i
i
50
(d)
Figure 4.10 - Step responses for C2 distillate composition for example problem in Sec.4.2. (a)-Full-order models,
(b)--TDMH full and reduced-order, (c)-CMH full and reduced-order, (41)-CM0 full and reduced order.
60
0.25
0.25
TDMH
CMH
0.20
0.20
CM0
0.15
0.15
;
0.10
0.10
0.05 1
0.05
TDM H
0.00
(
0
I
I
v
i
10
20
'
I
'
I
30
40
TIME (hours)
'
I
50
4+2
3+2
2+2
0.00
I
1
60
0
20
10
(a)
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0.00
20
30
40
TIME (hours)
(c)
I
30
40
TIME (hours)
60
50
(b)
0.25
10
I
50
60
Cm0
4+2
3+2
2+2
...................... 4.7WM.
I
10
I
r--I
20
1
1
1
30
40
TIME (hours)
1
1
1
50
(d)
Figure 4.11 - Step responses for C3 distillate composition in example problem in Sec.4.2. (a)-Full-order models,
(b)--TDMH full and reduced-order, (c)-CMH full and reduced-order, (d) -CMO full and reduced-order.
60
0.015
0.015
TDMH
TDMH
CMH
4*2
3*2
2*2
CM0
0.005
0.005
0.005
0.005
0
10
20
40
30
TIME (hours)
50
0
60
10
20
40
30
TIME (hours)
50
60
(b)
(a)
0.015
0.015
CM0
CMH
4*2
3*2
2*2
4*2
3*2
2*2
0.005
0.005
0.005
0.005
0
10
20
40
30
TIME (hours)
(c)
50
60
.....
0
10
20
40
30
TIME (hours)
50
(d)
Figure 4.12 - Step responses for C4 distillate composition in example problem in Sec.4.2. (a)-Full-order models,
(b TDMH full and reduced-order, (c)-CMH full and reduced-order, (1:1)-CM0 full and reduced-order.
60
TDMH
CMH
CMO
4x2
3x2
2x2
Full
4x2
3X2
2x2
Full
4X2
3X2
2x2
Relative Time
0.3753
0.3108
0.2516
0.4849
0.1957
0.1645
0.1333
0.1710
0.1602
0.1333
0.1086
T (deg. F x h)
30.48
141.875
520.15
73.445
47.19
120.63
557.67
990.04
48.09
141.775
635.77
C2 (h)
7.80E-2
3.83E-1
1.9057
4.40E-1
1.76E-1
3.12E-1
2.12
5.09
1.16E-1
2.58E-1
2.15
C3 (h)
4.86E-2
2.57E-1
1.5128
4.40E-1
1.46E-1
2.33E-1
1.73
5.09
9.37E-2
1.55E-1
1.8750
C4 (h)
2.96E-2
1.27E-1
3.93E-1
0.0
2.98E-2
1.26E-1
3.88E-1
0.0
3.0E-2
1.04E-1
2.77E
Table 4-6 - Relative computing times and integral absolute errors for distillate product in example problem
of Sec.4.2. Note: Computing time for full-order TDMH is 15'30". Errors computed for full-order models
are relative to the TDMH model and errors for reduced-order models are relative to the base model.
74
using an Integral Absolute error norm (IAE) given by
t=60
IAE = f jA(t)-A(t) I dt
t=o
where A(t) and A(t) represent respectively the "true" and approximated step
responses for the variable under consideration.
Figure 4.9 demonstrates the dynamic response for the distillate
product temperature. As in Sec.4.1, TDMH and CMO models yield identical
initial and final steady-state values. However, according to Fig.4.9(a), the
speeds of response for each model are clearly different. Comparison of the
IAE's for temperature responses in Table 4-6 for full-order CMH and CMO
models with those using 4x2 TDMH demonstrates the advantages of the
orthogonal collocation technique where, with an order reduction of 50%, the
error obtained by the reduced-order solution (IAE=30.48) is much smaller than
that from either full-order CMH or CMO models (IAE=73.45 and 990.04
respectively). Errors in the CMH and CMO models can be explained by
inspection of Figure 4.13, where initial and final molar holdup profiles along
the column are presented. Since both CMH and CMO models use the constant
molar holdup assumption, variations in molar holdup that occur after the
introduction of the step in the distillate flowrate are ignored causing the
differences observed in the dynamic behavior of the simpler model.
75
5
IrIT
10
rill-Ili
15
20
STAGE NUMBER
Figure 4.13 Initial and final molar holdup profiles
for example problem in Sec.4.2. (Solid lines represent
initial steady-state conditions and dashed lines represent
conditions after 60 process hours).
76
4.3-Pulse Tests
In this section, frequency response analysis is utilized to determine
whether reduced-order models preserve the dynamic characteristics of the
original full-order system. To do so, pulse tests have been performed using the
distillation column treated in example problem 2.
Pulse responses for the distillate mole fraction of component C2 are shown
in Figure 4.14 using a deviation variable as follows
YD,;(t) = YD,;(t) -YD,c(to)
These responses have been obtained by starting from the same initial steadystate conditions as in the previous section. Then a rectangular pulse (from 300
down to 285) is introduced in the distillate flowrate and sustained for 0.215
hours after which the distillate flowrate is returned to its initial value.
Pulse response input/output data carry important information regarding
the dynamic characteristics of the system and can be numerically analyzed to
generate frequency response Bode diagrams for the system. These diagrams
are used in the analysis of linear characteristics and design of linear
controllers. Even though the distillation models are nonlinear, the design of
linear controllers for distillation columns is usually based on linear dynamic
models. For the purpose of controller design, if the collocation solutions result
in the same Bode diagrams as the full-order solution, then the collocation
technique can be used for controller design.
0.003
0.003
TDMH
TDMH
CMH
CMO
4.2
3«2
0.002
0.002
0.001
0.001
0.000
0.000
10
15
20
25
2.2
IIIIIII
5
30
10
(a)
tr-
25
30
20
25
30
(b)
0.003
0.003
0.002
0.002
0.001
0.001
0.000
0.000
15
TIME (hrs.)
(c)
I
20
TIME (hrs.)
TIME (hrs.)
10
15
20
25
30
0
5
15
10
TIME (hrs.)
(d)
Figure 4.14 Pulse responses for C2 distillate composition for example problem in Sec.4.3. (a) Full-order models,
(13J - TDMH full and reduced-order, (c) - CMH full and reduced-order, (d) CMO full and reduced-order.
78
The Bode diagram for full-order pulse responses is presented in
Figure 4.15(a) where at higher frequencies, the amplitude ratio (AR) follows a
straight line with slope -1 and the phase angle (PA) asymptotically
approaches -90° indicating first-order dynamics for the systems under
investigation. Model comparisons are based on residual amplitude ratio
(defined as the ratio between the AR of the simplified and true models) and
residual phase angle (defined as the difference between the PA of the
simplified and true models) which, for perfect approximation of the "true"
solution, should equal 1.0 (dimensionless) and 0.0 degrees respectively
throughout the range of frequencies analyzed. In Figure 4.15(b), the residual
AR and PA demonstrate that full-order TDMH, CMH and CMO models differ
with regard to their dynamic characteristics. Inspection of Figures 4.16(a) and
(b) on the other hand, demonstrate that reduced-order techniques applied to
the TDMH model preserve the dynamic characteristics of the full-order system
with remarkable fidelity, yielding even better results with the lower order 2x2
approximation than those obtained using full-order solutions for CMH or
CMO models.
4.4-Coupled Columns
In order to demonstrate the versatility of orthogonal collocation models in
simulating more complex separation systems, its modular structure has been
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FREQUENCY (rod/hr.)
IMO 1
FREQUENCY (rod/hr.)
(a)
Figure 4.15 - (a) Bode diagram for full-order pulse test responses in Sec.4.3. (b)
from the physical simplification of the model.
14111=1.
0
0 01
01
1
FREQUENCY (rod/hr.)
(b)
Residual dynamics resulting
1
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1 111111
0.01
0 1
1
10
0.001
0 01
0 1
10
1
FREQUENCY (rod/hr.)
FREQUENCY (rod/hr.)
0
W
r
rJ
0
0.001
0 01
0 1
FREQUENCY (rod/hr.)
(a)
1
10
0 01
0 1
1
10
FREQUENCY (rod/hr.)
(b)
Figure 4.16 - (a) Bode diagram for the TDMH model pulse test responses in Sec.4.3. (b) - Residual dynamics
resulting from the collocation approximation.
co
0
0
0
O
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0
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1..411 %EP
4*2
0
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0
0.001
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0.1
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1
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0.1
10
1
FREQUENCY (rod/hr.)
(rod/hr.)
0
J
ta
ZO
O
a.
0
0
0
O
0.001
0 1
0 01
FREQUENCY (rod/hr.)
(a)
1
10
0.001
0 1
0 01
1
10
FREQUENCY (rod/hr.)
(b)
Figure 4.17 - (a) Bode diagram for the CMH model pulse test responses in Sec.4.3. (b) - Residual dynamics
resulting from the collocation approximation.
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83
applied to the dynamic simulation of an interacting columns scheme where
recycle streams present within the system generate off-band elements in the
Jacobian matrix. Since the resulting Jacobian does not present any particular
structure, numerical shortcuts for its evaluation are not possible and
consequently, an extensive computational effort is required for integrating the
system of equations arising from full-order models.
The specifications for the coupled columns scheme devised for illustration
purposes in this section are presented in Figure 4.19, where a reboiled stripper
has been connected to a single distillation column in order to provide an
improved separation (purity of outlet streams > 90%) between components C2,
C3 and C4 present in the feed used in example problem 2. The schematic
diagram for the corresponding reduced-order system is presented in Figure
4.20 where, since withdrawal and internal feed are respectively saturated
liquid and vapor, only a single stage is necessary for treating each of these
locations. Only the TDMH model has been used in this example. Collocation
points utilized for the order reduction technique are presented in Table 4-7 and
provide a reduction in the total number of ODE's from 85 to 45 and 65
corresponding to an order reduction of 47 and 24% respectively. Figures 4.21,
4.22, 4.23 and 4.24 present respectively the initial steady-state temperature and
C1, C2 and C3 mole fraction profiles for the main column and reboiled stripper.
In Figures 4.25 and 4.26, step responses for C2 and C3 distillate compositions to
a 10% increase (from 2.5 to 2.75 kBTU/hr) in the side column's reboiler duty
84
i=1
1=2
1=3
.411
i=4
i=8
C/ = 3 OOMOVh
C2 = 300MOIA
C3 = 300MOVh
1=12
1=1
F= 900mol/h
@ 170 deg.F
i=13
i=2
U=0.6643
i=3
B2
i=4
QR2=2500kBTU/h
Figure 4.19 - Specifications for the coupled
columns scheme utilized in Sec.4.4.
85
MI =?
M3 =3
B2
Figure 4.20 - Schematic diagram utilized when the
order reduction technique is applied to
the coupled columns problem in Sec.4.4.
86
are presented. Computing times required for simulating 60 process hours were
25min3Os for the full-order model and 8min33s and 15min44s (66 and 38%
reduction in computing time) for 1 and 2 collocation points per module
respectively.
Module # 1
0=condenser
3=recycled vapor stream return
Module # 2
8=liquid withdrawal
Module # 3
12=vapor feed stage
Module # 4
13=liquid feed stage
17=reboiler # 1
Reboiled Stripper
0=liquid entry point
4=reboiler # 2
1x1x1x1x1
2x2x2x2x2
0.0000
1.5000
3.0000
0.0000
1.0000
2.0000
3.0000
3.0000
5.5000
8.0000
3.0000
4.3820
6.6180
8.0000
8.0000
10.0000
12.0000
8.0000
9.1835
10.8165
12.0000
13.0000
15.0000
17.0000
13.0000
14.1835
15.8165
17.0000
0.0000
2.0000
4.0000
0.0000
1.1835
2.8165
4.0000
Table 4-7 - Collocation points utilized in the
solution of the example problem discussed in Sec.4.4.
87
300
250 MAIN COLUMN
6 200 REBOILED STRIPPER
LAJ
2
100 -w
I-
woo FULL ORDER
50
00000 2 POINT COLLOCATION
00000 1
0
I
0
t
1
3
I
r
I
6
II-III-1-i
POINT COLLOCATION
9
12
1
I
15
18
STAGE NUMBER
Figure 4.21 - Steady-state temperature profiles for the
example problem discussed in Sec.4.4.
88
1.0
co. FULLORDER
0.9
00000 2 POINT COLLOCATION
00000 1 POINT COLLOCATION
0.8
0.7
z0
i= 0.6
(4.
0.5
MAIN COLUMN
0 0.4
2
0.3
0.2 -
0.1 0.0
I
0
[I ri Iri
REBOILED STRIPPER
3
I
r
I
6
r
I
9
12
15
18
STAGE NUMBER
Figure 4.22 - Steady-state C2 liquid composition profiles
for the example problem discussed in Sec.4.4.
89
1.0 REBOILED STRIPPER
0.9
0.8
0.7 --
0.6 0.5
MAIN COLUMN
0.4 0.3 -
0.2 0.1
FULLORDER
00000 2 POINT COLLOCATION
uTilril
0000c 1 POINT COLLOCATION
0.0
0
3
6
ITI
I
9
I
12
I
I
15
18
STAGE NUMBER
Figure 4.23 - Steady-state C3 liquid composition profiles
for the example problem discussed in Sec.4.4.
90
1.0
0.9
FULLORDER
00000 2 POINT COLLOCATION
0.8
00000 1 POINT COLLOCATION
0.7
z
0
P. 0.6
U
0.5
MAIN COLUMN
cl 0.4
2
0.3
0.2
REBOILED STRIPPER
0.1
0.0
I
0
3
6
I
i
I
I
9
1
12
1
1
1
15
1
1
18
STAGE NUMBER
Figure 4.24 - Steady-state C4 liquid composition profiles
for the example problem discussed in Sec.4.4.
91
0.985
z
o
:4 0.980
z
481
12
18
24
30
36
42
48
54
60
TIME (hours)
Figure 4.25 Step response for C2 distillate composition
for the example problem discussed in Sec.4.4.
92
0.030
0.025
z0
cd
u_
0.020
Lu
...... - . .1
0
2
_-.-
--
0.015
FULL-ORDER
2 POINT COLLOCATION
1
0.010
0
POINT COLLOCATION
I
I
1
6
12
18
I
1
36
24
30
TIME (hours)
1
I
I
I
42
48
54
60
Figure 4.26 - Step response for C3 distillate composition
for the example problem discussed in Sec.4.4.
93
4.5-Conclusions
The use of three distillation models (CMO, CMH and TDMH) illustrated
that the orthogonal collocation technique is by no means limited to a particular
selection of state variables and thermodynamic or hydraulic relationships.
Steady-state results obtained demonstrated that the method is remarkably
accurate even at lower order approximations. Step and pulse responses served
to demonstrate that orthogonal collocation techniques can be useful in the
dynamic analysis of staged separation processes. Since conventional process
control techniques rely on linear models derived from step or pulse test
information, model parameters (time constants and steady-state gains) can
differ greatly depending on the physical model used: TDMH, CMH or CMO.
Reduced order solutions of the TDMH model are an attractive tool for
controller design and should be useful for on-line dynamic optimization.
The coupled columns scheme discussed in Sec. 4.4 demonstrated that the
method is flexible and robust enough that even complex separation networks
can be greatly simplified and analyzed.
94
NOTATION
Unless otherwise noted in the text, the definitions below express the
intended meaning for the following symbols
A
Cross sectional area for column, m2
B
Bottoms product flowrate, mol/h
D
Distillate product flowrate, mol/h
E1
Vaporization efficiency of species j
F
Feed stream flowrate, mol/h
FW
Flow parameter in Eq.(2-15)
hB
Molar enthalpy of bottoms product, BTU/mol
hi
Liquid molar enthalpy on stage i, BTU/mol
Height of vapor-free liquid over the weir on
stage i, m
Height of weir, m
HD
Molar enthalpy of distillate product, BTU/mol
HF
Molar enthalpy of feed stream, BTU/mol
Hi
Vapor molar enthalpy on stage i, BTU/mol
Hw
Molar enthalpy of withdrawal stream, BTU/mol
K..
zj
Equilibrium constant for component j on stage i
1(s,t)
Generic variable related to the liquid phase
1
Length of weir
95
Li
Liquid flowrate leaving stage i, mol/h
m
Number of collocation points in a module
M
Number of actual stages in a module
Total molar holdup on stage i, mol
nc
Total number of components
N
Total number of internal stages in a distillation column
P
Legendre and Jacobi orthogonal polynomials
Qc
Condenser duty, BTU/h
Qi
Liquid volumetric flowrate leaving stage i, GPM
QR
Reboiler duty, BTU/h
RB
Boilup ratio
s
Position variable (continuous)
t
Time
Temperature on stage i, deg. F
v(s,t)
Generic variable related to the vapor phase
Vapor flowrate leaving stage i, mol/h
W
Withdrawal stream flowrate, mol/h
Wi(s,t)
Lagrange interpolating polynomial for liquid phase
variables
Wrzp(s,t)
Lagrange interpolating polynomial for vapor phase
variables
xii
Liquid composition of species j on stage i
XB.4
Composition of species j on bottoms product
96
XF4
Composition of species j on feed stream
Xwi
Composition of species j on withdrawal stream
Vapor composition of species j on stage i
YD,j
Composition of species j on distillate product
Z.
Composition of species j on feed stream
Subscripts
c,j
Species j critical property
f
Feed stage location
B
Bottoms product
D
Distillate product
F
Feed stream
r
Rectifying section
s
Stripping section
Superscripts
Species molar thermodynamic property
Greek symbols
a, 13
Parameters in Hahn polynomials
Pi
Molar liquid density on stage i, mol/m3
Fraction of feed vaporized
97
BIBLIOGRAPHY
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Separation Processes I Development of the Model Reduction Procedure, AIChE
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Cho, Y.S. and B. Joseph, Reduced-Order Steady-State and Dynamic Models for
Separation Processes - II Application to Nonlinear Multicomponent Systems, AIChE
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Cho, Y.S. and B. Joseph, Reduced-Order Models for Separation Columns III
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Chemical Engineering, 8, 2, 81 (1984).
Finlayson, B.A., Nonlinear Analysis in Chemical Engineering, McGraw-Hill Book
Co., New York (1980).
Gani, R., C.A. Ruiz and I.T. Cameron, A Generalized Model For Distillation
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Henley, E.J. and J.D. Seader, Equilibrium-Stage Separation Operations in Chemical
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VI
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APPENDICES
99
APPENDIX A
Sequence of Computations
Figures A.1, A.2 and A.3 present the basic sequence of computations
utilized by the FORTRAN programs used for solving the example problems
discussed in Chapter 4. Even though these programs have a very similar
structure, different codes have been written for each physical model and for
the coupled columns scheme.
Copies of the source codes for these programs may be obtained by writing
to:
Dr. Keith L. Levien
Chemical Engineering Department
Oregon State University
Gleeson Hall 103
Corvallis, OR 97331-2702
100
Read inputs from
Misfile
(Calculate feed)
conditions
(Set liquid composition at
each stage
Read change in
manipulated variable
(Calculate liquid and vapor
flowrates
(Calculate vapor compositions
for each stage
Reset ODE's and call
integrating package
NO
1
YES
YES
NO
Interpolate liquid and vapor
compositions entering
collocation points
_
YES
( STOP >if NO
YES
Figure A.1 - Flowsheet diagram demonstrating the sequence of computations
utilized for solving a distillation problem using the CMO model.
101
Read inputs from
datafile
(Calculate feed )
conditions
47
(Set liquid composition at
each stage
4--
'Calculate bubble point temperatures,
vapor compositions and liquid and
vapor enthalpies at each stage
Calculate liquid and vapo9
flowrates
(Calculate differential term for
energy balances
NO
11
YES
YES
Interpolate liquid and vapor
compositions and enthalpies for
streams entering collocation points
NO
_
[
YES
( STOP) NO
YES
Figure A.2 - Flowsheet diagram demonstrating the sequence of computations
utilized for solving a distillation problem using the CMH model.
102
Read inputs from
datafile
(Calculate feed )
conditions
Set liquid composition and )
(molar
holdups at each stage
Calculate bubble point
temperatures, vapor compositions,
liquid and vapor enthalpies and
liquid flowrates at each stage
'(Calculate vapor flowrates
Calculate differential term for
energy balances
(Reset ODE's and call
integrating package
lir
Calculate differential term for mass"
balances for condenser and reboiler
NO
NO
YES
YES
NO
V
Interpolate liquid and vapor compositions
and enthalpies and liquid flowrates for
streams entering collocation points
( STOP) NO
YES
YES
Figure A.3 - Flowsheet diagram demonstrating the sequence of computations
utilized for solving a distillation problem using the TDMH model.
103
APPENDIX B
The Polynomial Interpolation Technique
According to Fig.B.1 and Eq.(1-14), in order to calculate the C2-species
material balance around location s1=2.5, it is necessary that compositions at
locations (s1-1)=1.5 and (s1+1)=3.5 for liquid and vapor streams entering
location si be determined by interpolation. Concentrating this discussion into
the liquid stream, Eq.(1-11) is used to calculate the Lagrange interpolating
polynomials for liquid phase variables within the module:
INn,=
1,1
(1.5-2.5)
=0.4
(0-2.5)
(1.5-0)
(2.5-0)
and Eq.(1-9) can be used to obtain
f(1.5) =0.4*0.9347+0.6*0.6042 =0.7364
Similarly, C2 liquid composition leaving stages 1, 2, 3 and 4 can also be
determined by interpolation yielding respectively 0.8025, 0.6703, 0.5381 and
0.4059 which reasonably compare with the values obtained by the full-order
solution (0.8545, 0.7282, 0.5669 and 0.4283).
104
I
II
s o=0
xcAso>=0.9347
x c2(si-1)
V
s 1=2.5
xc2(si)=0.6042
s 2=5
11.1
OL
x c2(s2)=0.3221
Figure B.1 - C2 steady-state composition profile for the rectifying
module in the column treated in Section 4.1 obtained using
the lx1 reduced-order TDMH model.