Redacted for Privacy - CORE · other parameters are assumed constant within sections of the column. Treybal (1985) and McCabe and Smith (1985) present detailed treatments on graphical

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

Redacted for PrivacyAbstract approved:

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 theDynamic Simulation of Staged Separation Processes

by

Marcio Matandos

A THESIS

submitted to

Oregon State University

in partial fulfillment ofthe requirements for the

degree of

Master of Science

Completed December 12, 1991

Commencement Tune 1992

APPROVED:

Redacted for PrivacyAssistant Professor of Chemical Engineering in charge of major

rm i n

Redacted for PrivacyHead of Depa4t#nent of Chemical Engineering

Redacted for PrivacyDean 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 41.2.1-Equilibrium-Staged Separation Operations 41.2.2-Systems of Differential and Algebraic Equations 81.2.3-Orthogonal Collocation 9

1.3-Literature Survey 13

2-DEVELOPMENT OF THE MODELS 18

2.1-General Assumptions 18

2.2-Full-Order Models 192.2.1-The Constant Molar Overflow (CMO) Model 252.2.2-The Constant Molar Holdup (CMH) Model 282.2.3-The Time-Dependent Molar Holdup (TDMH) Model 32

2.3-Reduced-Order Techniques 362.3.1-Constant Molar Overflow 412.3.2-Constant Molar Holdup 412.3.4-Time-Dependent Molar Holdup 41

3-NUMERICAL IMPLEMENTATION 43

3.1-Thermodynamic Properties 433.1.1-Enthalpies 433.1.2-Equilibrium Constants 44

3.1.2.1-Bubble Point Temperatures 453.1.2.2-Fraction of Feed Vaporized 46

3.1.3-Liquid Densities 47

3.2-Collocation Points 48

3.3-Software for Initial Value Problems 49

4-EXAMPLE PROBLEMS 51

4.1-Steady-State Results 51

4.1.1-Temperature Profiles 574.1.2-Composition Profiles 574..3-Liquid and Vapor Flowrate Profiles 64

4.2-Step Tests 66

4.3-Pulse Tests 76

4.4-Coupled Columns 78

4.5-Conclusions 93

NOTATION 94

BIBLIOGRAPHY 97

APPENDICES

A. Sequence of Computations 99

B. The Polynomial Interpolation Technique 103

LIST OF FIGURES

Figure Page

1.1 - Schematic diagram and notation for an equilibrium stage. 5

1.2 - Schematic diagram illustrating thecontinuous position variable s withina 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 thederivation of the equations for calculating liquid andvapor flowrates. 31

2.6 - Geometry utilized for calculating liquid flowratesleaving an equilibrium stage using the Francis weirformula. 33

2.7 - Schematic diagram and notation utilized when theorder-reduction technique is applied to a distillationcolumn. 37

4.1 Steady-state temperature profiles for exampleproblem in Sec.4.1 58

4.2 - Steady-state C1 composition profiles for exampleproblem in Sec.4.1. 59

4.3 Steady-state C2 composition profiles for exampleproblem in Sec.4.1. 60

4.4 - Steady-state C3 composition profiles for exampleproblem in Sec.4.1. 61

4.5 - Steady-state C4 composition profiles for exampleproblem in Sec.4.1. 62

Figure Page

4.6 - Steady-state C5 composition profiles for exampleproblem in Sec.4.1. 63

4.7 Steady-state liquid and vapor flowrate profiles for exampleproblem in Sec.4.1. 65

4.8 - Initial and final temperature profiles for example problemin Sec.4.2. 68

4.9 - Step responses for distillate temperature for exampleproblem in Sec.4 2 69

4.10 - Step responses for C2 distillate composition for exampleproblem in Sec.4.2 70

4.11 - Step responses for C3 distillate composition for exampleproblem in Sec.4.2 71

4.12 Step responses for C4 distillate composition for exampleproblem in Sec.4 2 72

4.13 - Initial and final molar holdup profiles for exampleproblem in Sec.4.2 75

4.14 - Pulse responses for C2 distillate composition for exampleproblem 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 responsesin Sec.4 3 80

4.17 - Bode diagram for the CMH model pulse test responsesin Sec.4.3 81

4.18 - Bode diagram for the CMO model pulse test responsesin Sec.4.3 82

4.19 - Specifications for the coupled columns scheme utilizedin Sec.4.4 84

4.20 - Schematic diagram utilized when the order reduction techniqueis applied to the coupled columns problem in Sec.4.4. 85

Figure

4.21 - Steady-state temperature profiles for theexample problem discussed in Sec 4 4

Page

87

4.22 - Steady-state C2 liquid composition profilesfor the example problem discussed in Sec.4.4. 88

4.23 - Steady-state C3 liquid composition profilesfor the example problem discussed in Sec.4.4. 89

4.24 - Steady-state C4 liquid composition profilesfor the example problem discussed in Sec.4.4. 90

4.25 - Step responses for C2 distillate compositionfor the example problem discussed in Sec.4.4. 91

4.26 - Step responses for C3 distillate compositionfor the example problem discussed in Sec.4.4. 92

LIST OF APPENDIX FIGURES

Figure Page

A.1 - Flowsheet diagram demonstrating the sequenceof computations utilized for solving a distillationproblem using the CMO model. 100

A.2 - Flowsheet diagram demonstrating the sequenceof computations utilized for solving a distillationproblem using the CMH model. 101

A.3 - Flowsheet diagram demonstrating the sequenceof computations utilized for solving a distillationproblem using the TDMH model. 102

B.1 - C2 steady-state composition profile for the rectifyingmodule in the column treated in Section 4.1 obtainedusing the lx1 reduced-order TDMH model. 104

LIST OF TABLES

Table Page

2-1 Design variables defining the distillation models on Sec 2 2 27

3-1 - Coefficients for liquid and vapor enthalpies 44

3-2 Coefficients for equilibrium constants. 45

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-stateconditions for the example problem of Sec.4.1 usingthe orthogonal collocation technique. 53

4-3 - Steady-state conditions for distillate and bottomsproducts for example problem discussed in Sec.4.1. 55

4-4 - Mean squared errors for steady-stateprofiles of example problem in Sec.4.1. 56

4-5 - Collocation points utilized in the solutionof the example problem discussed in Sec.4.2. 67

4-6 - Relative computing times and integral absolute errorsfor distillate product in example problem of Sec 4 2 73

4-7 - Collocation points utilized in the solution ofthe example problem discussed in Sec.4.4. 86

USE OF ORTHOGONAL COLLOCATION IN THEDYNAMIC 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

1

Li-1

hi -1

xi-ij

1

14

Hiyij

Mi, hi, xii

Li Vi i-1

hi Hi+1

xii yi+i,i

Figure 1.1 - Schematic diagram and notationfor 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.

dtz =V. +L. -V.-L.,-1

Balance for species j (j=1,...,nd

Energy balance

dt

dM.h.

dt=VI. +1H1 . +1 . -1 -V

iH

i-L

ihi

Equilibrium relationships

Summation relations

(1-2)

(1-3)

y.. = K; x.. (1-4)

n,

Ex1.4=1 ; E y = 1j=i

By combining Eqs (1-2) and (1-5), Eq.(1-1) can be obtained and

(1-5)

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

dudt

=f(u,v,t)

0 =g(u,v,t)

(1-6)

(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 backward-

differentiation (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 .8vSu 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

(1-8)

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.. 11

to

i=0

i=1

VI

s=0

s =1

s=2

iv 11

i=2

t V2

i=M

VM

MkA1 i =M +1

s =M-1

s=M

s=M+ I

Figure 1.2 - Schematic diagram illustrating thethe continuous position variable s within a

separation module.

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+1ne,1

where a tilde (-) represents an approximation to the variable of interest

and the W-functions are Lagrange interpolating polynomials calculated

using the formulas

11

(1-9)

(1-10)

m ssW =11 k n =0,...,m (1-11)1c=0 Sn Skhen

m+1 ssW,(s)

Ti= n =1,...,m+1 (1-12)

k=i sn-skir..

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

the collocation points.

dia(si,t)

dt= V(s i+1,t) + 1:(s i-1,t) -1-7(si,t) [(set)

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)

dt17(si,t)g i(si,t) L(s i,t)ii(si,t)

d i,t)Ii(s i,t)= V(s i+1,014(s i+1,t) + i-1,t)

dt17(s i,t)14(s i,t) 1.:(s i,t)li(s i,t)

12

(1-13)

(1-14)

(1-15)

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 m*nx=0

x=s-1 ; N=M-1

(1-16)

where the weighing function w(x;a,(3,N) is given by

(a+1) ((3 +1)N-xW(X;a,(3,N) =

x!(N-x)!

13

(1-17)

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=00

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

''''...._

Ili=f-1

i=f

D

Figure 2.1 - Schematic diagram and notationfor a distillation column.

IP]

FXF jHF

Li -1

j

Mi, hi, xii

ILihi Hi+ixii yi+li

Figure 2.2 - Schematic diagram and notationfor equilibrium stage.

XWHw

21

i=1

D,HD, YD,j

Figure 2.3 - Schematic diagram and notationfor condenser.

22

i=N

Figure 2.4 - Schematic diagram and notationfor reboiler.

23

Condenser

Internal Stages

Reboiler

dM0=V1 -L0 -D

dt

dMox

= DYDJViYij Loxoidt

dMoho=Villi-Loho-DHD-Qc

dt

dM.

dt=Vi+1+Li_1 +F-Vi-Li-W

dM.x..Vi+iyi+14 +Li_ixi_i +FXFi- V iyij- WXwj

dt

dM.h.

d;

dMN+1

dtB

dA4N+1xN+1,1.- V

N ÷iyN+1,;-BXB4dt

dMN+ihN+1 = LNhN Vs+1HN+1 BhB + QRdt

24

(2-1.a)

(2-1.b)

(2-1.c)

(2-2.a)

(2 -2.b)

(2-2.c)

(2-3.a)

(2-3.b)

(2-3.c)

Liquid and vapor enthalpies are given by

hi=h({xid,Ti) ;

and vapor-liquid equilibria are represented by the following functional

relations

25

(2-4)

yii = E. Ki4( )x.4 (2-5)

nc

E yi4 =1J=1

where i=0,1,...,f,...,N+1, is the stage number.

2.2.1-The Constant Molar Overflow (CMO) Model

(2-6)

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), (2-

2.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

Rectifying section

B=F-DVs=RB*BL =V +B

S S

LT=Ls-(1-11)*FVF =VS 4-11*F

(2-7)

(2-8)

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

Distillate rate, D R* R*

Boil-up ratio, RB R* NR NR

Reboiler duty, QR NR R* R*

Plate geometry, cd, lw, hw NR NR R

Stage molar holdup, M. R R NR

Table 2-1 - Design variables defining the distillation modelspresented 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 reduced-

order 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 =mdhidt ' dt

As demonstrated by Howard (1970) and under the simplifying

(2-9)

assumption of negligible mixing effects by Cho and Joseph (1983b), the

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 ahi

dt aT dt dt

29

(2-10)

Differentiating the bubble point relation given by Eqs (2-5) and (2-6)

d c dK..(EE.K..x..)=EE.(x1. +K

dt j=1 14 14 j1 4 dt dt

n dK.. dT.=E

dT dt dt

Which can be solved for dTildt

nc dx14..E E K14

dtdTi j=i

dt nc dK..E E x 14

11 4 dT

Substituting back into Eq.(2-10) yields

nc dx..14E K n

dhi ahii71 1 74 dt c ah1. dx..14

dt aT nc dK.. j.i axis dtE Ei x. 14

j.1 4 dT

(i =0,1,...,N +1)

(2-11)

(2-12)

(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 dhk

L =

B(hB-Hi+1)+F* (Hi+1-HF*" 2R

EMkk.N +1 dt

hi-Hi+1

(2-14)

Vi+1=Li+F* -B (2-15)

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 ; HF. =0 i>f

= (1-W)F ; 11; = hF i=f

F* = F ; 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 thederivation 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°'

)3"0.48Fw

(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)

Figure 2.6 - Geometry utilized for calculating liquid flowratesleaving an equilibrium stage using the Francis weir formula.

hOW

hw

33

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.) dpi7

dt dt 1 dt

The derivative dpi/dt above can be further expanded into

dpi ap dTi+

nc ap ddt aT dt j.1 axii dt

Substituting the term dT1/dt from Eq.(2-12) yields

nc dx..Edpi

P

I 14 dt

dt

appE K

j:c1 DK..E Ex..1

14

I 14 aTJ

i=0,N+1

n Avpi

j=i ax dt

(2-17)

(2-18)

(2-19)

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

QR + LN(hN /113 )+MN+1dt

VN+1 HN+1-hB

dhN

B =LN 17N+1dMN+1

dt

Vapor flowrates leaving internal stages are calculated using

(2-20)

(2-21)

I dM hL1.-1h2.-1 +Q

R+F HP. -Bh

B- E "

dt (2-22)V.= k-N

2

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

dM N+1 dMF HF + Q

R+ h

°(

dt°-D)+BhB+E k

dt. k1v1= =

H1 -h0

L0=V1-Ddt

F" and liFt follow the same definition presented in Sec.2.2.2 and the

dM0

36

(2-23)

(2-24)

term dMkhK/dt is calculated using the following formula

dMkhk dhk dM

dt k dt 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 whenthe 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

Stages within a module

dMo=Vo-Lo-D

dt

dMoxoiDYD,i

dt

dMoho=V0H0-Loho-DHD-Qc

dt

dM.=Vi+Li-Vi-Li

dt

dMixijViyii-Lixii

dt

dMihi=ViHi+Lihi-ViHi-Lihi

dt

(2-26.a)

(2-26.b)

(2-26.c)

(2-27.a)

(2 -27.b)

(2-27.c)

Reboiler

dMm +m +31 2

= L , Vni

+m +3 Bml +M2 +a l 2dt

dMmi+m2+3 Xm, +m2+34

BXBidt ,-,, +m2+3 m, +m2 ,j 1 2 3.A7

39

(2-28.a)

(2-28.b)

dMmi+m2+3

hml+m2+3 (2-28.c)

dt=L

m l+m2 +3 rm1

+M2 +3 Vm1

+m2 +3Hml +m2 +3 BhB 4- QR

Stage Above Feed Location

dMm+1

1 =V. 1: +WF-Vm +i-Lm1

2 +1 1 1dt

dM +ixm +1.mi 1 d

dt= Vnii+2Ymi +24. +(m1+1imi+1,j

,+ qi F Y.F.4 Vm, +1 Y M, + i,jL

M, +1 X M, + Li

dM _ohm +1

m1 1 =1/dt

m1+2Hrni +2 + 1,- m + 1h.m 1 + 1

+TFHF-V H -L hm1+1 m1+1 m1+1 m1+1

Stage Below Feed Location

dMm 1V-m1+2 + Lm1+1+ (1 -W )F Vmi + 2 m,-L +2dt

(2-29.a)

(2-29.b)

(2-29.c)

(2-30.a)

dMrn +2Xrn +2;

dt= I ni,+2gm1+2,j + Lmi+1Xm,+14

+ (1 -1" )FXF4 -V m,.2ymi+24 L Xmi

+2

dt+Lnyihrni+1

+ (1-110F hF V mi+2Hmi+2 - Lm1+2

hm1+2

40

(2-30.b)

(2-30.c)

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

hi= EJ.1

nc

Hi = yiiHi*(Ti)

(3-1.a)

(3-1.b)

Species molar enthalpies, hi' for the liquid and Hi* for the vapor, are

calculated using

hi*(T) = a1+b 1T +c 1T.

2

44

(3-2.a)

H.*(T)=A.1 +B.T+CT2 (3-2.b)

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 Bi C.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 psiaand 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 cci 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 and0 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

Newton's method for finding a root for such functions is given by

F(Tk)Tk., = Tk-

F (Tk)

(3-4)

(3-5)

where subscript k indicates the trial number and the derivative F'(T) is

calculated by

nc dK.(T)Fi(T)=Ex.E

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 z (1 -K.(TFOP) =E I F

))=0

1 -411(Ki(T F) -1)(3-7)

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:

XI .=Z1.

IP=0

z, 0 <'F <1 1x.= ; y.=K.x.1 1 +41(K-1) ' I I

I

y.=z. =1

(3-8)

3.1.3-Liquid Densities

Liquid molar densities are given by

nc

pi =Ei =i

and species molar densities, pi, for saturated liquids are calculated

using the following formula

pis(T)= Pcd

(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.

47

(3-9)

(3-10)

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 andVan 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)

with initial members given by

and coefficients

(3-11)

(20(x;c0,N)=1 (3-12)

(21(x;a,(3,N).1_ (a+i3+2)xN(a+1)

A = (N-n)(n+cc+1)(n+a+(+1)(2n+a+(3+1)2

B=(2n +a +(3)2

n(n+13)(N+n+a+13+1)

(3-13)

(3-14)

(3-15)

where: (a)k=0 k=0

(4=(a)(a+1)...(a+k+1) 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 Problem 1 Problem 2Total number of stages 11 18

Location of feed stage 6 14

Number of components 5 3

Feed temperature, deg. F 105.0 170.0

Feed flowrate, lbmol/hr 800.0 900.0

Species feed molar flowrate, lbmol/hr, and vaporizationefficiency

C1C2C3C4C5

160.01.0370.0// 1.0240.0/1.025.0/1.05.0/1.0

-300.0/1.0300.0/1.0300.0/1.0

Distillate rate, lbmol/hr 530.0 300.0

Boil-up ratio 2.7037 0.4073

Reboiler duty, kBTU/hr 3,199 6,000

Column diameter, m 0.5 0.5

Weir length, m 0.1 0.1

Weir height, m 0.05 0.05

Liquid level on condenser and reboiler drums, m 0.15 0.15

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.

1x1 2x2

Rectifying Module 0.0000 0.00002.5000 1.3820

0=condenser 5.0000 3.61805=vapor feed stage 5.0000

Stripping Module 6.0000 6.00009.0000 7.5858

6=liquid feed stage 12.0000 10.414212=reboiler 12.0000

Table 4-2 - Collocation points utilized fordetermining steady-state conditions for the

example problem of Sec.4.1 using theorthogonal 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)2MSE = 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 problemdiscussed 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

120

90

60

30

0

180

150

oa.8 120

90

60

30

38

A

0Ei

0 0 0 0 TDMH00000 CMH00000 cmo

0I I I I 1 1 I 1 1 1 1

1 2 3 4 5 6 7 8 9 10 11

STAGE NUMBER

(a)

12

150

cit.8 120

90

tq] 60a.

180

150

120

90

60

30

4 5 6 7 8 9 10 11 12

STAGE NUMBER

(b)

4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 B 9 10 11 12

STAGE NUMBER

(c)

STAGE NUMBER

(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. oo

ui

0.10

0.09 - - 0 0 0 TDMH0.080.07

00000 CMH[moon CMO

0.06

0.05

0.04

0.03 U

0.020 a it

0.01

0.00

0'1

I

2'3

l

4l 1 T ? I I?5 6 7 8 9 10 11 12

STAGE NUMBER

(a)0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

z

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

0.10

0.09

o 0.08

y 0.07

IA- 0.06

Ili 0.0502 0.04o5 0.03a=, 0.02

0.01

0.00

4 5 6 7 8 9 10 11 12

STAGE NUMBER

(b)

4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12

STAGE NUMBER

(c)

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

0.9

o 0.8

5 0.7

0.6

I 0.50x 0.4

nu..,/

E. 0.2

0.1

0.0

0 0 0 TDMH00000 CMH

O Docioo CMO

4 5 6 7 8 9 10 11 12

STAGE NUMBER

(a)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

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

4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12

STAGE NUMBER STAGE NUMBER

(c) (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.

4 5 6 7 8 9 10 11 12

STAGE NUMBER

(b)

1.0

0.9 -0.8 -0.7

0.6 ---,

0.5 - o

ki ilo li@

W

a

0.4 - @

0.30

0.2 -0.1 -

19

@o

a.. 0 0 0 TDMH00000 CMH0000 CMO

0.0 111111[1 IIII0 1 2 3 4 5 6 7 8 9 10 11 12

STAGE NUMBER

(a)1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

4 5 6 7 8 9 10 11 12

STAGE NUMBER

(c)

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

4 5 6 7 8 9 10 11 12

STAGE NUMBER

(b)

0 1 2 3 4 5 6 7 8 9 10 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

0.09

0.080.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

0.10

0.09

o 0.080.07

Li- 0.06

ti 0.0502 0.04

5 0.030.02

0.01

0.00

0

0 0 0 0 TDMH00000 CMH0000 CM()

111111 2 3 4 5 6 7

STAGE NUMBER

(a)

8I I I

9 10 11 12

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

0.10

0.09

o 0.08g 0.07E 0.06t2, 0.050

0.04

0.03

0.02

0.01

0.00

4 5 6 7 8 9 10 11 12

STAGE NUMBER

(b)

4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12

STAGE NUMBER STAGE NUMBER

(c) (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 0 0 0 0 TOMH

0.08 00000 CMHEmoo CMO

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00I I I lit I

0.10

0.09zo 0.08

0.07

6- 0.060.050

m 0.04o5 0.03o=, 0.02

0.01

0.00

0 1 2 3 4 5 6 7 8 9 10 11

STAGE NUMBER

(a)

12

0.100.09

0.08

0.07

0.06

0.050.04

0.03

0.02

0.01

0.00

0.10

0.09

0.080.07

0.06

0.050.04

0.03

0.02

0.01

0.00

4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12

4 5 6 7 8 9 10 11 12

STAGE NUMBER

(b)

STAGE NUMBER

(c)

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. wa.

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

.--.1500 .E 1500..

Zo -6

E Liquid E

F 1000 .iii 1000

& &3 30 0iL..,

_t500 1,_ 500

4 5 6 7 8 9 10 11 12

STAGE NUMBER

(a)

0 11111IIIIII0 1 2 3 4 5 6 7 8 9

STAGE NUMBER

(C)

10 11 12

0

2000

0111111111111 2 3 4 5 6 7 8

STAGE NUMBER

(b)

9 10 11 12

2 1500O

1.7.W 1000

0500

Vaporo*

Liquid

oo 000 CMO00000 2*2

1*1

0

1 I I I I I I I I 1 I

1 2 3 4 5 6 7 8

STAGE NUMBER

(d)

9 10 11 12

Figure 4.7 - Steady-state liquid and vapor flowrate_profiles for example problem in Sec.4.1. (a) - Full-orderUmodels, (b) TDMH full and reduced-order, (c) - .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.

2x2 3x2 4x2

Rectifying Module 0.0000 0.0000 0.00003.0479 1.8902 1.4067

0=condenser 9.9521 6.5000 4.503413=vapor feed stage 13.0000 11.1098 8.4966

13.0000 11.593313.0000

Stripping Module 14.0000 14.0000 14.000015.3820 15.3820 15.3820

14=liquid feed stage 17.6180 17.6180 17.618019=reboiler 19.0000 19.0000 19.0000

Table 4-5 - Collocation points utilized in thesolution 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

200

.. 1606.

CSw0Weg 120

igo.zW1

80

40

10 15 20STAGE NUMBER

Figure 4.8 - Initial and final temperature profiles forexample problem in Sec.4.2. (Solid lines represent

initial steady-state conditions and dashed linesrepresent conditions after 60 process hours).

68

c: 90

00

65 ; ----

40

C 900

TDMHCMHCM0

111i1j1111110 20 30 40 50

TIME (hours)

(a)

60

LaetD .

ii.., 65 ...a. .....2WI--

CMH

5 - -- 4+2-.1 3+2

2.28 40 f

1

11 ' I ' 1 I I 1

11

O 10 20 30 40 50 60TIME (hours)

C 90oI

V

:011 6o.

52

40

...

TDMH4+23+22.2

1 I 1

0 10 20 30 40TIME (hours)

C 906-o

ik 65a.2

5

8 40

(b)

50 60

CM04«23+22+2

....1 I I I 1

10 20 30 40TIME (hours)

1

50

(c) (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

0.90

........

.."-''''

...................................

OP

- ..../

..

.- .0.80 '

TDMH, CMHCM0

0.70 T1 II II I I I

10 20 30 40 50 60TIME (hours)

(a)1.00

0.90 1

0.80 CMH4.23*22*2

0.70 Ii

1 II 11 T

10 20 30 40 50 60TIME (hours)

(c)

1.00

0.90

0.80

0.70

1.00

0.90

0.80

0.70

...................................

TDMH4*23*22*2

I I I I

0 10 20 30 40TIME (hours)

(b)

I

50 60

....................................r--,

CM04*23.22*2

I 1 I i I 1 I

0 10 20 30 40TIME (hours)

(d)

i

50i

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

0.20

0.15;

0.10

0.05 1

0.00 ( Iv

I i ' I ' I ' I I

0 10 20 30 40TIME (hours)

(a)

TDMHCMHCM0

0.25

0.20

0.15

0.10

0.05

0.00

50 60

10 20 30 40 50 60TIME (hours)

(c)

0.25

0.20

0.15

0.10

TDM H4+23+2

0.05

2+20.00

1 I I

0 10 20 30 40 50 60TIME (hours)

(b)0.25

Cm04+20.203+22+2

0.15

0.10

0.05 ...................... 4.7WM.

0.00I I r--I 1 1 1

11

1

10 20 30 40 50 60TIME (hours)

(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.

0.015

0.005

TDMHCMHCM0

0.0050 10 20 30 40 50

TIME (hours)

(a)

60

0.015

0.005

CMH4*23*22*2

0.0050 10 20 30 40

TIME (hours)

(c)

50 60

0.015

0.005

TDMH4*23*22*2

0.0050 10 20 30 40 50

TIME (hours)

(b)

60

0.015

0.005

CM04*23*22*2

.....

0.0050 10 20 30 40

TIME (hours)

(d)

50

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 problemof 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 dtt=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.

IrIT rill-Ili5 10 15

STAGE NUMBER

20

Figure 4.13 Initial and final molar holdup profilesfor example problem in Sec.4.2. (Solid lines represent

initial steady-state conditions and dashed lines representconditions after 60 process hours).

75

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 steady-

state 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

0.002

0.001

0.000

0.003

0.002

0.001

TDMHCMHCMO

10 15 20TIME (hrs.)

(a)

25 30

0.003

0.002

0.001

0.000

0.003

0.002

0.001

0.000 0.00010 15 20 25 30 0 5 10 15 20 25 30

TDMH4.23«22.2

IIIIIII I tr-5 10 15 20 25 30

TIME (hrs.)

(b)

TIME (hrs.)

(c)

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

o00

0 -=00=

m.111::: ::::====Z...:...11. a .1601Ismomilot om......:1;;ammo milMANNINOMilii

Immo 011110111311111110111111 MINIM INIIIIIIIIIIII111111 111111=111111111111111

imimmoffiel

1.11111

TO,-.ECE!11i111

c-Eic3c:===........_4, 1111 MIIINI111111

IIIIIII IIIIIIIKMIIIIIIIIIIIII=:::::::==:::::::.=......L;T.::!=.===:::::=IIIIMMIIIMMIII =IIIMIIIINIM111,1111111111111111MIIIIIIIIIMIIIIIIIIIMM111111101111M111111MMOIIIM111111111111111IMMEN6IIIMYKIIIIIIIMM111111111111111111111111111111111MM11111:1MKINE11111111111111niiimumummulin

1111 111111111111111111111111111014111111

IIIIIIIIIIIIIIIIINMIITDMHCMHCMO

1111111111111111111 =111811111111M111111111111111111=11111=M111111111111=111M:1111111111111111111111U1111111111111M MINIENI

1 1 1n1111

0.001 0.01 0.1 1

FREQUENCY (rod/hr.)10

L.)

,.....

init

....,:.... . .

'

",'1 -I

..0 ..4o)

.,%

,.

......" .

1

0.001 0 01 0 1 1 11

FREQUENCY (rod/hr.)

(a)

0.001 0 01. 0 1 1

FREQUENCY (rod/hr.)

0

Ocsi

O

0

0

10

i/ -'11/46/

/I

II .

e." .

I

% ......

.,

ow. 4......... sr.

I......g 11 '

II..

1

/ ... ,...,. . s

14111=1.

IMO 1

Mil 0

0 01 01 1 1

FREQUENCY (rod/hr.)

(b)

Figure 4.15 - (a) Bode diagram for full-order pulse test responses in Sec.4.3. (b) Residual dynamics resultingfrom the physical simplification of the model.

W

r

00

000

1.111111.110.11 MIMM1141111111111. MIAMMINIIIMI_.. 1111 IMM111111111111111111111111 111Man1111 MIN11111MMONIMai11 NINSINII

111111111 11111311111 11111mel.... =:::::=%":::::: ==:...1 M.1 INIMMEMISI111111111111111111 MIIIII111111111111111111111111111111011OIPMINIM11111 IMMI1I1MINNIMMIIIN11 1111111111111=/l1111 1111MMENEVI: 1i

T MH4*23*22*2

II 111111

IP1111111101111M1111IM1= IMIIIMINI=111MINII1111111.11.. 1=1,11=11.1.11.1111111/.1.11.11.1.1.411 Ellab,111111M11.1.1111111.1.1..111. II.1111.111.11111011111 MEOW!

1 1 1 111111

0.001 0.01 0 1 1

FREQUENCY (rod/hr.)

0

0

10

0.001 0 01 0 1 1

FREQUENCY (rod/hr.)

(a)

10

0.001 0 01 0 1 1

FREQUENCY (rod/hr.)

10

0 01 0 1 1

FREQUENCY (rod/hr.)

(b)

rJ

10

Figure 4.16 - (a) Bode diagram for the TDMH model pulse test responses in Sec.4.3. (b) - Residual dynamicsresulting from the collocation approximation. co

00

OO0

080

CMH4*23*22*2

0.001

0

ta

ZO

O

O

I 1111111

0.01 0.1

FREQUENCY

I

1

(rod/hr.)

10

0.001 0 01 0 1 1

FREQUENCY (rod/hr.)

(a)

10

0

1..411 %EP .1

0.001 0 01 0.1 1

FREQUENCY (rod/hr.)

J

a. 0

0

0

0.001

10

0 01 0 1 1

FREQUENCY (rod/hr.)

(b)

10

Figure 4.17 - (a) Bode diagram for the CMH model pulse test responses in Sec.4.3. (b) - Residual dynamicsresulting from the collocation approximation. co

..="Tir;;;;;===;;;;;;=ral;;;;;;==1;;;;;I1111111111181111M11111MINMOIIIIMI11111111111111111111111111111111111111111111111111111111111IIIMMEMillIMMUM1111MONIIIMOMMOMIII

';'-'7;;;;74:17!FillillIIIII.11111111

IM=111,11 IM111111111111111 1111WEI INNI11111111111MM11111111 IM111011111111 ,=111111111111

11=111111111111111111=1111111111111MNIII11111111111iniiiii/1111111111111111111111111111E1111111011111

=2:11Onl:liarziaMM1111.." =II: EN 11%

; . .

.

.

111111111111111111111111111111111111111111

1011 11111111 11111111 1111111

111111111111111111101 111111

mainiiiirraiNIII=E11111111111111111

Impirsmunisnminimmmil

IAA

- II

1111110 11111111 11111 1111110

1111111 1111110 11111111 111111

11111111111111111 1111111

11111111 IN 111N111111MMERIIILIIMMANI1111=i111111

5 5 t 5

I

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

C/ = 3 OOMOVh

C2 = 300MOIA

C3 = 300MOVh

F= 900mol/h@ 170 deg.F

U=0.6643

i=1

1=2

1=3.411

i=4

i=8

1=12

i=131=1

i=2

i=3

Figure 4.19 - Specifications for the coupledcolumns scheme utilized in Sec.4.4.

84

B2

i=4

QR2=2500kBTU/h

85

MI =?

M3 =3

B2

Figure 4.20 - Schematic diagram utilized when theorder reduction technique is applied tothe 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.

1x1x1x1x1 2x2x2x2x2

Module # 1 0.0000 0.00001.5000 1.0000

0=condenser 3.0000 2.00003=recycled vapor stream return 3.0000

Module # 2 3.0000 3.00005.5000 4.3820

8=liquid withdrawal 8.0000 6.61808.0000

Module # 3 8.0000 8.000010.0000 9.1835

12=vapor feed stage 12.0000 10.816512.0000

Module # 4 13.0000 13.000015.0000 14.1835

13=liquid feed stage 17.0000 15.816517=reboiler # 1 17.0000

Reboiled Stripper 0.0000 0.00002.0000 1.1835

0=liquid entry point 4.0000 2.81654=reboiler # 2 4.0000

Table 4-7 - Collocation points utilized in thesolution of the example problem discussed in Sec.4.4.

I-

300

250 -

50

0

6 200 -

LAJ

2 100 -w -

REBOILED STRIPPER

MAIN COLUMN

woo FULL ORDER00000 2 POINT COLLOCATION00000 1 POINT COLLOCATION

0I t 1 I r I II-III-1-i

3 6 9 12

STAGE NUMBER

1 I

15 18

Figure 4.21 - Steady-state temperature profiles for theexample problem discussed in Sec.4.4.

87

88

1.0

0.9

0.8

0.7z0i= 0.6

(4. 0.5

0 0.420.3

0.2 -

0.1 -

0.00

co. FULLORDER00000 2 POINT COLLOCATION00000 1 POINT COLLOCATION

MAIN COLUMN

REBOILED STRIPPER

I I r I r I [I ri Iri3 6 9 12

STAGE NUMBER

15 18

Figure 4.22 - Steady-state C2 liquid composition profilesfor the example problem discussed in Sec.4.4.

1.0 -

0.9

0.8

0.7 --

0.6 -

0.5

0.4 -

0.3 -

0.2 -

0.1

0.0

REBOILED STRIPPER

MAIN COLUMN

FULLORDER00000 2 POINT COLLOCATION0000c 1 POINT COLLOCATIONuTilril

0 3 6ITI I I I

9 12

STAGE NUMBER

I

15 18

Figure 4.23 - Steady-state C3 liquid composition profilesfor the example problem discussed in Sec.4.4.

89

1.0

0.9

0.8

0.7z0P. 0.6U

0.5

cl 0.420.3

0.2

0.1

0.0

FULLORDER00000 2 POINT COLLOCATION00000 1 POINT COLLOCATION

MAIN COLUMN

REBOILED STRIPPER

0 3I I i I I 1 1 1 1 1 1

6 9 12 15 18

STAGE NUMBER

Figure 4.24 - Steady-state C4 liquid composition profilesfor the example problem discussed in Sec.4.4.

90

91

0.985

zo

:4 0.980

481z

12 18 24 30 36 42 48 54TIME (hours)

60

Figure 4.25 Step response for C2 distillate compositionfor the example problem discussed in Sec.4.4.

92

z0

cdu_

0.030

0.025

0.020Lu

02...... - . .1 _ - . -

- -

0.015

FULL-ORDER2 POINT COLLOCATION1 POINT COLLOCATION

0.010I I 1 1 I 1 I I I

0 6 12 18 24 30 36 42 48 54 60

TIME (hours)

Figure 4.26 - Step response for C3 distillate compositionfor 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 onstage 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 phasevariables

Wrzp(s,t) Lagrange interpolating polynomial for vapor phasevariables

xii Liquid composition of species j on stage i

XB.4 Composition of species j on bottoms product

XF4

Xwi

YD,j

Z.

Composition of species j on feed stream

Composition of species j on withdrawal stream

Vapor composition of species j on stage i

Composition of species j on distillate product

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

Greek symbols

a, 13

Pi

Species molar thermodynamic property

Parameters in Hahn polynomials

Molar liquid density on stage i, mol/m3

Fraction of feed vaporized

96

97

BIBLIOGRAPHY

Cho, Y.S. and B. Joseph, Reduced-Order Steady-State and Dynamic Models forSeparation Processes I Development of the Model Reduction Procedure, AIChEJournal, 29, 2, 261 (1983).

Cho, Y.S. and B. Joseph, Reduced-Order Steady-State and Dynamic Models forSeparation Processes - II Application to Nonlinear Multicomponent Systems, AIChEJournal, 29, 2, 270 (1983).

Cho, Y.S. and B. Joseph, Reduced-Order Models for Separation Columns IIIApplication to Columns with Multiple Feeds and Sidestreams, Computers &Chemical Engineering, 8, 2, 81 (1984).

Finlayson, B.A., Nonlinear Analysis in Chemical Engineering, McGraw-Hill BookCo., New York (1980).

Gani, R., C.A. Ruiz and I.T. Cameron, A Generalized Model For DistillationColumns I Model Description and Applications, Computers & ChemicalEngineering, 10, 3, 181 (1986).

Gani, R., C.A. Ruiz and I.T. Cameron, A Generalized Model For DistillationColumns II - Numerical and Computational Aspects, Computers & ChemicalEngineering, 10, 3, 199 (1986).

Henley, E.J. and J.D. Seader, Equilibrium-Stage Separation Operations in ChemicalEngineering, 1st ed., John Wiley & Sons, Inc., New York (1981).

Holland, C.D. and A.I. Liapis, Computer Methods For Solving Dynamic SeparationProblems, 3rd ed., McGraw-Hill Book Company, Inc., New York (1983).

Howard, G.M., Unsteady State Behavior of Multicomponent Distillation Columns,AIChE Journal, 16, 6, 1023, (1970).

McCabe, W.L., J.C. Smith and P. Harriot, Unit Operations of ChemicalEngineering, McGraw-Hill Book Co., New York (1985).

Osborne, A., The Calculation of Unsteady State Multicomponent Distillation UsingPartial Differential Equations, AIChE Journal, 17, 3, 696 (1971).

Petzold, L.R., A Description of DASSL: A Differential / Algebraic System Solver,SAND82-8637, Sandia National Laboratories (1982).

98

Pugkhem, C., Steady-State Simulation of Distillation Columns Using OrthogonalCollocation, M.S. Project, Oregon State University, Corvallis, OR, (1990).

Smith, J.M. and H.C. Van Ness, Introduction to Chemical EngineeringThermodynamics, McGraw-Hill Book Co., New York, (1987)

Srivastava, R.K. and B. Joseph, Reduced-Order Models for Separation Columns IVTreatment of Columns with Multiple Feeds and Sidestreams Via Spline Fitting,Computers & Chemical Engineering, 11, 2, 159 (1987).

Srivastava, R.K. and B. Joseph, Reduced-Order Models for Separation Columns VSelection of Collocation Points, Computers & Chemical Engineering, 9, 6, 601(1985).

Srivastava, R.K. and B. Joseph, Reduced-Order Models for Separation Columns VIColumns with Steep and Flat Composition Profiles, Computers & ChemicalEngineering, 11, 2, 165 (1987).

Stewart, W.E., K.L. Levien and M. Morari, Simulation of Fractionation byOrthogonal Collocation, Chemical Engineering Science, 40, 3,409 (1985).

Swartz, C.L.E. and W.E. Stewart, A Collocation Approach to Distillation ColumnDesign, AIChE Journal, 32, 11, 1832, (1986).

Treybal, R.E., Mass Trasfer Operations, McGraw-Hill Book Co., New York (1981).

Villadsen, J. and M.L. Michelsen, Solution of Differential Equation Models byPolynomial Approximation, 1st ed., Prentice-Hall, Inc., New Jersey (1978).

Wankat, P.C., Separations in Chemical Engineering : Equilibrium Staged Separations,1st ed., Elsevier Science Publishing Co., Inc., New York (1988).

Wilkinson, W.L. and W.D. Armstrong, An Approximate Method of PredictingComposition Response of a Fractionating Column, Chemical Engineering Science, 7,1, (1957).

Wong, K.T. and R. Luus, Model Reduction of High-Order Multistage Systems bythe Method of Orthogonal Collocation, The Canadian Journal of ChemicalEngineering, 58, 382 (1980).

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. LevienChemical Engineering DepartmentOregon State UniversityGleeson Hall 103Corvallis, OR 97331-2702

(Calculate vapor compositionsfor each stage

Read inputs fromMisfile

(Calculate feed)conditions

(Set liquid composition ateach stage

(Calculate liquid and vaporflowrates

Read change inmanipulated variable

100

YES

Interpolate liquid and vaporcompositions entering

collocation points

( STOP >if NO

Reset ODE's and callintegrating package

YES

YES

NO 1

NO _

YES

Figure A.1 - Flowsheet diagram demonstrating the sequence of computationsutilized for solving a distillation problem using the CMO model.

(Set liquid composition ateach stage

Read inputs fromdatafile

(Calculate feed )conditions

47

'Calculate bubble point temperatures,vapor compositions and liquid and

vapor enthalpies at each stage

(Calculate differential term forenergy balances

YES

[Interpolate liquid and vapor

compositions and enthalpies forstreams entering collocation points

( STOP) NO

Calculate liquid and vapo9flowrates

101

4--

NO11

YES

YES

NO _

YES

Figure A.2 - Flowsheet diagram demonstrating the sequence of computationsutilized for solving a distillation problem using the CMH model.

Read inputs fromdatafile

(Calculate feed )conditions

( )Set liquid composition andmolar holdups at each stage

Calculate bubble pointtemperatures, vapor compositions,

liquid and vapor enthalpies andliquid flowrates at each stage

Calculate differential term forenergy balances

lirCalculate differential term for mass"balances for condenser and reboiler

NO

YES

VInterpolate liquid and vapor compositions

and enthalpies and liquid flowrates forstreams entering collocation points

( STOP) NO

'(Calculate vapor flowrates

(Reset ODE's and callintegrating package

NO

YES

YES

NO

YES

102

Figure A.3 - Flowsheet diagram demonstrating the sequence of computationsutilized 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,=(1.5-2.5)

=0.4(0-2.5)

(1.5-0)1,1 (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

x c2(si-1)

V

11.1 OL

s o=0xcAso>=0.9347

s 1=2.5xc2(si)=0.6042

s 2=5x c2(s2)=0.3221

Figure B.1 - C2 steady-state composition profile for the rectifyingmodule in the column treated in Section 4.1 obtained using

the lx1 reduced-order TDMH model.