numerically optimized diabatic distillation columns

8/13/2019 Numerically Optimized Diabatic Distillation Columns

1/86

Numerically Optimized

Diabatic Distillation Columns

von der Fakultat fur Naturwissenschaftender Technischen Universitat Chemnitz

genehmigte Dissertation zur Erlangung des akademischen Grades

doctor rerum naturalium

(Dr. rer. nat.)

vorgelegt von Dipl.-Ing. Markus Schallergeboren am 19. Juni 1973 in Vorau (Osterreich)

eingereicht am 4. Mai 2007

Gutachter: Prof. Dr. Karl Heinz Hoffmann (TU Chemnitz)Prof. Dr. Michael Schreiber (TU Chemnitz)Prof. Dr. Peter Salamon (San Diego State University)

Tag der Verteidigung: 10. Juli 2007


2/86

2


3/86

3

Bibliographische Beschreibung

Schaller, MarkusNumerically Optimized Diabatic Distillation ColumnsTechnische Universitat Chemnitz, Fakultat fur Naturwissenschaften, 2007Dissertation (in englischer Sprache)86 Seiten, 35 Abbildungen, 2 Tabellen, 57 Literaturzitate

Referat

Im Gegensatz zur konventionellen adiabatischen Destillation erfolgt beider diabatischen Destillation Warmeaustausch nicht nur am Kondensatorund Verdampfer, sondern auch innerhalb der Kolonne an den einzelnenSiebboden, was die Entropieproduktion (=Exergieverlust) des Destillations-prozesses stark reduziert. In dieser Arbeit werden Modellsysteme zur di-abatischen Destillation von idealen binaren Gemischen mittels numerischerOptimierung untersucht.

Das Ausgangsmodell beschrankt sich auf die Minimierung der Entropiepro-duktion verursacht durch Warme- und Massentransport im Inneren der di-abatischen Destillationskolonne. Im zweiten Modell wird das diabatische

Modell um die Irreversibilitat bedingt durch den Warmeaustausch mit derUmgebung erweitert. Im dritten Modellsystem wird anstelle der bis dahinvoneinander unabhangig geregelten Bodentemperaturen eine diabatische Im-plementierung mit seriellen Warmetauschern untersucht, die nur mehr vierKontrollvariablen besitzt und besonders zur praktischen Anwendung geeignetist.

Fur alle diabatischen Modelle werden die minimale Entropieproduktion undoptimalen Betriebsprofile numerisch ermittelt, und mit konventionellen Des-tillationskolonnen verglichen. Alle Ergebnisse zeigen eine deutlich Reduktionder Entropieproduktion fur den diabatische Fall, besonders bei Kolonnen mit

vielen Boden.

Schlagworter

Binares Gemisch, Destillationskolonne, Diabatische Destillation,Entropieproduktion, Equal Thermodynamic Distance, Exergie,Irreversibilitat, Nichtgleichgewichtsthermodynamik, Optimierung,Warmeaustauscher


4/86

4


5/86

Contents

1 Introduction 7

2 The Concept of Diabatic Distillation 11

2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Material Balance . . . . . . . . . . . . . . . . . . . . . 14

2.1.2 Equilibrium State Equations . . . . . . . . . . . . . . . 15

2.1.3 Heat Balance . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.4 Entropy Production . . . . . . . . . . . . . . . . . . . . 16

2.2 Equal Thermodynamic Distance . . . . . . . . . . . . . . . . . 17

3 Numerical Optimization 21

3.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Diabatic Column . . . . . . . . . . . . . . . . . . . . . 22

3.1.2 Conventional Column . . . . . . . . . . . . . . . . . . . 23

3.2 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Numerical Performance . . . . . . . . . . . . . . . . . . 25

3.2.2 Total Entropy Production . . . . . . . . . . . . . . . . 26

3.2.3 Operating Profiles . . . . . . . . . . . . . . . . . . . . 28

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Heat Transfer Irreversibilities 39

4.1 Entropy Production due to Heat Exchange . . . . . . . . . . . 40

4.1.1 Fourier Heat Conduction . . . . . . . . . . . . . . . . . 40

5


6/86

6 CONTENTS

4.1.2 Newtonian Heat Conduction . . . . . . . . . . . . . . . 41

4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Influence of Heat Transfer Law . . . . . . . . . . . . . 43

4.2.2 Influence of Different Column Length . . . . . . . . . . 43

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Sequential Heat Exchangers 53

5.1 Heat Exchanger Model . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Computational Method . . . . . . . . . . . . . . . . . . . . . . 57

5.2.1 Entropy Production and Steady State Condition . . . . 57

5.2.2 Pseudo-dynamic Algorithm . . . . . . . . . . . . . . . 58

5.3 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . 58

5.3.1 Operation Profiles . . . . . . . . . . . . . . . . . . . . 59

5.3.2 Total Entropy Production . . . . . . . . . . . . . . . . 62

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Conclusions 65

A Standard Optimization Methods 67

A.1 Powells Method . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.2 Downhill Simplex Method . . . . . . . . . . . . . . . . . . . . 68

Bibliography 70

List of Figures and Tables 77

Zusammenfassung 81

Selbstandigkeitserklarung gema Promotionsordnung 6 83

Lebenslauf und Publikationen 84


7/86

Chapter 1

Introduction

A major application of thermodynamics is the provision of in-principle per-formance bounds for any kind of energy conversion process. The oldest andmost prominent of such performance limits is the Carnot efficiency givingthe maximum efficiency of any reversible heat engine working between twoinfinite heat reservoirs. Much later, in the second half of the 20th century,when the oil crisis of the 1970s required a stronger awareness of limited en-ergy resources, performance limits have been extended to thermodynamic

processes subject to finite time or finite rate constraints. Among such stud-ies, the Curzon-Alborn-Novikov efficiency [1] is a remarkable result and earlymilestone of this emerging thermodynamic research area, which is known asfinite-time thermodynamics [2, 3] or, more general, as control thermodynam-ics [4].

Naturally, such problems lead to the use of mathematical optimization. Aperformance objective is optimized subject to constraints imposed by thethermodynamic process under consideration. As a result, the optimal con-trols of the process are determined that achieve the optimal value of theobjective. Weaker, but generally more tractable is the question for bounds

on the optimal values of the objective. Hence, several ways to simplify andgeneralize problems in control thermodynamics have been proposed. Sala-mon [4] categorizes them into principles of problem simplification, principlesrelated to maximum power, and principles related to minimum entropy pro-duction. With the help of such principles, one can find performance boundsfor a particular class of thermodynamic processes aside from design detailsor other engineering aspects. Furthermore, such generalized models are anorientation and starting point for more detailed further analyes, e. g. nu-merical investigations. A mentionable successful tool in this context is the

7


8/86

8 CHAPTER 1. INTRODUCTION

concept of endoreversibility [510], where all irreversibilities are located at

the couplings of an reversible subsystem to its surroundings.

There has been plenty of research activity in the field of control thermo-dynamics during the last decades. Many publications focus on the poweror efficiency optimization of all kinds of heat engines (e.g. [1113]), or theminimization of entropy production in thermal engineering processes [14, 15].Solar energy conversion has been investigated [16, 17], and we also find earlycontrol thermodynamics approaches to chemical processes [18, 19] and distil-lation [20]. In this thesis we focus on the optimization of the latter.

Distillation

In many chemical production processes raw materials or end products aregiven as liquid mixtures which have to be separated into their components.Fractional distillation is the most important method used for the separationof liquid mixtures or liquified gaseous mixtures. Among numerous industrialapplications, distillation is particularly important for the petrochemical in-dustry. In oil refineries, crude oil is distilled to yield various commercial oilproducts.

Since distillation is a heat driven separation process, it significantly con-

tributes to the energy consumption. In the USA about 10% of the industrialenergy consumption accounts for distillation [21, 22]. More than 70% of theoperation costs are caused by the energy expenses [23]. But the second-lawefficiency of conventional distillation is very low, only around 520% [24, 25].This means that distillation is associated with a high entropy production(= exergy loss) and hence degradation of energy. To reduce the exergywasted it is recommended to alter design and operation of the distillationprocess. This is achieved by spreading the heat requirements over the wholelength of a distillation column. Such design consideration is referred to asdiabaticdistillation.

This document presents a numerical investigation of diabatic distillationmodels with emphasis on the determination of minimum entropy produc-tion and the corresponding optimal operating characteristics.

Document Structure

Chapter 2 introduces the concept of diabatic distillation which allows enor-mous reduction of the entropy production compared to conventionally de-


9/86

9

signed distillation columns. A mathematical model for diabatic tray distil-

lation is systematically built up for ideal binary mixtures. From this modelexpressions for the heating profile and total entropy production are obtainedas functions of the temperature profile inside the distillation column. Forfurther comparison, the asymptotic theory of equal thermodynmic distance(ETD) is outlined.

In Chapter 3 the entropy production in a diabatic distillation column is min-imized by applying Powells algorithm to the temperature profile inside thedistillation column. From the optimal temperature profile the minimal totalentropy production is obtained and compared to the entropy production pre-scribed by the ETD theory and to the entropy production of a conventional

column. The comparisons are performed for columns of different length andfor different purity requirements. Additionally, the temperature profiles, theheating requirements for each tray and the entropy production per tray areevaluated for numerically optimized, conventional and ETD columns.

In Chapter 4 the distillation model is extended to include the heat transferirreversibilies arising from the heat coupling of the column to the surround-ings. Two different heat transfer laws, Newtons linear law and Fouriersinverse law are investigated. For both heat transfer laws the minimum totalentropy production is determined by numerical optimization for varying heatresistance. For three column lengths, the optimal operation profiles (heat re-

quirements and entropy production per tray) are computed for low, high andindustrially relevant values of heat resistance.

In Chaper 5 the concept of independently adjustable tray temperatures isreplaced by a heat exchanger installation that only requires four controlvariables. For a sample column with this particular design, the optimaloperation profiles are determined and compared to a conventional column.Furthermore we focus on how much more irreversibility one must pay for thereduction of control variables.

Chapter 6 summarizes the core results of this thesis and gives an outlook on

potential open questions for further research.


10/86

10 CHAPTER 1. INTRODUCTION


11/86

Chapter 2

The Concept of Diabatic

Distillation

Distillation makes use of the differences in volatility of the components of amixture. The more volatile component has the lower boiling temperature.Hence, when the liquid mixture is heated up to its boiling temperature,the resulting vapor is enriched with the more volatile component. Aftercondensing this vapor one obtains a liquid with a higher concentration of

that component. This process is repeated until the components are separatedinto specified purities.

Fractional distillation is performed in vertical columns divided into trays.On each tray one stage of purification is carried out. The reboiler at thebottom serves as heat source, the condenser at the top serves as heat sink.According to the desired purity, heat QB is delivered to the bottom and heatQD is removed from the top. This creates a temperature gradient decreasingvertically along the column. The feed flow F carrying the mixture to beseparated is introduced near the middle of the column. On each tray, themixture boils resulting in vapor entering the tray above. The tray has an

overflow tube permitting the liquid to flow to the tray below. The morevolatile component of the binary mixture is removed at the top as distillateD, the other is removed at the bottom as bottom product B (see figure 2.1,on the left).

The thermodynamic inefficiency of conventional distillation has the followingreason: heat is added only at the highest temperatureTBin the column, whilethe heat removal takes place only at the lowest temperature TDin the column.Hence, the energy is degraded over the whole teperature range TB TD ofthe column. This is the reason for the high entropy production associated

11


12/86

12 CHAPTER 2. THE CONCEPT OF DIABATIC DISTILLATION

xD, D

xB, B

F, xF

V

V

V

V

V

V

V

L

L

L

L

L

L

1

1

3

4

5

6

7

3

4

5

6

2

L 7

L

2

V8

0

QD

QB

xD, D

xB, B

F, xF

V

V

V

V

V

V

V

L

L

L

L

L

L

1

1

3

4

5

6

7

3

4

5

6

2

L 7

L

2

V8

0

QD

QB

Additional

heat sinks

and

heat sources

Figure 2.1: Sketch of a conventional adiabatic distillation column and a

diabatic column with additional heat exchange. Both columns have N = 8trays including the reboiler as tray 8. The Ls andVs denote the liquid andvapor flows, respectively.

with distillation.

From that it can be concluded that a redistribution of the heat requirementsover the whole length of the column will result in a lower entropy production[26]. Then most of the heat would be used over a smaller temperature rangethanTBTD[27]. A column designed in such a way is referred to completelydiabatic distillation column because heat exchange between column and sur-roundings occurs on each tray of the column (figure 2.1, on the right). Otherterms than diabatic used in literature are thermally controlled or heat inter-grated column. The idea goes back to the work of Fonyo [28, 29] in the early1970s but recently has been explored by Rivero [3032], Salamon, Nultonand Andresen [22, 27], and Kjelstrup, Sauar, and De Koeijer [24, 25, 3336].

Now, the interesting question is how to determine the optimal tempera-ture and heating profiles ensuring operation at minimum entropy produc-tion. Various theoretical approaches exist to estimate optimal performance


13/86

2.1. MODEL DESCRIPTION 13

of diabatic distillation columns, but so far there is only one systematic exper-

imental study done by Rivero [30]. Salamon, Andresen and Nulton [22, 27]use a thermodynamic metric based on the entropy state function of the con-sidered mixture, which leads to the theory of equal thermodynamic distance(ETD). Kjelstrup et al. [25, 33, 34] use a method that is based on Onsagersirreversible thermodynamics, in which the entropy production is representedin terms of thermodynamic forces and fluxes. From this theory the principleof equipartition of thermodynamic forces (also called isoforce principle) isderived [37, 38]. Tondeur and Kvaalen [39] suggest a different equipartitionprinciple as optimality criterion, the equipartition of entropy production. Forsummaries and general discussions on equipartition principles in thermody-

namics the reader should refer to [4, 40, 41].

Because of the additional heat exchangers, the investment costs for a dia-batic column are significantly higher than for a conventional one. So-calledpartially diabatic columns are considered to allow a trade-off between extraequipment costs and minimizing entropy production. In these columns onlyselected trays are thermally controlled. For example, De Koeijeret al. usethe isoforce principle to find optimal locations of the trays to be heated orcooled [24]. However, this thesis only considers fully diabatic columns, be-cause they give the ultimate limit of what is possible in reducing the entropy

production.

Each of these theoretical approaches bases on a particular thermodynamicprinciple which gives an approximation to the optimum. To assess the va-lidity of these approximations, numerical minimization has shown to be veryeffective [35, 42]. Before introducing a fully numerical optimization schemein the next chapter, a mathematical model for binary distillation is presentedhere.

2.1 Model Description

The distillation column is considered to be operating at steady-state so allextensive quantities are per unit time. For convenience, only binary mixturesare considered and the pressure is assumed to be constant throughout thecolumn.


14/86


n

n+1

V

LV

n-1

nL

n

n+1

QHn H

H

n

vapliq

vap

Hn-1liq

n-1

H

n+1

n

Figure 2.2: Definitions of quantitiesaround tray n appearing in the bal-ance equations (2.3), (2.4) and (2.8).

2.1.1 Material Balance

In steady state operation, the feed, distillate and bottoms obey the flowbalance equations

F = D+ B, (2.1)

xFF = xDD+ xBB, (2.2)

where xF, xB and xD denote the corresponding mole fractions of the morevolatile component (lower boiling point) in the liquid phase. Similarly, theamount of material flowing out of a tray must be equal to the amount ofmaterial flowing into a tray. Hence, vapor coming up from trayn+ 1 andliquid flowing down from tray n have to balance the distillate D above thefeed and the bottoms B below the feed respectively (see figure 2.2),

Vn+1 Ln =

D above feed

B below feed , (2.3)

yn+1Vn+1 xnLn = xDD above feedxBB below feed . (2.4)

On the uppermost tray (n= 1) the balance equations simplify toV1=D +L0and y1 =xD; for the lowest tray (n= N, the reboiler) one obtains LN =Band xN=xB. For our purposes of looking for optimal diabatic columns, therefluxL0is taken equal to zero. Its purpose in an adiabatic column is to helpcarry heat out of the column. This function is not needed in the diabaticcolumn since heat can be taken directly from tray one.


15/86

2.1. MODEL DESCRIPTION 15

2.1.2 Equilibrium State Equations

It is assumed that each tray is in thermodynamic equilibrium. The tempera-ture dependencies of the molar fractionsxandyin the liquid and vapor phaserespectively, are given by the state equations for an ideal solution model [43]

y = xexp

Hvap,1(T)

R

1

Tb,1 1

T

, (2.5)

1 y = (1 x) exp

Hvap,2(T)

R

1

Tb,2 1

T

. (2.6)

HereTb,1andTb,2denote the boiling points of the two pure components. Theenthalpy differences Hvap,1(T) and Hvap,2(T) are calculated as

Hvap,i(T) = Hvap,ib + (T Tb,i)(cvap,ip cliq,ip ), (i= 1, 2) (2.7)

where Hvap,ib are the heats of vaporization of the pure components andcliq,ip

and cvap,ip are the corresponding heat capacities. Equation (2.7) requires theheat capacities to be temperature independent.

2.1.3 Heat Balance

In order to calculate the heat required at each tray to maintain the desiredtemperature profile, the energy balance has to be maintained for each trayn:

Qn=VnHvapn + LnH

liqn Vn+1Hvapn+1 Ln1Hliqn1. (2.8)

For conventional adiabatic distillation columns, equation (2.8) would be equalto zero (for 1 n < N), and there would be no control parameters over whichto optimize. The enthalpies Hvap and Hliq carried by the vapor and liquidflows are determined by

Hliq(T) = x cliq,1p (T Tref) + (1 x) cliq,2p (T Tref), (2.9)Hvap(T) = y

cliq,1p (T Tref) + Hvap,1(T)

+

+(1 y)cliq,2p (T Tref) + Hvap,2(T)

(2.10)

Trefis an arbitrary reference temperature whose value drops out of the cal-culations. It represents the temperature at which the (relative) enthalpyof the pure liquid is zero. Here we assumed constant heat capacities andnoninteracting mixture of ideal gases for the vapor phase.


16/86


For the lowest tray the energy balance reduces to

QN=VNHvapN + BH

liqN LN1HliqN1. (2.11)

For condenser and reboiler we obtain

QD = (D+ L0)(Hvap(T1) Hliq(TD)), (2.12)

QB = QN. (2.13)

For reflux L0 = 0, equation (2.12) becomes

QD=D(Hvap(T1) Hliq(TD)), (2.14)

while for the uppermost tray equation (2.8) turns into

Q1 =DHvap1 + L1H

liq1 V2Hvap2 . (2.15)

On the feed traynF, the enthalpy of the feed flow has to be explicitely addedin equation (2.8):

QF = QnF F Hliq(TF). (2.16)Note that it is assumed that the feed enters as liquid at its boiling tempera-ture TF. The feed tray nFis chosen such that the inequalityTnF1< TF< TnFholds.

2.1.4 Entropy Production

Equations (2.1)(2.16) allow us to evaluate the entropy production of thedistillation process. The entropy production per tray can be determined

from an entropy balance for each tray analogous to the heat balance (2.8).In order to focus on the separation process proper, unobscured by issuesof heat exchange, we define our system to be the interior of the column.This makes the irreversibility associated with the heat transfer in and out ofthe column extraneous to the subsequent optimization. Hence, we take thesource temperatures of the Qn to be equal to the tray temperatures Tn. Forthe entropy production per tray one then obtains

Sun =Vnsvapn + Lns

liqn Vn+1svapn+1 Ln1sliqn1

QnTn

. (2.17)


17/86

2.2. EQUAL THERMODYNAMIC DISTANCE 17

Thesliq andsvap are the entropies per mole of liquid and vapor flow, respec-

tively,

sliq(T) = x

sref,1+ cliq,1p ln

T

Tref

+ (1 x)

sref,2+ c

liq,2p ln

T

Tref

R[x ln x + (1 x)ln(1 x)], (2.18)

svap(T) = y

sref,1+ cliq,1p ln

T

Tref

+ (1 y)

sref,2+ c

liq,2p ln

T

Tref

+

+ y

Hvap,1b + (c

vap,1p cliq,1p ) ln

T

Tb,1

+

+ (1 y)

Hvap,2b + (cvap,2p cliq,2p ) ln

TTb,2

R[y ln y+ (1 y) ln(1 y)]. (2.19)

Summing over all trays yields the total entropy production

Su =Nn=0

Sun = Smassflows

Nn=0

QnTn

, (2.20)

where n= 0 refers to the condenser and Smassflows is given by

Smassflows = F sliqF + DsliqD + BsliqB . (2.21)

sliqF ,sliqD ands

liqB denote the entropies per mole of feed, distillate and bottoms

mass flow, respectively. Hence, the heat exchange between column and sur-roundings and the mass flows of distillate, bottoms and feed contribute tothe entropy production Su. Since the column is in steady state, its entropyis constant. This implies that the entropy production equals the change inentropy of the columns surroundings.

Note that Smassflows is fixed by the specifications of the process and is there-

fore not part of the optimization.

2.2 Equal Thermodynamic Distance

In the following the asymptotic theory of equal thermodynamic distance(ETD) is briefly described. This theory provides a lower bound on the en-tropy production (2.20) of a fully diabatic distillation column, and will belater used to compare with the numerical analysis.


18/86


The concept of ETD uses a thermodynamic metric based on the entropy

state function of the mixture to be separated. The thermodynamic length Lof a process is given by the line element [22]

dL =

dZt D2SdZ. (2.22)where Z = (U , V , . . .) is the vector of extensive variables and D2S is thematrix of partial derivatives 2S/ZiZj.

The distillation process is modeled as an N-step process [44], with N cor-responding to the number of trays in the distillation column. There is anasymptotic theory bounding the entropy production for such processes in thelimit ofN

. Asymptotically, the total entropy production Su of an

N-step process is bounded by

Su L2

2N (2.23)

This is a result from the horse-carrot theorem [22, 44]. The thermodynamiclength of an N-step process can be written as

L =Nn=1

Ln, (2.24)

where Ln is the length of the n-th step. Asymptotically, for minimal en-tropy production the lengths of the steps have to be equal, i.e.

L1=. . .= Ln =. . .= LN, (2.25)hence the name, equal thermodynamic distance.

For the distillation model described in the previous section, the thermody-namic length element equation (2.22) is given by [22]

dL =

CT

dT, (2.26)

where C is the total constant pressure coexistence heat capacity of the bi-

nary two-phase mixture in equilibrium [45]. This is the heat capacity of aconstant pressure system consisting ofL moles of liquid coexisting in equi-librium with V moles of vapor. Its mathematical representation is givenby

C = L

x cliq,1p + (1 x) cliq,2p + R Tx (1 x)

dx

dT

2

+ V

y cvap,1p + (1 y) cvap,2p + R Ty (1 y)

dy

dT

2 . (2.27)


19/86

2.2. EQUAL THERMODYNAMIC DISTANCE 19

350 360 370 380 390

T [K]

0

10

20

30

40

C

[kJmole1

K1]

xD/xB= 0.90/0.10x

D/x

B= 0.95/0.05

xD/x

B= 0.99/0.01

Figure 2.3: Heat capacity in equation (2.26) as a function of temperaturefor three different purity requirements. The vertical dotted lines show thecorresponding highest and lowest temperatures in the column. The break atT= 366 K shows the feed point.

As such a system is heated, the amounts of liquid and vapor change andthe compositions readjust in such a fashion as to maintain equilibrium. Thequantities of liquid and vapor that need to be counted in C are the flows Land V between trays. To giveV(T) and L(T) irrespective ofN, these needto be taken equal to the limiting (infinite N) values which are given abovethe feed by

V(T) = xD x(T)y(T) x(T)D, (2.28)

L(T) = xD y(T)

y(T) x(T)D, (2.29)

and below the feed by

V(T) = x(T) xBx(T) y(T)B, (2.30)

L(T) = y(T) xBx(T) y(T)B. (2.31)

In figure 2.3,C of a benzene/toluene mixture is depicted for different purityrequirements.


20/86


In order to establish an ETD path from the condenser T0 to the reboiler TN

in a column with Ntrays, one has to determine temperatures Tn such that

Tn+1Tn

CT

dT = 1

N

TNT0

CT

dT, n= 0, . . . , N 1 . (2.32)

The temperature profile obtained with the iteration above allows to calcu-late the entropy production (2.20) and all other relevant quantites. ETD isa first-order asymptotic theory [22, 27] for minimum entropy production in1/N. This gives rise to the question of how reliable ETD is for columns withfew trays. In order to answer this question, a fully numerical, multidimen-sional optimization routine is applied to minimize the entropy production in

a diabatic distillation column and to further compare the results with ETD.


21/86

Chapter 3

Numerical Optimization

Previous studies of diabatic distillation columns using the ETD approachhave shown enormous exergy savings compared to conventional distillationfor columns with a large number of trays (e. g. [21]). But because of theasymptotic nature of ETD, the true minimum will be lower for fewer trays.Consequently, one is interested in the difference between ETD and optimaloperation. This motivates a fully numerical optimization which will find thetrue minimum for any feasible number of trays.

The entropy production (2.20) is minimized using a multidimensional opti-mization routine. The tray temperaturesTn, n = 1, . . . , N are used as thecontrol variables, thus the optimal temperature of each tray in the column isdetermined without applying any thermodynamic principle like ETD. Onceknowing the optimal temperature profile, one can easily compute all thecorresponding quantities like heat demand per tray, liquid and vapor flowsentering/leaving each tray and molar fractions of the liquid and vapor phase.

Calculating the gradient of equation (2.20) is rather costly due to the struc-ture of the state equations (2.5) and (2.6). For this reason, Powells method

[46] is chosen to perform the minimizations since it does not require gradientinformation. A more detailed description of Powells algorithm is given inAppendix A.

This chapter gives a detailed comparison of ETD columns, numerically op-timized and conventional columns. Besides looking at the total entropy pro-duction of each column type, the resulting operation profiles of the columns(e. g. heat requirement per tray) are studied as well. The optimal profiles al-low to give design recommendations for diabatic columns. Results presentedin this chapter are also published in [35, 42].

21


22/86

22 CHAPTER 3. NUMERICAL OPTIMIZATION

3.1 Problem Definition

Appropriate penalty functions have to be added to the entropy productionin order to take into account the physical constraints on the optimizationproblem. In this section it is also shown how the entropy production of aconventional column can be expressed as an optimization problem.

3.1.1 Diabatic Column

The entropy production (2.20) consists of terms of the form Qn/Tn which

in turn are functions of the liquid and vapor flows Vn and Ln. An explicitrepresentation of the vapor flow above the feed using the material balance(2.3) and (2.4) is given by

Vn(Tn, Tn1) = xD xn1(Tn1)yn(Tn) xn1(Tn1) D. (3.1)

Analogous expressions exist for the liquid flows and the trays below the feed.Equation (3.1) has a singularity, the flow becomes infinite for yn = xn1.This implies the physical constraint

xn1 < yn, (3.2)

i. e., the molar fraction in the vapor coming up from tray n has to be largerthan the molar fraction in the liquid going down from tray n 1. This inturn means that the difference between the temperatures of two adjacenttrays must be restricted so that the constraint above is fulfilled. Violatingthis constraint leads to unphysical results, e. g. negative values of liquid andvapor flows.

The temperatureT1 at the uppermost tray as well as the temperature TN atthe reboiler are fixed by the given distillate and bottoms purity requirements

xD and xB respectively. This reduces the number of control variables toN 2. Defining the control vector T = (T2, . . . , T N1), the minimizationproblem then takes the form

Su,opt = minT

Nn=0

QnTn

+ P(T)

+ Smassflows. (3.3)

Again, here n= 0 refers to the condenser.P(T) denotes a penalty functionwhich is added to the entropy production when constraint (3.2) is not fulfilled.


23/86

3.1. PROBLEM DEFINITION 23

0 0.2 0.4 0.6 0.8 1

x [] and y []

350

360

370

380

390

T

[K]

x(T)

y(T)

Largest achievabletemperature step

Unachievably largetemperature step

Figure 3.1: Description of an unphysical situation in the liquid-vapor-diagram. When the temperature difference between two adjacent trays istoo large, constraint (3.2) is not fulfilled.

The penalty is written as

P(T) =MN1n=1

(xn yn+1)(xn yn+1)2, (3.4)

where (.) denotes the Heaviside function and Mis a sufficiently large num-ber. Powells method is then applied to equation (3.3) to obtain minimumentropy production and the corresponding optimal temperature profile.

3.1.2 Conventional Column

A calculation method to determine the temperature profile of a conventionalcolumn is required in order to compare the performance of a diabatic columnwith its adiabatic counterpart. A common way to obtain the temperatureprofile are shooting methods. But here a novel approach is given by formu-lating the entropy production of a conventional column as a minimizationproblem. One still uses the model for diabatic distillation given in section2.1, but takes the reflux L0 to be a positive quantity, as required for theconventional column.


24/86


In a conventional column, the amount of condenser heatQDcan be controlled

by the refluxL0(see equation (2.12)). For this reasonL0is added to the con-trol vector Tand a second term is added to the penalty function (3.4). Thisterm prevents the reflux from becoming negative during the optimization.Hence, the new penalty takes the form

P(T) =M1[1 (L0)]L20+ M2N1n=1

(xn yn+1)(xn yn+1)2, (3.5)

where M1 and M2 denote sufficiently large numbers. We introduce anotherpenalty function of the form

Q(T) =M3N1n=1

Q2n, (3.6)

where M3 is some constant turning the squared heats into the dimension ofan entropy. The latter penalty causes the intermediate heat requirements ofthe diabatic column to be turned off during the optimization. The penaltiesvanishing, only the entropy flows of condenser and reboiler account for theentropy production which we then can express as

Su,conv = minT

P(T) + Q(T) QDTD QBTB + Smassflows. (3.7)Thus, it is now possible to use Powells method for both diabatic and con-ventional column.

3.2 Optimization Results

A benzene/toluene mixture is chosen as our system to be separated. The

most important thermal properties are listed in Table 3.1. The entropy pro-duction for the separation of a 50/50 mole fraction benzene/toluene mixtureis minimized by applying ETD and numerical optimization and comparedto the corresponding conventional column. The number of trays and purityrequirements are varied to show differences in the performance of ETD andnumerical minimizations. The comparisons are always between columns withthe same material flows in and out. Notably the feed, bottoms and distillateflows match not only in magnitude but also in composition and temperaturein the columns compared.


25/86

3.2. OPTIMIZATION RESULTS 25

Table 3.1: Important thermal properties of pure benzene and toluene

Property Benzene Toluene

Boiling temperatures [K] 353.25 383.78

Heat capacity, vapor phase [J mole1K1] 81.63 106.01

Heat capacity, liquid phase [J mole1K1] 133.50 156.95

Heat of evaporization [J mole1] 33600 38000

Reference entropies at 298 K [J mole1K1] 269.20 319.74

3.2.1 Numerical Performance

Powells minimization routine only allows to find local minima. Thereforevarious runs with a wide range of different initial guesses are necessary forcalculating the minimum entropy production of a distillation column. Thus,one can test whether different starting values lead to different local minima.Random starting vectors are produced the following way: the temperaturein both conventional and diabatic columns is bounded by the fixed valuesT1 and TN, so N

2 random temperatures within the range [T1 . . . T N] are

determined. Subsequent sorting of these temperatures then yield a randominitial temperature vector for the optimization procedure.

Initial guesses as described above are applied to the objective functions (3.3)and (3.7). Multiple minimization runs for a given number of trays do notreveal different local minima. Hence, the resulting minimum value for theentropy production is most probably the global minimum. All what could beobserved was a slight dependence of the number of Powell iterations on theinitial guess. A good guess, i.e. close to the optimum control vector, reducedthe amount of computations.

Throughout all minimizations, a relative accuracy of 109

was used. Forshorter columns (N < 30) it takes less than 10N Powell itertions to getto the minimum. But the computational effort increases considerably oncelonger columns (N >50) are optimized. In these cases it needed up to several100NPowell iterations until the minimum was found.

Surprisingly, the evaluation of the entropy production for the conventionalcolumn (equation (3.7)) is much more costly than the optimization of thediabatic counterpart. Even for short columns it takes several 100N Powelliterations until the minimum. For longer columns the effect is even more


26/86


dramatic, requiring more than 105NPowell iterations. Apparently, the slow

convergence is due to the

n Q2n term in equation (3.7).

3.2.2 Total Entropy Production

First, we look at the entropy production as a function of the number of traysfor all three types of distillation columns. The results for the simulations areshown in figures 3.23.4 for three different purity requirements. The figuresinclude curves for

a conventional column calculated with equation (3.7) the corresponding ETD column determined by equation (2.32) the numerically optimized column (equation (3.3)) and the aymptotic lower boundL2/(2N) for the entropy production

based on the ETD calculation.

For all three purity requirements the optimal diabatic columns are far moreefficient than their conventional adiabatic counterparts. In the smallN re-

gion, the entropy production is reduced by about 1530%, for columns withmany trays one can find reductions of 6580%. For short columns, the numer-ical optimization results predict slightly less entropy production; the optimalvalues are up to 10% lower than the corresponding ETD values. For largerNvalues ETD agreed very well with the numerical minimization. The optimalresults also were above the ETD lower boundL2/(2N), but approached theETD bound asNwas increased. The largeNsimulation for the 99/01 purityrequirement had the closest values to the ETD bound as was expected dueto the asymptotic nature of the ETD theory.

For a certain total thermodynamic lengthL and number of trays N, eachETD step must go a distance L/N. WhenNis below a certain value, condi-tion (3.2) does not hold anymore because the thermodynamic distance (andhence the temperature difference) between two adjacent trays is too large.Since this leads to unphysical results ETD and its lower bound is not plottedfor N < 13 for the 95/05 separation and N < 32 for the 99/01 separation.This is a serious limitation of the ETD theory and makes it only applicablefor sufficiently long columns.

TheL2/2Nvalues are suprisingly far from the ETD curves. The reason forthis comes from the fact that the flow rates V and L enter the expression


27/86


8 12 16 20

N []

0

1

2

3

4

5

Su[

Wm

ole

1K

1]

conventional column

ETD

numerical optimum

L2/(2N)

xD/xB=0.90/0.10

Figure 3.2: Minimal entropy production for varying number of trays de-termined with ETD and numerical optimization. The required purity isxD= 0.90, xB = 0.10. For comparison, the entropy production for a conven-tional column and the lower bound for ETD,L2/(2N) is included.

10 15 20 25

N []

0

2

4

6

8

10

Su[

Wm

ole

1K

1]

convectional column

ETD

numerical optimum

L2/(2N)

xD/xB=0.95/0.05

Figure 3.3: Minimal entropy production for xD= 0.95, xB = 0.05.


28/86


10 30 50 70

N []

0

1

2

3

4

5

6

Su[

Wm

ole

1K

1]

conventional columnETD

numerical optimum

L2/(2N)

xD/xB=0.99/0.01

Figure 3.4: Minimal entropy production for xD= 0.99, xB = 0.01.

for C. The values ofV and L are the continuous values given in equation(2.26) which corresponds to an infinite number of trays. The continuous pathis needed to define thermodynamic distance along the column. When thetemperatures found from equation (2.32) are used to calculate the actual flowrates with the given number of traysN, the flow rates are significantly abovethe minimum reflux levels and account for the difference1. The surprisinglygood match between Su,opt and Su,ETD leads to a deeper explanation [47].There, it turns out that the difference between these two quantities is alwaysof order 1/N3.

3.2.3 Operating Profiles

Next we investigate the operating profiles of three columns with differentnumber of trays and purity requirements. Temperature profiles, heat re-quirements per tray, and the distribution of entropy production along thecolumn, as well as the liquid and vapor flows, and the molar fractions ofliquid and vapor phase, are shown for a 15 tray columns with a 90/10 sepa-ration (figure 3.5 and 3.6), a 25 tray columns with a 95/05 separation (figure

1This reflux rate refers to the value ofVL for that tray and should not be confused

with the reflux rate L0 for the column which is needed to be non-zero for the conventional

column but is zero for the ETD and the optimal columns.


29/86


3.7 and 3.8), and a 70 tray column with a 99/01 separation (figure 3.9 and

3.10). In each figure, the profiles are plotted for an optimized column, anETD column and the corresponding conventional column.

Tray Temperatures

The temperature profiles for both ETD and optimal show a characteristicmild S curvature found in optimally operating column where the separationis symmetric. This shape of the temperature profiles is probably due to theeffective heat capacity C. The temperature difference from tray to tray inthe column is smaller in the regions where the heat capacity is large. As can

be seen in figure 2.3, the 99/01 separation is the most dramatic example. Bycontrast, the temperature profile of the conventional column is much steeperin the top and bottom part but has a plateau in the middle of the columnaround the feed point. For longer columns and higher purity requirement thisshape becomes more distinct, as can be seen from the temperature profile ofthe 70 tray column (figure 3.9).

Material Flows and Concentration Profiles

The liquid and vapor flows are roughly constant over the column length inthe conventional column, apart from the jump of the liquid flow curve onthe feed tray. In the diabatic case, the flow rates decrease from the feedpoint towards the top and bottom part of the column. As the flow rates aresignificantly smaller than in the conventional column, some of the columnscross sectional area can be alloted for the heat exchangers to be installedwithout interfering with the material flows.

The molar fractions follow the shape of the temperatur profiles, indicating amore evenly distributed separation inside the diabatic column, as opposed tothe plateau in the middle of the conventional one, where no separation takes

place.

Entropy Production per Tray

The temperature profiles also reflect the distribution of the entropy produc-tion along the column. While the entropy production per tray is roughlythe same for all trays in the diabatic case, there are two huge peaks in thetop and bottom part of the conventional column, where all the energy degra-dation takes place. This situation becomes more dramatic with increasing


30/86


0 5 10 15n [-]

350

360

370

380

390

T

[K]

conventional columnETDnumerical optimum

0 5 10 15n [-]

-40

-30

-20

-10

0

10

20

30

40

Q[kWm

ole-1]


0 5 10 15n [-]

0

0.1

0.2

0.3

0.4

0.5

Su[

Wm

ole-1K

-1]

conventional column

ETDnumerical optimum

Figure 3.5: Temperature profileT, heating requirement Q, and entropy pro-duction Su per tray for a conventional column, an ETD column and a nu-merically optimized column with 15 trays. The required purity is xD = 0.90,xB = 0.10.


31/86


0 5 10 15n [-]

0

0.5

1

1.5

2

L

[-]


0 5 10 15n [-]

0

0.5

1

1.5

2

V[

-]


0 5 10 15n [-]

0

0.2

0.4

0.6

0.8

1

x

[-]

conventional column


0 5 10 15n [-]

0

0.2

0.4

0.6

0.8

1

y

[-]


Figure 3.6: Liquid flowL, vapor flowV, molar fractionx of liquid phase, andmolar fractiony of vapor phase, for a conventional column, an ETD columnand a numerically optimized column with 15 trays. The required purity isxD= 0.90, xB= 0.10.


32/86


0 5 10 15 20 25n [-]

350

360

370

380

390

T

[K]


0 5 10 15 20 25n [-]

-40

-30-20

-10

0

10

20

30

40

Q[kWm

ole-1]


0 5 10 15 20 25n [-]

0

0.1

0.2

0.3

0.4

0.5

Su[

Wm

ole-1K

-1]

conventional column




33/86


0 5 10 15 20 25n [-]

0

0.5

1

1.5

2

L

[-]


0 5 10 15 20 25n [-]

0

0.5

1

1.5

2

V[

-]


0 5 10 15 20 25n [-]

0

0.2

0.4

0.6

0.8

1

x

[-]


0 5 10 15 20 25n [-]

0

0.2

0.4

0.6

0.8

1

y

[-]


Figure 3.8: Liquid flowL, vapor flowV, molar fractionx of liquid phase, andmolar fractiony of vapor phase, for a conventional column, an ETD columnand a numerically optimized column with 25 trays. The required purity isxD= 0.95, xB= 0.05.


34/86


0 10 20 30 40 50 60 70n [-]

350

360

370

380

390

T

[K]


0 10 20 30 40 50 60 70n [-]

-40

-30

-20

-10

0

10

20

30

40

Q[kWm

ole-1]


0 10 20 30 40 50 60 70n [-]

0

0.1

0.2

0.3

0.4

Su[

WK

-1m

ole-1]




35/86


0 10 20 30 40 50 60 70n [-]

0

0.5

1

1.5

2

L

[-]


0 10 20 30 40 50 60 70n [-]

0

0.5

1

1.5

2

V[

-]


0 10 20 30 40 50 60 70n [-]

0

0.2

0.4

0.6

0.8

1

x

[-]


0 10 20 30 40 50 60 70n [-]

0

0.2

0.4

0.6

0.8

1

y

[-]


Figure 3.10: Liquid flowL, vapor flow V, molar fraction x of liquid phase,and molar fraction y of vapor phase, for a conventional column, an ETDcolumn and a numerically optimized column with 70 trays. The requiredpurity isxD = 0.99, xB = 0.01.


36/86


column length. In the region of the temperature plateau there is no entropy

production. It is also remarkable that the numerically optimized diabaticcolumn produces more entropy in the middle section than the ETD column.But in the top and bottom part the optimal column is more efficient than theETD column. In the optimal column the thermodynamic distance betweentwo adjacent trays is less than the ETD step (L/N) near the condenser andthe reboiler, whereas in the middle section it is larger thanL/N.

Heating and Cooling Duties

This is also the reason for the differences between the optimal and the ETDheating profiles. The figures show that in general the ETD column requireslarger condensers and reboilers than the optimal column but smaller heatexchangers on intermediate trays. Since squeezing large heat exchangers intodistillation trays has proved to be a difficult task, this may be a desirablefeature for some installations. Another thing to note is that the heat demandsfor the intermediate trays are fairly small and roughly the same for all thetrays. On closer examination of the heat demand for the 70 tray column, wenote that the demands near the feed and near the reboiler and condenser aresignificantly higher than the demands on the trays in between. This gives the

familiar (inverted-u)-u shape shown in figure 3.11 previously noted in otherstudies [21, 27, 35, 36]. A similar examination of the 15 tray column showsonly a slight tendency toward this behavior and keeps|Qn| constant for1< n < N. Nearly constant heat demand makes Riveros design for diabaticcolumns quite attractive [30]. His design employs two heat exchanger circuits:one above and one below feed. Each heat exchanger winds its way throughthe column and this arrangement can probably approximate the optimalheating/cooling profiles found here.

3.3 Summary

The entropy production associated with diabatic distillation is minimized us-ing Powells method. This allows a detailed comparison with results obatinedfrom the asymptotic ETD theory and conventional adiabatic distillationcolumns. The entropy production of the latter is also formulated as a min-imization problem. An optimal diabatic column offers enormous reductionof entropy production compared to its adiabatic counterpart. Some strik-ing values are listed in table 3.2. For columns with fewer trays, numerical


37/86

3.3. SUMMARY 37

0 10 20 30 40 50 60 70n []

2

1

0

1

2

Q[kWm

ole

1]

Figure 3.11: The famil-iar (inverted-u)-u shape ofthe optimal heating profile.Data shown are the sameas in the heat duty profileof figure 3.9 with end traysomitted.

optimization yields slightly better results than ETD. In this case there are sig-nificant deviations, particularly in the heating profiles. For longer columns,the optimal profile calls for a nearly constant heat demand, which works wellwith the Rivero implementation of diabatic columns [30, 32].

Table 3.2: Comparison of entropy production for conventional and optimizeddiabatic distillation columns.

column xD/xB Suconv S

uopt savings

15 trays 0.90/0.10 2.57 1.14 56%

25 trays 0.95/0.05 2.90 1.09 62%

70 trays 0.99/0.01 3.01 0.62 79%


38/86



39/86

Chapter 4

Heat Transfer Irreversibilities

The numerical study presented in Chapter 3 has focused on the entropyproduction insidethe column while relegating to the environment the entropyproduction associated with getting the requisite heat exchanges to happenat desired rates.

This chapter considers the question of how this heat exchange, when includedin the entropy production minimization, affects the optimal heating profilefor fully diabatic columns. In the absence of counting irreversibilities due to

heat exchange between the column and its surroundings, the optimal profileis independent of rate of operation in the sense that the optimal profile scalesdirectly with the feed rate. Usually the purities of the feed, distillate andbottoms are specified so that a given feed rate F determines the rate ofdistillate and bottoms production through

D = xF xB

xD xB F def

= d F (4.1)

B = xF xD

xB xDF def

= b F . (4.2)

Letting

qn=Qn/F (4.3)one can rewrite the entropy production (2.20) using the definitions ofd andbfrom equations (4.1) and (4.2)

Su,sep =F

sliqF + dsliqD + bsliqB

Nn=0

qnTn

. (4.4)

All the flows in the column are proportional to the feed rate F. This makesit natural that current studies [21, 25, 35, 42] all optimize the entropy pro-duction per feed rate. But the scaling behavior above is no longer the case

39


40/86

40 CHAPTER 4. HEAT TRANSFER IRREVERSIBILITIES

once the irreversibilities of the heat coupling to the outside of the column

are included.

To explore the effects of non-vanishing heat exchange which must in factproceed across a finite temperature difference and therefore produce entropy,we introduce two simple models for the heat transfer into the trays, theNewton and Fourier laws.

4.1 Entropy Production due to Heat

Exchange

The entropy production due to the heat exchange to the n-th tray is

Su,hxn =Qn

1

Tn 1

Texn

. (4.5)

For a given amount of heat transferred Qn, the required external temperatureTexn depends on our assumed heat transfer law. This law relates the heattransferred to the tray to the temperature Tn inside the tray and to thetemperatureTex in the heat exchange fluid outside the column. With our

choice of the internal temperature profile as the control parameter, it isconvenient to eliminate this dependence on the external temperatures bysolving for the external temperature in terms of the heat transferred and theinternal temperature and to use the resulting expression to eliminate Texn inequation (4.5). This is carried out below for both of our heat transfer laws.

Note that with this procedure the entire optimization, including the losses inthe heat exchangers, is still parametrized solely by the internal temperature.The external temperatures do not add a new degree of freedom but areconsequences of the internal profile.

4.1.1 Fourier Heat Conduction

The first model is Fouriers law of heat transfer. Here the transfer is takento be proportional to the difference of the inverse temperatures, i. e. to thethermodynamic force,

Qn=FL

1

Tn 1

Texn

, FL > 0. (4.6)


41/86

4.1. ENTROPY PRODUCTION DUE TO HEAT EXCHANGE 41

where FL is the conductance of the contact between tray n at temperature

Tn and a bath at temperature Texn .

Solving for 1/Texn in equation (4.6) one finds

1

Texn=

1

Tn Qn

FL, (4.7)

which is substituted into equation (4.5) to yield the very simply expression,independent of temperatures,

Su,hxn

= Q2n

FL. (4.8)

To make the feed rate dependence explicit, the quantity

gFL = F

FL, (4.9)

is introduced, which describes the relative rate of mass and heat flow. Onethen can express equation (4.8) with q=Q/F as

Su,hxn =F gFLq2n. (4.10)

Summing this over Ntrays, the total entropy production can be written as

Su = Su,sep + Su,hx (4.11)

= F


Nn=0

qnTn

+ gFLNn=0

q2n

. (4.12)

Here it is evident that only the internal temperatures Tn are needed toparametrize the full operation of the column. They appear only in the en-tropy production term due to the internal separation, not in the term from

the heat exchangers.

4.1.2 Newtonian Heat Conduction

In the second more conventional model, Newtonian heat conduction, the heattransfer is taken to be proportional to the difference of the temperatures,

Qn= NL(Texn Tn) , NL > 0, (4.13)


42/86


Proceeding in the same fashion as in the previous subsection by eliminating

Texn in equation (4.5) one finds1

Texn=

1

Tn+ QnNL

, (4.14)

and thus

Su,hxn = F2

NL

q2n

Tn

Tn+

F

NLqn

(4.15)

= F gNLq2n

Tn(Tn+ gNLqn)

, (4.16)

with gNL= F /NL.

This results in a total entropy production of

Su = Su,sep + Su,hx (4.17)

= F


Nn=0

qnTn

+ gNLNn=0

q2nTn(Tn+ gNLqn)

.

(4.18)

Note that for qn < 0 (heat being withdrawn from the column), the valueofgNL is limited to be in the range between 0 and minn

(Tn/qn). The limit

gNL = 0 corresponds to vanishing feed rate or infinitely fast heat exchangewhile the limit gNL= min

n(Tn/qn) corresponds to the highest possible rate of

heat removal for which Texn = 0.

4.2 Numerical Results

As model systems, three columns of different length (25, 45 and 65 trays) are

chosen to separate an ideal 50/50 benzene/toluene mixture. The requiredpurity are 95% for the distillate and 5% for the bottoms, respectively.

Powells routine (see appendix A) is applied to minimize the total entropyproductions (4.12) and (4.18). Like in chapter 3, random temperature profilesare used as initial guesses for the optimizations. It was observed that theconvergence of the algorithm is slower for larger g-values. For large g-valuesthe

q2nterm starts to dominate the behavior of equations (4.12) and (4.18).

In this case the objective function converges rather slow, like the conventionalcolumn (3.7) which also contains a

q2n term.


43/86

4.2. NUMERICAL RESULTS 43

4.2.1 Influence of Heat Transfer Law

The optimal temperature profiles and corresponding heating requirementsare determined for a 25-tray column with Fourier heat conduction (figures4.1 and 4.2) and Newtonian heat conduction (figures 4.3 and 4.4). The threecurves in each frame are calculated for g= 0 and for two increasingly severetransfer resistances. The value of zero represents perfect heat conduction andis therefore identical to the previous studies of optimal distillation consideringonly internal losses. The intermediate value corresponds to realistic heatconductance in a commercial heat exchanger, while the largest value ofg isincluded to show the strongly resistive regime. The values ofgNL andgFLare

chosen to correspond to roughly the same temperature differences across theconductances and thus differ by a factor of 1 .4 105 K2 due to the forms ofthe two transfer equations (4.13) and (4.6). It is clear that the form of thetransfer law has very little effect on the optimal temperature sequence andhence the heating demand. This speaks in favor of using the simpler Fourierexpression (4.12) for the entropy production where temperature appears onlyin the heating terms, not in connection with the heat exchangers.

A further comparison between the two heat transfer laws is provided byfigure 4.5, a log-log plot of the total entropy production relative to reversibleheat transfer for a variation of gNL and gFL over 6 orders of magnitude.

The realistic value of gNL = 3 104 moleK/J also used in figures 4.14.4and the corresponding gFL are marked on the curves. The two curves areremarkably similar. We see that heat resistance is insignificant for g 103 with normal operation right in the middle ofthe transition interval. The type of heat transfer (Newtonian or Fourier) isimmaterial.

4.2.2 Influence of Different Column Length

Figure 4.6, 4.8 and 4.10 show the optimal heating demands for columns with25, 45 and 65 trays using Fourier heat conduction. As indicated in the pre-vious subsection the corresponding Newtonian results are indistinguishableand are therefore not shown here. The shorter column displays the character-istic (inverted-u)-u shape of the heating curves observed in the last chapterfor near reversible heat exchange. This is quickly smoothed out with increas-ing heat resistance where the internal losses, relatively speaking, become lesssignificant.

The longest column by contrast, even for very slight heat resistance is oper-


44/86


0 5 10 15 20 25

n []

350

360

370

380

390

T

[K]

gFL= 0

gFL= 2.1*109

gFL= 7*108

Figure 4.1: Optimal temperature profiles for a 25 tray column with a Fourierheat law connecting the heat exchangers to the supply fluids. The relativeflow parameter gFL is as indicated on the figure.

0 5 10 15 20 25

n []

10

5

0

5

10

qn

[kWm

ole

1]

gFL= 0

gFL= 2.1*109

gFL= 7*108

Figure 4.2: Corresponding optimal heating profiles


45/86


0 5 10 15 20 25

n []

350

360

370

380

390

T

[K]

gNL= 0

gNL= 3*104

gNL= 102

Figure 4.3: Optimal temperature profiles for a 25 tray column with a Newtonheat law connecting the heat exchangers to the supply fluids. The relativeflow parameter gNL is as indicated on the figure.

0 5 10 15 20 25

n []

10

5

0

5

10

qn

[kWm

ole

1]

gNL= 0

gNL= 3*104

gNL= 102

Figure 4.4: Corresponding optimal heating profiles


46/86


8 7 6 5 4 3 2

log(gNL

) or log(gFL

*1.4*105)

0

0.4

0.8

1.2

1.6

2

log[Su(

gNL,F

L)/Su(0

)]

Fouriers lawNewtons law

Figure 4.5: Log-log plot of the total entropy production relative to reversibleheat transfer for a variation ofgNL andgFL over 6 orders of magnitude. Thepoint marked corresponds to industrially realistic values of heat conductance.

ated optimally with an essentially flat heating profile; the same rate of heattransfer on every tray in each section of the column. This resistive effectis quite surprising, quickly washing out the structure of the optimal uncon-strained column. It promises well for the simplified diabatic column designadvocated by Rivero [30, 32] in which the external heat exchange medium ispassed in sequence through heat exchangers on one tray after another in thestripping and rectifying sections, respectively. This design obviously allevi-ates the need for costly independent heat supply circuits for each tray. Atthe same time all heating curves show a marked reduction in heat duty forthe reboilers and condensers, making it possible to include them inside thecolumn rather than as separate exterior units.

The corresponding curves for the entropy production on each tray are evenmore illuminating for the effect of heat resistance. The entropy productiondue to heat and mass transfer on the trays alone is by no means flat, rather itfollows the general shape originally developed in interior optimization. Theentropy production due to heat exchange, is not flat either, but togetherthe internal and external entropy productions add up to an almost constantentropy production on each tray, a kind of equipartition often claimed tobe a good design principle [39]. Optimal heat exchange on the feed tray is


47/86


0 5 10 15 20 25

n []

10

5

0

5

10

qn

[kWm

ole

1]

gFL= 0

gFL= 2.1*109

gFL= 7*108

Figure 4.6: Optimal heat duties for the individual trays in a 25 tray columnwith a Fourier heat transfer. The relative flow parameter gFL is as indicatedon the figure.

0 5 10 15 20 25

n []

0

0.02

0.04

0.06

0.08

0.1

Su[

WK

1

mole

1]

Su separation

Su heat exchange

Su total

Figure 4.7: Optimal distribution of entropy production on the individualtrays for a 25 tray column. Here gFL = 2.1 109 mole/JK (industriallyrealistic value).


48/86


0 5 10 15 20 25 30 35 40 45

n []

4

2

0

2

4

6

qn

[kWm

ole

1]

gFL= 0

gFL= 2.1*109

gFL= 7*108

Figure 4.8: Optimal heat duties for the individual trays in a 45 tray columnwith a Fourier heat transfer. The relative flow parameter gNL is as indicatedon the figure.

0 5 10 15 20 25 30 35 40 45

n []

0

0.01

0.02

0.03

0.04

Su[

WK

1

mole

1]

Su separation

Su heat exchange

Su total



49/86


0 10 20 30 40 50 60 70

n []

4

2

0

2

4

qn

[kWm

ole

1]

gFL= 0

gFL= 2.1*109

gFL= 7*108

Figure 4.10: Optimal heat duties for the individual trays in a 65 tray columnwith a Fourier heat transfer. The relative flow parameter gNL is as indicatedon the figure.

0 10 20 30 40 50 60 70

n []

0

0.005

0.01

0.015

0.02

Su[

WK

1

mole

1]

Su separation

Su heat exchange

Su total



50/86


25 45 65

N

2

4

6

8

SuW

K1mole1

A B

C

D

E

Figure 4.12: Entropy production in adiabatic and diabatic columns of length25, 45 and 65 trays. The leftmost histogram (A) shows the ultimate lowerlimit for a diabatic column with g = 0. The center histogram shows theentropy production due to the internal heat and mass flows (B), the heatexchange (C) for a diabatic column with gFL = 2.1109 mole/JK. Theright histogram shows the separation (D) and heat exchanger losses (E) for acorresponding conventional adiabatic column.

usually quite small and can be made zero by appropriately balancing of the

vapor/liquid composition in the feed.

The entropy savings by using a diabatic column compared to a conventionaladiabatic one are evident from figure 4.12. Even for the shortest column of25 trays, the adiabatic entropy production (including heat exchanger losses)is four times as large as the losses in the diabatic column, with the heatexchangers accounting for about two thirds of the losses. In the diabaticcolumn, the contribution of heat exchange varies from about half for theshortest column to about two thirds for the longest column. To indicate theultimate lower limit of entropy production without regard to heat exchangelosses, the curve for g= 0 is shown as the leftmost histogram.

4.3 Summary

In the absence of counting heat transfer irreversibilities due to heat exchangebetween column and its surroundings, the optimal heating profile is indepen-dent of rate of operation in the sense that the optimal heating profile scalesdirectly with the feed rate. This is no longer the case once the irreversibilities


51/86

4.3. SUMMARY 51

of the heat coupling to the outside of the column are included. For slow oper-

ation (smallg-values) the solution is essentially unchanged from the solutionfor the problem without irreversibilities due to the heat transfer. For fastoperation (largeg-values) the entropy production inside the column becomesnegligible compared to the irreversibilities due to the heat exchange whichthen dominate the behavior of the optimum. The optimal heating profilessmooth out with increasing heat resistance. Particularly, long columns areoptimally operated with essentially flat heating profiles.

For industrially relevant g-values, the entropy production of an adiabaticcolumn (including heat exchanger losses) is several times larger than theentropy production in the optimized diabatic column. The contribution of

the heat exchange to the entropy production increases with column length.The results from chapter 3 as well as the extended study including heattransfer irreversibilities in this chapter reveal that the simplified diabaticcolumn design suggested by Rivero [30] is a promising attempt to implementdiabatic distillation columns.

The results of this chapter have been published in [48].


52/86



53/86

Chapter 5

Sequential Heat Exchangers

In the previous two chapters we found the minimum irreversibility for di-abatic distillation cloumns, where each tray was supplied by independentlyadjustableheat exchangers. However, such a design could be difficult to im-plement in practise. For each tray, two piercings in the column containmentare required, and each tray needs an individual heat exchange circuit. Theseextra investment costs may outbalance the savings achieved by reducing theexergy losses.

Rivero [30] proposed a simplified diabatic column design which is particularsuitable for retrofitting applications and therefore more promising than a di-abatic column with independently controlled tray temperatures. The Riveroimplementation uses a single heating fluid circulating in series from one trayto the next below the feed tray and a single cooling fluid circulation in seriesabove the feed tray, as shown in figure 5.1.

Such a sequential heat exchanger design has only four control variables:

1. The temperature of the sequential heat exchanger fluid entering therectifier.

2. The flow rate of the sequential heat exchanger fluid in the rectifier.

3. The temperature of the sequential heat exchanger fluid entering thestripper.

4. The flow rate of the sequential heat exchanger fluid in the rectifier.

The temperature profile of the distillation column is determined by thesefour parameters.

53


54/86

54 CHAPTER 5. SEQUENTIAL HEAT EXCHANGERS

Stripper

Rectifier

Condenser

Reboiler

QD

D, xD

B, xB

QB

F, xF

Figure 5.1: Conventional column retrofitted with sequential heat exchangersas proposed by Rivero. The sequential heat exchanger fluid flow is indicatedby the dashed lines.

In this chapter, numerical optimization is used to determine the optimaloperation of a diabatic design with sequential heat exchangers. In particular,we focus on how much extra irreversibility one has to pay for the loss ofindependent control of the external temperatures.

5.1 Heat Exchanger Model

Consider first a model of a heat exchanger on a single tray (see figure 5.2),where the heat exchanger model is taken from standard thermal engineeringliterature, e.g. [14, 15]. Let m be the flow rate of the heat exchanger fluid,Tinex,1 and T

outex,1 be the temperatures of the heat transfer fluid coming in and


55/86

5.1. HEAT EXCHANGER MODEL 55

mT1

UAToutex,1T

inex,1

m

0

Figure 5.2: Model of a single heat exchanger on one tray. The spatial pa-rameter runs from 0 to .

going out of the tray, T1 be the temperature of the tray, cp be the specific

heat capacity of the heat transfer fluid, be the length of the contact area ofthe heat exchanger, be the position along the heat exchanger, and U Abethe product of the heat exchanger area A and the heat conductivityUgivingthe total conductance of the heat exchanger unit. For convenience, the flowrate m, and thermal properties cp and UA are assumed to be temperatureindependent.

The heat flow in a small portion d is then

mcpdTex,1 = UA

(Tex,1 T1)d. (5.1)

Separating variables and integrating along the length of the heat exchangergives Tin

ex,1

Toutex,1

dTex,1(Tex,1 T1)=

UA

mcp

0

d. (5.2)

Solving the resulting equation results in

Toutex,1 =Tinex,1+ (T1 Tinex,1)

1 exp

UA

mcp

. (5.3)

The rate of heat transferred in the heat exchanger is given by

Qhx1 = mcp(Tinex,1 T1)

1 exp

UA

mcp

, (5.4)

and the entropy production associated with the heat transfer is calculated as

Su,hx1 = mcp ln

Toutex,1Tinex,1

+

Q1T1

. (5.5)

Figure 5.3 shows a schematic representation of a serial heat exchanger mean-dering throughn trays. Using equation (5.3) and knowing the temperatures


56/86


of the trays and the flow rate of the heat exchange fluid entering the sequen-

tial heat exchangers, the amount of heat transferred can be calculated ateach tray along the heat exchanger.

m

UA

m mT1

UA

m m

UA

m

T2

mTi

UA

m

Tinex,1 T

out

ex,1

Toutex,2 T

in

ex,2

Tinex,i T

out

ex,i

Tn

Toutex,n T

in

ex,n

Figure 5.3: Serial heat exchanger meandering through tray 1 to trayn of thedistillation column


57/86

5.2. COMPUTATIONAL METHOD 57

5.2 Computational Method

The heating stream of the sequential heat exchanger design enters the strip-per section of the column at tray N 1 with a mass flow rate mB and atemperature ofTinex,B, and leaves the column one tray below the feed tray ata temperatureToutex,B. The cooling stream enters the rectifier section at tray 1with a mass flow rate mD and a temperature ofT

inex,D, and leaves the column

one tray above the feed tray at a temperature Toutex,D. Feed tray, reboiler, andcondenser are not supplied by the seqential heat exchanger fluid.

Our aim is to find the optimal heat exchanger fluid parameters ( mB, mD,

T

in

ex,B, T

in

ex,D) to achieve minimal entropy production.

5.2.1 Entropy Production and Steady State Condition

With equation (2.21) for the entropy production associated with the sepa-ration process inside the column, and summing up expression (5.5), over alltrays supplied by the heat exchanger fluid, the overall entropy productionfor the sequential heat exchanger design is

Su

( mB, mD, Tin

ex,B, Tin

ex,D) = mB cp,B lnToutex,B

Tinex,B

+ mD cp,DlnToutex,D

Tinex,D

+ SB + SD + Smassflows. (5.6)

Here SB and SD denote the entropies entering reboiler and condenser,respectively. Equation (5.6) holds for steady state condition, when the col-umn temperature profile is such that the heat duty (2.8) on each tray n ofthe column matches the heat (5.4) delivered by the heat exchanger fluid onthat tray,

Qn =Qexn, n= (2, . . . , N

1). (5.7)

This steady state condition has to enter the optimization problem as anadditional constraint for the tray temperatures. In principle, the constraintcan be added to the entropy production (5.6) as penalty function of the formP=Mn(Qn Qexn)2, where M is a large constant.Unfortunately, standard minimization methods applied to this penalized ob-jective function fail to find a minimum as the procedure invariably gets stuck.This is due in part to the fact that this is a simulation where only narrowregions of parameter values make physical sense. It is further complicated


58/86


by numerous local minima and by the high sensitivity of the entropy produc-

tion on the temperature profile. Therefore a different approach based on analgorithm using a pseudo-dynamic is developed for finding the steady statetemperature profile.

5.2.2 Pseudo-dynamic Algorithm

In each iteration of the minimization procedure with respect to the controlvector v = ( mB, mD, T

inex,B, T

inex,D), we have to find the corresponding steady

state temperature profile for the current value of v. Since the operatingcharacteristics of the column are conveniently expressed in terms of its tem-perature profileT, it is natural to start the search for the steady state profileby guessing an initial temperature profile.

We define the vector of deviation from the steady state condition (5.7) by

Qexcess(T, v) :=Qex(T, v) Q(T, v). (5.8)

Further we evaluate the error expression

= Qexcess Qexcess , (5.9)

Now the temperature of each tray is changed by a small multiple of theexcess heat on that tray according to a discrete pseudo-dynamics,

T(t + 1) = T(t) + Qexcess(t), (5.10)

where t denotes the time iteration.

Starting with the initially guessed temperature profile, (5.8)-(5.10) are eval-uated iteratively. The error evaluation allows us to adjust the size of ineach ireration. If the error of one iteration is bigger than the one of theprevious iteration, ist reduced by an empirical factor, whereas is slightlyincreased ifbecomes smaller.

This heuristic scheme converges to the steady state temperature profile forthe current value v in the main optimization routine.

5.3 Optimization Results

The examples presented here use a conductanceU A= 500 W/K on each trayand ten times this value for the reboiler and the condenser. The specific heat


59/86


capacity of the heat exchange fluid used is the same for both the rectifier and

stripper section, and is taken to be cp = 300J(Kmole)1. The calculationscompare a 20 tray diabatic column and a 20 tray adiabatic column, bothseparating an ideal 50/50% benzene/toluene mixture to yield a purity of95% distillate and 5% bottoms. The results discussed include

the heating and cooling duties the tray-by-tray entropy production liquid and vapor flow rates

the temperature profiles the sensitivity of entropy production as the heat transfer fluid inlet

temperatures deviates from their optimal values

the amount of exergy savings compared to an adiabatic column and the minimum entropy production as a function of the total number

of trays

5.3.1 Operation Profiles

The left graph in figure 5.4 shows the heating and cooling duties for a se-quential heat exchanger column and its conventional counterpart. It turnsout that since the sequential heat exchangers reduce the duty that the re-boiler and condenser must perform, a diabatic column with these parametersrequires a reboiler that is only half the size of a conventional column and acondenser 2/3 the size of a conventional column. As already found in ourprevious studies, the heating and cooling loads are essentially constant overthe trays.

Figure 5.4 shows that the total entropy production rate is significantly lower

in the diabatic column and that it is more uniformly distributed; only thefeedtray deviates noticeably. This indicates that all the trays carry more orless the same burden of separation, quite distinct from the performance inthe adiabatic column. When comparing the entropy production of the con-densers (tray 0), it is observed that the entropy production in the adiabaticcolumns condenser is approximately twice as large as the corresponding en-tropy production in the diabatic column. Similarly, the entropy productionin the adiabatic columns reboiler (tray 20) is four times larger than theentropy production of the diabatic columns reboiler.


60/86


0 5 10 15 20n [-]

-40

-30

-20

-10

0

10

20

30

40

Q[kWm

ole-1]

diabatic: intermediate trays

diabatic: condenser/reboiler

conventional: condenser/reboiler

0 5 10 15 20n [-]

0

0.5

1

1.5

2

2.5

Su [WK

-1m

ole-1]

conventional column

diabatic column

Figure 5.4: Heating and cooling duties (left) and entropy production per tray(right) for a 20 tray sequential heat exchanger column.

Except for the feed tray, the diabatic column flow rates are noticeably lowerthan those of the adiabatic column as seen in figure 5.5. The significance oflower flow rates is that the column allows cross sectional area for the serialheat exchangers to be installed without interfering with the material flowsin the column, implying that a conventional column can be retrofitted withthe serial heat exchangers without adverse effects.

In figure 5.6 the middle solid line of the diabatic column chart shows thetemperature of each tray in the column; the line below the middle line is the

temperature profile for the sequential heat exchangers in the rectifier, whilethe one above is the temperature profile for the sequential heat exchanger inthe stripper. As for all optimized diabatic columns, the tray temperaturesvary almost linearly along the column, indicating that each tray is doingits share of the separation process. The heat exchanger temperatures areroughly a constant difference away from the tray temperatures, again in-dicating a uniform dissipation. By contrast the adiabatic column has verylittle temperature change across trays 7 through 12. Thus the serial heatexchangers help the diabatic column to utilize the trays better.


61/86


0 5 10 15 20n [-]

0

0.5

1

1.5

2

L

[-]and

V[

-]

L

V

diabatic

0 5 10 15 20n [-]

0

0.5

1

1.5

2

L

[-]and

V[

-]

L

V

adiabatic

Figure 5.5: Liquid and vapor flow profiles for a diabatic column with sequen-tial heat exchangers and a adiabatic reference column.

0 5 10 15 20n [-]

340

350

360

370

380

390

T

[

numerically optimized diabatic distillation columns

Documents