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A method for order reduction of rigorous dynamic models of distillation columns is presented. The approach is a modification and extension of the compartmental models presented in 1986 by Benallou et al. Formulating the algebraic equations and the differential equations for total holdup for the compartments, but keeping the differential equations for concentration dynamics for each tray, Murphree trayefficiencies can be used and response accuracy increased. The resulting models of substantially lower order show good dynamic and steady-state conformity with the complete rigorous model. The example of an industrial binarydistillation column willdemonstrate the effectiveness of this approach. KEYWORDS

Distillation columns; order reduction; compartmental models; dynamic simulation; differential-algebraic equations SCOPE

A typical rigorous model of a singletrayof a distillation column (Table 1)consists of equations describing holdups (concentrations), vapor flow (energy balance), liquid flow, bubble point (tray temperature), pressuredrop and vapor/liquid relationships (Gani et al., 1986; Rovaglio et al., 1990). Considering the dimensions of many industrial columns (often more than 50 trays), a rigorous simulation of the dynamic behaviour requires the simultaneous solution of a system of several hundred equations. The solutionof these differential-algebraic equations (DAE) with specially adapted integrators is usually successful for index 1 problems'. However, the integration of such large systems consumes a lot of computer time. A reduction of modelorder is thus verydesirable. Thispaperdescribes the development of reduced-order dynamic models for distillation columns, basedon a compartmental approach, withthe following properties: small dynamic and steady-state errorscompared to the complete rigorous model, including flow dynamics, andapplicability to any distillation column.

1. The differential index rn of the system F (t, Y (t), y' (t)) = 0 is the rninirnalnumber rnsuch that the system of F (t, y (t) ,y' (t)) = 0 and of the analytical differentiations dF (t, y (t), y' (t)) d'" F (I, Y (t). y' (I) ) dt

= 0, ".,

df'

= 0

can be transformed byalgebraic manipulations into anexplicit ordinary differential system (Hairer et al., 1991) 5311

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Table 1:Equations for a single trayj of the complete rigorous model fora distillation column NC material balances (k = I, ..., NC; j = 0, .... NT+1) dnk ·

d/

=

d(n.x t ·)

~t

,J

= FlF,k,j+Lj_1xt,j_l- (Lj+SI,j) xk,j+ (Vj+1-S.,j+l)Yk,j+l- VjYk,j

(1)

One energy balance 0= Fjh'F, j+Lj_1h'j_l - (Lj+SI)h'j+ (Vj+I-S.,j+l)h" i- s: Vjh " j+ Qr ~ (nF)

(2)

One bubble point equation NC NC ~ y~~i1ibrium -1 = ~ K• .(x. , T ., P,)xk .- 1 = 0 k ,J k -,J -,/ J J ,J

=1 Two fluid dynamic relationships k

k

=!(nl' Vj + l _dPj _ 1' geometry)

(4)

= Pj_ 1 + dPj_ 1 (n j _ 1' VI' geometry)

(5)

Lj Pj

(3)

=I

Vapor/liquid relationships (6)

ORDER REDUCTION BY IDEALIZING ASSUMPTIONS

Thecomplete rigorous model (Table 1)requires thesolution of a system of differential equations andalgebraicequations of high order. Toavoid mostof the algebraic equations, one or more of the following idealizing assumptions are frequently usedin theliterature to simplify the dynamic models: • constant relative volatility of substances • equal heat of evaporation of substances • feeds and reflux at bubble point • constant pressure drop • constant total holdup on eachtray(equimolal overflow) These assumptions imply constant vapor and liquid flows in each section of a distillation column. This approach of model reduction allows the formulation of dynamic models of distillation columns of low order, but contains severe limitations. The first assumptions require substance mixtures exhibiting ideal behaviour. Neglecting energy balances, small differences in heat of evaporation can cause large errorsin composition profile. The assumptions of bubble point temperature for feeds and reflux as well as the assumption of constant pressure drop are often not permissible. Assuming constant total holdup implies the neglect of flow dynamics. However, the important roleof flow dynamics in the dynamic behaviour of distillation columns has beenrecognized overthepastseveral years (Haggblom, 1991). It is illustrated in Fig. 1 by the simulation withthe complete rigorous model of a stepresponse to a 10% increase in reboiler heatsupply for an industrial binary distillation column, equipped with50 sieve trays. A model neglecting flow dynamics showslarge errors in dynamic behaviour. Therefore models formed by the mentioned idealizing assumptions cannot meet the specifications wanted. Better results are obtained by the reduced model developed in the following section. COMPARTMENTAL MODELS

Oneof the mostpromising approaches, proposed e.g. byBenallou et al. in 1986 usescompartmental models. Due to the factthatdifferences of flows andtemperatures between adjoining traysarerelatively small, several trays are bound together into a compartment Special material balances, energy balance, bubble pointequation and fluid dynamic relationships are developed for these compartments, but only for a special tray (sensitive tray) in thatcompartment, withthe assumption of uniform tray temperature and flows within. Concentrations offiowsleaving thecompartment areexpressed bysteady-state relationships called separation functions. These compartmental models proposed havetwodisadvantages, however:

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European Symposium on Computer Aided Process Engineering-2

0.050 , \

_ _ _ _ Completerigorous model

,

i \

\.\i

0.040

1 g

0.030

- _ _ _ _ _ _ Reduced model

" \, '\ \

,

_ __. _ . _ . _ . _ . Rigorous model with equimclal overflow

\,

u

.,

\,

0.020

\

" "

0.010

0,." <,

o.oooL_--'---_-'-_~_--=L.:.::=:::~~======'l

o

2

4

Time [hj

Fig. 1: Stepresponseof concentration on tray 47 to a 10% increase of reboilerbeat supply

• With regard to tray efficiency, Benallou (19H2) usesthe formula Yk . .j

= Ek.rr. .Kk r« ,xk ).

(7)

to calculate vaporconcentrations withthe stage vaporization efficiency E. In the separation functions, it is not feasible to replace the stage vaporization efficiency E by the frequently used Murphree tray efficiency 11 without solving a nonlinear equation system. The computational effort necessary wouldthen be comparable to that of solving an additional differential equation system of the sameorder. • By use of steady-state relationships for flow composition leaving the compartment, an inverse response occurs frequently where none is present in the rigorous full-order model (Horton et al., 1991). That effect must be minimized by a careful definition of compartment positions. Modification and Extension of Benallou's Compartmental Models

We suggestto avoidthese disadvantages by keeping NC-l differential equations for each tray describing holdup of substances. yet formulating only one differential equation for total holdup in the compartment. Thereby it is possible to use Murphree trayefficiency as in the complete rigorous model. The problem of a wrong inverse response vanishes as well. Nevertheless some differential equations can be avoided for eachcompartment. At a higherlevelof accuracy. thecomputational effort is equivalent to the introduction of Murphree trayefficiencies into separation functions. The assumption that the temperature profile between reference trays is linearallows the further improvement that any tray temperatures may be used with small error and without a new selection of reference trays. While keeping the fundamental concept of compartments. this method can be characterized by the following assumptions: • the total holdup is the sameon alI trays in a compartment • the vapor flows and liquid flows within a compartment are uniform • the temperature profile between adjoining reference trays is linear • the pressure dropover each trayin a compartment is equal to the pressure dropover the sensitive trayof thatcompartment Withtheseassumptions. the reduced model consists of theequations listedin Table 2. While NC-I differential equations for each tray must be solved, the equations for total holdup. bubble point, vapor flow,

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liquidflow and pressure drophave to be calculated onlyonceper compartment ratherthan for everytray. For this method to be applied, a distillation column must be divided into compartments with a sensitive tray (Fig. 2a). Since the reduced model proposed includes flow dynamics and energy balances, feed trays andtrays with sidestreams mustbe handled as "one-tray compartments" (Fig. 2b). VetYk,q

Le-1' Xk,q-l

FI" XF,k,r

r5

Sl,r Sv,r+1

_t_ L~,

Ve, Yk,r

Le-l, Xk,r-l

-q-

L~. xk,r

Ve+1' Yk,r+l

V~+l' Yk,I+1

Xk,l

b)

a)

Fig, 2: Compartmental representation of a columnsection

The relative accuracy of the reduced model as compared to that of the complete rigorous model will be determined primarily by the number of compartments. A special source of errorsis the algebraic equation system: The concentration dynamics responds sensitively to errors in internal flows, Theseerrors can be Table 2: Equations for the reduced-order model of a distillation column l. Equationsfor tray j (j = 0, ... , NT+1) NC-I materialbalances (k = I. ,.., NC-1) dn k

,

_,J

dt

= FX .)X ,J (V.J + l-SV,)'+l)Yk.J'+l-VYk' F.kJ,+L'_IX k,J'-1- (L,-S/ J, J J ,J k,+ J.J

(8)

Vapor/liquid relationship

(9)

Yk,j = T1Kk.l k,j+ (l-T1)Yk,j+l

II. Equationsfor compartmentSwith reference tray 1'; One material balancefor totalholdup dn~

dt

= F~+L~_l-L~+ V~+l- V~-S/.~-Sv.~+l

(10)

One energybalance

o = F~h'F ~+L~_lh'r ~-, - (L~+S/ ~)h'r~ + (V~+l-Sv ~+l)h" r~+l" -V~h" r~ +Q~-!!..(n~h·r) "dt " ~ ..

.....

"

."

"

(11)

."

One bubble point equation NC ~

NC

K 1 -L -

L. Ykequilibrium r .~ k=l

(

k,~ r Xk r' ,~

T r' P) r~ Xk ,~ r - 1 - 0 ~

(12)

k=l

Twohydrodynamic relationships Lr~ AP~

= L~ = f(n~, V~, APrG, geometry)

= (Numberof traysin compartment) * AP rG(n~,

(13) V~,

geometry)

(14)

Linear interpolation of tray temperatures between sensitive trays

t, -t; T j

=

G

G+'U'-r)+T

r~ - r~ + 1

~

rG

(15)

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minimized through careful selection of compartments andreference trays. Large gradients in internalflows causedby different heat of evaporation have to be expected in areas with large concentration gradients. Therefore in sections with large mass transfer (i.e. sharp concentration profiles), a selection of smaller compartments is preferable. NUMERICAL TREATMENT AND RESULTS

The equations for both the complete rigorous model and the reduced model constitute differential-algebraic equations (DAE). Due to the specific properties of these DAE (e.g. extremely stiff behaviour) the integration ofsuch systems requires specially adapted, implicit integration methods (Brenan et al. , 1989; Harrer et al., 1991). The application of the methods developed during the last decade (e.g. DASSL, LSODI, RADAU5) is usuallysuccessful forindex1 problems. Inorderto simplify the numerical treatment of the model, we propose also to neglect the differential termd(njh' j)/dt in the energybalanceof the models. The error introduced by this simplification is small sincethe energy transport by vapor flow is much largerthan the differential term. In a comparison of different integrators, the best results were obtained using the DASSL integrator containedin the NAGFortranlibrary. Table 3 shows a comparison between a reduced (21 compartments) and a full-order model (50 trays) for computing a step response to a 10% decrease in feed composition. ComTable 3: Calculation efforts for full-order and reduced-order model Full-order model Reduced-order model Number ofequations 317 159 Number ofJacobian evaluations 21 18 Number of steps 1086 339 CPU time (s) 2130 188 Integratorsettings (DASSL): Scalar error control withmaximum norm, RTOL=1O·6, ATOL= 10'8, numerical evaluation ofJacobian matrix, full matrix algebra Computer: SUNSPARC station 1+ (RISe architecture) positions were controlled by a (DIV, V) control configuration (Ryskamp, 1987), with temperature measurement points on trays 8 and 44 and using identical controller settings (Fig. 3). A comparison of the numerical resultsobtained with the twomodels shows the gooddynamic and steady-state agreement. Specialsmallereffectssuchasoscillation of bottom product impurity cannot be reproduced sincethe compart-

- - - - Complete rigoro us model

0.018

- - - - - - - Reduced mode l

" " bottom product

/

top prod uct

Time Ih]

Fig. 3: Step response to a 10% decreasein feed composition

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mental approach smooths hydrodynamics. Thissmall disadvantage is acceptable, however, in lightof the significant reduction of computer timeandconsidering theerrors introduced by idealising assumptions. CONCLUSIONS

The proposed method allows the formulation of dynamic models for distillation columns whose orderis significantly reduced compared to thoseof complete rigorous models. Through choice of numberof compartments, an easy adaptation to any distillation column and any accuracy is possible. The comparison between the complete andreduced models shows thatthereduced models reproduce allimportant dynamic effects without introducing any wrong dynamics. Dynamic and staticerrorby model reduction is small. Since all concentrations andpressures arecalculated andtraytemperatures interpolated, no restriction for control designcanbe observed. Thelarge reduction of computer timeespecially makes this method a powerful,time-and-money-saving tool. REFERENCES

Benallou, A. (1982), Dynamic Modelling andBilinear Control Strategies for Distillation Columns, Dissertation, University of California SantaBarbara Benallou, A., D. E. Seborg andD.A.Mellichamp (1986), Dynamic Compartmental Models for Separation Processes, AIChEJ., 'J2:., 1067-1078 Brenan, K. E., S. L. Campbell and L. R Petzold (1989), Numerical solution Ofinitial-value problems in differential-algebraic equations, North-Holland, New York Gani, R, C. A. Ruiz and I. T. Cameron (1986), A Generalized Model for Distillation Columns-I, Compo Chem. Eng., ro, 181-198 Haggblom, K. E. (1991), Modeling of FlowDynamics for Control of Distillation Columns, Proceedings of the 1991 American Control Conference, pp. 785-790, Boston Hairer, E. and G. Wanner (1991), Solving Ordinary Differential Equations 11, Stiff and Differential-AlgebraicProblems, Springer-Verlag, Berlin Horton, R R, B. W. Bequette and T. F. Edgar (1991), Improvements in Dynamic Compartmental Modeling for Distillation, Compo Chem. Eng.,.ti, 197-201 Rovaglio, M., E. Ranzi, G. Biardi andT.Faravelli (1990), Rigorous Dynamics andControl of Continuous Distillation Systems - Simulation andExperimental Results, Compo Chem. Eng., 14, 871-887 Ryskamp, C. J. (1987), Dual Composition Column Control. In: Handbook of Advanced Process Control Systems and Instrumentation (LesKane, 00.), pp. 158-168, GulfPublishing Company, Houston NOTATION E F h' h" K L

n

NC NT P

Q r

s

stage vaporization efficiency flow of feed specific enthalpy of liquid phase specific enthalpy of vapor phase equilibrium constant liquidflow holdup number of components in thecolumn number of trays in thecolumn pressure heat flow number of thesensitive tray of the compartment flow of sidestream

T

traytemperature

t

time

vapor flow liquid phase composition y vapor phase composition Greek letters 11 Murphree trayefficiency ~ comparlffientnumber V x

Subscripts F j

k 1 v

feed number of tray number of component liquid vapor

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