Role of energy balances in dynamic simulation of multicomponent distillation columns

Role of energy balances in dynamic simulation of multicomponent distillation columns

Comjwr. them. Engng. Vol. 12, No. 8, pp. X3-786, Printed in Great Britain. All rights reserved 1988 00!%1354/88 Copyright 0 1988 $3.00 + 0.00 Pe...

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Comjwr. them. Engng. Vol. 12, No. 8, pp. X3-786, Printed in Great Britain. All rights reserved

1988

00!%1354/88 Copyright

0

1988

$3.00 + 0.00

Pergamon

Press

plc

ROLE OF ENERGY BALANCES IN DYNAMIC SIMULATION OF MULTICOMPONENT DISTILLATION COLUMNS E. ~NZI, Dipartimento

M.

ROVAGLIO,

T.

FARAVELLI

and G.

di Chimica Industriale c lngegneria Chimica, Polite&co da Vinci 32, I-20133 Milano, Italy (Received

5 Nozlrmber

1987; received for publication

BIARDI

di Milano, Piazza Leonardo

16 December

1987)

Abstract-The role of energy balances in simulating the transient behaviour of multicomponent distillation columns is analyzed and discussed in this note. Despite the usual approaches referred to in the recent literature, a few examples show how big discrepancies can lx observed as a result of neglecting the time derivative of the energy holdups. In the meantime, a few comparisons are presented in order to show the possible important benefits related to a simultaneous solution of the whole system of algebraic and differential equations.

INTRODUCTION

There is a general agreement in recent literature (Tyreus et al., 1975; Boston et al., 1980; Gallun and Holland, 1982; Feng et al., 1984; Ranzi et al., 1985; Gani et al., 1986; Cuille and Reklaitis, 1986) that implicit integrators, specifically involving backward differentiation formulas, are highly effective and reliable for the dynamic simulation of distillation columns. On the contrary, there is disagreement and very little knowledge about the effects of the simplifying assumptions made in attempting to solve the whole set of differential algebraic equations. The goal of this note is to show the practical importance of the time derivatives of enthalpy, in the intermediate stages of a distillation column as well as in the reboiler and the condenser. In this concern, the low-order difference approximation employed by Boston er al. (1980) in the BATCHFRAC paykage and the simplified version of the more general model proposed by Gani et al. (1986) do not seem completely satisfactory when using a rigorous model for the simulation of the transient behaviour of distillation columns. Moreover, as far as computing time is concerned, the suggested and adopted simplifications seem to be of little benefit when a simultaneous solver of algebraic and differential equations can be adopted. The type of dynamic problems considered are those referred to as “slow dynamics”. In other words, the disturbances dealt with can be as large as they are compatible with the continuity and the representativeness of the system of differential and algebraic equations. However, very quick phenomena, such as startup and shutdown of a distillation tower, cannot be easily accounted for. Of course the purpose which a simulation is performed for will determine the accuracy requirements

of the results. But, not only for academic purposes, when the model is referred to as a general and rigorous one, the enthalpy balance equations have to be taken into account.

Model

equations

and numerical

solution

The model for a distillation column consists of a set of mass and enthalpy differential balances coupled to a set of algebraic equations (sometimes referred to in terms of constraints or procedures by a few authors). We can write the above set of equations in the following form: Y’ =f(y.

2, t).

g(v.

t, t) = 0

where y stands for the differential variables and z for the algebraic ones. The detailed description of the model equations is beyond the scope of this note; extensive details on this point can, however, be found in the previously cited papers. The various ways to rearrange and/or to combine the different equations depends on the limitations imposed by the integration solver. Therefore, starting from the pioneering work of Di Stefano (1968), several approximations have mainly in order to reduce the been suggested, complexity of the overall problem and, partially, to respect the requirements of the available numerical methods. As a matter of fact, the general approach is to partition the overall system of differential and algebraic equations. Boston et uf. (1980) suggest a variation of the algorithm previously applied for the “inside-out” solution of steady-state problems; in this way, it is possible to distinguish between outside loop variables (differential equations) and inside loop variables (algebraic equations). Moreover, Gani et al. (1986) simultaneously solved all of the ODE system while subsets of algebraic equations are solved separately. 783

E. RANZI et al

784

402.0

-

0

I

I

4

6

I 12

I 76

0.02445

-

0.02440

-

0.02435

4

0

6

12

16

Fig. 1. BTE column: comparison among different modelsreboiler temperature (K) vs time (min).

Fig. 3. BTE column: comparison among different modelsreflux flow rate (kmol s-’ vs time (min).

As a consequence, in order to deduce vapor flowrates from the energy balance equations, sometimes simplifications or even approximations of steady-state conditions for energy are suggested and/or adopted. This was also the conclusion of the work of Howard (1970) where a convenient transformation of the energy balance is suggested. In fact, starting from the usual differential balance, enthalpy derivatives can be converted into algebraic terms, making use of component balance equations and bubble-point relationships. In order to show the relative importance of enthalpy derivatives, we adopted a model discussed elsewhere (Rovaglio et al., 1986) consisting of (2NC + 5) equations for each stage, where NC stands for the number of components. The differential variables are the NC liquid holdups of the different components and the energy holdup, whilst algebraic variables are the NC equilibrium compositions, liquid and vapor flowrates, tray temperature and pressure. As for the numerical solution, we used the routine LSODES from the package “ODEPACK” (Hindmarsh, 1982) but we have also modified the original source code in order to simultaneously solve differential and algebraic equations following the modifications presented and discussed by Feng er al. (1984).

Thus, let’s say, the complete model (C) has been compared with two simplified models: Sl = Enthalpy derivatives are disregarded. S2 = Enthalpy derivatives are accounted only for the reboiler and the condenser. A simple comparison among the three different models is reported in Figs 14. These results are referred to a ternary (benzene, toluene and ethylbenzene) distillation column, whose characteristics are summarized in Table 1, when a ramp disturbance is imposed to the feed flowrate (- 10%) between 20 and 80 s. This comparison clearly indicates the importance of the proper evaluation of time derivatives of enthalpy in the columns; in fact, when taking into account the differential energy balances, only on the top and the bottom of the column a large deviation still exists during the transient approach to the new steady-state solution. Similar results were also observed by Gani er al. (1986), when comparing responses from simple and more complex versions of his model As a general remark on these comparisons, we can observe that the initial slopes of the different curves, mainly useful for control purposes, are strongly affected by the presence of time derivatives of enthalpy; moreover, in agreement with previous experiences (Ranzi et al., 1986), the most general and

--=t

0

4

e

12

36

Fig. 2. BTE column: comparison among different modelsreboiler pressure (atm) vs time (min).

0

4

e

12

16

Fig. 4. BTE column: comparison among different modelsethylbenzene fraction on tray 11 vs time (min).

Energy

Table

1. Geometrv

and

owratina

balances

conditions

lay out 16 sieve trays Tower dia Tray spacing Hole dia Pitch (equilater)

in simulation of multicomponent distillation columns of the BTE

column

Tray

Reflux accumulator Eff. section Height Cubic dome side Kettle reboiler Free volume Dia Length

1.17” 0.4 m 0.005 m 0.015 m

377 376

geometry 0.24 m* 0.8 m 0.1 m

geometry 0.22 m3 0.5” 1.2” 0.026 kmol s370.0 K



0. I 5 0.50 0.35 295.8 kcal s-

Ethylbenzene Reboiler duty Partial condenser Reflux ratio Duty PreSStIre



3.0 279.5 kcal s-’ 1.01 at”

complete model reduces and smoothes the inverse responses predicted by the simplified models. These deviations are also confirmed in a second example related to a deethanizer column (26 real trays and 9 components) whose characteristics are

2. Geometry

and

operating

Reflux accumulator EK. section Height

conditions

column

Tray lay-out-lower section 16 valve trays (split-flow) Tower dia _ . Tray spacing 303 valves (VI ballast) Upper section 8 valve trays (cross-flow) Tower dia I . Tray spacing 140 valves (VI ballast)

Kettle reboiler Free volume Dia Length

r

379 378

Operating conditions Feed flowrate on tray IO Feed temperature Feed composition (mol fractions) Benzene Toluene

Table

785

of

the deethanizer

2.43 m 0.61 m

I .35 m 0.61 m

geometry 3.2 m2 0.72 m

Fig. 5. Deethanizer column: comparison between C and S2 models-reboiler temperature (K) vs time (min).

summarized in Table 2. In this case, the effect of an increase of 10% of feed flowrate is examined. Figures 5 and 6 show the temperature behaviours at the top and the bottom of the column. Due to the larger relative importance of energy holdups inside the column, the simplified model S2 attains the new steady-state conditions in about one half of the time req.uired by the complete model. The apparent opposite behaviour of transient responses of model C in the previous example has been already observed and justified on the basis of the smoothing effect on the reverse response of the system in the case of the more complete model (Ranzi et al., 1986). In simulating this column we have observed that the transformation of the energy differential equation into algebraic form, as suggested by Howard (1970), improves the robustness of the whole numerical integration. As a mater of fact, in this way it becomes possible to associate the vapor flowrate to the energy balance and the temperature to the bubble-point relationship whilst the original attributions were the temperature to the total energy definition, the pressure to the bubble-point equation and finally the vapor flowrate to the pressure drop across the stage. The improved robustness and efficiency can be mainly demonstrated by the lotier number of integration steps and Jacobian evaluations and by the more

geometry

Operating conditions Feed flowrate on tray 17 Feed temperature _ Feed composition (mol fractions) Methane Ethane Propylene Propane i-butane I -butene

n-butane i-pentane

Hydrogen sulfide Reboiler duty Partial condenser Total hauid reflux Duty . pressure

1.33 m3 1.0” 5.5 m 0.26 kmol s 306.0 K 0.020 0.192 0.001 0.427 0.120 0.003 0.229

b

’ 314

312

t

0.001

0.007 912.4 kcal s-’

t

a,

156.7 kcal s-’ 30.01 atm

I 10

I

I

20

30

Fig. 6. Deethanizer column: comparison between C and S2 models-ondenser

temperature

(K)

vs time

(min).

786

E.

bNZ1

regular behaviour of the vapor flowrate along the integration time. As a final remark, we can observe that in our experience, the solution of the overall problem (198 differential-algebraic equations in the former example and 598 equations in the latter one) when adopting the sparse matrix technique, requires less computational efforts than alternative approaches of algebraic procedures. In fact, for the BTE column, where ideal K-values and theoretical trays are assumed, the overall system can be reduced to 144 differential-algebraic equations by directly substituting equilibrium relationships into the differential material balances. Despite the sharp reduction in system dimension, the benefit was only a negligible saving of lO-15% of computing time. On the contrary, for the deethanizer column, where real trays and nonideal K-values (Grayson-Streed) have been considered, the reduction from the original 598 to 364 equations does require the solution of internal loops (or procedures) and the computing time increased, in our experience, by a factor higher than 2. A similar ratio in computing time has also been observed in the BTE column when nonideal K-values are considered. Obviously these comparisons have been made with proper initial conditions obtained always through steady-state simulations. Due to the plain procedures utilized here, this fact leads to the conclusion that, in a simultaneous approach, marginal savings can be obtained only through direct substitutions, while nonlinear algebraic procedures make the problem heavier despite the dimension reduction. Such a situation can be easily explained on the basis of the strong importance of thermodynamic calculations. In fact, K-values justify up to 35% of computing time per function evaluation and, even in the case of only two or three iterations inside the internal loop, the increase in computing time becomes relevant and cannot be compensated by the reduction in system dimension. This situation has been already observed in the case of steady-state simulation (Buzzi Ferraris, 1978). On the contrary, algebraic procedures and/or decomposition approaches simplify the problem of ini-

et

al.

tial estimates of algebraic variables and make the analysis of possible discontinuities easier-like in the case of startup and shutdown operations.

REFERENCES

Boston J. F., H. 1. Britt, S. Jirapongphan and V. B. Shab, An advanced system for the simulation of batch distillation operations. Foundation of Computer Aided Chemical Process Design, Vol. 2, pp. 203-237. Engineering Foundation New York (1980). Buzzi Ferraris G., Advances in computation of stages separation system. Use of Computers in Chemical Engineering Congress, Paris (1978). Cuille P. E. and G. V. Reklaitis, Dynamic simulation of multicomponent batch rectification with chemical reactions. Comput. &em. Engng 10, 389-398 (1986). Di Stefano G. P., Stability of numerical integration

techniques. AIChE

JI 14, 946955

(1968).

Feng A., C. D. Holland and S. E. Gallun, Development and comparison of a generalized semi-implicit Runge-Kutta method with Gear’s method for system of coupled differential and algebraic equations. Comput. them. Engng 8, 51-59

(1984).

Gallun S. E. and C. D. Holland, Gear’s procedure for the simultaneous solution of differential and algebraic equations with application to unsteady state distillation problems. Comput them. Engng 6, 231-244 (1982). Gani R., C. A. Ruiz and I. T. Cameron, A generalized model for distillation columns--I. Model description and applications. Comput. them. Engng 10, 181-198 (1986). Hindmarsh A. C., ODEPACK: a systemized collection of ODE solvers. Lawrence Livermore National Laboratory Report UCRL-88007 (1982). Howard G. H., Unsteady state behavior of multicomponent distillation columns. AIChE JI 16, 1022~1033 (1970). Rovaglio M., G. Biardi, E. Ranzi, R. Binda and P. Rizzetto, Multicomponent distillation columns under transient conditions--L. modelling approach. lg. C&m. Ital. 22, 3-7

(1986).

Ranzi E., M. Rovaglio, G. Biardi, P. Rizzetto and R. Binda, Multicomponent distillation columns under transient conditions--II. Comparison among different modelling schemes. Ing. Chim. Ital. 22, 3-9 (1986). Ranzi E., M. Rovaglio, S. Pierucci, G. Bussani and T. Faravelli, Dinamica degli impianti chimici: apparecchiature di equilibrio liquidwvapore. Ing. Chim. Ital. 21, 2943 (1985). Tyreus B. D., W. L. Luyben and W. E. Schiesser, Stiffness in distillation models and the use of an implicit integration method to reduce computation times. Ind. Engng Chem.

Process

Des.

Drv.

14, 427

(1975).