Modeling of batch distil~ation by the pulse-response

SPAIN

J. R. González-Velasco M. A. Gutiérrez-Ortiz J. M. Castresana-Pelaya J. A. González-Marcos

Modeling of batch distil~ation by the pulse-response method Threedifferent mathematical models are proposed for modeling the dynamic response of a batch-distillation column to a step change in the reflux ratio: 1) ideal (no holdup); 2) condenser holdup only; and 3) liquid holdup on the pIates. These models, deueIoped for the case of no distillate remo val, can be extended to other operating conditions. Dynamic data from an experimental pack"ed column, operating on the methanol-water system, are analyzed with these models.

Introduction Batch distillation is an unsteady-state process in which aH of the compositions, as weH as the amount of liquid in the reboller, vary with time. Strictly speaking, one should also include the consideration that the liquid holdup in the condenser and on the plates could vary with time, and that there exist vapor holdups in the reboiler on the plates and in the condenser, holdups whose dynamics should be described [1 l. Since consideration of aH of these aspects involves excessive complication of a mathematical model, however, the latter is normally based on assuming a series _ of simplifications, standard in the literature [2-4]. Owing to the intrinsically unsteady-state characteristics of a batch distillation system, the response to any stimulus includes both the dynamics induced by the stimulus and those of the system itself, and it is necessary to eliminate From Anales de Química, Serie A, Química Física y Química Técnica 84, No. 3, pp. 361-367 (1988),

with permission. J. R. González-Velasco, M. A. Gutiérrez-Ortiz, J. M. Castresana-Pelayo, and J. A. González-Marcos are associated with the Departamento de Ingenieria Quimica, Facultad de Ciencias, U ni versidad del Pais Vasco, Apartado 644, 48080 Bilbao. 0020 -6318/90/3014-0568/$05 .00 of Chemical Engineers . 568

July 1990

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1990, Am e rican Inst itute

the changes corresponding to the transitory nature of the process in order to estimate the effect resulting strictly from the perturbation. For this reason, it is pro po sed in this work to perform a perturbation of the system when this latter is the steady state, recycling the distillate to the reboiler and bringing the system to a new steady state. There is thus produced a jump from one steady state to another, and the system response is due solely to the stimulus, which simplifies analysis of the latter Two variables can be employed to implement a stimulus in a simple manner: the boilup and the reflux. A change in the boilup does not alter the steady state, but only modifies the rate of approach to the latter. A change in the reflux ratio, however, produces an aIteration in the steady lte, and the equipment passes from the one state the other at arate dependent upon the llldgnitude, distribution and characteristics ofthe holdups. A jump from one steady state to another can also be produced by removal of an amount of material from the equipment, but this type of stimulus has not been considered because of the practical difficuIties which it presents and the great variety of possible system responses, the diversity of which derives from the large number of equipment points at which removal ofmaterial could be implemented.

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Modeling of batch distillation equipment is considered in this work. After achievement of an initial steady state, the equipment was subjected to a stimulus in the reflux ratio, which led the equipment to a new steady state. Comparison between the actual response of the equipment to this stimulus with the theoretical ones corresponding to different systems led to knowledge of the holdup characteristics in the equipment studied, as well as the equations which best model it.

Mathematical models In the derivation of the mathematical models, the following simplifications [1-5] are considered: • Binary system with constant relative volatility; • Negligible vapor holdups; • Perfect rnixing in the liquid holdups; • Constant boilup rate; • McCabe-Thiele hypothesis; • Total condenser; • Negligible fluid dynarnic delays (quasisteady state). This last assumption is important, since it implies that an instantaneous variation in the liquid flow rate is produced from the column to a change in the reflux ratio by not admitting the need to perform calculation of the fluid dynamic aspects associated with the problem and centering attention on mass transfer. The models which are obtained by adopting these considerations consist of a system of differential equations, characterized by being nurnerically unstable and analytically intractable [4,5]. Thus, the model must be solved numerically with a high-speed computer and with programs which keep the CPU times within operationally acceptable values. The typical operation oí a batch distillation ,paratus, as represented in Figure 1, consists of two well-differentiated phases: (a) Placement in operation, or startup, which is cornprised of heating the reboiler charge up to the boiling point and subsequent operation without distillate removal, which can be realized at total reflux or at a finite reflux ratio. If the unit has appreciable holdups in addition to that of the reboiler, the startup has to include an intermediate stage for filling up these holdups. (b) Production of distillate. During this phase the equipment can be operated at constant reflux ratio, at constant product composition, or with sorne optimal reflux-time trajectory [5-7] as well. Below are presented mathematical rnodels INTERNATlONAL CHEMICAL ENGINEERING

Fig. 1. Sketch of a batch distillation unit.

corresponding to the startup phase, and which describe the equipment behavior during the passage from one steady state to another for three different systems.

Ideal rnodel (negligible holdup) Since no holdup exists, the startup phase consists solely of heating the reboiler liquid up to its boiling point, with whatever steady state being achieved instantaneously. The equations for this model are (1)

(2)

for p

= s, 1, 2, ...

, n.

Model with appreciable holdup in the condenser In this system the approach to steady state is retarded because ofthe condenser holdup, and the dynamics of the concentrations in this holdup are included in the model equations. This model is described by Equation (1), together with the following ones:

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dx"

(3)

M - - = V (y -x.,,)

dt

n

R Yp-I=

0.96 Xc.

R+1

-

R=2

mole fraction

(4)

(X¡,-x.)-Yn

0.92 tdS

S =100 moles XS =0.5 mole fraclion V =150 moles/h n= 8 plates

(5)

--=0

dt

0.66 (6)

0(=2

0.64 r-

ModeI with appreciable holdup on the pIates and in the condenser 0~80

Whereas in the two preceding cases, employment of the number of theoretical pI ates was sufficient in order to express the separation efficiency of the column, in this model the individual efficiencies of the plates must be considered because of the liquid presence on each rtage, i.e., the equations for equilibrium are expressed as (7)

1+(