= 10 kmol/hr
No. of components,
Initial charge composition,
= <0.25,0.25, 0.25,0.25>
= <10.2,4.5, 2.3, 1.0>
B,_4 = 2.5 kmol xB1A = <0.0, 0.0, 0.004,0.996>
Figure 4.20. MultiBD Column - Product Amounts and Compositions
Extensive literature survey shows that only a few considered modelling, simulation and optimal control of such column (Hitch and Rousseau, 1988; Hansen and Jorgensen, 1986). Nad and Spiegel (1987) simulated their experimental packed column using the methods developed for tray columns by estimating the number of trays representative of the packed height. The reasons for overlooking this area are deficiencies remain evident in several key areas such as (a) the inability of current numerical scheme to solve the complicated systems of nonlinear model equations (partial differential and algebraic equations, PDAEs) required for a rigorous treatment of simultaneous mass and heat transfer phenomena, (b) lack of flexibility of most current packed column algorithms as they are typically process dependent with narrow operating regions. In addition, inherent dynamic nature of batch distillation makes PDAEs more complicated.
Wajge et al. (1997) attempted to develop rigorous PDAE model for packed batch distillation with and without chemical reaction and used finite difference and orthogonal collocation techniques to solve such model. The main purpose of the study was to investigate the efficiencies of the numerical methods employed. The authors observed that the collocation techniques are computationally more efficient compared to the finite difference method, however the order of approximating polynomial needs to be carefully chosen to achieve a right balance between accuracy and efficiency. See the original reference for further details.
Unlike continuous distillation, batch distillation is inherently an unsteady state process. Dynamics in continuous distillation are usually in the form of relatively small upsets from steady state operation, whereas in batch distillation individual species can completely disappear from the column, first from the reboiler (in the case of CBD columns) and then from the entire column. Therefore the model describing a batch column is always dynamic in nature and results in a system of Ordinary Differential Equations (ODEs) or a coupled system of Differential and Algebraic Equations (DAEs) (model types III, IV and V).
System of ODEs and DAEs can be classified according to their index (Morison, 1984). The model equations presented in the previous sections mostly constitute a coupled system of DAEs of index one (type III) or two (Type IV and V). Index is simply defined by the maximum number of differentiations required to reduce a DAE system to an ODE system. DAE system of index exceeding unity occurs in many areas of chemical engineering modelling. For example if the assumption of total condensation in the condenser is removed and material and energy balances are written for a fixed condenser volume, the resulting DAEs will be of index two (type IV and V - CVH models). This was shown in detail by Pantelides et al. (1988). Solution of such DAE systems is sometimes difficult. Classification of DAE system according to their index is given in some detail in Morison (1984).
Systems of index zero or one are ODE systems or simple DAE systems, and should cause no problem when integrated by well known existing methods. Integration of higher index DAEs requires special treatment (Gritsis, 1990; Bosley and Edgar, 1994a,b). See the original references for further details.
Depending on the numerical techniques available for integration of the model equations, model reformulation or simplified version of the original model has always been the first step. In the Sixties and Seventies simplified models as sets of ordinary differential equations (ODEs) were developed. Explicit Euler method or explicit Runge-Kutta method (Huckaba and Danly, 1960; Domenech and Enjalbert, 1981; Coward, 1967; Robinson, 1969, 1970; etc) were used to integrate such model equations. The ODE models ignored column holdup and therefore non-stiff integration techniques were suitable for those models.
However, in batch distillation, the system is frequently very stiff, owing either to wide ranges in relative volatilities or large differences in tray and reboiler holdups. Therefore, if methods for non-stiff problems are applied to stiff problems (ODE models but having column holdup and/or energy balances), a very small integration step must be used to ensure that the solution remains stable (Meadow, 1963; Distefano, 1968; Boston et al., 1980; Holland and Liapis; 1983, etc.).
In the Eighties and Nineties ODE models were discretized as algebraic equations using orthogonal collocation on finite elements techniques and solved (Hansen and Jorgensen, 1986; Christensen and Jorgensen, 1987; Logsdon and Biegler, 1993; Li et al., 1998). Diwekar and co-workers (1986, 1987, 1991a, 1991b, 1992, 1995) and Sundaram and Evans (1993a,b) assumed that the batch distillation column could be considered as continuous column with changing feed. That is, for small interval of time the batch column behaviour is analogous to a continuous column and they employed widely used Fenske-Underwood-Gilliland shortcut method to integrate the model equations. Galindez and Fredenslund (1988) also considered a similar approach but employed rigorous continuous distillation model and modified Naphtali-Sandholm method (Christiansen et al., 1979; Naphtali and Sandholm, 1971) to integrate the model equations. Naphtali and Sandholm method requires grouping of the model equations by stages (number of plates) and the use of Newton-Raphson method. Mori et al. (1995) reformulated DAEs model of Distefano (1968) as ODEs and used the generalized implicit Euler's method to integrate the model equations.
Rigorous and stiff batch distillation models considering mass and energy balances, column holdup and physical properties result in a coupled system of DAEs. Solution of such model equations without any reformulation was developed by Gear (1971) and Hindmarsh (1980) based on Backward Differentiation Formula (BDF). BDF methods are basically predictor-corrector methods. At each step a prediction is made of the differential variable at the next point in time. A correction procedure corrects the prediction. If the difference between the predicted and corrected states is less than the required local error, the step is accepted. Otherwise the step length is reduced and another attempt is made. The step length may also be increased if possible and the order of prediction is changed when this seems useful.
Cuille and Reklaitis (1986), Ruiz (1988), and Mujtaba and co-workers (1989-2003) used this method in batch distillation studies.
Bosley and Edgar (1994a,b) developed techniques to calculate directly dynamic distillation (DDD) derivatives (non iterative), which were then employed to integrate rigorous DAEs (index 2) based batch distillation model.
Seader and Henley (1998) provided a detailed account of the merits and demerits of different integration methods with typical conventional batch distillation examples. They have noted three particular issues with the integration methods applied in batch distillation calculations that require attention. These are the truncation error, stability and stiffness ratio. These will be briefly discussed in the following.
Local truncation errors are the result of approximating the functions on the right hand side of the ODEs at each time step. Although locally small, these errors can propagate as calculation proceeds to subsequent time steps and can result to large global truncation errors. The net effect is the gradual but significant loss of accuracy in the computed dependent variables. Use of smaller time step will reduce truncation errors.
Consider the following equation (y is dependent variable, x is the independent variable, h is the integration step size), dx
If explicit Euler's approximation is implemented then,
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