Modeling batch distillation

The suitability of the different operating modes can be determined with the aid of programs for modeling batch distillations. The treatment of batch distillations is much more laborious than that for continuous distillations. However, the decision can be simplified by using Equations (2.3.2-29) and (2.3.2-30), which were determined for the different operating modes with the aid of differential equations. The minimum vapor quantity G is used for comparison ... 

In terms of downstream processes, the flow-rates, compositions, and so on, dictate the size and number of each unit operation for example, while a batch distillation may be used to separate a single feed into a number of different product streams, a continuous distillation train would in general require N columns for N different product streams. The fact that a high degree of modeling is used in the design of each MPI, results in the generally held belief that continuous processes... 

Distillation is a well-known process and scale-up methods have been well established. Many computer programs for the simulation of continuous distillation columns that are operated at steady state are available. In fine chemicals manufacture, this concerns separations of products in the production of bulk fine chemicals and for solvent recovery/purification. In the past decade, software for modelling of distillation columns operated at non-steady state, including batch distillation, has been developed. In the fine chemicals business, usually batch distillation is applied. 

Figure 3.58. Model representation of a batch distillation column and typical plate n, as per Luyben (1973).

Relaxation methods are not competitive with the steady-state methods in the use of computer time, because of slow convergence. However, because they model the actual operation of the column, convergence should be achieved for all practical problems. The method has the potential of development for the study of the transient behaviour of column designs, and for the analysis and design of batch distillation columns. 

Other synonyms for steady state are time-invariant, static, or stationary. These terms refer to a process in which the values of the dependent variables remain constant with respect to time. Unsteady state processes are also called nonsteady state, transient, or dynamic and represent the situation when the process-dependent variables change with time. A typical example of an unsteady state process is the operation of a batch distillation column, which would exhibit a time-varying product composition. A transient model reduces to a steady state model when d/dt = 0. Most optimization problems treated in this book are based on steady state models. Optimization problems involving dynamic models usually pertain to optimal control or real-time optimization problems (see Chapter 16)... 

Changes in the hydraulic hold-up of liquid on the column plates is known to have a significant effect on the separating efficiency of batch distillation columns, and may be relatively easily incorporated into the batch simulation model. The hydraulic condition of the plates is represented in Fig. 3.52. 

If necessary the hydraulic relationships, previously derived for batch distillation, are also easily implemented into a continuous distillation model. 

Derive a mathematical model of this batch distillation system for the case where the tray holdups cannot be neglected. 

The model of a multicomponent batch distillation column was derived in Sec. 3.13. For a simulation example, let us consider a ternary mixture. Three products will be produced and two slop cuts may also be produced. Constant relative volatility, equimolal overflow, constant tray holdup, and ideal trays are assumed. 

The embedded model approach represented by problem (17) has been very successful in solving large process problems. Sargent and Sullivan (1979) optimized feed changeover policies for a sequence of distillation columns that included seven control profiles and 50 differential equations. More recently, Mujtaba and Macchietto (1988) used the SPEEDUP implementation of this method for optimal control of plate-to-plate batch distillation columns. 

For the synthesis of heterogeneous batch distillation the liquid-liquid envelope at the decanter temperature is considered in addition to the residue curve map. Therefore, the binary interaction parameters used in predicting liquid-liquid equilibrium are estimated from binary heterogeneous azeotrope or liquid-liquid equilibrium data [8,10], Table 3 shows the calculated purity of original components in each phase split at 25 °C for all heterogeneous azeotropes reported in Table 1. The thermodynamic models and binary coefficients used in the calculation of the liquid-liquid-vapour equilibrium, liquid-liquid equilibrium at 25 °C and the separatrices are reported in Table 2. 

Schneider R, Noeres C, Kreul LU, Gorak A. Dynamic modeling and simulation of reactive batch distillation. Computers Chem Eng 1999 23 S423-S426. 

For a given Rexf, the vapour rate profile is averaged to obtain an average Vexp to be used in the batch distillation model developed by Greaves et al.. Figure 3.12... 

In a steady state continuous distillation with the assumption of a well mixed liquid and vapour on the plates, the holdup has no effect on the analysis (modelling of such columns does not usually include column holdup) since any quantity of liquid holdup in the system has no effect on the mass flows in the system (Rose, 1985). Batch distillation however is inherently an unsteady state process and the liquid holdup in the system become sinks (accumulators) of material which affect the rate of change of flows and hence the whole dynamic response of the system. 

A summary of several example cases illustrated in Mujtaba and Macchietto (1998) is presented below. Instead of carrying out the investigation in a pilot-plant batch distillation column, a rigorous mathematical model (Chapter 4) for a conventional column was developed and incorporated into the minimum time optimisation problem which was numerically solved. Further details on optimisation techniques are presented in later chapters. 

Greaves, M. A., Hybrid Modelling, Simulation and Optimisation of Batch Distillation Using Neural Network Techniques. Ph.D. Thesis, (University of Bradford, Bradford, UK, 2003). 

Robinson and Gilliland (1950), Abrams et al. (1987) and Hasebe et al. (1992, 1995), Sorensen and Skogestad (1996a), Barolo et al. (1998), Furlonge et al. (1999) reported simulation results using inverted, batch distillation columns with middle vessel and multi-effect batch distillation, respectively. Mujtaba (1997) simulated batch distillation operation using a continuous distillation model. 

The Rayleigh model was developed for a single stage batch distillation-where a liquid mixture is charged in a still and a vapour is produced by heating the liquid. At any time, the vapour on top of the liquid is in equilibrium with the liquid left in the still. The vapour is removed as soon as it is produced but no part of the vapour is returned as reflux to the still after condensation. 

The shortcut model is developed based on the assumption that batch distillation operation can be represented by a series of continuous distillation operation of short duration and employs modified Fenske-Underwood-Gilliland (FUG) shortcut model of continuous distillation (Diwekar and Madhavan, 1991a,b Sundaram and Evans, 1993a,b). Starting with an initial charge (B0, xB0) at time f=fo and for a small interval of time At = t, - t0, the batch distillation column conditions at to and ts is schematically shown in Figure 4.1 (Galindez and Fredenslund, 1988). 

Note that Sundaram and Evans (1993a,b) used FUG method of continuous distillation directly and developed time explicit model, while Diwekar (1992) developed modified FUG method as described above and time implicit model for batch distillation. Sundaram and Evans used time as an independent variable of the model while Diwekar (1992) used reboiler composition as independent variable. Both models are based on zero column holdup and does not include plate-to-plate calculations. See the original references for further details. 

Seader and Henley (1998) considered the separation of a ternary mixture in a batch distillation column with B0 = 100 moles, xB0 = = <0.33, 0.33, 0.34> molefraction, relative volatility a= <2.0, 1.5, 1.0>, theoretical plates N = 3, reflux ratio R = 10 and vapour boilup ratio V = 110 kmol/hr. The column operation was simulated using the short-cut model of Sundaram and Evans (1993a). The results in terms of reboiler holdup (Bj), reboiler composition profile (xBI), accumulated distillate composition profile (xa), minimum number of plates (Nmin) and minimum... 

Converse and Huber (1965), Robinson (1970), Mayur and Jackson (1971), Luyben (1988) and Mujtaba (1997) used this model for simulation and optimisation of conventional batch distillation. Domenech and Enjalbert (1981) used similar model in their simulation study with the exception that they used temperature dependent phase equilibria instead of constant relative volatility. Christiansen et al. (1995) used this model (excluding column holdup) to study parametric sensitivity of ideal binary columns. 

Figure 4.3. Batch Distillation using Simple Model...

Referring to Figure 4.5 and the assumptions listed in section 4.2.4.1 the rigorous model for batch distillation with chemical reactions is presented below. Further assumptions are no chemical reactions in the vapour phase and in the condenser accumulator. 

Attarwala and Abrams (1974) and Mujtaba (1997) used the above model for batch distillation task using a continuous column (see section 2.2.4). 

Referring to Figure 4.12 for inverted batch distillation column, the intermediate plate equations in model types III, IV and V presented in section 4.2 will remain the same. The model equations for the condenser and for the reboiler for types III, IV and V models are presented below. 

A liquid binary mixture with B0 = 10 kmol (Hc) and xB0 = <0.6, 0.4> (xj) molefraction is subject to inverted batch distillation shown in Figure 4.12. The relative volatility of the mixture over the operating temperature range is assumed constant with a value of (a-) 2. The number of plates is, N= 10. The vapour boilup rate is, V = 10.0 kmol/hr. The total plate holdup is 0.3 kmol and the reboiler holdup is 0.1 kmol. The total batch time of operation is 4 hr with two time intervals. The first interval is of duration 1 hr and the column is operated with a reboil ratio of 0.8. The second interval is of duration 3 hrs when the column is operated with a reboil ratio of 0.9. The column operation is simulated with the type III model (section 4.3.2.1). 

Referring to Figure 2.2 for MVC column configuration, the model equations for the rectifying section are the same (except the reboiler equations) as those presented for conventional batch distillation column (Type III, IV, V in section 4.2). The model equations for the stripping section are the same (except the condenser equations) as those presented for inverted batch distillation column (Type III, IV, V in section 4.3.2). However, note that the vapour and liquid flow rates in the rectifying and stripping sections will not be same because of the introduction of the feed plate. 

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). 

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.). 

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