10.1 INTRODUCTION
Distillation is the most common unit operation in the chemical industry and understanding its behaviour has been a defining characteristic of a good chemical engineer. The key objective of developing a mathematical process model is to predict the dynamic characteristics of a plant, whether the model is used for the advanced model-based controller synthesis or for the soft sensor design. When developing a model of a distillation column, several unacceptable assumptions are sometimes adopted aiming to simplify the theory. Unfortunately, the resulting simplified model may fail to capture precisely the nonlinear and interactive distillation dynamics.
Realistic performance of an actual column can seldom be predicted satisfactorily by excluding the simultaneous effects of heat transfer and fluid flow on the trays. The liquid hydraulic in a tray is an important factor in predicting the dynamic performance. The plant dynamics are also greatly influenced by the phase nonidealities. In addition, accurate estimation of the physical properties is also a vital issue.
This chapter presents the development of a dynamic model for a realistic alcohol distillation column taking into consideration all the factors described above, including liquid-phase nonideality, nonequimolal overflow, rigorous tray hydraulics and inefficient trays. The model structure of such a practical column contains a large number of ordinary differential equations (ODEs) along with nonlinear algebraic equations (AEs) that must be solved. Keeping this objective in mind, the dynamic simulation of the example distillation model has also been included with computer program in this chapter.
10.2 THE PROCESS AND THE MODEL
The present study concerns a continuous distillation column with one input feed stream and two liquid products (distillate and bottoms). The example column is actually used to distil ethanol from water. A schematic representation of the distillation column is shown in Figure 10.1.
FIGURE 10.1 Schematic representation of the alcohol distillation column example.
The fractionating column is made up of 20 inefficient trays, a reboiler and a total condenser.
Numbering of trays is started from the bottom of the column: tray 1 is the bottom tray above the reboiler, tray 2 is the next, etc. The feed (saturated liquid) enters at tray 10. The condensation of overhead vapour is total and the condensed liquid is accumulated in the reflux drum. No vapour distillate is produced here.
Some of the condensed liquid (not subcooled) is removed from the reflux drum as the distillate product, and some of it is sent back as a reflux stream to the column to provide liquid flow on the trays. At the base of the column, the bottoms is withdrawn as a liquid product. A reboiler heated by steam, generates the boil-up vapour and the vapour produced flows back to the bottom plate. The reboiler runs like a theoretical tray.
In order to describe detailed distillation operation, we will attempt to develop a rigorous distillation model. The mathematical model, which can represent a binary column, is a large structured system of differential–algebraic equations supported by vapour–liquid equilibrium (VLE) and physical properties.
The dynamic distillation simulator involves the computations of composition (or mole fraction), flow rate, tray holdup, enthalpy, average molecular weight and density, and VLE.
The following assumptions have been used to develop the distillation model.
The liquid is perfectly mixed on each tray. This assumption implies that the liquid on any tray n has the uniform composition xn.
The liquid and vapour leaving each tray are in thermal equilibrium.
The operating pressure is constant (one atmosphere) and pressure drops are negligible.
The tray efficiency is defined from the Murphree relation and 70% vapour-phase Murphree efficiency has been assumed for every tray.
The vapour-phase holdup is negligible with regard to the liquid-phase holdup. This assumption is quite reasonable since in most systems, the vapour density is much smaller than the liquid density.
Algebraic forms of equations are used to calculate the liquid-phase and vapour-phase enthalpies.
The energy balance equations (enthalpy derivatives with respect to time) are not used to compute enthalpies; they are generally employed for vapour flow rate calculations.
The tray hydraulics are modelled with Francis weir relationship (details in Chapter 9).
Variations of liquid holdups are considered in each tray excluding reflux drum and column base.
Indeed, the reflux drum and column base holdups in most of the industrial columns are held almost constant by employing level controllers.
Coolant and steam dynamics are negligible in the condenser and reboiler, respectively. However, to examine the condenser and reboiler dynamics, an example of a heat exchanger is discussed with the development of dynamic model in Chapter 1.
Wilson thermodynamic model is used (details in Chapter 8) for VLE predictions.
The thermal losses are assumed to be negligible.
10.2.1 Material and Energy Balance Equations
A set of ordinary differential equations consisting of total (or overall) continuity, component (or partial) continuity, and energy (or enthalpy) balance equations is given in the following descriptions.
Reboiler–Column Base System (subscript ‘B’)
Total Continuity:... ...(10.1) Component Continuity:... ...(10.2) Energy equation:... ...(10.3) Bottom Tray (subscript ‘1’)
Total continuity:... ...(10.4) Component continuity:... ...(10.5) Energy equation:... ...(10.6) nth Tray (subscript ‘n’, where n = 2 to nF – 1 and nF + 1 to nT – 1)
Total continuity:... ...(10.7)
Component continuity:... ...(10.8)
Energy equation:... ....(10.9)
Feed Tray (subscript ‘nF’)
Total continuity:... ...(10.10) Component continuity:
...(10.11)
Energy equation:
...(10.12)
Top Tray (subscript ‘nT’)
Total continuity:... ...(10.13)
Component continuity:... ...(10.14)
Energy equation:... ...(10.15)
Condenser–Reflux Drum System (subscript ‘D’)
Total continuity:... ...(10.16) Component continuity:... ...(10.17)
In the above modelling equations, xn is the mole fraction of a more volatile component (here ethanol) in a liquid stream leaving nth tray, yn the mole fraction of ethanol in a vapour stream leaving nth tray, xF the mole fraction of ethanol in the feed stream, xD the mole fraction of ethanol in the liquid distillate, xB the mole fraction of ethanol in the bottom product, Ln the liquid flow rate leaving nth tray (gmol/min), Vn the vapour flow rate leaving nth tray (gmol/min), F the feed flow rate (gmol/min), R the reflux flow rate (gmol/min), D the distillate flow rate (gmol/min), B the bottoms flow rate (gmol/min), VB the vapour boil-up rate (gmol/min), mn the liquid holdup on nth tray (gmol), mD the liquid holdup in the reflux drum (gmol), mB the liquid holdup in the column base (gmol), HDL the enthalpy of distillate (J/gmol), HBL the enthalpy of bottom product (J/gmol), HFL the enthalpy of feed stream (J/gmol), HnL the enthalpy of a liquid stream leaving nth tray (J/gmol), HnV the enthalpy of a vapour stream leaving nth tray (J/gmol) and QR the heat input to the reboiler (J/min). The dot symbol (.) on a variable is used to denote the time derivative of that variable. Also, represents the time derivative of mx, i.e., .
It is well-known that the dynamic changes in internal energies on the trays are much faster than the composition or total holdup changes. Therefore, the energy balance Equations (10.3), (10.6), (10.9), (10.12), and (10.15) with putting zero in the left hand sides are commonly (and here also) utilized to compute the vapour flow rates (VB, V1, Vn, ).
We must remember that it is not necessary to include Equation (10.3) in the reboiler–column base system model if VB be the manipulated input instead of QR for the bottom loop. It is also worthy to mention at this moment that since no vapour flow rate calculation is involved in the condenser–reflux drum system, there is no need to make an energy balance equation for that system. As stated previously, the liquid as well as vapour phase enthalpy is estimated using the algebraic forms of equations.
10.3 DYNAMIC SIMULATION
As mentioned, the dynamic distillation simulator (to be developed), computes the tray holdup, phase composition, flow rate, enthalpy, average molecular weight and density, and vapour–liquid equilibrium.