Reactive Distillation Model
3.2.1 Column Configuration
Reactive distillation columns for fuel ether synthesis are actually hybrid columns consisting of three distinct sections. The reactive section is located between two nonreactive column sections that separate the reactants from the product. This configuration is shown in Figure 3.1. The stripping section purifies the ether product and recycles unreacted reactants to the reactive section. The rectifying section is required to prevent the loss of ether in the distillate. The combined action of the nonreactive sections produces a composition within the reaction section that promotes the forward (synthesis) reaction in favour of the backward (decomposition) reaction by Le Chatelier's principle.
CONDENSER
CONDENSER
Figure 3.1  Hybrid Reactive Distillation Column for Ether Synthesis 3.2.2 Reactive Stage Model
Distillation systems have traditionally been modelled as a series of equilibrium stages using the MESH (Material balance, vapourliquid Equilibria, mole fraction Summations and Heat balance) equations. The main alternative is the rate based mass transfer model that is implemented with the MERQ (Material balance, Energy balance, Rate equations for mass transfer and phase eQuilibrium at the vapourliquid interface) equations. The ratebased approach is gaining in popularity but it requires the estimation of a large number of empirical and semiempirical parameters and the complex array of equations the model generates demands significant computational power. These models are not yet any more accurate than simulations using the equilibrium model although there is undoubted potential to reflect reality more closely. Where experimental data is readily available to estimate mass transfer coefficients and the column geometry is well known and easily definable algebraically, the ratebased approach is a viable alternative. However, the equilibrium stage model (modified with efficiency parameters, if necessary) remains the best choice in most cases and is still the clearly preferred modeling technique in both industry and academia.
The absence of the requisite data for rate based modelling compelled the development of a reactive distillation model using the equilibrium stage approach. Equations (3.12)(3.26) describe the reactive stage shown in Figure 3.2. The reaction is modelled rigorously using the equilibrium and kinetic data from equations (3.3), (3.7) and (3.8). The construction of the equations is such that the heat of reaction is calculated implicitly and does not need to be separately specified. The principal sidereaction (dimerisation) is modelled as an equilbrium reaction while the other sidereactions are considered negligible. The ETBE reaction is fast and proceeds close to equilibrium wi(h only a modest catalyst loading so that the assumption of reaction equilibrium introduces little error in many simulations. A simplified model that assumes reaction equilibrium (or an infinite supply of catalyst) replaces equations (3.16), (3.18) and (3.19) with equation (3.27).
reactions
VAPOUR
LIQUID
Lout Xoul
Figure 3.2  A Reactive Equilibrium Stage
4, + k,  LoU¡  y o,+IX+IX, = o ri.ETBE ~ ri,EtOH ~ ri,iBui
a aiBut
ETBE
Jf n
lnKETBE =10.387 +
4060.59
+ V,„X,Jn ~ LoutXl.ou, ~ VoutXl,oul + r,J + r2.i = 0 (3.21)
aiBu, a EiOH
It is important to note that the pressure drop across the equilibrium stage is not calculated. Therefore, the column is assumed to operate with a fixed pressure profile. Similarly, these equations assume no accumulation on each stage so that liquid (or vapour) holdup is not calculated. However, constant molal overflow has not been assumed so that the composition changes resulting from VLE and the reactions will affect the flows of both liquid and vapour.
3.2.3 NonReactive Stage Model
Nonreactive stages (i.e. stages in the rectifying and stripping sections) can be modelled on the same basis as the reactive stages except that the reaction inputs to the liquid phase (ry and r2) do not need to be considered. The simplified set of equations resulting from setting rtJ = 0 are given by equations (3.28)(3.34).
4. + v„  Loui  VM = 0 
(3.28)  
^ttt^ij» 1 in i,in out t,out r oui out u 
(3.29)  
KHL + ~ L ou, H tout ~ VoutHlout = 0 
(3.30)  
Py^YrtfPr 
à. 
3.2.4 Feed Stage Model The feed stage is simply a nonrcactive (or reactive) stage with an additional input that alters the form of the mass and energy balances. The resulting equations are: FZ + L>n*i>n + y.^i.in ~ Lour\.out " VoutX,out = 0 (3.36) FHf + LmH^n +   V0UiHlu, = 0 3.2.5 Condenser Model The condenser model can be considered a special case of the nonreactive stage model. The configuration in Figure 3.1 shows a total condenser (i.e. no vapour phase product) and two liquidphase products (i.e. reflux and distillate), and the material balance equations (3.42 and 3.43) must be modified to reflect this. The heat removal rate must be accounted for in the model and the energy balance equation (3.44) was modified accordingly. In order for the column pressure to be linked to the condenser duty, the equilibrium relationship (equation 3.45) was also modified. This allows the pressure to be fixed at the condenser and for this value to be propagated down the column without creating a numerical problem which can arise with other approaches. 3.2.6 Reboiler Model The reboiler was also modelled as a modified equilibrium stage. Figure 3.1 shows a partial reboiler with both a vapour and a liquid product and this is reflected in the material balance equations (3.463.47). The energy balance equation (3.48) must be altered to include the reboiler duty but the equilibrium relationships (equations 3.493.52) are the same as the nonreactive stage model.
3.2.7 Thermodynamic Methods Equations (3.12)(3.52) fully describe the reactive distillation column but additional equations and numerical routines are required to determine the parameters of this model. Geometrical parameters are not required as the flow properties (i.e. pressure drop and holdup) are not modelled but several physical properties are required. Vapour pressures were estimated from liquid temperatures using Antoine equations fitted to recent experimental data (Krahenbuhl and Gmehling, 1994; Gmehling and Onken, 1977; Dean, 1992; Reid et al., 1987). The UNIFAC model was used to estimate activity coefficients in order to be consistent with the reaction model and to provide the most accurate representation of the phase behaviour. Enthalpies and densities were estimated from liquid or vapour phase temperatures using the SoaveRedlichKwong (SRK) method. The SRK equations are recommended for hydrocarbon systems at moderate temperature and pressure although similar equationofstate methods (eg. PengRobinson) would have been equally acceptable. Although these methods often provide inaccurate density predictions, this was of little consequence here because mass and molar flow rates are used exclusively in the subsequent analyses. An ideal vapour phase was assumed due to the modest pressure. 3.2.8 Simulation PackagesThe reactive distillation model described above was implemented directly in SpeedUp (Aspen Tech 1993) and indirectly in Pro/11 (SimSci, 1994), both of which are commercial process simulators. SpeedUp is an equationbased simulator that is sufficiently flexible to permit equations to be input in any form provided that consistency constraints are honoured and the model conforms to the simulator's syntax and structure requirements. The global system of equations for the full model contained a total of 578 variables and 504 linear and nonlinear equations. Importantly, this model could be updated and modified for the analysis of dynamic responses. However, the numerical solution of this set of equations required very good initial estimates of the system outputs. This reduces the robustness of the solution method (but not of the model itself) and requires the initial solution to be found by slowly building the full model in stages, saving the entire output matrix after each stage for use in solving the next stage. Pro/II is a sequential modular simulation package that contains an extensive model library that allows most unit operations to be simulated easily and accurately. It allows the property methods and reaction details to be specified and provides some control over solution methods but requires the use of the default unit operation equations. This simplifies the simulation task but somewhat restricts its range of applicability. The ease of development and robustness of the model were distinct advantages of using Pro/II but its deficiencies compelled the use of SpeedUp in some cases. Specifically, convergence was found to be unlikely when the reaction kinetics were modelled fully rigorously although SpeedUp can handle this case provided the equation structure is correct. This facility of SpeedUp was used to test the assumption of chemical equilibrium in the Pro/II simulations. A moderate catalyst loading was found to produce an isobutene conversion of only 0.20.3% less than the equilibrium conversion  an acceptable result. On the other hand, Pro/II was the better tool for checking the assumption of an ideal vapour phase. A fugacity coefficient model can be easily implemented within Pro/II by specifying a change in the thermodynamic routines being used. The fugacities were found to be generally within 0.5% of the partial pressures  an acceptable result which again confirms the validity of the original assumption. 
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