(26d) Modeling and Optimization of a Bubble Column Reactor and Benzene Chlorination Process
A BCR model used in this work is that first the component mole balances in the liquid phase are represented by an axial dispersion equation with Danckwerts boundary conditions. Also, it is assumed that reactions take place in the only liquid phase. On the other hand, the component and total mole balances in the gas phase are considered as plug flow with initial values. Second, the energy balance is considered as an axial dispersion with Danckwertsâ boundary condition. The temperature of both liquid and gas is assumed as being the same under adiabatic mixture temperature. Third, the momentum balance which is obtained with hydraulic head and wall friction. The pressure at the top of the reactor is provided as the boundary condition. Finally, empirical flow dynamics equations for dispersion coefficient and gas hold up, and mechanical energy balance equations for recirculation rate are used. Hence, the model consists of ordinary differential equations (ODEs) which include both initial and boundary value problem, and nonlinear algebraic equations.
For optimization of the BCR, the differential algebraic equations (DAE) are discretized with orthogonal collocation finite element (OCFE) method. Then, the discretized ODEs and other nonlinear algebraic equations are simultaneously solved with NLP solver. This is implemented in Pyomo with PyomoDAE. IPOPT is also used as the NLP solver.
The model is validated and sufficiently agree with the actual plant data. Then, the optimization has been done with the chlorine feed rate as the degree of freedom. In the optimization, the yield of chlorobenzene, p-dichlorobenzene, and o-dichlorobenzene is respectively maximized. From the optimization result, the estimated dispersion coefficient shows the flow regime is in between plug and perfect mixed flow for all cases. It is, however, noted that the optimized yield tends to correspond to the ideal CSTR model. From the analysis of the recirculation rate in the BCR, the recirculation rate is always more than 100 times of feed rate of liquid benzene. This high recirculation condition leads to almost perfect mixed condition for over the BCR which includes external loop even though the part of the bubble column portion does not show perfect mixed condition. Although the detailed DAE model is necessary for design optimization such as column size, this result could allow us to reduce the detailed DAE model into simple CSTR model in case only the operational condition such as flow rate is concerned.
Next, the optimization of the whole process sheet is carried out as the equation oriented (EO) model in Pyomo. The process consists of several separators after BCR such as distillations, strippers, absorbers and crystallizer. The aforementioned simplified CSTR model is used for the BCR model. The distillations, strippers and absorbers are modeled as tray models with MESH equations or flash models with ideal thermodynamic property and empirical partial vapor pressure for HCl absorber. The crystallizer is modeled as the separator with component fractions which are determined with ideal solid-liquid equilibrium. For constraints for the demand of products, the excess or short of demand is considered because the production balance is not feasible for all cases of the demand one. The excess production is counted on as waste. Moreover, the objective function is the profit of the production, which obtained by subtracting operation cost such as raw material, utility and waste treatment from sales of products. Then, the optimization has been done by maximizing the profit.
The optimization of the results gives us appropriate operating conditions such as feed rate, reflux ratio and recycle ratio for changing demands. It is interesting that the solution suggests manufacturers could make more profit by producing the amount of o-dichlorobenzene less than the demand of one in a case. It is not a trivial business option for manufacturers.