A MILP approach to the optimization of the operation procedure of a fed-batch emulsification process in a stirred vessel
Introduction
Emulsification is a key manufacturing technology in the food industry. Examples of emulsions2 are mayonnaise and many kinds of dressings. For the manufacturing of these products oil and water (or more general an oily and an aqueous phase), surfactants, ingredients and energy are needed. Surfactant molecules have both hydrophilic and lipophilic molecular groups. Based on these properties surfactant molecules absorb at interfaces between oil and water, facilitating the formation of emulsions by lowering the interfacial tension. Surfactants are also responsible for short- and long-term stability by preventing coalescence.
Equipment as used for the production of oil-in-water (o/w) emulsions is shown in Fig. 1. It consists of a stirred vessel in combination with a colloid mill and a recirculation loop. The vessel is equipped with a scraper stirrer: a device consisting of several blades rotating at low speed at a small distance from the vessel wall in order to achieve mixing and breaking of the oil drops. The colloid mill consists of a stator and a rotor. In the narrow gap between these the intensity of the hydrodynamic forces acting on the drops is very high, which causes the breakage of the oil drops. The colloid mill also acts like a pump resulting in a recirculating flow to the vessel. This pump action is due to the high rotating speed of the rotor and the conically shaped form of the mill. The process is operated fed-batch wise and typical production times are in the order of minutes.
In industrial practice the operation procedure is conventional in the sense that the oil flow addition rate and the stirrer and rotor speed (the input variables) have constant values in time. After the oil addition the process is continued for a certain time to ensure a sufficient drop size reduction. For profit maximization it is desirable to decrease the production time while maintaining the product quality specifications. Experiments are expensive and time-consuming; therefore, a model-based optimization approach is followed here. The emulsion quality is strongly affected by the drop size distribution (DSD). The desired DSD is often multi-modal and/or asymmetric. This makes the control of the moments of the DSD inadequate and creates the need for the control of the full distribution. One possible control configuration is as follows: an off-line dynamic optimization problem is solved to calculate the trajectories of the input variables such that a certain predefined, terminal, DSD is reached in minimum time. Then a controller is designed to track the computed optimal DSD trajectory. As a first step towards the optimization of the actual emulsification process only the off-line optimization of emulsification in the stirred vessel is considered. The optimization of the operation procedure of emulsification has received little attention in the literature, but there is extensive literature on numerical strategies for the solution of dynamic optimization problems (see, for example, Bryson & Ho, 1975, Vassiliadis, Sargent & Pantelides, 1994, Yen & Nagurka, 1992). However, due to the specific model structure, these methods are not well suited for the solution of the optimization problem as considered. Therefore, a new approach is suggested here.
The structure of this paper is as follows. A description of the model is given in Section 2 and in Section 3 the model structure is analyzed. The mathematical formulation of the optimization problem is presented in Section 4. In Section 5 it is explained why often used optimization methods are inappropriate and a new optimization approach is presented. Several optimization studies are performed to choose the stirrer speed and the oil flow rate as function of the time such that a certain predefined, terminal, DSD is reached in minimum time in the stirred vessel. The results of these optimization studies are shown in Section 6. Finally, concluding remarks are stated in Section 7.
Section snippets
Modeling of emulsification
Several models have been published that deal with emulsification in a stirred vessel equipped with a turbine or propeller stirrer. These models are derived for emulsification under turbulent flow conditions and only a few of these models predict the evolution of the DSD in time (see, for example, Coulaloglou & Tavlarides, 1977, Tsouris & Tavlarides, 1994, Chen, Prüs & Warnecke, 1998). Food emulsions are typically highly viscous and the vessel is equipped with a scraper stirrer that rotates at a
Analysis of the model structure
The model consists of a coupled set of non-linear integro-differential equations. Analytical solutions of the model are not feasible, hence to arrive at a time domain solution, numerical methods are needed. The continuous PBE is discretized according to the method as described in Kumar and Ramkrishna (1996). The discretized states are denoted as Ni,b(t) (m−3) and Ni,l(t) (m−3), respectively. Ni,b(t) describes the number of drops with volume vi (m3) per emulsion volume in the Bulk compartment. Ni
Formulation of the dynamic optimization problem
The goal is to choose the stirrer speed Nst(t) and the oil flow addition rate Fin(t) as a function of the time such that a certain predefined, terminal, DSD and volume fraction is reached in minimal time. Hence the objective J is to minimize the final time tf (s) subject to the following constraints:
- 1.
The dynamic model describing the evolution of the DSD as function of time. It is given bywhere and are, respectively, the state and the input variables. The vector
Numerical solution of the dynamic optimization problem
There is extensive literature available on numerical strategies for the solution of dynamic optimization problems. An often used method is the so-called direct method where the infinite dimensional dynamic optimization problem is approximated as a finite dimensional non-linear program (NLP). Within the framework of the direct method, there are two general strategies: the sequential method (Edgar & Himmelblau, 1988) and the simultaneous or collocation method (Biegler, 1984). The sequential
Optimization of the operation procedure
In this study the MILP approach is used to choose the stirrer speed and oil flow rate trajectories such that a certain predefined DSD is reached in minimum time. First, some implementation issues are discussed. Then the results of two optimization studies are presented.
Conclusions and future work
The suggested optimization approach approximates the original minimum time optimization problem as a MILP. The MILP can be solved for its global optimum which is a good solution of the original optimization problem. The MILP approach enables to choose the stirrer speed and oil flow rate as a function of time such that a certain predefined, terminal, DSD is reached in minimum time in the stirred vessel. The feasibility of the approach is illustrated by means of several optimization studies.
References (16)
- et al.
Modelling and optimization of general hybrid systems in the continuous time domain
Computers and Chemical Engineering
(1998) - et al.
A population balance model for disperse systems: drop size distributions in emulsion
Chemical Engineering Science
(1998) - et al.
Description of interaction processes in agitated liquid-liquid dispersions
Chemical Engineering Science
(1977) - et al.
On the solution of population balance equations by discretization—I. A fixed pivot technique
Chemical Engineering Science
(1996) - et al.
Control of systems integrating logic, dynamics and constraints
Automatica
(1999) Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation
Computers and Chemical Engineering
(1984)- et al.
Applied optimal control
(1975) - et al.
Optimization of chemical processes
(1988)
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- 1
Present address: Philips Centre for Industrial Technology, Kastanjelaan 2, 5600 MD Eindhoven, The Netherlands.