Mixture designs problem

In some mixture design problems (such as formulations), it may not be necessary to consider processing issues and hence we would not have the process model constraints. In this case the problem becomes a simple mixing problem, which would already have been addressed by the miscibility criteria in sub-problem 4M. Hence, for these problems, we will not need sub-problem 5M. Also in some cases we might have to identify a mixture whose constituents perform different functions such as solvents and anti solvents for crystallization. In such cases we would have to formulate and solve more than one single compound design problems to identify the constituents and then solve the final two sub-problems to identify the optimal mixture. In certain cases we may not have process model constraints, however, we may still have to solve an optimization problem with other constraints, in sub-problem 4 and sub-problem 5m respectively. 

Since in this mixture design problem we have to identify a mixture whose constituents perform different functions, i.e., the solvent needs to have high solubility for the solute while the anti-solvent needs to reduce the solubility, we have to solve two different single compound design problems (involving subproblem 1M, 2m and 3M) to identify the candidate solvents and anti-solvents. The mutually miscible pairs are identified in sub-problem 4M and the final optimisation problem is solved in sub-problem 5M. 

J.H. de Boer, A.K. Smilde and D.A. Doombos, Introduction of a robustness coefficient in optimization Procedures Implementation in mixture design problems. Part I Theory, Chemometrics and Intelligent Laboratory Systems, 7 (1990) 223-236. 

Before discussing more complicated models and systems with more than three components, we present a mixture design problem with real data, performed in the laboratory of Professor G. Oliveira Neto (Chemistry Department, Campinas State University). 

A common problem in pre-formulation of the cosmetic product including lipstick is the optimisation of the mixture composition aimed to obtain a product with the required characteristic. Mixture design represents an efficient approach for solving such optimisation problem [10]. It has been proved to be an effective tool to select the best lipstick formulation [11]. 

If the design problem in the absence of significant constraints can be decoupled in this way, there must be some mechanism behind this. Take two different sequences for the separation of a four-component mixture, Figure 21.103. Summing the feed flowrates of the key components (see Chapter 9) to each column in the sequence, the total flowrate is the same in both cases, Figure 21.10. However, the flow of nonkey components is different, Figure 21.10. 

In many design problems, the determination of a wall heat-transfer coefficient or the heat flux between the tube wall and the fluid mixture is only part of the required information. The pressure drop within the system, the rate of phase change at the gas-liquid interface, the point at which the tube walls become dry, and the holdup of the fluids at each point in the pipe must all be determined. 

Considering the various types of constraint equations, a general CAMD problem (Molecular and Mixture design) can be formulated as the following MINLP. 

The constraints in the general CAMD problem are first decomposed into two parts, namely pure component design part and mixture design part as shown in Figure 2. If we are interested in single compound solvent design, only the first part is needed while if we are interested in mixture design both parts are needed. 

The mixture design CAMD problem is also solved as a series of sub-problems. Here promising pure component solvents are designed first and then candidate solvent mixtures are identified. The first three sub-problems deal with the design of pure component solvents. 

Sub-problem 3M considers the mixture property constraints. The molecules from sub-problem 2M are considered in this sub-problem. The starting point is a list of promising solvents. From this list of solvents, the optimal mixture and the compositions of the constituents are identified by solving sub-problem 4M and sub-problem 5M. Since the first three sub-problems in the mixture design involves designing pure component solvents, these sub-problems are essentially the same as the first three sub-problems in single compound design. 

This CAMD mixture design (solvent/anti-solvent) problem is formulated as an MINLP model as shown below. The objective function and the various property constraints in the model are discussed subsequently... 

This problem encompasses two single compound CAMD problems, namely design of solvent and anti-solvent and then identification of optimal mixture pair and its composition. The single component solvent design problem is the same as in case study 1 (Sub-problems 1, 2 and 3). The 10 molecules that are designed in the first case study are considered here. The single component antisolvent design proceeds as follows... 

Newman, S.A., "Novel Applications of Phase Equilibria Methods to Process Design Problems." Paper presented at National Physical Laboratory Conference on Chemical Thermodynamic Data on Fluids and Fluid Mixtures, Teddington. Middlesex, U.K., September 11-12, 1978. Conference Proceedings published by IPC Science and Technology Press. 

Calculations of the relations between the input and output amounts and compositions and the number of extraction stages are based on material balances and equilibrium relations. Knowledge of efficiencies and capacities of the equipment then is applied to find its actual size and configuration. Since extraction processes usually are performed under adiabatic and isothermal conditions, in this respect the design problem is simpler than for thermal separations where enthalpy balances also are involved. On the other hand, the design is complicated by the fact that extraction is feasible only of nonideal liquid mixtures. Consequently, the activity coefficient behaviors of two liquid phases must be taken into account or direct equilibrium data must be available. 

Provide clear definition of the design problem. Collect sufficient engineering data. Get a comprehensive picture of chemistry and reaction conditions, thermal effects and chemical equilibrium, as well as about safety, toxicity and environmental problems. Examine the availability of physical properties for components and mixtures of significance, identify azeotropes and key binaries. Define the key constraints. 

For zeotropic mixtures the problem is not the feasibility of separation, because a suitable column can always be designed, but the optimal sequencing of splits. The optimality criterion should be the total cost of separations, in term of investment and operation. 

Problem 2.11 Use of a Constrained Mixture Design to Investigate the Conductivity of a Molten Salt System... 

Problem 2.15 A Four Component Mixture Design Used for Blending of Olive Oils... 

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