A product of production in ethanolamine synthesis joint is a reaction mixture of three components: monoethanolamine (MEA), diethanolamine (DEA), and triethanolamine (TEA) that later is exposed to division into separate components through rectification and evaporation that is linked to significant energetic costs. In fact, a part of DEA in the reaction mixture does not depend on parameters of synthesis process in the studied area of their alterations and is supported at the level of average value of a selection. Parts of MEA and TEA are controlled factors and are adequately defined by the process parameters according to basic control channels.
Target products are realized in terms of market, and their realization volume is defined by a demand for separate types of products. Depending on a demand, an objective arise – receive a reaction mixture with a maximum content of the required component in order to decrease costs at the division stage, particularly:
– receive a reaction mixture with a maximum content of MEA№
– receive a reaction mixture with a maximum content of TEA.
Therefore, optimization criterion is a content of the required component in the reaction mixture: part of MEA in the reaction mixture (Y1) – for the first problem, and part of TEA (%) in the reaction mixture (Y3) for the second problem. The prepared regression models describe dependences of the composition of the reaction mixture on the initial data and parameters of the production process condition:
(1)
where Y1 is a part of MEA in the reaction mixture at its discharge from the synthesis joint (%), Y2 is a part of TEA (%) in the reaction mixture, X1 is a consumption of ethylene oxide (m/hr), Х2 is a consumption of NH3 (m3/hr), Х3 is a temperature in the upper part of the synthesis reactor (°С).
We should consider that regression model is real only within the studied range of alteration of equation parameter X1, X2, and Х3 that can often limit the search for optimal solutions. Besides, we have established a highly-predictable and statistically-adequate relation between Y1 and Y3 (Fig. 1).
Here we can see a problem of search for an optimal solution with limitations. In this case optimization problem is solved via methods of mathematical programming, in other words, methods of solving problems of finding a function extremum at a number of final vector space that is be defined by limitations, such as equalities and (or) inequalities.
The problem of mathematical programing can be generally presented as:
(2)
Where Z is the optimization criterion (target function), where Z – optimization criterion (objective function), Fi(Х) and Hj(X) are limitations, X is a vector of n-dimensional vector space of the equation parameters.
It is a standard form of putting down a problem of mathematic equation.
The number of solutions of limitation system in this case can be called acceptable multiplicity of solutions. Solving an optimization problem at a number of acceptable solutions is the multiplicity of optimal solutions.
Setting a problem in case when we need to achieve a maximum part of MEA in the reaction mixture at its discharge from the synthesis angle, looks as:
Fig. 1. Relations between parts of TEA and MEA in the product before the division. Equation of regression: YЗ = 63,847 – 0,8512·Y1; Coefficient of determination D = 0,8476; Fisher criterion F = 934,14
Limited by the following equalities:
Limited by the following inequalities:
Inequality limitations set an area, in which regression model of the process of receiving ethanolamine is adequate, and are defined by an initial statistic selection.
In a similar way we can form the second problem – search for an optimal conditions of achieving maximum part of TEA in reaction mixture before the division:
Limited by the following equalities:
Limited by the following inequalities:
Both problems are characterized by setting a target function and limitations by linear algebraic functions, in other words, these problems refer to the class of problems of linear programming. For linear problems, optimal results are usually achieved at the border of the acceptable solutions. The range of acceptable solutions is defined by solving the system of limitations, in other words, it is limited by planes in the area (Х1, Х2, Х3):
And planes:
Both problems share the same range of acceptable solutions, but their optimal solutions will differ, as there are different target functions.
Target function for the problem of optimizing the part of MEA in reaction mixture at its discharge from the synthesis joint is:
We can see that the maximum value of Y1 will be achieved under the smallest of all possible values of X1 (optimal EO) and Х3 (temperature at the top of the synthesis reactor). Within the range of acceptable solutions the smallest values of X1 = 0,5 and the smallest value of Х3 = 30,0. These limits are set by the conditions of technological process at the available equipment and are defined by the initial statistic selection. The value of factor Х2 (supply of NH3) should be found from the condition that an optimal point is located on a limitation plane
Particularly – at its crossing with planes X1 = 0,5 and the minimum value Х3 = 30,0
The optimum is reached at the point: supply of OX (X1) equal 0,5 m3/hr, supply of ammonia (X2) equal 7,3 m3/hr, and temperature of synthesis reactor (Х3) = 30 °С. The predicted composition of the reaction mixture in this case: part of MEA (Y1) equal 61,9 %, part of DEA (Y2) equal 27.0 %, and part of TEA (Y3) equal 11,1 %.
For the problem of optimizing a part of TEA (Y3) in reaction mixture before the division, the target function (optimization criterion) is:
Y3 obtains its biggest value at the line of crossing of planes that limit the range of accepted solutions from the side of the biggest values of X1 = 3,0 and Х3 = 80,0 at the point that lies on the limiting plane
Specifically, at its crossing with planes X1 = 3,0 and Х3 = 80,0.
The extremum takes place at the point: supply of OX (X1) equal 3,0 m3/hr, supply of ammonia (X2) equal 15,6 m3/hr, and temperature of synthesis reactor (Х3) = 80 °С. The predicted composition of the reaction mixture in this case: part of MEA (Y1) equal 43,59 %, part of DEA (Y2) equal 29,7 %, and part of TEA (Y3) equal 26,8 %.
Other variant of extremum problems are possible within this process, for example: finding an equation that provides for receiving the minimal part of MEA (or TEA) in the reaction mixture at its discharge from the synthesis joint.
If it is necessary to find conditions of receiving a reaction mixture with the miimum part of MEA, the target function (optimization criterion) looks as
And the system of limitation remains the same. It is obvious that within the range of the accepted solutions, the minimum value of Y1 will take place under the biggest border values of EO supply (X1 = 3,0 m3/hr) and the temperature at the top of the synthesis reactor (Х3 = 80 °С), and supply of ammonia is defined by the point of crossing between the plane
and other limiting planes. Calculations show that such point coincides with a point that provides for a maximum value of Y3 – part of TEA in the reaction mixture.
While searching condition of receiving the reaction mixture with the minimum content of TEA, our target function is:
With a set range of acceptable solutions. Supply of ammonia (Х2) should be found as a point of crossing between limiting planes Х1 = 0,5; Х3 = 30,0, and
Such point will coincide to the one that provides for the maximum content of MEA in the reaction mixture.
Extreme points are found in any definition of this problem on borders of the range of accepted solutions. We have found local extremums, and, in order to find global extremums, one should scan the whole surface of the accepted range of existing solutions. In case of necessity, this procedure must be realized within the process of optimal control over the object.
Fig. 2. Algorithm of an optimal control over the process of receiving ethanolamine
Limitations in forms of inequalities in this definition of the problem are defined by the composition of the initial statistic selection, according to which, the regression model of the control object has been received. The model is adequate within the studied area, but gives an increasing number of prediction errors as it remotes from the center of the studied area. In order to search for optimal regimes of control within a wider range of alteration in parameters of production, it is necessary to develop a determined mathematic model of the object.