Industrial Applications of Genetic Algorithms:Hydrocyclone Model Using Genetic Programming

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Fuzzy mathematics offers a nice alternative to neural network models. Fuzzy mathematics provides a mechanism by which the “rule-of-thumb” approach often used by humans to solve difficult problems can be incorporated into a set of traditional production rules [6]. The resulting production rules are linguistic in nature, and thus, can be easily comprehended and interpreted by a human. The linguistic rules and membership functions can then potentially be used to understand the mechanics of the system being modeled. The development of a fuzzy linguistic model is complicated only by the difficulty associated with selecting efficient rules and membership functions. There are however, automated approaches for tuning fuzzy systems.

The rules and membership functions required by a fuzzy linguistic model can be discovered using a genetic algorithm. Genetic algorithms are search algorithms based on natural genetics, and these innovative techniques have been used successfully to solve a wide variety of search problems [7]. Genetic algorithms possess a number of qualities that make them inviting for use in a self-generating fuzzy linguistic modeling system [8]. Despite the relative success of using genetic algorithms to evolve the rules and membership functions required in fuzzy systems, effectively tuning a fuzzy system can be a daunting and time-consuming task.

In this chapter, an alternative approach is considered. Here, genetic programming is used in an attempt to solve a system identification problem related to a hydrocyclone separation device. Genetic programming [9] is a genetic algorithm-based approach to machine learning in which entire programs are generated based exclusively on data. In this way, they are similar to neural networks. However, there is an important fundamental difference in the results of neural networks and genetic programming. Genetic programming results in a definite equation containing function calls described by the user. The previous chapter provides an overview of genetic programming, thus such a description is not repeated here. From the previous chapter, it is clear that complex functions developed by the user and specific to the application at hand can be incorporated into the genetic programming solution. This capability can be quite important and can help prevent the “re-invention of the wheel,” so to speak.

In the current chapter, the hydrocyclone system identification problem is described. Next, the key characteristics of the genetic programming search are described. Then, results are presented demonstrating the effectiveness of genetic programming in solving the particular system identification problem. Finally, conclusions are presented.

HYDROCYCLONE DATA FOR FUZZY MODEL DEVELOPMENT

Hydrocyclones are commonly used for separating slurries in the mineral processing industry [10], and are used extensively to perform separations in other industries as well. Hydrocyclones utilize centrifugal forces to accelerate the settling rate of particles. Since their mechanical structure is simple, durable, and relatively inexpensive, hydrocyclones are one of the most popular mineral separating devices found in industry. They are used in closed-circuit grinding, desliming circuits, de-gritting procedures, and thickening operations. Figure 15.1 shows a schematic of a typical hydrocyclone. The lower part of the hydrocyclone has a conical shape with an opening at the apex, allowing the coarse or heavier particles to be removed via the underflow. The top section of the hydrocyclone is cylindrical and is closed with the exception of an overflow pipe called a vortex finder. This vortex finder prevents the mineral sample from going directly into the overflow, while allowing the fine particles to remain in the hydrocyclone. The actual mineral separation occurs in the cylindrical section due to the existence of a complex velocity distribution that carries the fine particles out the top and the coarse particles to the apex.

Figure 15.1 A typical hydrocyclone has a conical lower section and a cylindrical upper portion.

There have been numerous efforts to model hydrocyclones. Plitt [11] developed a statistical model to predict the split size of the hydrocyclone, d50. The split size is that size particle that has a 50 percent chance of exiting in either the overflow or the underflow. Plitt’s model has proven to be quite robust and is often cited in the literature. Other fairly effective models have been developed by various researchers [12,13,14]. Despite the numerous efforts to model hydrocyclones, none of the models have been universally adopted. Most of the models are either too computationally intensive or are applicable to a limited range of hydrocyclone designs. Therefore, the intent here is to develop a robust, fuzzy-based hydrocyclone model. To facilitate this hydrocyclone modeling effort, hydrocyclone performance data has been acquired from the U.S. Bureau of Mines. This data pertains to a wide range of hydrocyclones used on a variety of mineral samples including iron, phosphate, and copper. A computer model of a hydrocyclone must consider several input parameters to accurately compute a value of d50. Figure 15.2 is a schematic of a hydrocyclone model where:

Dc | = diameter of the hydrocyclone,
Di | = diameter of the slurry input,
Do | = diameter of the overflow,
Du | = diameter of the underflow,
h | = height of the hydrocyclone,
Q | = volumetric flow rate into the hydrocyclone,
φ | = percent solids in the slurry input,
ρ | = density of the solids,
d50 | = hydrocyclone split size.

Figure 15.2 A typical hydrocyclone computer model receives various input parameters in an attempt to predict the d50 size.

The data available for the model development consists of input-output pairs. The inputs consist of the variables shown entering the model in Figure 15.2. The output is the associated d50. The data is segregated into two distinct groups: (1) a training set, and (2) a test set. In the training data set, the desired output data are provided to the fuzzy system so as to update the fuzzy system parameters in attempts to develop an accurate model. The desired output is not presented to the fuzzy model in the test set, so this test set serves as a test suite for measuring the performance of a trained system.

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