Conventional or electrochemical bioleaching of copper from concentrates in stirred tank reactors is one of the most complex and difficult processes in hydrometallurgy. It not only is nonlinear and time variant, but also is hardly defined. The bioleaching process has been constituted of two interacting subsystems: an abiotic system, which is a mineral suspension in a solution of chemical and electrochemical compounds and gases as well as a biological system, composed of a singular or mixed culture of microorganisms. For mathematical modeling of this process, mass transfer between three different phases that is too complicated, must be taken into account [1] . On the other hand, the presence of several species of acidophilic bacteria, which have different mechanisms to dissolve minerals (contacting and non-contacting mechanisms), has complicated the nature of the biological subsystem. So the system not only is difficult to formulate mathematically, but also is even more difficult to validate experimentally due to oversimplifications of the conditions, as a consequence, the process cannot be programmed in a precise way.

The use of fuzzy logic, which reflects the qualitative and inaccurate nature of human reasoning, can enable expert systems to be more flexible [2] . It was initiated in 1965 by Lotfi A. Zadeh [3] . In a fuzzy logic model, language terms (linguistic variables) are used to convey concepts relating to the system’s components and language instruments (linguistic operators) are used to convey concepts relating to the interrelationship and dynamics of these components. Fuzzy logic systems are widely used for control, system identification, and pattern recognition problems. The main advantage of fuzzy interference with respect to traditional mathematical models lies in the fact that the relationship between inputs and outputs is not determined by complex equations, but by a set of logical rules, reflecting an expert’s knowledge [4] . In complicated process control systems, fuzzy logic is integrated with conventional PID (proportional-integral-derivative) systems and is a very useful approach in pro- cess automation [5] . Fuzzy logic can be used in order to conveniently incorporate the practical in-house operating knowledge into the control solution [6] . During the last decade, researchers have attempted to predict both physical and chemical processing of ores and concentrates using fuzzy logic systems [5] [7] -[10] . Bioleaching process has also been modeled by different methods [11] -[15] . Pazuki et al. [16] used Artificial Neural Network to optimize the bioleaching of iron from a Kaolin ore, and a good agreement was obtained between the model and experimental data. A reaction-based kinetic model for conventional and electrochemical bioleaching of copper concentrates was previously published [13] .

Considering, the multiplicity of various chemical, biological, electrochemical and operational parameters, nonlinear behavior of metal extraction in bioleaching processes, and the high ability of knowledge based systems in such complex media, in this research, a multi input-multi output fuzzy logic model was defined to predict copper and iron recovery from a flotation copper concentrate in a stirred electro-bioreactor. The proposed model predicts the nonlinear behavior of conventional and electrochemical bioleaching processes successfully.

Data used in this fuzzy logic modeling was obtained from an experimental work performed previously by the author and his coworkers [17] . In that research, a chalcopyrite copper concentrate from the Sarcheshmeh Copper Mine (Kerman, Iran) was used to perform conventional and electrochemical bioleaching processes. X-ray fluorescence (XRF) and X-ray diffraction (XRD) analyses of the sample showed 27.7% Cu, 24.6% Fe, 14.8% S, and chalcopyrite (CuFeS₂) as the major mineral and pyrite (FeS₂) as the minor one. Experiments were carried out in a three compartment electro-bioreactor at 10% (w/v) solid content. A mixed culture of mesophilic bacteria and a mixed culture of moderately thermophilic bacteria were used at 35˚C and 50˚C, respectively. Experiments were conducted in nutrient medium, 9 K; stirring rate, 450 rpm; applied potential, 420 mV (in electrobioleaching tests); initial pH, 1.8; and aeration rate, 1.3 L∙min⁻¹. The potential of the working electrode was controlled with respect to the reference electrode using a Solartron Sl 1287 potentiostat. The details of apparatus and techniques used have been previously described [17] .

A fuzzy logic model was obtained to predict the recoveries of copper and iron from a chalcopyrite copper

concentrate by conventional and electrochemical bioleaching processes. The recoveries of these metals are targeted functions for hydrometallurgical extraction of copper from copper flotation concentrates. The input variables were the method of process (bioleaching or electrobioleaching), the type of bacteria (mesophilic and moderately thermophilic bacteria) and time (day). A fuzzy relationship was developed between stated inputs and the outputs by means of “if-then” rules. The comparison of the experimental data and predicted values of the model showed a good match between them, in which the R-squared of the model was more than 0.97 in each series of data. The results showed the capability of the fuzzy model to flexibly predict complex and nonlinear bioleaching processes and it is a powerful tool for metal extraction in such a complicated system.

The authors would like to thank the National Iranian Copper Industry Company (NICICO) that allowed them to use experimental data for conducting this research.