Development of a kinetic model of magnetite leaching

The object of the research is the process of magnetite leaching with nitric acid solutions, and the subject of the research is the mathematical justification of the kinetic model and the calculation of kinetic parameters. The article considers the case of leaching in the kinetic region, while magnetite is considered as a polydispersity material of spherical shape. It is proposed to use the distribution function of the number of particles N by their radius r in the form N=a∙rb, where a and b are constants. This distribution was used to derive the equation for the rate of the process W, taking into account the change in the surface of the particles depending on the degree of leaching α: W=dα/dτ=K*∙(1–α)m∙((С0(γ–α)/γ))n, where K* is the rate constant; m and n are the order of solid material and nitric acid, respectively; C0 and γ are the initial concentration of nitric acid and its stoichiometric excess. The order m is defined as m=(b+2)/(b+3), at b→∞ the order m→1. When b=0, m=2/3 is the case of the equation for a shrinking sphere. An algorithm for calculating kinetic parameters in the Excel is proposed. Experimental dependences of the degree of transformation α on time τ are approximated by a third-order equation; by differentiating the obtained equation, the values of the velocity Wexp=dα/dτ at individual points are calculated. After the logarithm of the above equation, there is the expression (γ=1): ln(Wexp)=ln(K*)+m∙ln(1–α)+n∙ln(С0(1–α)). With the help of the «LINEST» function in Excel, for a temperature of 373 K, the values of the order of m=0.93 and n=1.29 and the rate constant K*=0.08 were obtained. Calculation of kinetic parameters for different temperatures takes into account the dependence of the rate constant on temperature: ln(Wexp)=lnk0–E/R∙1/T+m∙ln(1–α)+n∙ln(C0(1–α)). As a result of the calculations, the values n=0.83; m=1.2; E/R=–10402; lnk0=25.09 were obtained. The value of the multiplier k0=exp(lnk0)=7.88∙1010, the activation energy E=–8.31∙E/R=86440 J/mol, the total reaction order n+m=2.03 was calculated. The obtained kinetic parameters were used to determine the calculated values of the rate W. The average relative error between the experimental and calculated values of the leaching rates is 10 %. The proposed method of processing experimental data using a mathematical leaching model can be used for any leaching process.