Energy and population balances in comminution process modelling based on the informational entropy Henryk Otwinowski ⁎ Department of Boilers and Thermodynamics, Faculty of Mechanical Engineering and Informatics, Czestochowa University of Technology, al. Armii Krajowej 19C, PL-42 218 Czestochowa, Poland Received 30 March 2005; received in revised form 4 April 2006 Available online 6 June 2006 Abstract The results of theoretical and experimental studies of a comminution process are presented. There are two random functions: the selection function and the breakage function in the stochastic model based on a population balance. This model enables prediction of particle size distributions of comminution products after determination of both random functions. Maximum entropy method is used in the entropy model for determining the breakage function. Two cases are analysed, based on continuous and discrete particle size distribution functions of the fed material. Apart from mass balance, the energy balance of comminution process is also used. Searched form of breakage function is determined with the application of methodology of calculus of variations. The results of experimental identification of both models are presented. The parameters that occur in the discrete form of the selection and breakage functions were the identification objects. The results of experimental investigations of quartz sand single comminution in a laboratory jet mill provided an identification base. The experimentally identified results of the entropy model confirmed the adequacy of the theoretical analysis and demonstrated the possibility of adequate prediction of particle size distributions resulting from single comminution. © 2006 Elsevier B.V. All rights reserved. Keywords: Comminution modelling; Particle size distribution; Energy balance; Population balance; Breakage functions; Informational entropy 1. Introduction The rapid development of the chemical, cement and mining industries as well as the power industry has focused attention on the problem of the efficiency and energy-consumption of commi- nution. The mechanism of comminution and the various systems of forces and stresses present in the particles of any comminuted material are very complicated. As a result, there is currently no universal theory of comminution. The extensive literature deal- ing with this problem presents results of research and mea- surements. The formulation and modification of adequate mathematical models of the phenomena occurring during the comminution process is the best way of gaining an understanding of comminution. 2. A review of selected methods of comminution modelling Technical development has made the application of highly comminuted materials increasingly necessary. The comminution of brittle solids is a highly energy-consuming process. So, in the second half of the 19th century, the first theories aimed at estab- lishing the relationship between solid comminution effect and energy used during comminution were formed. These theo- ries, also called hypotheses, are known by the surnames of their authors, such as: Rittinger [1], Kick [2–4], Bond [5–7] and Charles [8–10]. An alternative formulation of the energy hypo- theses was proposed by Djingheuzian [11–13]. Djingheuzian's hypothesis, known as the thermodynamical theory of comminu- tion, was developed by Guillot [14] and Mielczarek [15,16]. The main task of research into loose material comminution is to determine general laws concerning the evolution of the particle size distribution during comminution. Mathematical models pre- dicting the particle size distribution of the comminution product Powder Technology 167 (2006) 33 – 44 www.elsevier.com/locate/powtec ⁎ Tel.: +48 34 3250583; fax: +48 34 3250579. E-mail address: otwinowski@kkt.pcz.czest.pl. 0032-5910/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2006.05.011