Csaba Mihálykó

The hyperbolic tangent distribution family

A general description and analysis of the hyperbolic tangent distribution family are presented. Analytical formulae are given for both the cumulative and frequency functions of the distribution, as well as for the ordinary moments of whole order. A linear transformation of the size coordinate axis is applied by means of which the distribution function can be located appropriately over the interval between the observed smallest and largest sizes of particles during the fitting procedure. Including the parameters of this transformation, the distribution family possesses four parameters, which allows for effective fitting to experimental size distribution data. A two-level fitting procedure has been developed which was tested by using simulated noisy data. A number of distribution functions (log-normal, Rosin-Rammler, and beta distributions) may also be described well by hyperbolic tangent distribution functions, so that it provides a convenient method for comparing measurement data described quantitatively in different ways.

A simulation model for analysis and design of continuous grinding mills

Abstract A computer model and simulation program were elaborated for studying the stationary state processes of grinding mills. The computer model was developed on the basis of the axial dispersion model taking into consideration also the effects of mixing of the ground material. The final form of the model was expressed as a set of recursive equations, the successive solution of which converges to the stationary state of the mill. By means of the newly developed simulation program, the effects of parameters of a continuous grinding mill on the size distribution of the particulate product were studied for selection and breakage distribution functions of the form S(L) = KsLα and B(L, l) = (L/l)β, respectively. The program, however, can be utilized in the simulation-based analysis and design of both continuous and batch grinding devices by selecting the respective input—output as well as the mixing conditions of the computer model.

Modeling of heat transfer processes in particulate systems

Abstract A population balance model, taking into account the particle-particle and particle-wall heat transfer by collisions is presented for modelling heat transfer processes in fluid-solid systems. The spatial distribution of the temperature is described by the compartments-in-series with back-flow model. An infinite hierarchy of moment equations, describing the time evolution of the moments of particle temperature in cells is derived that can be closed at any order of moments. The properties of the model and the effects of parameters are examined by numerical experiments using the moment equation model. The simulation results indicate that the population balance model provides a good tool for describing the temperature inhomogeneities of the particle populations in particulate systems, and can be used efficiently for analysing the heat transfer in fluid-solid energy conversion processing.

Axial dispersion/population balance model of heat transfer in gas-solid turbulent fluidized beds

Abstract An axial dispersion/population balance model is presented for describing heat transfer processes in gas–solid turbulent fluidized beds. In the model, the gas and particle transport is described by the axial dispersion model, while the particle–particle and particle–wall heat transfers are modeled as collisional random events, characterized by the collision frequencies and random variables with probability density functions determined on interval [0,1]. An infinite hierarchy of moment equations is derived from the population balance equation, which can be closed at any order of moments. The properties of the model and the effects of process parameters are examined by numerical experimentation.

Axial dispersion/population balance model of heat transfer in turbulent fluidization

Abstract An axial dispersion/population balance model is presented for describing heat transfer processes in gas-solid turbulent fluidized beds. In the model, the gas and particle transport is described by the axial dispersion model, while the particle-particle and particle-wall heat transfers are modeled as collisional random events, characterized by the collision frequencies and random variables with probability density functions determined on interval [0,1]. An infinite hierarchy of moment equations is derived from the population balance equations, which can be closed from the first order of moments. The properties of the model and the effects of process parameters are examined by numerical experimentation.

A generalization of the Thurstone method for multiple choice and incomplete paired comparisons

A ranking method based on paired comparisons is proposed. The object’s characteristics are considered as random variables and the observers judge about their differences. The differences are classified. More than two classes are allowed. Assuming Gauss distributed latent random variables we set up the likelihood function and estimate the parameters by the maximum likelihood method. The rank of the objects is the order of the expectations. We analyse the log-likelihood function and provide reasonable conditions for the existence of the maximum value and the uniqueness of the maximizer. Some illustrative examples are also presented. The method can be applied in case of incomplete comparisons as well. It allows constructing confidence intervals for the probabilities and testing the hypothesis that there are no significant differences between the expectations.

Incomplete paired comparisons in case of multiple choice and general log-concave probability density functions

A scoring method based on paired comparison allowing multiple choice is investigated. We allow general log-concave probability density functions for the random variables describing the difference of the objects. This case involves Bradley–Terry models and Thurstone models as well. A sufficient condition is proved for the existence and uniqueness of the maximum likelihood estimation of the parameters in case of incomplete comparisons. The axiomatic properties of the method are also investigated.

Application of Advanced Integrodifferential Equations in Insurance Mathematics and Process Engineering

In this paper we consider a dual risk model with general inter-arrival time distribution and general size distribution. A special Gerber–Shiu discounted penalty function is defined and an integral equation is derived for it in the case of dependent inter-arrival times and sizes. We prove the existence and uniqueness of the solution of the integral equation in the set of bounded functions and we show that the solution tends to zero exponentially. If the density function of the inter-arrival time satisfies a linear differential equation with constant coefficients, the integral equation is transformed into an integrodifferential equation with advances in arguments and an explicit solution is given without any assumption on the size distribution. We also present a link between the Lundberg fundamental equations of the Sparre Andersen risk model and the dual risk model.