The flotation rate constant is the value characterizing the kinetics of cyclic flotation. In the statistical theory of flotation its value is the function of probabilities of collision, adhesion and detachment of particle from the air bubble. The particle -air bubble collision plays a key role since there must be a prior collision before the particle -air bubble adhesion happens. The probability of such an event to occur is proportional to the ratio of the particle diameter to the bubble diameter. When the particle size is given, it is possible to control the value of collision probability by means of the size of air bubble. Consequently, it is significant to find the effect of physical and physicochemical factors upon the diameter of air bubbles in the form of a mathematical dependence.In the pneumo-mechanical flotation machine the air bubbles are generated by the blades of the rotor. The dispergation rate is affected by, among others, rotational speed of the rotor, the air flow rate and the liquid surface tension, depending on the type and concentration of applied flotation reagents.In the proposed paper the authors will present the distribution of air bubble diameters on the grounds of the above factors, according to the laws of thermodynamics. The correctness of the derived dependences will be verified empirically.Keywords: flotation, dispersion of bubbles, bubble size distribution, Rayleigh distribution Flotacja jest procesem masowym, o którego przebiegu decyduje szereg zdarzeń losowych. Są nimi zderzenia ziarna z pęcherzykiem powietrza oraz trwała adhezja ziarna do powierzchni pęcherzyka. Ze względu na losowy charakter wymienionych wyżej zdarzeń można mówić jedynie o prawdopodobieństwie zajścia zdarzenia. Opisując subprocesy flotacji określonymi prawdopodobieństwami można wyznaczyć wartość stałej prędkości flotacji. Warunkiem koniecznym do zajścia adhezji jest uprzednie zderzenie ziarna z pęcherzykiem. W modelu opracowanym przez zespół Yoon i Luttrell (1989) W tym przypadku do wyliczenia średniej wartości stałej prędkości flotacji niezbędna jest znajomości rozkładu wielkości pęcherzyków i wielkości ziarna. Również dla wyliczenia całkowitej powierzchni pęcherzyków przepływających przez jednostkę powierzchni przekroju poprzecznego w jednostce czasu, konieczna jest znajomość rozkładu wielkości pęcherzyków.W prezentowanej pracy przedstawiono sposób określenia rozkładu wielkości pęcherzyków w pneumo-mechanicznej maszynie flotacyjnej, oparty na rozważaniach heurystycznych.W procesie dyspergowania powietrza podawanego do komory flotacyjnej rośnie powierzchnia pęcherzyków powietrza, powstających z jednostki objętości gazu, co pociąga za sobą wzrost energii powierzchniowej na granicy faz ciecz-gaz. Założono, że duże bańki powietrza dzielą się na mniejsze pęcherzyki według prawa Boltzmanna rozkładu energii (wzory 10, 12 i 13).Przy założeniu, że energia pęcherzyków jest zmienną losową o rozkładzie ciągłym, wyliczono funkcję gęstości rozkładu tej zmiennej, wyrażoną wzorem (16). Średnica pęcherzyka powietrza, a zaraze...

The particle settling velocity is the feature of separation in such processes as flowing classification and jigging. It characterizes material forwarded to the separation process and belongs to the so-called complex features because it is the function of particle density and size. i.e. the function of two simple features. The affiliation to a given subset is determined by the values of two properties and the distribution of such feature in a sample is the function of distributions of particle density and size. The knowledge about distribution of particle settling velocity in jigging process is as much important factor as knowledge about particle size distribution in screening or particle density distribution in dense media beneficiation.The paper will present a method of determining the distribution of settling velocity in the sample of spherical particles for the turbulent particle motion in which the settling velocity is expressed by the Newton formula. Because it depends on density and size of particle which are random variable of certain distributions, the settling velocity is a random variable. Applying theorems of probability, concerning distributions function of random variables, the authors present general formula of probability density function of settling velocity for the turbulent motion and particularly calculate probability density function for Weibull's forms of frequency functions of particle size and density. Distribution of settling velocity will calculate numerically and perform in graphical form.The paper presents the simulation of calculation of settling velocity distribution on the basis of real distributions of density and projective diameter of particles assuming that particles are spherical.Keywords: jigging process, settling velocity, turbulent motion, random variable, function of random variable Prędkość opadania ziarna jest cechą rozdziału w takich procesach przeróbki surowców jak klasyfikacja czy wzbogacanie w osadzarce. Cecha ta opisuje materiał kierowany do procesu rozdziału i należy do tzw. cech złożonych, ze względu na to, że jest funkcją dwóch cech prostych, którymi są: wielkość ziarna i gęstość ziarna. Przynależność do określonego podzbioru ziaren jest określona przez wartość dwóch cech, a rozkład tych cech w próbce jest funkcją rozkładów gęstości i wielkości ziarna. Znajomość rozkładu

The paper presents the results of analyze the particle size distribution function of comminution products of dolomitic type of copper ore. The breakage tests for single irregular particles were performed with using a hydraulic press device. The authors prepared five particle size fractions of each material, within ranges: 16-18 mm, 18-20 mm, 20-25 mm, 25-31,5 mm and 31-45 mm. The particle size distribution function of single-particle breakage test was calculated separately for each size fraction. In addition, the cumulative particle size distribution function for five particle size fractions was presented. In theoretical part the study of applied functions of particle size distribution for comminution a set of particles and models of crushing single particles was performed. In that paper the curves of the particle size distribution were approximated by the three-parameter function, which parameters depend on the particle strength and material type. For conformity assessment the model distribution function to the empirical distribution function a residual deviation and non-linear correlation coefficient were calculated. The three-parameter function approximating agrees well with the particle size distribution obtained from experimental data. The dependence of the parameters of a particle size distribution function on the dolomite particle strength was presented. The results indicate the identity of single particle grinding mechanism by slow compression of irregular particles of dolomitic type of copper ore, regardless of the initial particle size.

Beneficiation of chromium waste By means of magnetic and gravitational separation odzysk chromu z odpadów chromowych z wykorzystaniem separacji magnetycznej i grawitacyjnejThe paper presents the results of investigations of chromium recovery from the Cr waste mud, originating in the production of sodium dichromate. The differentiation of physical properties of particles of different contents of chromium and magnesium compounds is a premise for the use of physical separation methods. The investigations were performed on 2 waste samples, taken by random from the waste dump. The investigations of magnetic and gravity concentration for sample i and ii were made, respectively. The results of screen and float and sink analyses of sample ii indicate that it is possible to obtain Cr-and Mg-enriched products by means of a combined method of separation, i.e. hydraulic classification and, next, gravitational concentration on the shaking table. The concept of gravity concentration of the tested chromium waste was verified in the technical scale on multiboard shaking tables. The advantage of these fittings in relation to laboratory conditions is the possibility of separation of discussed waste without previous classification. The longer time of separation on the industrial shaking table as well as the respective amount of water directed to the process together with feed enabled an exact separation of initial material. The obtained concentrate in such conditions contains from 25 to 35% Cr 2 O 3 , depending on the chromium content in the initial waste material. it results from the fractional magnetic analysis of sample i that by means of the magnetic separation it is possible to obtain the magnetic concentrate of Cr 2 O 3 content minimum 20% and, simultaneously, to obtain the non-magnetic product of increased Mgo content. The optimum separation conditions regarding the yield and content of Cr 2 O 3 were determined. The highest value of criterion K for the grade of the concentrate containing over 20% of Cr 2 O 3 is obtained when the 10 mm diameter balls and the 1150 kA/m magnetic field intensity are applied. The yield of the concentrate equals about 9.5 % and recovery of Cr 2 O 3 almost 22%. in the non-magnetic product the content of Cr 2 O 3 equals 7.74% and Mgo 27.36%.

In continuous enrichment processes, in industrial conditions and due to the generally required high efficiency of machines, a relatively high concentration of feed is applied. Since it is necessary to liberate the useful minerals, the feed has to be ground thoroughly and this fact results in the percentage growth of the fraction of particles measuring a few tens or several micrometers. When the particles are so fine and the feed is so much concentrated, the magnetic particles interactions play a significant role and they lead to the phenomenon of magnetic flocculation. The flocs contain the particles of waste rock and intergrowths. In order to obtain the final product of high quality the material is subjected to multiple enrichment operations. This paper presents a mathematical model of multiple magnetic separation in which the grade of the magnetic concentrate was expressed as the function of multiplication ratio of separation and the value of external forces acting on a particle as well as interactions of the magnetic particles. The dependence of the content of magnetic component in the highgradient separation concentrates on the multiplication ratio of separation, flow velocity of the suspension through the separator working space, the intensity of matrix vibration and the concentration of solid particles in the feed for the separation of magnetite ore was verified. The verification was carried out according to the author's own research and the reference. Keywordswill denote the probability of the non-magnetic particle remaining in the main material stream to a point s or, in a different way, the probability of non-separation of the non-magnetic particle into the non-magnetic product on the separation path s when the conditionis fulfilled.A non-separation of the non-magnetic particle in two consecutive sections of the effective path s and u (s, u > 0) will be a pair of independent events. Therefore the probability of non-separating the 216 M. BROEK non-magnetic particle in the joint section of the effective path s + u will be, according to the formula of total probability, equal to [2] P(r > s + u) P(r > u)e(r > s),After differentiating expression (3b) in relation to u (assuming that there is a density f'(s + 0), s >_ 0), we can obtainDividing both sides of Eq. (3b) by Eq. (4) we obtainwhere -/z =f'(0) < 0 becausef(0) is the maximum value of function f(s) P(r > s), s > O. From Eq. (6) after considering condition (2) we obtain the following solution f(s) =e-"s.As it was said before, f(s) denotes a probability of non-separation of the non-magnetic particle of susceptibility n into the non-magnetic product. The measure of this probability will be constituted by a number of particles of susceptibility Xn, contained in the magnetic product after travelling the path s (number of non-separated particles)to the total number of particles of this susceptibility in the feedTherefore from expression (7) The balance equation for the non-magnetic component at the division of the feed into two products, i.e. magnetic and ...