UDC 6.7 Gravity separation V.I. KRIVOSHCHEKOV, PhD. (Ukraine, Dnepropetrovsk, National Mining University),.A. NOVIKOV (Ukraine, Dnepropetrovsk, Institte of Geotechnical Mechanics named by of N.S. Polyakov NAN of Ukraine) FEATURES OF ADJACENT STREAM HYDRODYNAMICS OF THE TWO-PHASE MEDIUM The analysis of the characteristics of local areas of the slrry flow in the hydrocyclone withot perechistki and wall-perechistkoy. Problem and its connection with scientific and practical tasks. The perspective direction on efficiency increase of water mineral sspension division in a hydrocyclone is a new flow strctre formation of the two-phase medim with se of these or another hydrodynamic effects. At the same time the whole crrent area can be relatively presented in the form of local zones with an individal speed field and firm particle concentration. The flow past hydrocyclone constrctive elements is accompanied by emergence of local pressre losses of the two-phase medim which vale is defined by geometrical parameters of a crrent, concentration of a firm phase in a local zone, intensity of trblence and interphase interactions. Despite of division efficiency increase, se of hydrodynamic effects can be accompanied by increase of local pressre losses and energy consmption on hydrocyclone acting. In particlar it belongs to detachable crrents and to opposite directed adjacent streams of the two-phase medim. Analysis of researches and pblications. One of research tasks at two-phase hydrocyclone crrent stdy is fnctional connection establishments between diffsion coefficients of firm particles and their concentration taking into accont intra phase interactions [, ]. Natre of these interactions is defined by the size and the direction of a liqid phase speed vector in adjacent layers, and also by their trblization degree [3]. Paper [4] is devoted to research of laminar two-phase crrents on an inclined srface. However the two-layer scheme of a laminar crrent offered in this paper isn't adapted for trblent flow, and also doesn't allow investigating featre of adjacent layers interaction with the different vale of speed gradient and its direction. The key hydrodynamic parameter defining interaction natre of adjacent the twophase layers in local crrent zones in a hydrocyclone [5] is the vale of laminar and trblent tangential stress [6]. Problem definition. The prpose of this paper is hydrodynamics featres research of the two-phase adjacent streams. Statement of a material and reslts. et s consider a conterflow and direct-flow crrent of the two-phase medim (the Newtonian liqid and the sspended firm particles) between two solid srfaces (fig., ). Dressing of minerals, 03.. 54(95)
a) b) Fig.. Speed diagram (a) and the circlation scheme (b) on relative border section of conter flow crrents of the two-phase medim: transitional zone of the slowed-down crrent; local zone of a conter flow crrent;, - the characteristic vales of local zones of conter flow crrents; λ,-λ characteristic diameters of whirlwinds in local zones;, average speeds of conter flow crrents a) b) Fig... Speed diagram (a) and the circlation scheme (b) on relative border section of the two-phase direct-flow crrents: transitional zone of the accelerated crrent; local zone of a direct-flow crrent;, average speeds of direct-flow crrents Speed increase of the two-phase medim at its conter flow (fig., a) will be observed in the range from х = -х to х = - and from х = х to х =. And on a local zone border (fig., b) at х = - and х = the speed vale accepts the maximm vale then it decreases as approaching to relative limit of streams section (axis 0Y). The last fact is connected with energy increase of losses on friction at interaction of conterflow crrents. Dressing of minerals, 03. 54(95)
As >, in a local zone of a conterflow crrent it is formed the bigger volme of circlation on relative limit of streams section (fig., b). At the same time tangent vale components of the Newtonian liqid strain tensors τ is directly proportional to its dynamic viscosity μ and the derivative speed of shift d/dx (a tangent component of deformation speeds tensor), i.e. d. () dx At a direct-flow crrent of the two-phase medim (fig., a) the increase in its speed will be observed in the ranges from х = -х to x = 0 and from х = х to x = 0, and along relative border of streams section (fig., b) it is formed the small circlation volme de to insignificant distinction of crrents speeds that leads to vale τ decrease in comparison with a conterflow crrent. et s examine a firm particle migration with an eqivalent diameter d 0 < λ (d 0 < λ ) in a local zone of a conterflow crrent (fig., b) in the range from х = - to х =. We will consider that the particle repeats the Newtonian liqid behavior. The observation time of a firm particle migration in a considered local zone is t >, () D where D diffsion coefficient of a firm particle [7]. Dring a period τ the firm particle is in area 0 and moves in the positive direction along an axis 0y with an average speed of, and dring a period τ it is in area - 0 and moves in an opposite direction with an average speed of. At the same time firm particle shift along ordinate axis dring t = τ + τ will be defined as y = τ τ. (3) et τ τ τ and τ τ τ, where τ, τ expectation τ and τ, and Δτ = Δτ deviations from them. Then at t = 0, y = 0 we will receive y. (4) τ τ τ τ As t and Dressing of minerals, 03.. 54(95) t two members in the eqation (4) are eqal zero. Then ;, nder a condition: = = const the first ( ) (5)
Dressing of minerals, 03. 54(95) Therefore, having determined the left part vale of the system second eqation (5), it is possible to calclate the average sqare vale in the particle spent time fnction in the crrent certain area. At firm particle migration in a local zone of a direct-flow crrent (fig., b) its movement happens in the positive direction along an axis 0y with average speeds of,, and shift vale along an coordinate axis dring time t = τ + τ is determined by a formla y = τ + τ. (6) Setting expectations τ, τ and deviations from them Δτ = Δτ by analogy to a case of a conterflow crrent, at t = 0, y= 0 we receive : τ τ τ τ y. (7) Then nder a condition: = = const the system of the eqations (5) has a form:. ) ( ; (8) Ratios (3), (6) for the firm particle shift vale along an axis 0Y don't consider vortex character of the two-phase medim near relative limit of streams section (fig., ). Accepting the movement speed of a stream vortex center eqal to the average speed, for a firm particle shift vales at a conterflow crrent of the two-phase medim we receive: ; sin ; sin y y (9) For a direct-flow crrent of the two-phase medim we write down:
y sin ; y sin ; y y y. Gravity separation (0) For probability density determination of a firm particle finding in a point with coordinates x, y in the timepoint t in a local zone of conterflow and direct-flow crrents of the two-phase medim it is possible to se the eqations of Fokker-Plank- Kolmogorov [8]: D U t x x D U t x x of x 0; of x 0, () where ω fnction of probability density; U, U firm particle speeds. Additional conditions to the eqation system () are: 0 x 0 x of of x ; x. () Formation of vortexes close to relative border section of the two-phase medim streams leads to appearance of longitdinal and cross plsation components of phase speed. At the same rime, the greatest these plsations vales are characteristic for a conterflow crrent as in this case an exchange process of an implse and energy between opposite directed streams are more intensive, than at a direct flow. Ths, even at a laminar movement mode of the two-phase medim, distinction of speed gradients and the direction of the crrents divided by relative border, can lead to appearance of trblent adjacent layer mixing. Then the parameters, and λ, λ will characterize macro - and trblence micro scale accordingly. Expression () represents laminar tangent tension. For trblent flow the vale of tangential tension are defined as [9] Dressing of minerals, 03.. 54(95) v. (3) t where ρ the Newtonian liqid density; ', v' longitdinal and cross plsation com-
ponents of the Newtonian liqid speed. The vale of trblence intensity in a local zone of the two-phase conterflow and direct-flow crrents is determined by formlas [3, 7]: I I 3 / 3 v ; 3 / 3 v, (4) (5) where v,, longitdinal mean sqare plsation components of stream speed; v cross mean sqare plsation components of stream speed. Conclsions and directions of frther researches: The tangential tension increase of viscos liqid on section relative border of conterflow crrents of the two-phase medim in comparison with a direct-flow crrent is connected with a bigger difference of streams speeds and more intensive vortex formation of a liqid phase accordingly; Considerable exchange intensity of an implse and energy between the adjacent two-phase layers at its conterflow crrent in a hydrocyclone leads to local loss increase of pressre and energy consmption for the hydrocyclone action. Frther researches of athors will be directed on research of fields of a concentration fineness of firm particles in conterflow and direct-flow streams of the twophase medim. References. Нигматлин Р.И. Основы механики гетерогенных сред [Текст] / Р.И. Нигматлин. М.: Нака, 978. 336 с.. Фортье А. Механика сспензий [Текст] / А. Фортье. М.: Мир, 97. 64 с. 3. Шлихтинг Г. Теория пограничного слоя [Текст] / Г. Шлихтинг. М.: Нака, 974. 7 с. 4. Проблемы гравитационного обогащения титано-цирконовых песков [Текст] / Б.А. Блюсс, А.М. Сокил, О.Г. Гоман. Д.: Полиграфист, 999. 90 с. 5. Кривощеков, В.И. К разработке пневмосепаратора с осевым вводом исходного продкта [Текст] / В.И. Кривощеков, Л.А. Новиков // Збагачення корисних копалин: Нак.-техн. зб. 0. Вип. 44(85). С. 86-96. 6. Лойцянский Л.Г. Механика жидкостей и газа [Текст] / Л.Г. Лойцянский. М.: Нака, 970. 906 с. 7. Кривощеков. В.И. Определение коэффициента диффзии твердых частиц в трблентном потоке пльпы [Текст] /В.И. Кривощеков // Збагачення корисних копалин: Нак.- техн. зб. 999. Вып. 4(45) С. 77-78. 8. Левич В.Г. Физико-химическая гидромеханика [Текст] / В.Г. Левич. М.: Физматгиз, 959. 5 с. Dressing of minerals, 03. 54(95)
9. Лойцянский Л.Г. Механика жидкостей и газа [Текст] / Л.Г. Лойцянский. М.: Нака, 970. 906 с. Krivoshchekov V.I., Novikov.A., 03 Надійшла до редколегії 7.09.03 р. Рекомендовано до пблікації д.т.н. Б.О. Блюсом Dressing of minerals, 03.. 54(95)