Gudret Kelbaliyev

DEVELOPMENT OF NEW EMPIRICAL EQUATIONS FOR ESTIMATION OF DRAG COEFFICIENT, SHAPE DEFORMATION, AND RISING VELOCITY OF GAS BUBBLES OR LIQUID DROPS

Some new correlations are derived to estimate the drag coefficient, the shape deformation, and the rising velocity of particles moving in an infinite liquid medium. The correlations are derived in terms of the dimensionless groups such as Reynolds number (Re), Morton number (Mo), and Weber number (We). The derivations are based on the experimental data or some other correlations given in the literature. A single statement is proposed to estimate the drag coefficient for the spherical solid particles that may be applicable in the range of 0.5 < Re < 105. Similarly, some other equations are also derived to estimate the drag coefficient, the shape deformation, or the rising velocity for gas bubbles or liquid drops. The drag equation is applicable in the range of 0.5 < Re < 100 and 9 × 10−7 ≤ Mo ≤ 7; the shape deformation equation is applicable in the range of 0.5 < Re < 100 and 1.1 × 10−5 ≤ Mo ≤ 7; and the rising velocity equation is applicable in the range of 0.1 < Re < 100 and 9 × 10−7 ≤ Mo ≤ 80. The model predictions are compared with the experimental data and with the other correlations given in the literature. The results indicated that the model predictions are in a good agreement with the literature data.

A new model for estimation of drag force in the flow of Newtonian fluids around rigid or deformable particles

Abstract A number of relationships are given in the literature for estimation of the drag coefficients; however, most of these are applicable only for a narrow range of Reynolds number ( Re ). Some new empirical relationships are proposed in this study to estimate the drag coefficients in the flow of Newtonian liquids around the solid spherical or cylindrical particles and the deformable particles. An approximation series approach is used to develop these empirical relationships. It is shown that the proposed relationships can be applicable for the solid spherical or cylindrical particles in the range of 0.1> Re >10 6 , and for the deformable particles (drops and bubbles) in the range of 0.1> Re >10 4 . The predicted coefficients are in a good agreement with the experimental data given in the literature.

The roughness effects on friction and heat transfer in the fully developed turbulent flow in pipes

Convective heat transfer coefficient is strongly influenced by the mechanism of flow during forced convection. In this paper, the effect of pipe roughness on friction factor and convective heat transfer in fully developed turbulent flow are briefly discussed. A correlation for the friction factor applicable in the region of transition to the fully developed turbulent flow regime is proposed. Using this relationship, some new approximation formulae are proposed to predict the convective heat transfer coefficients in the pipes with a relative roughness of e/D⩽0.05. The effectiveness parameter for the heat transfer is investigated as a function of the pipe roughness, Reynolds number and Prandtl number. The effect of fouling is also briefly discussed. The predictions of the proposed correlations are compared with the experimental data and with some other previous correlations given in the literature.

Eddy diffusivity of particles in turbulent flow in rough channels

Abstract The eddy diffusivity coefficient of particles depends on the flow characteristics and on the physical properties of the particles. Experimental data and theoretical considerations suggest that the ratio of the eddy diffusion coefficient of particles ( D TP ) to the eddy diffusion coefficient of fluid ( D T ) is dependent mainly on the friction velocity of fluid (U ∗ ) , the sedimentation velocity of the particles ( V s ), the Reynolds number ( Re ) of the flow and also the relative roughness of channel ( e / D ). It is shown here that this relation among these parameters may be represented as D TP /D T = μ R ∝(U ∗ m /V S ) n , where m and n are empirical constants. Some new empirical relationships are proposed for the eddy diffusion coefficient of particles in the turbulent flow in vertical or horizontal channels. The results compare favorably with experimental data from the literature.

Kinetics of Drop Breakage in Stirred Vessels under Turbulent Conditions

The emulsification of dodecane oil was studied in a standard stirred vessel filled with distilled water. The in situ size distribution of the drops was measured at certain time intervals by using a particle size analyzer while stirring. The results showed that an increase in the stirring time and stirring rate resulted in a decrease in the mean drop size of drops and shifted the drop size distribution toward smaller sizes. The minimum droplet size in the turbulent flow was found to be proportional to amin∼(WD)−1.75. Normalization of data exhibited that the size distribution of drops is self‐preserving. The distribution of mean drop size and distribution functions of drops were determined by using models developed from the Focker‐Plank equation regarding size distribution of particles in turbulent flows. The evaluation of droplet breakage with time in isotropic turbulent flow showed that the size distribution can be represented by the summation of log‐normal distribution function in series. It was found that the experimental data could be represented well by the empirical models.

Estimation of the maximum stable drop sizes, coalescence frequencies and the size distributions in isotropic turbulent dispersions

Abstract The process of coalescence or breakup of drops in turbulent flow is of importance in many technical applications. A new size distribution takes place due to the coalescence or the breakup of the drops during the motion of a dispersed system. Based on the experimental data given in the literature, some new empirical relationships are developed in this paper to evaluate the maximum stable drop sizes, the coalescence frequencies and the drop size distribution in an isotropic turbulent flow. The relationships are developed essentially in terms of the particle Reynolds number or of the physical properties of the system. The Focker–Planck equation is used to estimate the particle size distribution. The model predictions are compared with the experimental data given in the literature. The results indicated that the predicted values and the experimental data are in a good agreement.

Estimation of the Minimum Stable Drop Sizes, Break‐up Frequencies, and Size Distributions in Turbulent Dispersions

The breakage of drops or bubbles in isotropic turbulent dispersions has been investigated. Based on the experimental data given in the literature, some new empirical relationships are derived to evaluate the minimum stable drop sizes, the breakup frequencies, and the drop size distribution in turbulent dispersions. The solutions of the stochastic Focker‐Planck equation are used to estimate the particle size distribution. The relationships are expressed essentially in terms of the system properties. In addition, a new relationship was proposed to estimate the ranges of Reynolds, Weber, or Morton number, at which the drag coefficient is minimum. The model predictions are compared with the experimental data given in the literature.

Stochastical modeling of the granule size distribution in the agglomeration processes of powdered materials

Abstract Powders with the correct amount of a liquid binder can be formed into agglomerates by vibrating, shaking or paddle mixing. Rotating drums or disks are commonly used in industry. The break-up process of the drops of the binding liquid and the kinetics of the powder agglomeration affect the granule size distribution in the agglomerate. Generally, mathematical procedures are required to estimate such size distributions. A theoretical model is presented in this study to emphasize the major system parameters affecting the particle size distribution during agglomeration. First, an analytical relationship is derived to represent the changes in the droplet volume with time. Then, Focker–Planck equation is used to predict the granule size distribution in an agglomeration process. The model predictions are compared with the experimental data. The results indicated that the model predictions and the experimental data are in a good agreement.

A theoretical model for the particle distribution in a polydispersed solid mixture under hydrodynamic and gravitational effects

Abstract The particle size distribution in a polydispersed mixture significantly affects the technological behaviour of the mixture. Generally, mathematical procedures are required to estimate the particle size distribution or concentration distribution in a sorting process of such a mixture. A theoretical model is presented in this study to emphasize the major system parameters affecting the particle size distribution under hydrodynamic and gravitational effects. First, an analytical relationship is derived to represent the concentration profile in a sorting column during the classification of polydispersed mixtures. Then, the Focker–Planck equation is used to obtain the distribution function depending on the particle size. Some of the parameters of these analytical relationships are evaluated by comparing the equations with experimental data. The results indicated that the model predictions and the experimental data are in good agreement.

Modeling of Drop Coalescence in Isotropic Turbulent Flow

Coalescence of drops in fully developed turbulent flow depends on the size of drops and the properties of the flow. By comparing the size of drops with the Kolmogoroff length scale, collision frequencies of drops have been determined. If λ>λ0, then the collision frequency of drops in a gaseous medium is given as ωc∼(ϵR/a2)1/3 and for λ<λ0, it is given as ωc∼(ϵR/νc)1/2 in a liquid medium. New expressions for the fluctuation, thinning, and breaking of the intervening film between drops that is formed due to collisions were also suggested. At various Re and Mo numbers, for the calculation of maximum stable sizes of drops, some equations are suggested. In order to evaluate coalescence and break up rates a new dimensionless number, , is introduced. This number is defined as the ratio of the energy of the turbulent flow to surface energy. As a result of the coalescence of drops, evolution of the distribution function with time is determined from the solution of the Focker‐Planck equation. Comparisons of the calculated drop sizes with the experimental data reported in literature showed good agreement.