Computation on Turbulent Dilute Liquid-Particale. Flows through a Centrifugal Impeller*

Transcription

1 Computation on Turbulent Dilute Liquid-Particale Flows through a Centrifugal Impeller* Yulin WU** Risaburo OBA+ Toshiaki IKOHAGI + Abstract In present work, two-dimensional turbulent liquid- around a blade-toblade surface of a centrifugal impeller has been simulated by using a two- turbulence model, that is, between liquid solid has been considered in conservative momentum equations of two-. Calculated results show that turbulence model can predict essential features of complex liquid- through impeller. Keywords: Liquid-Particle Flow, Turbulence Model, Numerical Analysis, Centrifugal Impeller 1. INTRODUCTION Analyzing turbulent two- s has attracted considerable attention due to its numerous industrial applications. Recent progress in modeling turbulent fluid-particulate s was reported by Genchev&Karpuzov (1980), Elghobashi&Abou- Arab (1983), Pourahmadi&Humphery (1983), Besnard&Harlow (1985) Chen&Harlow (1985) [1]-[5]. Most of existing models are suitable only for relatively dilute mixtures, in which - collisional effects fluctuation energy interaction between liquid- particulate-s are negligible. During last decade, re has been an interesting development in modeling dense solid-liquid s as shown in References of Ma&Ahmadi (1988), Ahmadi&Abu-Zaid (1990) Asakura et al. (1991) [6]-[81]. In present work, refore, two-dimensional turbulent liquid- around a bladeto-blade surface of a centrifugal impeller has been simulated by using two- turbulence model, braic particulate turbulence model. In our numerical processing, a finite volume method is applied in a body-fitted coordinate system. Calculated results show that turbulence model can predict essential features of complex liquid- through a centrifugal impeller. 2. GOVERNING EQUATIONS For incompressible liquid-particulate through a centrifugal impeller, continuity-, momentum- turbulent character-equations of both liquid- particulate-s in a twodimensional cartesian coordinate system, fixed on follows: 2. 1 Liquid Phase (1) Continuity equation (1) *Received **Department of Hydraulic Engineering, Tsinghua University, Beijing, China 118 Japanese J. Multi Flow Vol.8 No.2 (1994)

2 (2) Momentum equations where (2) The algebraic turbulence model of particulate is used in this computation. So eddy viscos- (8) (9) where (3) (4) 3. TRANSFORMATION OF THESE EQUATIONS OF MOTION In many problems, boundary geometries are very complex, especially for internal problems with complicated boundaries, such as those of centrifugal impellers. So use of nonorthogonal body-fitted coordinate (BFC) systems can be beneficial in many aspects. It is not only why boundary geometries can be represented more closely using BFC systems, but also why grid-refined solution can be easily obtained. The governing equations of motion for twodimensional relatively steady incompressible liquid through such an impeller, expressed in 2. 2 Particulate Phase (1) Continuity equation (2)Momentum equations (6) where U1 U2 are contravariant velocity components in body-fitted coordinates. (11) (7) Eqs.(1) to (4) Eqs.(7) (8) respectively. Sd is extra source term resulting from non-orthogonality, that is, (12)

3 NE, NW, SE, SW. (13) where (14) 4. SOLUTION AND FINITE DIFFERENCE FORMULATION 4. 1 Discretization of Equations Discretization of Eq. (10) is performed by using finite difference approximations in transformed domain. The second order central differencing is used for approximating diffusion source terms. For convective terms, hybrid differencing scheme (Spalding, 1972) [9] is employed (i. e. using central differencing for cell Peclet number less than or equal to 2 switching to upwind differencing when cell Peclet number is greater than 2). The finite difference equation is arranged by collecting terms according to grid nodes around a central volume as shown in 1. The final expression is given by Eq. (14) in which A represents link coefficients between grid nodes B, E, W, N, S 1 Two-dimensional grid structure labeling around a grid node B Solution Procedures of Two-Phase Flows The present numerical method for solving liquidparticulate s contains following steps: (1) Guess initial velocity pressure fields of a single liquid in calculated domain. (2) Solve for velocity, pressure turbulent characters k e in single s, by using SIMPLEC method (proposed by Spalding 1980, improved by Chen Y. S. 1986) [10], [11] to get a coarse convergence solution of s. (3) Based on single solution, we solve for particulate turbulent field, by using algebraic turbulence model. (4) Calculate interaction terms between liquid particulate-s to get some sourceterms in control equations for liquid. (5) Solve again momentum equations, turbulent energy its dissipation rate equations for single but including interaction terms. (6) Set new variables as improved ones return to step (3); repeat process untill it converges SIMPLEC Solver for Incompressible Flows The governing equation solving incompressible turbulent s are nonlinear are strongly coupled. Iterative procedures are employed to drive equations to converge solutions. It is particu- 120 Japanese J. Multi Flow Vol.8 No.2 (1994)

4 larly important for such an incompressible to satisfy continuity equation momentum equations at same time. A velocity-pressure correction procedure is used in present study to drive pressure field velocity field to be divergence free. It requires a staggering grid system, in which velocity components are solved are stored at grid nodes, pressure p is located at corners of control volume of velocities. In this way, coupling velocities pressure can be also enforced. In SIMPLEC procedure, velocity corrections u'b v'b can be calculated equations: by following (15a) (15b) And pressure correction value p" can be obtained through solving a Poisson's equation (16) with source term equal to local divergence of field (u*, v*) specified. (3) Solid walls In near wall regions, to avoid need for detailed calculations wall functions are introduced are used in finite difference calculations. That is, total tangential velocity VB at point B near wall is corrected by velocity profile where YB is normal distance from point B to wall is formulated by neglecting convection diffusion of turbulent kinetic energy equation Then, in this region. we obtains (16) 5. BOUNDARY CONDITIONS 5.1 Flow in Liquid Phase (1) Inlet boundary The inlet values of radial velocity component ur1 circumferential near wall, length scale is assumed to be pro- portional to normal distance from wall, which altered. The inlet value of turbulent kinetic energy leads to kin adopts % of inlet mean kinetic energy of, that is, where l is specific length of inlet boundary, final results. (2) Outlet boundary The velocity components at outlet of calculated domain are deduced from ir immediately upstream values by adding a fixed amount, whose amount is calculated to ensure overall mass conservation. cause limited effects of upstream are known at a high Reynolds number zero gradient is stant, which implies that wall flux of k is zero. Accordingly zero normal gradient prescription for k is appropriate. For turbulent energy dissipation rate (4) Boundary condition of pressure The correction values p' of SIMPLE scheme possess zero gradient specification everywhere except inlet. Just one internal pressure p specification at a point is n required to all pressure to be calculated. 5.2 Particulate Phase Variables in particulate also possess zero normal gradient specification everywhere except inlet. At inlet velocity components are assumed to have same values as those in liquid, because re are not any experimental data on ir inlet velocity difference. Even if inlet velocity is taken as a slightly smaller value than that of liquid, calculated results may still indicate that outlet velocity is larger than liquid one, which has been verified by experiments in Refs. 121

5 [4], at [10]. And inlet known 6. can bulk be density easily volumetric slurry work, mensional, its which pump is have geometrical parameters inlet impeller outlet inlet density to condition. 11 contour to results Cv=10% s takes place mainly near outlet. That is, relative velocity components at outlet are larger than those in liquid. The calculated results agree with measured results by using LDV by Carder, et al.(1991)[13]. The results also agree with observation of individual movement by using high-speed photography by Herbich Christopher (1963)[14]. That is, s are not appreciably slowed down even at large cross-sectional area of impeller discharge ring, ir absolute tangential velocities do not increase towards tan- single show as those condition, 13 in as well 10 as results Cv=5%. of with 10 vectors its same Cv same at illustrate same velocity lines Pa) condition of relative relative contour water concentration for con- gential velocities of transporting liquid. (3) An increase in volumetric concentration (Figs ) results in a marked difference between velocities liquid ones, because particulate with a large concentration Cv as a pseudo-fluid in present model keeps more inertia from impeller inlet to dis- 15 Cv=2%. From marks se can (1) 2 be calculated results, following re- obtained. According to comparison between 2 Relative velocity field of single water (n=1450rpm,μ ~105 at 9 of 12 (2) The velocity field in particulate is different from that of liquid (Fig.10 to 15). The velocity difference between se two show lines two- show lines dition. when respectively value volumetric Its D2=330mm, contour energy for impeller. ur1=6m/s, 5 its 6 this follows: p=2.65t/m3. kinetic for dp=1.0mm, (maximum turbulent that D1=130mm, velocity vectors, so liquid- out as a centrifugal n=1450rpm, water carried of in two-di- 1993)[12], dilute diameter diameter pressure passage basically model et al. diameter 2 to are actual are speed of been impeller velocity an turbulent calculations shape (Wu, two-dimensional radial 6, 3 7, existence of solid s in impeller has a little influence on liquid at dilute concentration (Cv<10%). In liquid in impeller ( 2 6), circulation occurs near pressure surface of blade from mid span to outlet resulting from large expansion of passage sections as blades are only four as attack angle is largely positive at blade inlet at this operating condition. But existence of s affects to pressure developed by impeller. The denser concentration is in impeller, less pressure of liquid field is at impeller outlet. according because blade impeller rotating particulate RESULTS present impeller concentration. CALCULATED In in determined 3 Japanese Velocity contours of single water (Wmax=13.27m/s) r1=6m/s). J.Multi Flow Vol.8 No.2 (1994)

6 4 Pressure contours of single 5 Turbulent energy contours of single water 6 Water velocity of two- 7 Water velocity contours of two- 8 Pressure contours of two- 9 Turbulent energy of two-

7 10 Velocity field of 11 Velocity of 12 Velocity field of 13 Velocity of 14 Velocity field of 15 Velocity of 124 Japanese J. Multi Flow Vol. 8 No. 2 (1994)

8 charge ring. With an increase in concentration, velocities along blade pressure surface will be higher than those along suction surface at outlet ring. This phenomenon also agrees with observation of moving individual s in centrifugal impellers by Wu, et al. (1993) [12], suggests existence of distorted particulate distributions in impeller as shown by Roco Reinhart (1980) [15]. 7. CONCLUSIONS (1) The turbulent liquid- model used in this paper can predict essential features of this in centrifugal impellers at dilute concentrations. (2) The velocity field of particulate is different from that of liquid in centrifugal impellers, especially near outlet ring of impeller. Relative velocities of s are not slowed down near outlet. (3) At dilute concentrations, existence of s scarcely affects liquid- in impeller. But it makes pressure at impeller oulet lower than that of single a little. The denser concentration is, more impeller outlet pressure decreases. NOMENCLATURE A: link coefficients [-] C: constants : volumetric concentration dp: diameter of [m] eijk: levy-civita symbol [-] G: general generation [-] J: Jacobian of transformation [-] k: turbulent kinetic energy of liquid : turbulent kinetic energy of particulate [m2/s2] [m2/s2] w: velocity produced by rotation [m/s] : cartesian coordinates [m] GREEK LETTERS : rate of turbulent energy dissipation [m2/s3] : transformation coefficients : general diffusion coefficient : liquid viscosity : effective liquid viscosity of liquid : eddy viscosity of particulate : body-fitted coordinates [m] : density of liquid [kg/m3] : bulk density of particulate [kg/m3] : material density of [kg/m3] : constant : constant [s] : dynamic response time of : general variable : rotating speed [rad/s] REFERENCES [1] Genchev, Z. D. Karpuzov, D. S., J. Fluid Mech., Vol.101, (1980). [2] Elghobashi, S. E. Abou-Arab, T. W., Phys. Fluids, Vol.26, (1983). [3] Pourahmadi, F. Humphery, J. A. C., Physco Chemical Hydrodynamics, Vol.4, (1983). [4] Besnard, D. C. Harlow, F. H., Los Alamos National Laboratory, Report No.LA-10187MS (1985). [5] Chen, C. P. Harlow, F. R, Canadian J. Chem. Engng., Vol.63, (1985). [6] Ma, D. N. Ahmadi, G., Powder Tech., Vol.56, (1988). [7] Ahmadi, G. Abou-Zaid, S., Int. J. Non-Newtonian Fluid Mechanics, Vol.35, (1990). [8] Asakura, K. et al., ASME FED, Vol.118, Liquid-Solid Flows, (1991). [9] Spalding, D. B., Int. J. Num. Methods Eng., Vol.4, (1972). [10] Spalding, D. B., Imperial College, London, HTS/80/1 p: pressure [Pa] (1980). [11] Chen, Y. S., NASA CR (1986). p": correction value of pressure p' [Pa] r : radius [m] [12] Wu, Y. L. et al., Proc. of Int. Symp. on Aeropace Fluid Science, Sendai, (1993). Sd: general additional source term [-] [13] Cader, T. et al., ASME FED, Vol.118, Liquid-Solid Flows, 101 (1991). [14] Herbich, J. B. Christopher, R. J., Proc. of IAHR Congress, London, (1963). U1, U2: contravariant velocities in body-fitted [15] Roco, M. Reinhart, E, BHRA, Hydrotransport, Vol.7, (1980). coordinates [m/s] 125