SD Numerical Simulation Technique for Hydrodynamic Flow Gas-Solids Mixing
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1 SD Numerical Simulation Technique for Hydrodynamic Flow Gas-Solids Mixing Presented by: Irla Mantilla N. National University of Engineering, Lima - Perú
2 Abstract We formulate a new mathematical model for a combustion chamber hydrodynamic fluid bed system CFB in thermal coal or solid waste power plants. This mixture model is based in conservation equations mass and momentum. This model gas - solid is obtained from two-phase hydrodinamic model, which takes into account a parameter ε ratio densities gas/solid, it generates a free boundary problem. Making an asymptotic adjustment and uncoupling of the dependent variables, then this problem has solution. The numerical simulation in 2D is implemented with COMSOL Multiphysics.
3 Content 1. Problem Formulation 2. Theoretical Analysis-Contribution 3. Numerical Resolution: Using COMSOL MULTIPHYSIC 4. Results
4 INTRODUCTION SQUEME SYSTEM CFB [6]
5 1. Problem Formulation Antecedent: The interphase momentum transfer between the two phases represented by the drag force, play an important role in any multiphase flow approach. Due to its high relevance, this phenomenon was frequently investigated in the literature. The ultimate goal of these work was to get an optinum drag model for betters fluidized bed hydrodynamics. The volume fractions conservation equations are related as: s g 1
6 Equations two phases of Gidaspow, Syamlal & O Brien Mass conservation equations Momentum conservation equations The simulation results showed that the drag models of Gidaspow and Syamlal & O Brien highly overestimate the gassolid drag force for the CFB the particles could not predict the formation of dense phase in the fluidized bed [2].
7 The conditions are characteristic of fast fluidization [1 ], [4]
8 A gas injection grid of Chamber CFB Grid of pipes Inlet
9 Two Phases Model Drew [2] Phase Gas: t n+divnu = 0 1 t nu+divnuu+p g I = div2 g ndu+ng-qmu-v 2 Phase Particle t m+divmu = 0 3 t mv+divmvv+p p I = div2 p mdv+mg+qmu-v 4 = g / p, = p ; n = n, m = m, u = u, v = v Dw = ½[gradw+gradw T ]
10 2. Theoretical Analysis and Contribution Assuming the existence of an indicator that measures the ratio of proportionality between the densities of the two phases, in particular the parameter such that 0<<<1. = g / p, = p ; n = n, m = m, u = u, v = v. When 0, result the following mathematical model which is compressible apparently. t +divv = 0 5 t v+divvv + P = divdv+ g 6 p h = -qu-v 7 div1-u+v = 0 8 where P = p c +p h.
11 Dw = ½[gradw+gradw T ] 9 Equation of state p c = o exp[k/ * -], o 1, 0 * <1 10 Equation for the drag force between phases: q = C q /1- s, s >0, s[1.4, 3.6] 11
12 Let t an open subset of [R 3 + x [0,>], 0 = {x = x 1, x 2, t R 3 / t > 0}, 2.1 t = { x 1, x 2, tr 3 / x 1, x 2 Ω, 0 t<}, R The problem is to find the volume fraction of particles C 1 t C 0 t, velocity of the
13 solid particles velocity vc 2,1 t [C 1,0 t ] d, and gas velocity represented by u[c 1,0 t ] d [C 1,0 t ] d. The problem is considered hydrodynamic pressure p h C 1,0 t C 0 t, from the state equation 5-9, to d= 2, d is the dimension of the space of the dependent variables, this vector functions that vary in space and time, which satisfy the system of equations
14 Conservative Form Two phase Compresible Model 1 v1 2 v 2 3 v1 v 2 2 F pc v1 v1v 2, 2 vv 1 2 pc v v1 v2 v1 v 2 G 2 2 3Re x y y x v1 v2 v2 v y x y x 0 q S ph M t g x 1 x q ph M t g y 1 y T
15 If v=v1,v2=0 p c = g colitional pression gradient p h = -qu hydrodinamic pression gradient div1-u = 0 M = 1-u P = 0
16 Contribution 1: Non conservative of the mixture model Ux,y,t = u 1 =, u 2 = v, u 3 = u 1 0, 0, ; 1 ; p g p p g g p p g g t t R R Rg v u v u div div p p grad U RU div RU divru R
17 Boundary conditions [1-u+v]. n = M>0 C 0 0 x [0, > 2.3 [v].n= m 0 C 0 0 x [0, > 2.4 [vv + PI - Dv].n = 0 C 0 0 x [0, > Is this a boundary free problem Initial conditions x,0 = 0 x,y C 0 R 2 + x {0,T} 2.5 vx,0 = v 0 x,y [C 0 R 2 + x {0,T}] Cauchy problem
18 Contribution 2: Conditions to solve 0 1, v M div v u R R U M Rg Fr Rg v u Slip u R R Fr p p grad P grad m p p g g h
19 3. Numerical Analysis The work consists of the construction of a numerical model for the quantitative study of the problem. This includes formulation of decoupling techniques. The solution of the variational problem in space-time, singularized the discreta inestability in time during the process computational. To overcome this difficulty we have used the Galerkin method with a numerical technique to capture the discontinuities in the Stream Lines Difussion SD with finite elements of type P1 + P2 [5], [6].
20 In the two-dimensional case, after a process dimensionless introducing a vector function of states, thus the Conservative system in variational form convective-diffusive-reactive flow in the domain located in a rectangular geometry region Ω = 0,Lx0,Hx[0,T Boundary condition: Inlet imput and Wall Initial condition: Step Stabilization : SD Numerical Method, this is expressed by:
21 Application: Discretization Stream Diffusion capturing Method [3], [6] h n K K n i i h n s n N n i i n h i n h R n h h n n h i t K y x P U H u U dt U U L s t y x U U dt U U U U U f U U B L U B S N n U Find t y x t y x SetU H,, /,,, lim ;, v v v v,,, 1 0 : v v v v,, v,, , 1 0 n n n t n ,
22 Numerical Resolution: Using COMSOL Multiphysics In the two-dimensional case approximates the solution of the problem, then the method Galerkin stabilized stream Difussion SD and a difference scheme BDF for the variable explicit Capture and temporal discontinuities of singularities in the streamlines of the convective flow, can be improved with a remesh evolutive with h= {10-4, 10-3 } side length element maximum and minimum and with a resolution of 0.25 of curvature.
23 4. RESULTS Parameters
24 Geometry
25 Hadamard R. Model Type
26 Finite Element type P1+P2
27 Finite Elements, shape function
28 Stabilization SD
29 Domain fixed CFB The axial section of the is represented in the XY plane.
30 Initial mesh
31 Remeshing finite Element
32 Pressure isolines
33 A nozzle
34 Multiple nozzles
35 Multiple nozzles
36 Conclusion 1. The spectrum of the color palette, particles red and only gas flows blue, a speed minimum fluidization of the results observed with the increase in the flow in bed, manifests a state of suspension caused by the upward flow gas by one and multiple nozzles. This flow creates drag force inertial force which balances gravity and terminal velocity which is manifested in the rate of free of the disperse phase. 2. The minimum speed is observed when bubbling the first bubble ppears, this is important because it causes the homogenity mixing Solid -Gas.
37 3. The convergence criteri is obtained when there expansion homogeneous mixture, ie. c Fr c * h, t U h h 0 1 Re 400,500
38 References 1. J.R. Grace, G. Sun, Influence of particle size distribution on the performance of fluidized bed reactors, Journal, Chem. Eng., Volume. 69 5, pages Drew, D.A, Mathematical modelling of two-phase Flow. Annual Review Fluid Mechanical, Volume 15, pages Claes Johnson, Numerical solution of partial differential equations by the finite element, pages , Cambridge University Press, Sweden, Zimerman, S and F. Taghipour. CFD Modeling of the Hydrodinamics and Reaction Kinetics of FCC Fluidized Bed Reactors. Ind. Eng. Chem. Journal. Volume 44, pages COMSOL MULTIPHYSICS, User s guide. Version 4.0, pages , Module CFD, pages Mantilla, Irla, Mathematical Contribution to Simulate the Numeric Behavior of the Mixture Flow Gas Solid, Doctoral Thesis in National University of Engineering, 2012.
39 Thank you very much
SD Numerical Simulation Technique for Hydrodynamic Flow Gas- Solids Mixing Mantilla Núñez, Irla* 1 S. De Vicente C. 2
SD Nuerical Siulation Technique for Hydrodynaic Flow Gas- Solids Mixing Mantilla Núñez, Irla* S. De Vicente C. National University of Engineering, Lia, Perú, Polytechnic University of Madrid. *Lia 5, irlan@uni.edu.pe.
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