SD Numerical Simulation Technique for Hydrodynamic Flow Gas- Solids Mixing Mantilla Núñez, Irla* 1 S. De Vicente C. 2

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1 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, Abstract: We forulate a new atheatical odel of Gas-Solids Mixing hydrodynaic flow [] in a cobustion chaber with a fluid bed syste used in the cobustion of ineral coal waste. This odel in study is called Model Gas- Solids Mixing and it is constructed by averaging the conservation equations (ass and oentu) for a two-phase flow, which takes into account the existence of a sall paraeter rho in the order of -4. This paraeter is related to the ratio of the ass densities of both phases and is a free boundary proble aking an asyptotic adjustent in the odel. This odel is iportant in the production of theral energybased solid waste. The siulation is ipleented in COMSOL Multiphysics. Keywords: Non linear eleent finite, Laws conservation, Strea lines Diffusion.. Introduction In this odel, a nonlinear ter is added to the Navier-Stokes equations to take into account the effect of nonlinearity in the copressible flow. In this proble in particular during the process epty areas of particles called singularities appear, ie. when the volue fraction of the dispersed phase is zero caused for the gas velocity. This is why there are serious theoretical probles upon convergence in the solution. Because of difficulties encountered in the odel, it there is not an exact solution. In this paper we have developed a weak solution based on the results found in the literature []. This contribution is copleented by the construction of a nuerical schee for the quantitative study of the proble. This includes the forulation of decoupling techniques. The solution of the variational proble in space-tie we realized the discreta inestability in tie during the process. To overcoe this difficulty we have used the Galerkin ethod with a nuerical technique to capture the discontinuities in the Strea Lines Difussion (SD) with finite eleents of type P + P. We have built a new nuerical odel evolutionary for hydrodynaic volue fraction of solids and speed-gas in the siulating of the reactor with COMSOL Multiphysics [3] for the effective siulation of the evolutionary proble.. Physical Model Assuing the odel of the proble of phase solid-gas two-phase flow is then isotheral fro a regularization free boundary proble which arises after a calculation odel asyptotic phases, due to the steep gradient of solids concentration representing fluidization process in all its agnitude, this eans building a whole atheatical ethodology and regularization of singularities during the developent of the proble of ixing. Fro a suitable proposal for the state equations and constitutive properties depend viscous stresses. In this way you can get the convergence of the approxiation schee. This phenoenon of hydrodynaic considered dense phase (solid particles) coexists with a continuous phase (gas), ay originate suspensions stationary (stationary cloud) of particles in the gas. The purpose of this section is to odel the transport of the solid particles fro one zone to another to generate a cloud of particles, which fact eans that the gas-solid syste behaves like a viscous fluid acroscopically high teperatures and isotheral ode. Studies on gas-solid syste in bed chaber, the foration and evolution of the bubble size is very iportant [] since it is an indication that there is instability in the fluid bed chaber. If we consider a reference volue D, then fro the laws of conservation of total ass and oentu for the solid and gas phases, we obtain a syste of equations Navier-Stokes copressible flow Non- linear. Gas phase: t n+div(nu) = (.) t (nu)+div(nuu+p g I)=div( g nd(u))+ngq(uv) (.)

2 Solid Phase: t +div(u) = (.3) t (v)+div(vv+p p I)=div( p D(v)) +g+q(u-v) (.4) where: n = g : specific ass of the gas phase = p (-): particulate phase specific ass, [ *,]: porosity (gas volue fraction), * (,), = -: Particle concentration, i.e volue fraction occupied by particulate atter, u and phase velocities of gas and particles, p h y p c : Hydrodynaic pressure for collisional pressure gas phase to particle phase. q=q():friction between phases g y p : kineatic viscosities of gas phase and particle phase, considered constant in this odel g y p : densities of the gas phase and particle phase Assuing Newtonian behavior of the biphasic ixture (valid for low voluetric concentrations of solid particles), is used the stress operator: D(w) = ½[grad(w)+(grad(w)) T ] (.5) Assuing the existence of an indicator that easures the ratio of proportionality between the densities of the two phases, in particular the paraeter such that <<<. = g / p, = p ; n = n(), = (), u = u(), v = v(). When, result the following atheatical odel which is copressible apparently. t +div(v) = (.6) t (v)+div(vv) + P = div(d(v))+ g (.7) p h = -q()(u-v) (.8) div((-)u+v) = (.9) siendo P = p c +p h.. Equation of state p c ()= o exp[k/( * -)] o, * < (.) Equation for the drag force between phases: q() = C q /(-) s, s >, s[.4, 3.6] (.). Stateent of Proble Let t an open subset of [R 3 + x [,>], = {x = (x, x, x 3 ) R 3 / x 3 = }, (.) L = {x = (x, x, x 3 )R 3 / x 3 = L, <L<}, (.) The proble is to find the volue fraction of particles C ( t )C ( t ), velocity of the solid particles vc, ( t )[C, ( t )] d, gas velocity represented by u[c, ( t )] d [C, ( t )] d, to solve the proble is considered hydrodynaic pressure p h C, ( t ) C ( t ), fro the state equation (.6)- (.9), assuing that d =, is the diension of the space of the dependent variables are vector functions that vary in space and tie, which satisfy the syste of equations (.6)-(.4).. Boundary conditions [(-)u+v)]. n = M C ( x [, >) (.3) [v]. n = C ( x [, >) (.4) [vv + PI - D(v)]. n = C ( x [, >). Initial conditions (x,) = (x) C (R 3 + x {}) (.5) v(x,) = v (x) [C (R 3 + x {}] d (.6) En un espacio unidiensional, introduciendo el vector de estado U(x,t) = such that: u u ( x, t) ( x, t) ; where u =, u = v P) U t = f(u) x +G(U) x = S(U) (.7) u f(u) = u, G(U) = u, p( u ) u ( ) x u S(U) = uq( u) u ug ( M ( t) ) u u u P) u = ( M ( t) ) (.8) u Where: u : Dynaic viscosity (considered constant)

3 M ( ) : represents the incoing gas flow (air) t in. This data is known. Air is passed through the botto cobustion chaber through a distribution plate with ultiple nozzles to be bubbled the bed holding in suspension. The residue is roughly -3% is the total bed weight. To establish a nuerical odel, the initial and boundary conditions in particular are given by:.4, u( x,), u ( x,) (, t) M t u ( L, t) x L / 4 x L / 4 4. Nuerical aproxiation Consider an weak one diensional forulation of the proble to obtain solutions with less regularity of the required. T T L U ( F ( U ) G ( U )) dxdt L t S ( U ) dxdt L x ( x,) U ( x,) dx Taking C (, L x, T ), and if choose x i-/, x i+/ <,L> and t j, t j+ <,T>, to T<. The spatial discretization of the coputational doain [, L] consists of M eleents. Between prototype odel and geoetric siilarity exists when the relationship between all diensions corresponding to the odel and prototype are equal kineatic siilarity exists if the path of the oving particles are geoetrically siilar counterparts and the relationship between the particle velocities are hoologous equal, between two kineatically siilar systes exist geoetric and dynaic siilarity if the relations between the forces counterparts in the odel and prototype are the sae. The forces ay be a cobination of the viscous forces, due to pressure forces, gravitational forces, forces due to surface tension and inertial forces. The proble in unidiensional space you can see that converges [5] with regularization of Harten Van Leer. Syste Diffusive convective flow diensional reagent [3]. Conservation syste copressible diensional Navier-Stokes can be expressed in its copact for and be alost linearized conservative as seen in the bidiensional case, introducing a vector function of states, v ( v, v ), thus the preservative syste in for convective-diffusive flow-reactive to and is expressed as: p, if ( x, t) x, x x t j, t j ( x, t) i i, another case. The discretization of the tie variable is done in [5] using the explicit Euler ethod. Spatial integration is approxiated by the idpoint ethod. Then the discrete proble, P is expressed: t x i x i ( U U ) dx( F i F i ) G i x i x i S( U, t>, t = t, U =U(t ) y F (t ) t H/ax{v+c, v-c } ) dx

4 v v 3 v v F ( ) p c v vv, vv p c v v v v v G ( ) 3 R e x y y x v v v v y x y x q ( ) S ( ) p h M ( t ) g x x q ( ) p h M ( t ) g y y T eshing sequence and considering the boundary layers and free triangular in the boundary free proble and refineent. For that, we use the esh toolbar. For the solution of syste PDE with COMSOL Multiphysics Solve and using the weak for using the finite eleent ethod P + P and stabilization techniques as Strealine Diffusion (SD) ([3]-[4]). Then, the proble tie dependent solver with IDA which use variable order variable step zise and rect, known Backward Differentiation Forulas (BDF) ethod. Finally, quase linear systes solvers with PARDISO ethod. The convergence of the nuerical schee paraeters depends preconditioned and the rearrangeent of the atrix, PARDISO factorization ethod. 6. Results The results shown are coal waste assuptions regarding the paraeter in the order of Use of COMSOL Multiphysics In the two-diensional case, after a process diensionless [5] and then being obtained its weak forulation, the doain is located in a rectangular geoetry region Ω T = ((,L)x(,H))x[,T), has: T L H T L H T L H ( U F ( U ) G( U )) dydxdt t S( U ) dydx dt ( x, ) U ( x,) dydx dt x (.9) as shown in Figure with diensions L as the length of the base and in particular H = is the height of the reactor. Initial conditions are considered in t = the step function and rect. The boundary conditions for input the gas is inhoogeneous Newan type and elsewhere of the reactor hoogeneous wall type in the rest of the border. The function for ixing flow, speed and pressure obtained with the sliding-type odel Hadaard and the condition is considered hoogeneous boundary Dirichlet output. Using COMSOL under study non stationary, i.e., tie dependent. The features odeling geoetry is building the Bézier Polygon. Adding an new Figure. The axial section of the fixed bed reactor is represented in the XY plane. Figure. Gas velocity through a nozzle in the bed

5 It has built an algorith of the solution process of a non-stationary atheatical odel on a regular doain spaces evolutionary one and two diensions of the proble of gas phase ixture a boiler solids fluid bed syste. Figure 3. Pressure strea lines in the bed Figure 4. Speed and volue fraction in a reactor of N-nozzles. 7. Conclusions Treatent was perfored for decoupling asyptotic variables syste, generating the forulation of a boundary value proble and value initial associated with a syste of nonlinear regularized conservation, expressed in ters of a set of partial differential equations of type Navier Stokes copressible viscous flow and appearance for the variables conservative such as speed of flow, bulk density and overall pressure. [5]. This work contributes to the knowledge of nuerical techniques that are very iportant in order to predict the optial size for fluidizing solid particles in ultiphase flow odels. The detailed inforation will be given at the conference [4], [5]. In the two-diensional case approxiates the solution of the proble with the ethod Galerkin (SD) and a difference schee (BDF) for the variable explicit Capture and teporal discontinuities of singularities in the strealines of the convective flow, can be stabilized with a esh refined tie step evolution with close to x -3 and side length eleent, axiu and iniu.5x -4 and with a resolution of.5 curvature. Can be interpreted four characteristics regarding speed, these are: A surface speed which occurs when in the vicinity of the colun The section shown in the spectru of the color palette, where no particles (no red) and only gas flows (presence of blue), a speed iniu fluidization of the results observed with the increase in the flow in bed, anifests a state of suspension caused by the upward flow and gas by ultiple nozzles. This flow creates drag force (inertial force) which balances gravity and terinal velocity which is anifested in the rate of free fall of the disperse phase through the fluid and when it stays away colun environent. The iniu speed is observed when bubbling the first bubble appears, this is iportant because it causes the ixing of particles and particle expansion chaber LF. 8. References. J.R. Grace, G. Sun, Influence of particle size distribution on the perforance of fluidized bed reactors, Journal, Che. Eng., Volue. 69 (5), pages 6-34 (99).. Drew, D.A, Matheatical odelling of twophase Flow. Annual Review Fluid Mechanical, Volue 5, pages 6-9 (983). 3. Claes Johnson, Nuerical solution of partial differential equations by the finite eleent, pages 8-87, Cabridge University Press, Sweden, (994). 4. COMSOL MULTIPHYSICS, User s guide. Version 4., pages 9-98, Module CFD, pages 6-38 (). 5. Mantilla, Irla, Matheatical Contribution to Siulate the Nueric Behavior of the Mixture

6 Flow Gas Solid, Doctoral Thesis in National University of Engineering,. 9. Acknowledgeents Acknowledgeents to Research Institute of the Faculty of Science of the National University of Engineering, to Departent of Applied Matheatics and Coputer Science fro the Polytechnic University of Madrid, COMSOL and the organizers for letting e participate in the event.