The effect of numerical diffusion and the influence of computational grid over gas-solid two-phase flow in a bubbling fluidized bed. (English) Zbl 1205.76259
Summary: The numerical diffusion effects appear due to the discretization process of the convective terms of the transport equations. This phenomenon takes place also in the numerical simulation of gas-solid two-phase flows in bubbling fluidized beds (BFB). In the present work a comparative analysis of the numerical results obtained using two interpolation schemes for convective terms, namely FOUP (First Order UPwind) and a high order scheme (Superbee) is presented. The equations are derived by considering the Eulerian-Eulerian gas-solid two-fluid model and the kinetic theory of granular flows (KTGF) for modeling solid phase constitutive equations. For that purpose the MFIX (Multiphase Flow with Interphase eXchanges) code developed at NETL (National Energy Technology Laboratory, US Department of Energy) is used. The numerical diffusion is analyzed by considering a single bubbling detachment and its hydrodynamic process in a two-dimensional BFB. The bubble shape is used as a metric for the description of the results. The influence of the computational grid is also analyzed. It is concluded that the Superbee scheme produces better results and this scheme is recommended for discretizations of the convective terms in coarse grids. The FOUP scheme can be used only in fine grids but it requires a high computational effort. In this study it is also verified that the analysis about estimating uncertainty in grid refinement can be applied in specific points of the grid when a monotonic convergence in time and space occurs.
MSC:
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
| 76S05 | Flows in porous media; filtration; seepage |
References:
| [1] | Davidson, J., Symposium on fluidization-discussion, Trans. Inst. Chem. Eng., 39, 230-232 (1961) |
| [2] | Jackson, R., The mechanics of fluidized beds: part 1. The stability of the state of uniform fluidization, Trans. Inst. Chem. Eng., 41, 13-21 (1963) |
| [3] | Murray, J., On the mathematics of fluidization: part 1. Fundamental equations and wave propagation, J. Fluid Mech., 21, 465-493 (1965) · Zbl 0125.44603 |
| [4] | Collins, R., An extension of Davidson’s theory of bubbles in fluidised beds, Chem. Eng. Sci., 20, 747-755 (1965) |
| [5] | Anderson, T.; Jackson, R., A fluid mechanical description of fluidized beds, Ind. Eng. Chem. Fund., 6, 4, 527-534 (1967) |
| [6] | Stewart, P., Isolated bubbles in fluidised beds-theory and experiment, Tran. Inst. Chem. Eng., 46, 60-68 (1968) |
| [7] | Gidaspow, D., B. Ettehadieh, Fluidization in two dimensional beds with a jet 2. Hydrodynamics modeling, Ind. Eng. Chem. Fund., 22, 193-201 (1983) |
| [8] | Gidaspow, D., Multiphase Flow and Fluidization (1994), Academic Press: Academic Press London · Zbl 0789.76001 |
| [9] | Bouillard, J.; Gidaspow, D.; Lyczkowski, R., Hydrodynamics of fluidization: fast-bubble simulation in a two-dimensional fluidized bed, Powder Tech., 66, 107-118 (1991) |
| [10] | Kuipers, J.; van Duin, K.; van Beckum, F.; van Swaaij, W., Computer simulation of the hydrodynamics of a two-dimensional gas-fluidized bed, Comput. Chem. Eng., 17, 8, 839-858 (1993) |
| [11] | Boemer, A.; Qi, H.; Renz, U., Eulerian simulation of bubble formation at a jet in a two-dimensional fluidized bed, Int. J. Multiph. Flow, 23, 5, 927-944 (1997) · Zbl 1135.76362 |
| [12] | Syamlal, M.; O’Brien, T., Computer simulations of bubbles in a fluidized bed, AIChE Symp. Ser. Fluidization. Fluid Particle Syst.: Fund. Appl., 85, 22-31 (1989) |
| [13] | Patil, D. J.; van Sint Annaland, M.; Kuipers, J. A.M., Critical comparison of hydrodynamic models for gas-solid fluidized beds-Part I: bubbling gas-solid fluidized beds operated with a jet, Chem. Eng. Sci., 60, 57-72 (2005) |
| [14] | Patil, D. J.; van Sint Annaland, M.; Kuipers, J. A.M., Critical comparison of hydrodynamic models for gas-solid fluidized beds-Part II: freely bubbling gas-solid fluidized beds, Chem. Eng. Sci., 60, 73-84 (2005) |
| [15] | M. Syamlal, W. Rogers, T. O’Brien, MFIX documentation: theory guide, Tchenical Note DOE/METC-95/1013, 1993.; M. Syamlal, W. Rogers, T. O’Brien, MFIX documentation: theory guide, Tchenical Note DOE/METC-95/1013, 1993. |
| [16] | Lun, C.; Savage, S.; Jeffrey, D., Kinetic theories for granular flow: inelastic particles in couette flow and slightly inelastic particles in a general flow field, J. Fluid Mech., 140, 223-256 (1984) · Zbl 0553.73098 |
| [17] | van Wachem, B.; Schouten, J.; Krishna, R.; van den Bleek, C., Eulerian simulations of bubbling behavior in gas-solid fluidized beds, Comput. Chem. Eng., 22, S299-S306 (1998) |
| [18] | van Wachem, B.; Schouten, J.; Krishna, R.; van den Bleek, C., Validation of the Eulerian simulated dynamic behaviour of gas-solid fluidized beds, Chem. Eng. Sci., 54, 2141-2149 (1999) |
| [19] | Soo, S., Fluid Dynamics of Multiphase System (1967), Blaisdell Inc.: Blaisdell Inc. Massachusetts · Zbl 0173.52901 |
| [20] | Drew, D., Mathematical modeling of two-phase flow, Ann. Rev. Fluid Mech., 15, 261 (1983) · Zbl 0569.76104 |
| [21] | Enwald, H.; Peirano, E.; Almstedt, A., Eulerian two-phase flow theory applied to fluidization, Int. J. Multiph. Flow, 22, 21-66 (1996) · Zbl 1135.76409 |
| [22] | Huilin, L.; Yurong, H.; Wentie, L.; Ding, J.; Gidaspow, D.; Bouillard, J., Computer simulations of gas-solid flow in spouted beds using kinetic-frictional stress model of granular flow, Chem. Eng. Sci., 59, 865-878 (2004) |
| [23] | Johnson, P.; Jackson, R., Frictional-collisional constitutive realtions for granular materials with application to plane shearing, J. Fluid Mech., 176, 67-93 (1987) |
| [24] | Johnson, P.; Nott, P.; Jackson, R., Frictional-collisional equations of motion for particulate flows and their application to chutes, J. Fluid Mech., 210, 501-535 (1990) |
| [25] | Schaeffer, D., Instability in the evolution equations describing incompressible granular flow, J. Differential Equations, 66, 19-50 (1987) · Zbl 0647.35037 |
| [26] | Wang, S.; Xiang, L.; Huilin, L.; Long, Y.; Dan, S.; Yurong, H.; Yonglong, D., Numerical simulations of flow behavior of gas and particles in spouted beds using frictional-kinetic stresses model, Powder Tech., 196, 184-193 (2009) |
| [27] | Makkawi, Y.; Ocone, R., A model for gas-solid flow in a horizontal duct with a smooth merge of rapid-intermediate-dense flows, Chem. Eng. Sci., 61, 4271-4281 (2006) |
| [28] | Jenkins, J.; Savage, S., A theory for rapid flow of identical smooth, nearly elastic spherical particles, J. Fluid Mech., 130, 187-202 (1983) · Zbl 0523.76001 |
| [29] | Tuzun, U.; Houlsby, G.; Nedderman, R.; Savage, S., The flow of granular materials-II, velocity distributions in slow flow, Chem. Eng. Sci., 37, 12, 1691-1789 (1982) |
| [30] | Jenike, A., A Theory of flow of particulate solids in converging and diverging channels based on a conical yield function, Powder Tech., 50, 229-236 (1987) |
| [31] | Wang, S.; Liu, G.; Huilin, L.; Yinghua, B.; Ding, J.; Zhao, Y., Prediction of radial distribution function of particles in a gas-solid fluidized bed using discrete hard-sphere model, Ind. Eng. Chem. Res., 48, 1343-1352 (2009) |
| [32] | Bagnold, R. A., Experiments on a gravity free dispersion of large solid spheres in a Newtonian fluid under shear, Proc. R. Soc. London, 49, A225 (1954) |
| [33] | Ma, D.; Ahmadi, G., An equation of state for dense rigid sphere gases, J. Chem. Phys., 84, 3449 (1986) |
| [34] | Carnahan, N. F.; Starling, K. E., Equation of state for nonattracting rigid spheres, J. Chem. Phys., 51, 635 (1969) |
| [35] | Garside, J.; Al-Dibouni, M., Velocity-voidage relationship for fluidization and sedimentation, IEEC Proc. Des. Dev., 16, 206-214 (1977) |
| [36] | Dalla Valle, J., Micromeritics (1948), Pitman: Pitman London |
| [37] | Patankar, S. V., Numerical Heat Transfer and Fluid Flow (1980), Hemisphere: Hemisphere New York · Zbl 0595.76001 |
| [38] | Courant, R.; Isaacson, E.; Reeves, M., On the solution of nonlinear hyperbolic differential equations by finite differences, Comm. Pure Appl. Math., 5, 243-255 (1952) · Zbl 0047.11704 |
| [39] | Spalding, D., A novel finite-difference formulation for differential expressions involving both first and second derivatives, Internat. J. Numer. Methods Engrg., 4, 551-559 (1972) |
| [40] | Price, H.; Varga, R.; Warren, J., Applications of oscillation matrices to diffusion-correction equations, J. Math. Phys., 45, 301-311 (1966) · Zbl 0143.38301 |
| [41] | Leonard, B., A stable and accurate convective modeling procedure based on quadratic upstream interpolations, Comput. Methods Appl. Mech. Eng., 19, 59-98 (1979) · Zbl 0423.76070 |
| [42] | van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136 (1979) · Zbl 1364.65223 |
| [43] | Sweby, P., High resolution schemes using flux limiters for hyperbolic conservative laws, SIAM J. Numer. Anal., 21, 995-1011 (1984) · Zbl 0565.65048 |
| [44] | Leonard, B., Simple high-accuracy resolutions program for convective modeling of discontinuities, Internat. J. Numer. Methods Fluids, 8, 1291-1318 (1988) · Zbl 0667.76125 |
| [45] | Gaskell, P.; Lau, A., Curvature-Compensated Convective Transport: SMART, a new boudedness preserving transport algorithm, International Journal for Numerical Methods in Fluids, 8, 617-641 (1988) · Zbl 0668.76118 |
| [46] | Zhu, J.; Leschziner, M., A local oscillation-damping algorithm for higher-order convection schemes, Comput. Methods Appl. Mech. Eng., 67, 355-366 (1988) · Zbl 0641.76082 |
| [47] | Varonos, A.; Bergeles, G., Development and assessment of a variable-order nonoscillatory scheme for convection term discretization, International Journal for Numerical Methods in Fluid, 26, 1-16 (1998) · Zbl 0906.76060 |
| [48] | Leonard, B.; Mokhtari, S., Beyond first-order upwinding: the ultra-sharp alternative for non-oscillatory steady-state simulation of convection, Internat. J. Numer. Methods Engrg., 30, 729-766 (1990) |
| [49] | Guenther, C.; Syamlal, M., The effect of numerical diffusion on simulation of isolated bubbles in a gas-solid fluidized bed, Powder Tech., 116, 142-154 (2001) |
| [50] | Roache, P. J., Perspective: A method for uniform reporting of grid refinement studies, J. Fluid Eng., 116, 405-413 (1994) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
