A Critical Investigation of High-Order Flux Limiters In Multiphase Flow Problems

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1 A Critical Investigation o High-Order Flux Limiters In Multiphase Flow Problems Chris Guenther Fluent In., 3647 Collins Ferry Rd., Morgantown, WV 26505, USA cpg@luent.com ABSTRACT. In recent years inite volume computations o two-luid hydrodynamic models have been used in a number o numerical investigations o gas-solids luidized bed and circulating luidized bed processes. To avoid numerical diusion rom dominating simulation results, proper discretization o convective terms is critical i e higher-order methods should be used in the computation o convective terms at cell aces. It is well known that higher-order methods may give rise to nonphysical spatial oscillations. This problem is routinely addressed in single-phase theory by using lux limiters that satisy certain constraints in order to obtain total variation diminishing schemes. However, applying these methods to multiphase low problems is not straightorward and a critical look at these methods in a multiphase context is needed. KEYWORDS: Flux limiters, high-order, multiphase, two-luid, circulating luidized beds. 1. Introduction This paper investigates the use o high-order lux limiters in a inite volume computation o a two-luid hydrodynamic model MFIX (Multiphase Flow with Interphase exchanges; developed at the US Department o Energy s National Energy Technology Laboratory (NETL). Finite volume computations o two-luid hydrodynamic models (TFM) have recently been used in a number o numerical investigations o gas-solids luidized bed and circulating luidized bed (CFB) processes [GUE 01a]-[GUE 02]. The low ields in these processes are transient in nature, dominated by convection, and the computational domain required or most industrial applications is itsel problematic. Under such conditions lux limited high-order methods should be used in the discretization o convective terms to minimize numerical diusion. There are numerous single-phase investigations in the literature that used high-order lux limiters constrained by [HAR 83] and additional restrictions o [SWE 84] to demonstrate advection o scalar proiles. Researchers and commercial codes have both assumed a natural extension o these results to their TFMs without

2 2 Finite volumes or complex applications any thorough investigation into how these methods might aect simulation results. The main objective o this paper is to take a critical look at the use o high-order lux limiters in multiphase low problems and demonstrate that these methods can have signiicant aect on simulation results. To demonstrate the diiculties in using lux limiters in multiphase problems a CFB currently in operation at the NETL is used to compare experimental and numerical results. Circulating luidized beds are commonly used in the power, chemical, and mining industries and numerical simulations using TFMs are slowly emerging as a possible tool to aid in the design and scale-up o uture CFBs. In many CFB applications solids, as they are vertically transported, tend to aggregate near the walls in the riser section o circulating luidized beds. This aggregation o solids is typically reerred to as clusters and the continuous ormation and breakup o these clusters strongly inluences the hydrodynamic behavior o a CFB. Clusters eect solids inventory, pressure drop in the riser, gas-solids mixing, chemical reactions, and heat transer. So, it is essential or any TFM to capture the ormation and transient behavior o these clusters. This investigation shows that the choice o a lux limiter can severely aect clustering characteristics in a CFB. It is shown that standard lux limiters such as Smart [GAS 88], Minmod [ROE 85], Superbee [ROE 92], and the Universal Limiter o [LEO 91] predict very dierent clustering characteristics and hence very dierent hydrodynamic behavior in a CFB. Flux limiters can be compared based on their tendency to compress or diuse scalar proiles and low ield gradients. The compressibility and diusivity o a lux limiter can be demonstrated using the normalized variable diagram [LEO 88]. In many applications (both single and multiphase problems) a compressible limiter is preerable [GUE 01a], [OUA 99], however, this investigation shows that compressible limiters adversely aect clustering by allowing unphysical solids compaction in cells where sharp gradients in the void raction exist and in cells near mass inlet boundary conditions. The latter problem can be easily avoided by turning o a compressible limiter near mass inlet boundary conditions. On the other hand, sharp gradients in the void raction dominate the computational domain in a CFB and a high-order lux limiter is needed which can prevent unphysical solids compaction and maintain a sharp interace between the gas and solids phases. Understanding the role high-order lux limiters play in a simulation is critical in separating numerical errors rom modeling errors and is an important irst step in deining and developing a high-order lux limiter explicitly designed or multiphase low problems. 2. Mathematical Model Two-luid hydrodynamic models, also reerred to as Eulerian-Eulerian models, treat the luid and solids as two continuous and ully interpenetrating phases. This approach results in mass, momentum, and energy balance equations or both the gas and solids phases. For isothermal conditions, the continuity and momentum balance,

3 Flux Limiters in Multiphase Problems 3 equations or two phases (gas and solids) are given below. Gas-phase continuity Solids-phase continuity Gas-phase momentum ε g ρ g ε g ρ g v g 0 [1] t ε s ρ s ε s ρ s v s 0 [2] t ε g ρ g v g ε g ρ g v g v g ε g P g τ g F gs v s v g ε g ρ g g [3] t Solids-phase momentum ε s ρ s v s ε s ρ s v s v s ε s P g S s F gs v s v g ε s ρ s g [4] t The eight dependent hydrodynamic variables in 3D: void raction ε g (the solids raction ε s 1 ε g ), pressure P g, and six velocity components are ound by using MFIX to numerically solve the coupled non-linear partial dierential equations [1]-[4]. Constitutive relations or the gas and solids stress tensors τ g S s, gas-solids drag term F gs, and solution procedure or the void raction can be ound in [SYA 93] and [SYA 98]. 3. Numerical Procedure The governing set o partial dierential equations [1]-[4] are solved on a computational domain with a inite number o cells or control volumes. Nodal points are located at cell centers where scalars (pressure, voidage, etc.) are stored. Velocities are deined at cell aces by using a staggered grid [PAT 80]. The inite volume method integrates the equations over each cell which, or a general property φ, produces a discretized set o equations o the orm a P φ P a nb φ nb S P [5] nb where the P is the nodal point and the subscript nb represent contributions rom surrounding cell aces. The partial elimination algorithm [SPA 80] is used to uncouple the discretized momentum equations due to the gas-solids drag. Using an extension o SIMPLE [PAT 80], a sequential iterative procedure is used to solve [5] and a solids volume raction correction equation [SYA 98] adjust velocities in order to satisy conservation o mass in both phases. One o the key eatures o the inite volume method is the need to calculate convective luxes at cell aces. Recent investigations [GUE 01a]-[GUE 01c] have shown

4 4 Finite volumes or complex applications that high-order lux limiters are crucial or accurately predicting bubble dynamics in gas-solids luidized beds. To summarize this important point, [GUE 01a] used the deerred correction method o [RUB 82] to implement lux limited high-order approximations o cell ace contributions. The lux limiters prevent non-physical spatial oscillations; the deerred correction method retains the desirable stability properties o a low order method, in this paper irst-order upwinding (FOU), to orm the algebraic system given by [5]. To capture the resolution o a higher order method an additional source term is added a P φ P a nb φ nb S P S DC [6] nb The additional source term is given by S DC ε m ρ m v m φ LO φ HO A [7] where A is the area o the cell ace and the subscript m phase. g(s) or the gas(solids) Equation [6] is derived by adding and subtracting FOU convection terms to the higher order convection terms. The dierence between the higher order convection terms and the FOU convection terms are combined to orm the source term S DC. The added FOU convection term appears implicitly in the coeicients a P and a nb. Since the added and subtracted FOU convection term cancel out, at convergence there is no net contribution to equation [6] rom φ LO. Thereore, at convergence the discretization used in equation [6] is purely higher order. 4. Numerical Results To demonstrate one o the problems associated with using high-order lux limiters in multiphase problems consider a low o solids moving rom let to right subscripts U,C, and D represent upstream, center, and downstream cell center nodes and the subscript represents the right cell ace o the center node. Let the variable φ represent the solids volume raction in this case assume a high solids raction is lowing into a region o low solids raction. In order to approximate φ HO in [7] MFIX ollows the lux limiting ormulation o [SWE 84] and [LEO 90] φ HO φ C 1 2 Ψ θ φ D φ C [8] where Ψ θ is the lux limiter. Using this approach a normalized value φ C is generated by the ratio φ φ C φ U C φ [9] D φ U and the argument θ is deined by θ φ C 1 φ C [10]

5 Flux Limiters in Multiphase Problems 5 The second order lux limiters used in this investigation are the ollowing Limiter Ψ θ Superbee max 0 min 1 2θ min 2 θ Smart max 0 min 4θ θ 2 Minmod max 0 min 1 θ An alternative way to generate φ HO is also included in MFIX which uses an highorder interpolation scheme, in this paper a our point ourth-order interpolation (FP- FOI) scheme by [SON 99], to approximate φ HO and then the universal limiter o [LEO 91] to constrain φ HO to prevent non-physical spatial oscillations. The problem, independent o which lux limiter is chosen, is when φ U!"! φ C φ D and φ C # φ D. Then φ C # 1 and the argument θ takes on large values eectively diminishing or even canceling upstream inormation in the approximation o φ HO. Hence, the cell ace approximation is dominated by downstream inormation and when the solids volume raction is small compared to φ U the lux o solids through φ HO is under predicted which results in solids accumulating in the center cell. Similar inconsistent luxes would also occur i the low o solids was reversed i e, φ U $"$ φ C φ D and φ C # φ D. In the case o a CFB clusters (regions o high solids raction) are continuously orming and lowing into regions o low solids raction. Based on the discussion above the density or solids volume raction o a cluster increases as the cells representing the cluster compact. Eventually these clusters reach a critical size and mass, which can no longer be supported by the momentum o the gas and all rapidly through the riser orming a dense bed at the bottom. To demonstrate the eect high-order lux limters have on the dynamics o a CFB three-dimensional simulations o the riser section o NETL s CFB were conducted. Details o the experimental set up and conditions can be ound in [GUE 02]. In each simulation the chosen limiter was used in all cell ace approximations or the convective terms required by the discretization o equations [1]-[4]. Table 1 compares the solids inventory in the riser predicted by the dierent limiters. Limiter Superbee FPFOI Smart Minmod Inventory (g) Superbee being the most compressive and hence produces the most compaction in a cell results in the highest solids inventory in the riser. Similarly, Smart being less compressive than Superbee and Minmod being the least compressive both predict lower solids inventory. Also shown are the results o using [SON 99] interpolation scheme (FPFOI) with the universal limiter o [LEO 91]. Unortunately, in each o these simulations the solids inventory in the riser was over predicted resulting in a large incremental pressure drop through the riser compared to experimental data. Further simulations were conducted including grid reinement and simulations o other experimental CFBs all resulting in the same conclusions as above. However, i the high-order lux

6 6 Finite volumes or complex applications limiter is not used in the continuity equation or the solids volume raction and a diusive scheme (in this paper irst-order upwinding) is used instead, upstream inormation dominates the approximation o φ and cell compaction is no longer observed. Using [SON 99] interpolation scheme FPFOI and the universal limiter o [LEO 91] or all cell ace approximations except those required in the discretization o equation [2] a solids inventory o 1520 g was predicted which is a 36% reduction compared to the results in Table 1. Similar results were also ound or the other lux limiters when irst-order upwinding was used in equation [2]. Under these conditions excellent agreement was observed with experimental incremental pressure drop data and solids lux measurements through out most o the riser [GUE 02]. However, near the base o the riser, where solids enter and solids volume raction is generally higher, the incremental pressure drop was under predicted which might indicate numerical diusion is adversely aecting results in the lower portion o the riser. Current research is investigating the use o dierent lux limiters in dierent regions o the CFB. However, this type o approach would not be practical in general and a lux limiter, which could adjust the amount o compaction or diusion, based on local hydrodynamic conditions would be preerred. Accurate prediction o clustering and solids inventory is critical in numerical investigations o CFBs because solids inventory is the dominant contribution to the pressure drop in the riser and experimental pressure drop data is generally the most reliable, easiest to obtain, and the data most requently used to compare numerical and experimental results. Also, it was shown in [GUE 02] that solids inventory should be used to determine when steady-state conditions have been reached and ield variables should be time averaged. Despite the improved results when using a low-order diusive scheme in equation [2] this approach when applied to luidized bed or spouting bed applications can adversely eect numerical results and should be avoided. In these applications maintaining a sharp interace between regions o high and low solids volume raction is critical. Low-order diusive schemes will not capture this aect and high-order compressive schemes are recommended [GUE 01a]-[GUE 01c]. 5. Conclusions The results o this investigation show that high-order lux limiters can have a signiicant eect on the numerical results o a multiphase problem and suggest the need or a lux limiter explicitly designed or multiphase problems. An important numerical issue in using high-order lux limiters in multiphase problems is the tendency or solids to compact in cells along sharp gradients in the void raction. However, how much solids should remain or leave a given cell even or a one-dimensional case can not be determined analytically. Hence, a lux limiter or multiphase problems must be developed with physical arguments and validated with experimental data. Based on the results o this investigation and other numerical investigations o gas-solids systems the ollowing list gives eatures, which would be desirable in a lux limiter or multiphase problems.

7 Flux Limiters in Multiphase Problems 7 The lux limiter should be applicable to a variety o multiphase applications. Universal in the sense o [LEO 91] i% e%, any high-order cell ace interpolant can be constrained to prevent non-physical spatial oscillations. The amount o solids compaction/diusion in a cell should be physically based and hydrodynamically adjustable rather than compaction/diusion determined by the limiters ixed position in the normalized variable diagram. It should be shape preserving even in the presence o sharp gradients. It should be able to distinguish between physical and numerical extrema. Current research is underway to develop a high-order lux limiter or multiphase problems based on the items above. Reerences [GUE 01a] GUENTHER C. AND SYAMLAL M., The Eect o Numerical Diusion on Isolated Bubbles in a Gas-Solid Fluidized Bed, Powder Technol., vol. 16, 2001, pp [GUE 01b] GUENTHER C. AND SYAMLAL M., The Eect o Numerical Diusion on Gas-Solids Fluidized Beds and the use o Deerred Correction in a Finite Volume Method to Stabilize High-Order Discretization o Convection Terms, Scientiic Computing and Applications, P. Minev and Y. Lin eds., Nova Science Publishers 2001, pp [GUE 01c] GUENTHER C., O BRIEN T., AND SYAMLAL M., A Numerical Model o Silane Pyrolysis in a Gas-Solids Fluidized Bed, Fourth International Conerence on Multiphase Flow, E. Michaelides eds., [GUE 02] GUENTHER C., SYAMLAL M., SHADLE L., AND LUDLOW C. A Numerical Model o an Industrial Scale Gas-Solids CFB Seventh International Conerence on Circulating Fluidized Beds, J. Grace, J. Zhu, and H. de Lasa eds., To appear May [HAR 83] HARTEN A., High Resolution Schemes or Hyperbolic Conservation Laws, J. Comput. Phy., vol. 49, 1983, pp [SWE 84] SWEBY P., High Resolution Schemes Using Flux Limiters or Hyperbolic Conservation Laws, SIAM J. Num. Anal., vol. 21, 1984, pp [GAS 88] GASKELL P. AND LAU A. Curvature-Compensated Convective Transport: SMART a New Boundedness Preserving Transport Algorithm, Int. J. Numer. Meth. Fluids vol. 8, 1988, pp [ROE 85] ROE B., Some Contributions to the Modeling o Discontinuous Flows, Lec. Notes Appl. Math., vol. 22, 1985, pp

8 8 Finite volumes or complex applications [ROE 92] ROE B. AND SIDILKOVER D., Optimum Positive Linear Schemes or Advection in Two and Three Dimensions, SIAM J. Num. Anal., vol. 29, 1992, pp [LEO 91] LEONARD B. The ULTIMATE Conservation Dierence Scheme Applied to Unsteady One-Dimensional Advection, Comput. Methods Appl. Mech. Engrg., vol. 88, 1991, pp [LEO 88] LEONARD B., Simple High-Accuracy Resolution Program or Convective Modeling o Discontinuities, Int. J. Numer. Methods Eng., vol. 8, 1988, pp [OUA 99] OUAHSINE A. AND SMAOUI H., Flux-Limiter Schemes or Oceanic Tracers: Applications or the English Channel Tidal Model, Comput. Methods Appl. Mech. Engrg., vol. 179, 1999, pp [SYA 93] SYAMLAL M., ROGERS W., AND O BRIEN T., MFIX Documentation: Theory Guide, Technical Note, DOE/METC-95/ [PAT 80] PATANKAR S., Numerical Heat Transer and Fluid Flow, Hemisphere Publishing Corporation, [SPA 80] SPALDING D., Numerical Computations o Multiphase Flow and Heat Transer Recent Advances in Numerical Methods in Fluid Mechanics, C. Taylor and K. Morton eds., Pineridge, Swansea, 1980, pp [SYA 98] SYAMLAL M., MFIX documentation numerical technique, EG&G technical report, DE-AC21-95MC31346, [RUB 82] RUBIN S. AND KHOSLA P, Polynomial Interpolation Method or Viscous Flow Calculations, J. Comput. Phys., vol. 27, 1982, pp [LEO 90] LEONARD B. AND MOKHTARI S., Beyond First-Order Upwinding The Ultra Sharp Alternative or Non-Oscillating Steady-State Simulations o Convection, Int. J. Numer. Methods Eng., vol. 30, 1990, pp [SON 99] SONG B., LIU G., LAM K., AND AMANO R., Four Point Interpolation Scheme For Convective Fluxes Approximation, Numerical Heat Transer, Part B, vol. 35, 1999, pp