International Journal of Chemical Reactor Engineering Volume 1 2003 Article A1 Numerical Convection Algorithms and Their Role in Eulerian CFD Reactor Simulations Hugo A. Jakobsen∗ ∗Norwegian University of Science and Technology, [email protected] Copyright c 2003 by the authors. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, bepress. Numerical Convection Algorithms and Their Role in Eulerian CFD Reactor Simulations Hugo A. Jakobsen Abstract In this paper a comparative convection algorithm study is presented. The performance of a large number of schemes is compared evaluating the predicted solutions for a standard benchmarking test problem. The nature of the errors caused by the numerical approximations to the convection term is highlighted. Although there is no algorithm that performs the best in general, several conclu- sions can be made. The tests performed show that the 1st order upwind scheme and several variations of this scheme are very diffusive and should be avoided. Most stable 2nd order schemes seem to be much more accurate, whereas the ac- curacy gained by higher order schemes (3rd order and 4th order) may be a little more costly. Implicit time integration schemes are usually not as efficient as the corresponding explicit schemes due to the computational time required on the it- erative process. With larger time steps the accuracy of implicit schemes decrease rapidly. The choice of proper higher order schemes (2nd order schemes) is then seemingly determined by the trade off between accuracy and computational time. The conservative methods like− the UTOPIA, the QUICK 1D combined with a limiter, and a limited number of FCT and TVD formulations− may be sufficient solving the multi fluid model equations. For advective terms (e.g., as occur in the temperature− equation) the non flux based modified method of characteristics is very fast, but also other higher− order− (2nd order) schemes performed KEYWORDS: multiphase reactors, Eulerian models, numerical diffusion, dy- namic flow patterns, numerical methods, convection Jakobsen: Numerical Convection Schemes 1 1. INTRODUCTION In the last two decades an increasing trend in applying computational fluid dynamics (CFD) to elucidate details of the reactor performance has been seen in the literature and recognized as very useful by the industry. Kuipers and van Swaaij (1997) provided a survey on the application of CFD to the field of chemical reaction engineering. Recent experimental studies on the flow structures of multiphase chemical reactors like bubble column, fluidized bed- and stirred tank reactors have provided insight and evidence of the dynamic nature of these systems. The instantaneous flow structures found in these reactors, are different from those inferred by utilizing time average data. Steady-state model computations can thus not provide a rational basis for the fundamental description of the interfacial mass, momentum, and energy transport processes. The transient multiphase flow models may apparently more realistically describe the multiphase flow structure. Furthermore, due to the relatively high holdup of the dispersed phases in operating reactors, the Eulerian modeling framework has to be adopted. In this paper we focus on an important aspect of dynamic Eulerian models, namely the errors caused by the numerical approximations to the convection terms. Very different numerical properties are built into the various numerical schemes proposed for solving the model equations of this type. Care has to be taken to make sure that the numerical algorithm chosen is consistent with and reflects the actual physics expressed by the theoretical model equations applied. The implementation of low accuracy convection schemes may totally destroy the physics reflected by the sophisticated multiphase CFD model formulations in use today (a typical two-fluid reactor model is given in appendix A). The objective of investigating this problem is to gain insight into the expected errors and the applicability of dynamic Eulerian methods to the CFD modeling of multiphase reactors. An ideal scheme should satisfy several criteria: (1) positiveness, (2) conservativeness, (3) shape preservation, (4) small numerical diffusion and dispersion, (5) accurate phase speed, (6) boundedness, (7) transportiveness, (8) monotonicity, (9) entropy-satisfying, (10) accurate implementation of the boundary conditions, (11) accurate resolution of discontinuities, (12) low computational costs, (13) low complexity (easy to implement), (14) efficient parallelization, and (15) generality. These numerical properties are not entirely independent, we merely want to highlight their importance. There is no single scheme that fulfil all these criteria completely, however many methods may meet some of these requirements. In practice, it is still an open question whether or not all of the listed properties could strictly be met by a single scheme. It this paper we compare the performance of several schemes considered good candidates for use in multiphase Eulerian reactor models. Most commercial multiphase CFD codes basically seeking steady state solutions still resort to the classical upwind (or donor cell) method due to it's stability properties. For many years the 3nd order QUICK scheme was considered favorable in single phase CFD because of the improved accuracy obtained by this scheme. Lately, certain Flux-Corrected-Transport (FCT) schemes and Total Variation Diminishing (TVD) schemes are claimed to be preferable in CFD. Other non-linear flux limiters have also been developed intending to improve the performance of the basic schemes. 2. THEORETICAL ASPECTS The equation describing the advection of a scalar variable, φ, yields ∂φ +⋅∇=v φ 0 (1) ∂t The conservative form of the above equation is derived by use of the continuity equation ∂()ρφ +∇⋅()0ρφv = (2) ∂t where φ denotes the scalar variable transported, v denotes the fluid velocity vector and ρ denotes the fluid density. Produced by The Berkeley Electronic Press, 2002 2 International Journal of Chemical Reactor Engineering Vol. 1 [2003], Article A1 The origin of the numerical errors involved solving the Eulerian model formulations is related to the discretization problem and the choice of approximations to the differential equations. Numerical methods constructed based on this advective form of the transport terms are shape preserving, but not conservative. Schemes constructed based on the conservative form (or flux form) of the transport terms are preferable when the model expresses a local conservation law for a conservative variable (i.e., in contrast to the temperature equation that should be solved on the non-conservation form). These flux based methods guarantee conservation of the transported variable φ (Roache, 1992), but are usually not shape preserving. Thuburn (1995) and Leonard, Lock and MacVean (1996) discuss these numerical issues in further detail. In multiphase flow calculations implicit upstream differencing is still a commonly used method for the convective terms in spite of the well-known and serious accuracy problems associated with the implicit artificial viscosity of the method. According to Roache (1992), a simple Taylor series analysis on the 1D transport equation shows that the transient artificial viscosity coefficients for explicit upwind differencing is given by wz∆ υ =−(1CFL ) (3) numerical 2 the corresponding implicit method gives wz∆ υ =+(1CFL ) (4) numerical 2 where νnumerical denotes the numerical or artificial viscosity, w denotes the z-component of the velocity vector, and CFL denotes the Courant number (based on the Courant-Friedrichs-Lewy condition). It can be noted that at least the explicit upwind method for the constant velocity model gives the exact answer for CFL=1, whereas the implicit upwind differencing method never does. The numerical viscosity of the implicit method may increase a lot for CFL >> 1, which is the argument that does not justify its use compared to explicit upwind differencing. This finding is the reason why we have primarily included explicit methods in our test program. The truncation error of advection and convetion schemes can be analyzed using the modified equation method (Warming and Hyett, 1974). The presence of ∆z (i.e. the grid spacing) in the leading error term indicates the order of accuracy of the scheme. The even-ordered derivatives in the error represent the diffusion error, while the odd-ordered derivatives represent the dispersion (or phase speed) error. Artificial diffusion is thus built into all 1st order upstream schemes. The exception is the special case when the Courant number is equal to 1, then the error term vanishes. Oscillations are produced if an odd-order derivative gives a weighty contribution to the truncation error of the scheme. Even order upwind methods tend to produce oscillations upwind of a change in gradient, while even order central difference methods give oscillations downwind of a change in gradient. Another method for analyzing the truncation error and the numerical stability properties of the schemes is the Fourier (or von Neumann) method (e.g. O'Brien, Hyman and Kaplan, 1951; Odman, 1997). Odman (1997) stated that all the 1st order upwind based schemes introduce some numerical diffusion, so methods with comparatively low numerical diffusion should