Drag Laws 102
By Liz Marshall and Sergio Vasquez, Fluent Inc., USA
In the last issue of FLUENT News, we presented an overview of the basic drag laws in FLUENT. These laws are appropriate for many applications of the discrete phase model (DPM) and the standard or granular multiphase models, using either the Eulerian or mixture formulation. Drag contributes to the momentum exchange between different phases in a system. The momentum exchange term, which appears in each component of the momentum equation, is proportional to the velocity differential between phases and a drag function,which is obtained empirically for a specific system and flow regime.
 A liquid fuel spray in an IC engine simulation |
Most drag laws make the assumption that the particles are spherical and that drag is largely dependent on the particle diameter and relative Reynolds number. There are many applications, however, where these simplifying assumptions cannot be made. In this article, we present some of the alternative drag laws available in FLUENT and the special circumstances that require them.
Non-spherical particles
For non-spherical particles, a special drag law is available for the DPM. Called the non-spherical drag law, it uses the diameter of a particle with the equivalent volume to compute the relative Reynolds number [1], which is computed from the relative velocity between the phases. The drag function makes use of this spherical relative Reynolds number and a shape function, which is defined as the ratio of the surface area of the sphere of equivalent volume to that of the actual particle. In a study of high aspect ratio switch grass in a co-fired coal burner [2],the results showed that by modeling the cylindrical particles using these modifications, more accurate predictions of particle trajectories and residence times were obtained, and predictions of other combustion-related aspects of the process were improved as well.
Sub-micron particles
Sub-micron particles are so tiny that they can respond to the random motions of a fluid, in addition to the bulk flow. In FLUENT, a drag law is available for sub-micron particles [3] that differs from most other drag laws in that it does not depend on the relative Reynolds number. Instead, it is a modified form of Stokes’drag law for flow over a stationary sphere that depends on the fluid viscosity, the particle diameter and density, and a Cunningham correction factor that depend son the mean free path. In laminar flows with sub-micron particles, a Brownian motion option is available for the DPM, and this so-called Stokes-Cunningham drag law is the most appropriate choice for modeling momentum exchange.
Liquid sprays
Numerical models for the spray of liquid droplets into a gas involve complex physics, such as droplet collisions and breakup as well as the formation of sprays through atomization. These phenomena can be modeled only through careful consideration of how a droplet or larger liquid region responds when it is subjected to the external forces applied by the surrounding gas. An accurate description of drag is also critical to the success of any spray model. In FLUENT, the dynamic drag law is available for this purpose. It computes the drag for a sphere and for a disk of equivalent volume. An equation is solved to assess the degree of distortion of the drop, and based on the result, a linear interpolation is done between the spherical and disk drag limits to obtain an appropriate dragon the drop [4]. This calculation is done dynamically during the trajectory calculation routine.
 Harvesting switchgrass for use in biomass boilers |
Solid-liquid mixtures
Multiphase mixtures of granular (particulate) material and a turbulent liquid can sometimes be modeled using the basic Schiller-Naumann (SN)drag law in FLUENT. The success of this model depends on how much the fluid turbulence influences the drag. The ratio of the particle diameter, dp, to the Kolmogorov length scale of the flow, λ, an approximate measure of the size of the smallest turbulent eddies, provides a good measure of the importance of this effect. For dp/λ< 0.2 (or thereabouts) the role of turbulence is weak and the SN drag law should suffice. For larger values of dp/λ,a modified drag law should be used that takes the turbulence into account.One law that was first proposed by Magelli [5], and later published by Pinelli [6] has demonstrated very good results. Whereas the SN drag law is derived from the settling velocity of a particle in a still fluid column, the Magelli law computes the settling velocity of a particle in a turbulent fluid,and uses this to compute a corrected drag function. While not available in the standard release of FLUENT, this custom drag law can be obtained by contacting your local Fluent office.
 The Magelli drag law outperforms the SN law for a turbulent stirred tank containing large (675 micron) particles; the Eulerian granular multiphase model was used for the simulation [7] |
References
- Haider, A.; Levenspiel, O.: Drag Coefficient and Terminal Velocity of Spherical and Non spherical Particles. Powder Technology 58, p. 63-70, 1989.
- Gera, D.; Mathur, M.P.; Freeman, M.C.; Robinson, A.: Effect of Large Aspect Ratio of Biomass Particles on Carbon Burn out in a Utility Boiler, Energy & Fuels, 16(6),p. 1523-32, 2002.
- Ounis, H.; Ahmadi, G.; McLaughlin, J.B.: Brownian Diffusion of Submicrometer Particles in the Viscous Sublayer. Journal of Colloid and Interface Science, 143(1),p. 266-277, 1991.
- Liu, A.B.; Mather, D.; Reitz, R.D.: Modeling the Effects of Drop Drag and Break upon Fuel Sprays. SAE Technical Paper 930072, SAE, 1993.
- Magelli, F.; Fajner, D.; Nocentini, M.; Pasquali, G.: Solid Concentration Profiles in Slurry Reactors Stirred with Multiple Impellers: Recent Results. Engineering Foundation Conference Mixing XI, Henniker, NH 1987.
- Pinelli, D.; Nocentini, M; Magelli, F.: Solids Distribution in Stirred Slurry Reactors:Influence of Some Mixer Configurations and Limits to the Applicability of a Simple Model for Predictions. Chem. Eng. Comm. 118, p. 91-107, 2001.
- Montante, G.; Rondini, A.; Bakker, A.; Magelli, F.: CFD Predictions of Solid Concentration Distributions in a Baffled Stirred Vessel Agitated with Multiple PBT Impellers. CHISA 2002, Prague, August 25-29, 2002.
 |
 |