(213c) Development of Drift Velocity Transport Equation for Filtered Drag Force Model

Topic: Computation Modeling and Validation for Fluidization Processes

Title: Development of Drift Velocity Transport Equation for Filtered Drag Force Model

Authors: Yundi Jiang¹, Ali Ozel², Jari Kolehmainen¹, Yannis Kevrekidis^1,3, Sankaran Sundaresan¹

¹Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08540, USA

²School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK

³Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, MD, 21218, Maryland, USA

Introduction: Gas-particles flows in industrial vessels such as circulating fluidized beds are inherently unstable and exhibit structures of various temporal and spatial scales. Filtered Euler-Euler models are introduced to simulate such flows so that we do not have to resolve the fine-scale flow structures, which are impractical to resolve due to computing limitations [1]. It solves filtered transport equations for both solid and gas phases and the fine-scale flow structures that are unresolved are taken into account through sub-grid modeling [2,3,4]. It was found that, among all the terms in the filtered transport equations, the sub-grid contribution of the drag term is the most significant [3,4]. Previous research has established that sub-grid drift velocity, defined as the difference between the filtered gas velocity and the filtered gas velocity seen by the particles, is an essential marker to model the sub-grid contribution of the drag term [3,4], and similarly in Reynolds-averaged kinetic theory models [5]. Although drift velocity can capture the sub-grid contribution of the filtered drag term, the accessibility of this sub-grid variable is lacking in coarse simulation and the estimation of it remains a challenge. In the present study, we seek to develop a transport equation to calculate drift velocity.

Methods and Results: A transport equation for drift velocity is derived from filtered and unfiltered solid continuity and gas momentum equations, yielding thirteen unclosed sub-grid scale terms. We perform budget analysis by taking the domain and time average of each term and four out of the thirteen terms are significant. These four terms are further combined and reduced to two: a sub-grid gas pressure gradient and a sub-grid void fraction scaled drag term. Constitutive models are required to close these two terms. Finding an explicit model with the guidance of physical intuition has been challenging, so we use data mining techniques to close the equation. Neural network models are built on the filtered variables computed from fine-grid Euler-Euler simulation results. Correlations coefficients of 0.8 and 0.91 are obtained for the two terms. After closing the transport equation, we assess its predictability by performing a coarse-grid Euler-Euler simulation of a 3-D dense fluidized bed. A comparison of the axial direction average solid volume fraction profile of fine- and coarse-grid simulation indicates that the sub-grid contribution of the drag term can be well captured by evolving the transport equation.

Conclusion: Sub-grid modeling of the drag term requires an accurate estimation of drift velocity, which needs a transport equation due to the significance of temporal effects. A transport equation is derived theoretically, simplified with budget analysis, closed with neural network models, and assessed with a coarse-grid simulation. The results from the coarse simulation indicate that by including a drift velocity evolution, we can effectively correct the drag term.

References:

[1] Agrawal, K., Loezos, P. N., Syamlal, M., & Sundaresan, S. (2001). The role of meso-scale structures in rapid gas–solid flows. Journal of Fluid Mechanics, 445, 151-185.

[2] Igci, Y., Andrews, A. T., Sundaresan, S., Pannala, S., & O'Brien, T. (2008). Filtered two‐fluid models for fluidized gas‐particle suspensions. AIChE Journal, 54(6), 1431-1448.

[3] Ozel, A., Fede, P., & Simonin, O. (2013). Development of filtered Euler–Euler two-phase model for circulating fluidised bed: high resolution simulation, formulation and a priori analyses. International Journal of Multiphase Flow, 55, 43-63.

[4] Parmentier, J., Simonin O., & Delsart,O. (2012) A functional subgrid drift velocity model for filtered drag prediction in dense fluidized bed." AIChE Journal 58.4: 1084-1098.

[5] Fox, R. O. (2014) On multiphase turbulence models for collisional fluid–particle flows. Journal of Fluid Mechanics ,742, 368-424.