Experimental validation of Lagrangian–Eulerian simulations of fluidized beds

doi:10.1016/S0032-5910(00)00389-2

Powder Technology

Volume 116, Issues 2–3, 23 May 2001, Pages 155-165

https://doi.org/10.1016/S0032-5910(00)00389-2 Get rights and content

Abstract

The present study aims to validate two-dimensional Lagrangian–Eulerian simulations of gas–solid fluidized beds by comparing these with dedicated experimental data obtained with polystyrene Geldart type D particles of 1.545 mm size.

Experimental data on pressure, voidage, and bed height fluctuations, and the power spectral density are compared with three different implementations of the Lagrangian–Eulerian model. Though qualitative trends found in the experiment are correctly reproduced by the simulations, it is found that the simulations are particularly sensitive to porosity estimation procedures used in the three different simulation strategies employed. Furthermore, the phenomenon of particle clustering predicted by the model does not conform to experimental observations; this is because the physics of the break-up of clusters is not properly captured in the Lagrangian-Eulerian model.

Introduction

Due to increasing computer power, discrete particle models, or Lagrangian models, have become a very useful and versatile tool to study the hydrodynamic behavior of particulate flows. In these models, the Newtonian equations of motion are solved for each individual particle, and a collision model is applied to handle particle encounters. Recently, such particle models have been combined with a Eulerian fluid model to simulate freely bubbling fluidized beds and circulating fluidized beds (e.g. [4], [12], [16]). Up to date, however, these models have not been properly validated by comparison with experiments.

Another approach in simulating the behavior of fluidized beds is through Eulerian–Eulerian modeling. In this approach the particle phase is averaged and thus the particles are not seen as separate objects, as in Jackson [6] (volume averaging) or Zhang and Prosperetti [17] (ensemble averaging). After the correct particle and gas governing equations are obtained, closure relations need to be applied to describe the particle–particle interactions and the gas–particle interactions. The conservation laws applicable during a hard sphere collision are volume averaged to describe the particle–particle interactions. This was done for an ideal gas consisting of fully elastic particles by, for instance, Chapman and Cowling [2]. For more realistic particles, Lun et al. [8] successfully derived the kinetic theory of granular flow (KTGF). This theory provides the particle–particle closure derived from first principles.

The drawback of volume averaging the particle phase, as in the Eulerian–Eulerian approach, is the loss of small-scale information. In the Eulerian–Eulerian approach, it is impossible to predict the paths of individual particles, while Lagrangian models can be used to study the motion of each individual particle. In the Lagrangian–Eulerian approach, in which the paths of individual particles are calculated, many researchers do employ a coarser grid resolution for the gas phase equations than the length-scale that is used in the particle-phase calculations. Hence, the gas-phase does not perceive, initiate, group, or break-up particle clusters when the clusters are smaller than the length scale of the gas-phase solution. As a result, micro-scale cluster formation is not driven by gas flow, but only by particle–particle collisions. Also, due to the different averaging scales, the gas does not flow around small clusters of particles, but flows through them as if the clusters are a fixed porous medium. Only when clustering is mainly initiated by particle–particle collisions and clusters are much larger than the gas-phase averaging scale, the use of a coarser gas-phase grid resolution is justified.

The goal of this work is to validate the predictions of the Lagrangian–Eulerian model with experiments of a small fluidized bed with Geldart D particles, to gain insight in the effect of the assumptions made in the Lagrangian–Eulerian model derivation. The small bed geometry and the large particles make the computationally expensive Lagrangian–Eulerian simulations of this system feasible. Results obtained from simulations of the pressure fluctuations, voidage fluctuations, bed expansion, and the visual representation of the location of the particles are compared to experiments with the same geometry, particle type, and superficial gas velocities.

Section snippets

Particle phase

We consider flows of homogeneous, inelastic, frictional spheres in a two-dimensional geometry. The path of each individual particle is calculated, this is called a Lagrangian calculation. The calculation of the paths of the particles consists of two steps: (i) calculating the motion of the particles, and (ii) the treatment of the collision of a particle with another particle.

Gas phase

The motion of the gas-phase is calculated from the volume averaged gas-phase governing equations as put forward by Jackson [6]. The continuity equation for the gas phase is $∂ ϵ_{g} ρ_{g} ∂ t +∇·ϵ_{g} v_{g} =0$ and the momentum balance is $∂ ϵ_{g} ρ_{g} v_{g} ∂ t +∇·(ϵ_{g} ρ_{g} v_{g} v_{g})=−ϵ_{g} ∇P+ϵ_{g} ∇· τ ̄ ̄_{g} +ϵ_{g} ρ_{g} g − ∑ i=1 K V_{s,i} β(v_{g} − v_{s,i}) ϵ_{s} δ(x − x_{s,i})$ in which the last term represents the interphase momentum transfer between the gas phase and each individual particle. δ represents a pulse function, which is one if its argument is zero and zero otherwise. The

Solution method

The gas-phase is calculated on a computational grid with individual grid size of four to eight particle diameters. The gas-phase governing equations have been solved on a staggered grid (e.g. Ref. [10]) employing the SIMPLE algorithm to determine the pressure of the gas phase. The discretization of the terms is done with a second order TVD scheme in space and the second-order Crank–Nicholson scheme in time. A preconditioned bi-conjugent gradient method was employed to solve the discretized

Experimental set-up

The experimental set-up consisted of a two-dimensional plexiglass rectangular column, 500 mm high, 90 mm wide, and 8 mm deep. A schematic representation of the set-up is given in Fig. 3. The gas flow was controlled with a variable area flow meter and a valve. The dry air from the compressor system was humidified to reduce static electricity build-up in the fluidized bed. The gas was only humidified partially to prevent condensation of water in the bed (operating temperature 17°C); polystyrene

Results and discussion

Fig. 5 shows a visual comparison of the location of the particles in three simulations at a superficial gas velocity at U=0.9 m/s and snapshots from the corresponding experiment covering a complete period from bubble formation to bubble eruption. Animations of the performed experiments and simulations can be viewed at the WWW address http://www.tcp.chem.tue.nl/~scr/wachem/particle.html.

In all three simulation strategies employed, large clusters of particles were found to be present at the top

Conclusions

The goal of this paper is to validate two-dimensional Lagrangian–Eulerian simulations of a gas–solid fluidized bed containing polystyrene particles with laboratory-scale experiments of the same geometry. One difficulty in the two-dimensional Lagrangian–Eulerian model is the translation of the two-dimensional porosity of the particles to a three-dimensional one, required by the gas-phase and the interphase momentum transfer. To tackle this problem, we have followed three strategies, A, B, and C,

Acknowledgements

These investigations were supported (in part) with financial assistance from the Netherlands Organization for Scientific Research (NWO). This support is gratefully acknowledged.

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¹: Present address: Laboratory of Chemical Reactor Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

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Experimental validation of Lagrangian–Eulerian simulations of fluidized beds

Abstract

Introduction

Section snippets

Particle phase

Gas phase

Solution method

Experimental set-up

Results and discussion

Conclusions

Acknowledgements

Int. J. Multiphase Flow

Chem. Eng. Sci.

Int. J. Multiphase Flow

Chem. Eng. Sci.

Wear

Powder Technol.

Powder Technol.

Chem. Eng. Sci.