Turbulent kinetic energy in a moving porous bed

doi:10.1016/j.icheatmasstransfer.2008.05.011

International Communications in Heat and Mass Transfer

Volume 35, Issue 9, November 2008, Pages 1049-1052

https://doi.org/10.1016/j.icheatmasstransfer.2008.05.011 Get rights and content

Abstract

This paper presents a set of transport equations for solving problems involving turbulent flow in a moving bed reactor. The reactor is seen as a porous matrix with a moving solid phase. Equations are time-and-volume averaged and the solid phase is considered to have an imposed constant velocity. Additional drag terms appearing in the momentum equation are assumed to be a function of the relative velocity between the fluid and solid phase. Turbulence equations are influenced by the speed of the solid phase in relation to that of the flowing fluid. Results show the decrease of turbulent kinetic energy levels as the solid speed approaches the speed of the moving bed.

Introduction

When analyzing turbulent flow in porous media, there are many situations of practical relevance in which the porous substrate moves along with the flow, usually with a different velocity than that of the working fluid. Several manufacturing processes deal with such configuration and applied computations can be found in the literature [1], [2], [3], [4]. Biomass pelletization and preparation for energy production may also consider systems having a moving porous bed [5], [6]. Therefore, the ability to realistic model such systems is of great advantage to a number of materials, food and energy production processes.

Accordingly, a turbulence model for flow in a fixed and rigid porous media has been proposed [7], [8], which is today fully documented and available in the open literature [9]. However, in all work presented in [9], no movement of the solid phase was considered. The purpose of this contribution is to extend the previous work on turbulence in porous media, exploring now configurations that consider the movement of the solid material.

Section snippets

Macroscopic model for fixed bed

A macroscopic form of the governing equations is obtained by taking the volumetric average of the entire equation set. In this development, the porous medium is considered to be rigid, fixed and saturated by the incompressible fluid. As mentioned, derivation of this equation set is already available in the literature [7], [8], [9] so that details need not to be repeated here. Nevertheless, for the sake of completeness, transport equations in their final modeled form are here presented.

The

Macroscopic model for moving bed

Here, only cases where the solid phase velocity is kept constant will be considered. The configuration analyzed can be better visualized with the help of the Representative elementary control-volume of Fig. 1. A moving bed crosses a fixed control volume in addition to a flowing fluid, which is not necessarily moving with a velocity aligned with the solid phase velocity. The steps below show first some basic definitions prior to presenting a proposal for a set of transport equations for

Basic definitions and hypotheses

The first step here is to defined velocities and their averages related to a fixed representative elementary control-volume.

A general form for a volume-average of any property φ, distributed within a phase γ that occupy volume ΔV_γ, can be written as [10], ${〈 φ 〉}^{γ} = \frac{1}{Δ V_{γ}} \int_{Δ V_{γ}} φ d V_{γ}$

In the general case, the volume ratio occupied by phase γ will be ϕ^γ = ΔV_γ/ΔV.

If there are two phases, a solid (γ = s) and a fluid phase (γ = f), volume average can be established on both regions. Also, $ϕ^{s} = Δ V_{s} / Δ V = 1 - Δ V_{f} / Δ V = 1 - ϕ^{f}$ and for

Transport equations

Incorporating now in Eq. (6) a model for the Macroscopic Reynolds Stresses $- ρ ϕ {〈 \overset{―}{u' u'} 〉}^{i}$ (see [7], [8], [9] for details), and assuming that a relative movement between the two phases is described by Eq. (16), the momentum equation reads, $\begin{matrix} ρ [\nabla \cdot (\frac{{\overset{―}{u}}_{D} {\overset{―}{u}}_{D}}{ϕ})] - \nabla \cdot {(μ + μ_{t_{ϕ}}) [\nabla {\overset{―}{u}}_{D} + {(\nabla {\overset{―}{u}}_{D})}^{T}]} = \\ - \nabla (ϕ {〈 \overset{―}{p} 〉}^{i}) - \underset{Viscous and Form drags due to {\overset{―}{u}}_{rel}}{\underset{︸}{[\frac{μ ϕ}{K} {\overset{―}{u}}_{rel} + \frac{c_{F} ϕ ρ | {\overset{―}{u}}_{rel} | {\overset{―}{u}}_{rel}}{\sqrt{K}}]}} \end{matrix}$

The last two terms in the above equation represent the drag caused by the relative movement between phases. When the two materials flow along with the

Application to a moving bed

A numerical example of the above is shown next. The flow under consideration is schematically presented in Fig. 2, where a channel is completely filled with a moving layer of a porous material. The channel shown in the figure has length and height given by L and H, respectively. A constant property fluid flows longitudinally from left to right permeating through the porous structure. Results at the channel center (y = H/2) are a representative of uniform one-dimensional fully developed flow after

Numerical details

The above transport equations are discretized in a generalized coordinate system using the control volume method [12]. Faces of the volumes are formed by lines of constant coordinates η − ξ. All computations were carried out until normalized residues of the algebraic equations were brought down to 10⁻⁷. Details of the discretization of all terms in the equations can be found in [8], [9]. Further, the height of the channel section was taken as H = 0.075 m and the length L was 0.75 m. For all runs

Results and discussion

The flow in Fig. 2 was computed with the set of Eqs. (1), (18), (21), (22) including constitutive Eq. (3) and the Kolmogorov–Prandtl expression (5). The wall function approach was used for treating the flow close to the walls.

Fig. 3 shows values for of the non-dimensional turbulent kinetic energy 〈k〉^v/|¯u_D|² along the channel mid-height. In all cases, inlet values for 〈k〉^v/|¯u_D|² were equal to 5.23 × 10^{− 4}. The figure indicates the damping of turbulence as the solid velocity approaches the fluid

Conclusions

Numerical solutions for turbulent flow in a moving porous bed were obtained for different ratios u_S/¯u_D. Governing equations were discretized and numerically solved. Increasing the solid speed reduces the interfacial drag, which minimizes the conversion of mean mechanical energy into turbulence. Results herein may contribute to the design and analysis of engineering equipment where a moving porous body is identified.

Acknowledgments

The author is thankful to CNPq and FAPESP, Brazil, for their invaluable support during the course of this research.

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^☆: Communicated by W.J. Minkowycz.

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Turbulent kinetic energy in a moving porous bed☆

Abstract

Introduction

Section snippets

Macroscopic model for fixed bed

Macroscopic model for moving bed

Basic definitions and hypotheses

Transport equations

Application to a moving bed

Numerical details

Results and discussion

Conclusions

Acknowledgments

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Modeling of heat transfer in a moving packed bed: case of the preheater in nickel carbonyl process

Journal of Applied Mechanics

Reduction of the volatile matter evolution rate from a plastic pellet during bubbling fluidized bed pyrolysis by using porous bed material

Chemical Engineering & Technology