Elsevier

Biosystems Engineering

Volume 198, October 2020, Pages 323-337
Biosystems Engineering

Research Paper
Analysis of air flow resistance through a porous stone bed

https://doi.org/10.1016/j.biosystemseng.2020.08.002Get rights and content

Highlights

  • Rock bed airflow resistance based on aerodynamic properties of stone layers.

  • Simplification of the Ergun's equation by omitting the viscosity term in estimating the flow resistance.

  • Dimensional analysis applied to determine the parameters of airflow resistance as a function of independent variables.

The results of analysis of airflow resistance through a porous stone accumulator bed are presented with studies conducted in laboratory and in a full-scale facility. The Darcy–Weisbach equation and Ergun's equation were used to determine the flow resistance coefficients. It was found that a simplification of the Ergun's equation by omitting the viscosity term did not result in a significant error in estimating the flow resistance. Dimensional analysis was applied to analyse the flow resistance in the full-scale facility and to determine the form and parameters of equation allowing a calculation of airflow resistance as a function of independent variables. The analysis showed a significant convergence between the friction factors and flow resistance calculated with the use of proposed equations and the values obtained from measurements. It is concluded that the following factors have the largest impact on the airflow resistance in order: bed particle diameter, followed by airflow velocity, effective section diameter and, to the same degree, layer height and bed mass.

Introduction

The fluid flow through porous media is a complex and a not fully known phenomenon, both in respect of its use in industrial technology and its occurrence in natural environment. Using a phenomenological approach this flow can be seen to be subject to various hydrodynamic criteria, depending on the medium structure, fluid type (single- and polyphase) and flow type (gravitational or pressure). The knowledge of flow resistance (pressure drop) through porous bodies is therefore a very important issue in many areas of technology. Gas flow through porous systems (modules) is very often used in many industrial processes. The choice of filling type and its internal structure has a significant impact on hydrodynamics of the entire system. Channel diameter is usually chosen based on the volumetric gas stream in order to prevent excessive local flow resistances. Other solutions are based on gap systems, and with gaps additionally filled with a porous material with pore size in the order of a few micrometres (Prasser et al., 2002). Another application area of porous materials are reactive catalytic afterburning processes. The internal structure of such systems can be diverse but the common feature is a very high degree of channel packing leaving a large area for free gas flow which is undoubtedly gives a large process advantage. Metal monoliths are less frequently used, mainly due to difficulty in deposition of catalytic substances on their surface (Cybulski & Moulijn, 1994). The knowledge of flow resistance is also necessary during the airflow through irregular-shaped wooden biomass particles (drying, pyrolysis, gasification). Pressure drop determination was investigated by Mayerhofer et al. (2011) with the fluid flow resistance through wooden particles investigated the impact of porosity, particle diameter and the influence of spatial orientation of particles on the pressure drop of flowing air determined. The results of studies on the airflow through a layer of stored rapeseed were presented in (Szwed & Łukaszuk, 2003) and Chung et al. (2001), studied the impact of moisture and grain size of sorghum and rough rice on the airflow resistance. An extensive review of experimental results on flow through porous media was given in (Wałowski, 2017) which included not only the results for biological materials, but substrates that are used in greenhouses such as gravel, wood chips, polydisperse materials, coal, coke and polystyrene.

The contemporary challenge is to implement the solutions that minimise the consumption of fossil fuels, and as a result reduce the emissions of hazardous substances to the atmosphere. One of such solutions is the use of heat accumulators in protected cropping facilities. The idea of such accumulator is heat storage in the bed and supply of stored heat to the inside of a building. The storage takes place during periods of high radiation, and the discharge is effected during the night time energy demand. The design details and energy effects of such system are presented in paper by Kurpaska et al. (2012). Such accumulators can be used for seasonal storage of heat generated in solar collectors (Duffie & Beckman, 2006). The authors discussed many design solutions along with engineering calculations for solar energy conversion, including heat storage in a stone bed.

An important issue during the forced air flow through a porous bed is pressure drop. This process is a consequence of flow resistance. Examining the problem using a phenomenological approach, the fluid flow through a porous medium can be subjected to various hydrodynamic criteria, depending on the medium structure, fluid type (single- and polyphase) and type of flow forcing. A description of the theory of fluid flow through a porous medium can be found in (Ichikawa & Selvadurai, 2012). The many cases of fluid flow through porous media are described and their basic relationships for saturated and unsaturated media presented, with and without the occurrence of chemical reactions.

As shown by Ichikawa and Selvadurai (2012), all attempts at theoretical description of flow resistance through a porous layer assume the resistance depend on parameters characterising frees spaces (pores), inter alia layer porosity, particle diameter, particle sphericity, particle roughness, etc. The theoretical description assumes thus an equilibrium between the kinetic and potential energy of the fluid, and the pressure difference in the system is used mainly to overcome the resistance which in hydrodynamics reflects the pressure drop. The theoretical analysis of fluid flow through a porous medium can also be made based on Bernoulli's equation. If it is assumed that the capillary is a straight section and the linear resistance coefficient and the capillary length and diameter are accounted for, then a relationship will be obtained that expresses the pressure drop as a function of fluid velocity and its density. This relationship is called the Darcy–Weisbach equation. This equation is the most frequent formula used as a basis for theoretical solutions of fluid flow through porous media. One of its modifications is the Kozeny equation which accounts for medium sphericity and porosity. The Kozeny capillary theory was further modified by Carman to take into account the tortuosity of pores. Assuming that the fluid flow through the medium pores is laminar, the Poiseuille's equation can be used to describe the pressure droop with Forchheimer's equation used to describe the pressure drop at higher flow velocities. The practical applicability of this equation was shown by Ergun who determined the constants for this equation during the analysis of flow of various gases (CO2, N2, CH4 and H) through coke and sand.

The results of studies in the literature usually describe the pressure drop as a function of particles being washed by flowing air for the various porous materials. Flow resistance problems were analysed, inter alia, by Zhong et al. (2016), who used the Ergun equation to describe the air pressure changes for air flow through a porous bed (i.e. a sintered metal). From their experimental investigations, they concluded that this method is useful and recommended the need for further studies. Sobieski and Zhang (2017) proposed an integrated method combining the discrete element method (DEM) and the computational fluid dynamics (CFD) for describing the pressure drop, and from the results of experiments they concluded that such an approach allows a determination of bed and process parameters. Vollmari et al. (2015) also used the integrated DEM and CFD method to determine the pressure drop of flow through a column filled with arbitrary-shaped particles. From experiments, they concluded that their results conformed with simulation results that depended on the shape of bed particles. They indicated a need for further studies of polydisperse systems. Hamad et al. (2017) analysed the pressure drop a multi-phase fluid (water and air) flowing through pipes with different diameters. Also Sabiri and Comiti (1995) investigated the flow of non-Newtonian purely viscous fluids flowing through beds of various structure. They proposed a model for pressure drop calculation which accounted for the structural parameters of the porous medium and its application covered a wide range of Reynolds number. Andrade et al. (1999) investigated the origin of deviations from Darcy's law using numerical simulations of the Navier–Stokes equations in a two-dimensional disordered porous media, concluding that in the range of low Reynolds numbers the flow of a Newtonian fluid through a porous medium is subject to the Darcy's law. Perovi et al. (2017) conducted simulation experiments of fluid flow through pores of different diameters and used the Darcy equation combined with a CFD technique developed. They concluded that the proposed approach (using a network of computers) was useful for diversified porous materials. Plessis et al. (1994) also showed a satisfactory conformity of the pressure drop description calculated from Darcy's law during experiments investigating the isothermal flow of a Newtonian fluid through a porous body. Allen et al. (2013, 2015) analysed the pressure drop and heat transfer of air flowing through rock particles packed into a measurement column. As a result of the analysis, they determined the friction coefficient for diversified airflow directions along the axis of rock particles. They concluded that the packing arrangement affected pressure drop but it did not affect heat transfer. Koekemoer and Luckos (2015) used an alternative approach to the Darcy's law and determined the parameters in the Ergun equation for determination of air flow resistance through particles with diverse external dimensions. Wang et al. (1999), however carried out an analysis directed at the understanding of the flow in anisotropic porous media. Three anisotropy models were proposed: induced by size, by connections induced, and by spatial correlation. They proposed and verified a tensor form of Forchheimer equation. Fourar et al. (2005) investigated fluid flow through bodies consisting of a large number of uniform blocks. In their theoretical description they used the Forchheimer equation with a constant coefficient of inertia which was calculated from numerical simulations. Depending on the spatial configuration of analysed bodies (parallel and serial connections were used) they determined the Reynolds number. Tian, Xua, et al. (2018) analysed the heat transfer intensity and flow resistance of a fluid of various temperatures and concentrations, which was classified as a non-Newtonian fluid, flowing through a packed porous bed. The authors used the Ergun equation to describe the pressure drop and concluded that the Reynolds number depended on the fluid parameters.

Bernsdorf and Durst (2000) used the fluid flow theory in a porous medium to describe the pressure drop. They used a computer tomography technique to describe the pores parameters (porosity, specific surface area) and concluded that CT is useful in both the analysis of resistance values and of deformation of the flowing fluid.

Summarising the results of the studies presented above, it can be concluded that they generally consider cases identified with laminar fluid flow where Darcy's law was applied for description. The basic problem, both theoretical and experimental, is the relationship between the permeability of the medium and porosity. Analysis of available research papers indicates that they relate to gas flow through porous media, and because the porous medium usually has a diverse configuration the use of standard Darcy–Weisbach relationship very often leads to large discrepancies between the theoretical and experimental results. It should be added that absence of data on relevant parameters makes it impossible to provide reliable engineering calculations which have an application significance (e.g. for fan choice). The quoted papers mainly focus on phenomena occurring during the flow through granular media (i.e. loosely packed) and most often investigate a column filed with a porous material. Only a few papers deal with gas permeability through porous materials of skeleton (rigid) structure. The stone accumulator bed used to support heating in horticultural facilities, operating in the charge–discharge cycle, is such porous material.

The presented paper has cognitive and application characteristics: cognitive because the flow resistance values are determined as a function of air velocity and parameters, and the particle size used in the accumulator bed. The application characteristics are directly related to a practical problem; the determination of values which must be known in order to choose the correct fan. Hence, such studies should be regarded as being both appropriate and justified.

Section snippets

Material and method

The studies were conducted in two stages, in the laboratory and in a full-scale plastic tunnel facility. The laboratory investigation included an analysis of the governing equations of porous flow and how they relate to the case of a porous stone bed.

Results and discussion

The laboratory experiments were conducted for the following input data: air temperature (t); 21.7≤ t ≤ 28.4 °C, air humidity (RH); 23.8 ≤ RH ≤ 31.9%, airflow velocity in the 0.11 m diameter measuring section (V); 1.3 ≤ V ≤ 2.8 ms−1. The calculated bed porosity was ε = 0.432, and the substitute diameter of bed elements dcz = 0.044 m.

Figure 5 presents the coefficient of friction factor calculated from Eq. (1) as a function of airflow velocity (calculated for empty column) - (a) and the Reynolds

Conclusions

  • 1.

    The following factors have the largest impact on the airflow resistance (assuming an equal percent change of independent variables): bed particle diameter, followed by airflow velocity, effective section diameter and to the same degree bed layer height and bed mass.

  • 2.

    The air flow resistance coefficient in the examined bed can accurately described by the relationship: λ=0.967Re0.0983; R2 = 0.9917 in application range: 214.4≤ Re ≤ 464.5

  • 3.

    The coefficients in the Ergun equation were determined as k1

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research was financed by the Ministry of Science and Higher Education of the Republic of Poland.

References (26)

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