Estimating pressure drop and Ergun/Forchheimer parameters of flow through packed bed of spheres with large particle diameters
Graphical abstract
Introduction
The behavior of fluid flow in porous structure of packed bed with large particle sizes is very important for a large variety of engineering applications, such as mining, geothermal, oil and gas and construction industries. For example, in oil and gas industry, the flow of the gas/oil is governed by the pressure and the porous structure of the broken rocks due to drilling and fracturing. In construction industry, rock pile for foundation and associated seepage flow can be very important for the structural strength of the infrastructure. In typical underground mining operations, the drilling and blasting creates broken rocks structure in the stope which drives ventilation air to leak through the broken rocks. Understanding the gas leakage characterization and pressure drop through this caved-zone is one of the most essential tasks which directly influences the safety in underground operations as well as the required fan power for circulating the air. The reliable estimation of the pressure gradient through a body of broken rock (i.e., porous structure) is fundamentally important to the design of the ventilation fans. In other words, the permeability of a packed bed of broken rocks significantly affects the ventilation conditions in the underground working environment. Although the permeability of porous media plays a fundamental role, its accurate prediction is still not well understood. The specific characteristics (e.g., permeability and porosity) of each structure of porous media must be carefully assessed to conduct a cost-effective analysis for these systems (i.e., the pressure gradient dictates the flowrate). Due to the importance of porous media and its numerous associated applications, fluid flow and heat transfer in porous media has been vigorously studied [1,2]. Many studies have shown that the pressure gradient of fluid across a porous media can vary significantly, depending on the characteristics of the solid particles (e.g., particle shape and size, porosity, and packing arrangement) and the fluid properties (e.g., density, viscosity, and velocity).
Essentially, the pressure gradient of flow through a porous medium is caused by the frictional drag [3] which can be characterized by the empirical Darcy law based on viscous force This assumption is typically valid within low fluid velocity (Eq. (1)).where μ, U and K are the dynamic viscosity, physical velocity, and permeability, respectively [2]. The ∇p shows the pressure gradient and is negative concerning the fluid flow direction. Although Darcy introduced a one-dimensional empirical model to calculate the pressure gradient through porous media for laminar flow [4], many studies have demonstrated that the Darcy law is only valid for estimating fluid behavior dominated by the viscous effect. Meaning that, the Darcy law is not valid at higher velocities, where the pressure drop becomes quadratic with velocity resulting from an increase in the inertia [βρU2] (Eq. (2)).where β and ρ are the inertial resistance coefficient (also known as the non-Darcy coefficient) and fluid density, respectively. Experimental data specify that the pressure drop in a porous medium is caused by a linear combination of fluid flow velocity [μU/K], namely Darcy term, and square of the fluid flow velocity [βρU2], namely Forchheimer term, whereas the square term is proportional to the inertial effects occurred through the porous structure [5], also, known as Forchheimer number. In addition, it is important to seek correlations between packed bed characteristics and the viscous and inertial resistance forces under various conditions. Allen et al. [6], Erdim et al. [7], Vollmari et al. [8], Anbar et al. [9], Partopour and Dixon [10], and the others [[11], [12], [13], [14]] reviewed and summarized models for spherical and non-spherical particles in fixed bed; however, none of these focused on large particle sizes.
In the flow through porous media, two well-known equations, Ergun correlation and Forchheimer equation are widely used to predict pressure drop as a function of velocity. Other different correlations basically have been developed based on these two equations [15]. While in fluidized bed, some other correlations are commonly used, such as Wen-Yu or De Felice based correlations [16].
The pressure gradient across a packed bed is generally estimated using the semi-empirical Ergun correlation (Eq.(3)) [[17], [18], [19]] matched to experimental data from packed beds that comprise small or micro-sized particles. First and second Ergun constants for Darcy-Forchheimer flow in Eq. (3), respectively, A and B vary with fluid velocity through porous media and may not accurately estimate the pressure gradient [[20], [21], [22]]. Thus, the Ergun correlation should be validated as a function of the Reynolds number [12]. The generality of constants A and B are widely debated. Many researchers claim that these values are different for each packed bed and should be empirically determined for each case [11,[23], [24], [25]].
From Eqs. (2) and (3), correlations for the viscous and inertial resistance coefficient can be obtained:
where the constants A and B are 150 and 1.75, respectively, in the original Ergun correlation. ∇p is the pressure loss per unit length (Δp/L). According to Eqs. (4), (5), the permeability (K) and the passability (η) parameters can be determined by the inverse of viscous resistance coefficient (α) and the inverse of inertial resistance coefficient (β), respectively. The square root of the permeability is assumed to be the proper characteristic length for defining the Reynolds number (Eq. (6)).
It should be noted that the Ergun correlation is only valid to precisely calculate the pressure gradient through porous media with a uniform particle size (UPS) packing arrangement and porosity (ɛ) (i.e, the volume available to the fluid for passing through the pores as a ratio of the total volume [23,26]) between 0.35 and 0.55; it is not able to predict the fluid pressure gradient through all kind of porous structures [6]. Numerous correlations have been established that are valid only for a packed bed with small particle size (e.g., 3.2 × 10−3 m [27], 1.59 × 10−3 m [28], 9.0 × 10−3 m [29]). For packed beds, discrepancies may be related to the implemented correlations for estimating pressure drop within the porous media.
For describing the pressure drop in porous medium, Forchheimer equation (Eq. (7)) is also commonly used. Forchheimer equation is an empirical correlation developed by using analogy with pipe flow with coefficients as correction factors to account for viscosity and inertial terms.
F is Forchheimer coefficient that accounts for inertia drag. Forchheimer equation uses the following equation (Eq. (8)) for the permeability as a function of particle size and porosity:where k is the Carman-Kozeny constant. The viscous resistance coefficient (α = 1/K) and inertial resistance coefficient (β) determine the porous structure and must be determined to calculate the pressure gradient through porous media analytically.where F is Forchheimer coefficient, Fo is the Forchheimer number. Although Kaviany [1] stated that k is identically 5, for spherical particles, the other researches [11,[23], [24], [25]] claimed that k is a function of tortuosity, porosity and particle size. It should be noted that the coefficients for Forchheimer correlation including F and k can be also obtained by utilizing the α and β from Ergun correlation.
A comprehensive literature review, comparison and reconciliation of Ergun and Forchheimer equations performed by Erdim et al. [7]. However, the validity of these correlations to a fixed packed bed of spheres with large, asymmetric, and irregular, particles, is questionable. Most published studies are based on packed beds with spherical particles [7,27,28]; few consider non-spherical particles [10,25,[30], [31], [32], [33]]. In their comprehensive review, Allen et al. [6] presented limitations of the Ergun and Forchheimer correlations for estimating the pressure gradient in porous media. It is also essential to predict the fluid flow behavior and associated pressure gradient through different porous media [34] since, to some extent, they determine the pressure required to maintain the constant mass flow rate and the rate of heat exchange between two phases in a porous medium in some applications. Several models have been developed to estimate the fluid pressure gradient in porous media composed of granular particles. Given that the permeability of a fixed packed bed influences the fluid pressure gradient, it is necessary to assess the variation among models for packed beds.
It is generally recognized that the pressure gradient through packed beds filled by spheres or semi-spheres should be calculated by means of semi-empirical models. Otherwise, the modified Ergun or Forchheimer correlations should be used by evaluating viscous and inertial resistance coefficient [35]. Several projects have been undertaken to study fluid flow behavior and heat transfer in porous structures at the nano- and micro-scale [[36], [37], [38]]. Models developed using the Ergun correlation are valid only for specific conditions and can differ significantly depending on the porous medium properties (Table 1).
Many researchers have asserted that the constants in these equations need to be calculated empirically for different packed beds because they depend on many factors, including the particle size and shape in addition to packing method and size distribution of particles in the packed bed [39]. The authors agree that unique constant value of Ergun and Forchheimer equations are not valid for all cases; however, the same values can be used for the similar beds. Large particles are common in the mining, geothermal, oil and gas and construction industries, yet no quantitative data have been published on porous media containing large particles (e.g., particle diameter > 10 cm) and macroscale particles (Table 1). For example, the properties of the caved zone with the particle size larger than 10 cm in the gob of longwall mining operation can directly affect air and methane leakage in the ventilation [40] is not directly available. Accurate measurement of porosity and permeability is not an easy task because the caved zone or other porous structure is not accessible to carry out conventional direct tests. Recent investigations by the authors assessed the seasonal thermal energy storage capacity of rock for supplying a portion of the energy demands in mine ventilation system in deep underground mines [[41], [42], [43], [44]]. Data pertaining to characteristics of the rock structure did not exist or were not accessible at most mine sites.
These limitations encouraged the authors to develop a numerical model in quantitative agreement with experimental results from the literature. There is a fundamental need to develop a pressure gradient model for porous media composed of large particles, which should be able to estimate the inertial and viscous resistance coefficients of a packed bed filled with large particles like rocks. Therefore, the objectives of this study are to: (i) generate a three-dimensional (3D) computational pore-scale model (PSM) of a packed bed of large spheres; (ii) evaluate the effect of packing arrangement (i.e., UPS versus non-Uniform Particle Size (n-UPS)) on the pressure gradient of the air passing through the packed bed of spheres; (iii) propose modified coefficients for the Ergun correlation and Forchheimer equation.
Section snippets
Model development
First, two sets of PSMs were generated, one using the UPS packing arrangement and one using the n-UPS packing arrangement (Fig. 1). Next, the analytical model was utilized to calculate the initial value for the viscous (α, k) and inertial (β, F) resistance coefficients. After that, for simplicity and saving times, a volume-averaged model (VAM) was created and the calculated value of α, k and β, F were entered into the model. Finally, the results of VAM and analytical model was compared to
Numerical methods
A large packed bed filled by spherical particles was created using EDEM software, then the model was developed by importing the bed geometry into Ansys 17.2 software. The influence of particle size and arrangement (UPS versus n-UPS) and porosity on the permeability of the packed bed was subsequently analyzed. To conduct the mesh independence test, the initial UPS and n-UPS models were created with approximately 6.5 × 105 and 1.1 × 106 nodes, respectively. Subsequent UPS and n-UPS models were
Validation of pore-scale CFD model
In spite of high-accuracy of CFD model output because of its significant advancement within the past few years, it is fundamental to validate the model results with the experimental data prior to applying them for further examination of the fluid flow behavior through the packed beds. Consequently, we created a packed bed of spherical particles imitating the experimental setup implemented by [53] whereby golf balls of diameter 42.6 mm were utilized to develop the packed bed (470 mm width by
Conclusion
It is essential to accurately predict the pressure gradient of air passing through a packed bed of large particles in various industry applications, such as mining, geothermal, oil and gas and construction. To the best of the authors' knowledge, this paper presents the first 3D PSM of flow through packed bed focused on large (in the scale of meters) particles. New constant values for the Ergun and Forchheimer correlations were proposed as functions of porosity and particle size. These modified
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