An evaluation of selected friction factor correlations and results for the pressure drop through random and structured packed beds of uniform spheres

Highlights

• Ergun correlation not applicable to packed beds consisting uniform sized spheres.

• KTA correlation should be used for packed beds consisting of uniform sized spheres.

• Eisfeld and Schnitzlein correlation should only be applied to packed beds with $D/d_p>2$.

• New friction correlation proposed for SC, BCC, OSC and HCP structured packed beds.

Abstract

The focus in the paper is on an evaluation of the friction factors as a function of the modified Reynolds number for the pressure drop through randomly and structured packed beds consisting only of uniform sized spheres. In the evaluation of the friction factor for large aspect ratio randomly packed beds it is confirmed that the use of the Ergun correlation should be avoided and that the KTA correlation should be employed to predict the friction factor. The effect of the wall for small aspect ratio randomly packed beds as predicted by the Eisfeld and Schnitzlein correlation is also evaluated and it is suggested that the use of the correlation be restricted to aspects ratios of $D/d_p>2$. The combination between the Eisfeld and Schnitzlein and KTA correlations should also be investigated. An evaluation is also provided for the friction factor results for selected structured packed beds. A correlation is proposed to predict the friction factor for offset simple cubic, hexagonal close packed, simple cubic and body-centred cubic packed beds. It is recommended that the friction factors for structured packed beds should be studied in more detail to develop a sound understanding of the effect of the packing structure on the pressure drop through a packed bed.

Introduction

Packed or granular beds can be found and play an important role in many practical applications such as amongst others water filtration plants, chemical reactors, heat storage in solar power plants, distillation columns, recuperator heat exchangers, nuclear fusion reactors and pebble bed high temperature gas-cooled reactors (HTGR) (Carman, 1937, Wu et al., 2002, Ribeiro et al., 2010, Allen et al., 2013, Klein, 2016, Guo et al., 2017, Lovreglio et al., 2018).

One of the important parameters in the performance and operation of packed beds that contributes to the pumping power required and plays a role in the system energy losses is the pressure drop through a packed (Ergun, 1952, Hassan and Kang, 2012). The pressure drop is amongst others affected by the void fraction or porosity of the packed bed, the size, shape and roughness of the particles, the packing arrangement of the particles and the geometry of the packed bed (Allen et al., 2013).

The results of many experimental studies and the evaluation of many experimental results have been published over time (Carman, 1937, Ergun, 1952, Wentz and Thodos, 1963, Susskind and Becker, 1966, Susskind and Becker, 1967, Metha and Hawley, 1969, Hicks, 1970, Macdonald et al., 1979, KTA, 1980, KTA, 1981, Raichura, 1999, Eisfeld and Schnitzlein, 2001, Ribeiro et al., 2010, Bai et al., 2011, Hassan and Kang, 2012, Allen et al., 2013, Erdim et al., 2015, Guo et al., 2017). The primary purpose of the studies was to derive or evaluate friction factor correlations for the prediction of the pressure drop through packed beds. Summaries and evaluations of a large collection of the friction factor correlations derived for the pressure drop through packed beds are given by amongst others Hassan and Kang, 2012, Erdim et al., 2015, and Guo et al. (2019). Numerical studies using Computational Fluid Dynamics (CFD) codes have also been performed to gain an understanding of the flow field in packed beds and evaluate the pressure drop through the packed beds(Calis et al., 2001, Reddy and Joshi, 2008, Reddy and Joshi, 2010, Eppinger et al., 2011, Wu et al., 2018, Guo et al., 2019, Gorman et al., 2019).

The focus in the current investigation is on the friction factor for the pressure drop through random and structured packed beds consisting of uniform sized spheres. Only a brief overview is given of a number studies that are of direct relevance to the current evaluation.

Carman (1937) evaluated data on the pressure drop through packed beds consisting of spheres that was obtained from various published studies. The modified Reynolds numbers for the studies were $0.06⩽{Re}_m⩽60000$, the porosities (or void fractions) were $0.3⩽\epsilon ⩽0.42$ and sphere diameters were $0.0025\text{m}⩽d_p⩽0.0905\text{m}$. The form of the correlation derived by Carman (1937) for the friction factor as a function of the modified Reynolds number for large aspect ratio packed beds is referred to as the Carman-type correlation or equation and account for the viscous and the inertial regimes. Carman (1937) also proposed an adaptation to the correlation to account the effect of the wall in packed beds with a smaller aspect ratio. Carman (1937) states that when the aspect ratio $D/d_p>10$ the wall effect is negligible. $D$ is the diameter of the cylindrical packed bed. The friction factor correlations derived by Bauer, 1960, Barthels, 1972; the KTA, 1980, KTA, 1981, and Erdim et al. (2015) are of this type.

Ergun (1952) also derived a correlation for the friction factor as a function of the modified Reynolds number based on data collected from published studies, as well as data obtained by him and colleagues. The modified Reynolds numbers for the data were $1⩽{Re}_m⩽2500$ and the data was for spheres, sand and pulverized coke of various sizes and also accounts for the viscous and inertial regimes. The form of the correlation or equation is referred to as the Ergun-type and is more commonly used compared to the Carman-type. Although the Ergun correlation was not strictly derived for packed beds consisting purely of uniform-sized spheres, it has been applied in the numerical analysis of packed pebble beds, for example Becker and Laurien, 2003, Visser et al., 2007, Suikkanen et al., 2014, Latifi et al., 2016, Burström et al., 2018, and Mardus-Hall et al. (2020).

Wentz and Thodos (1963), based on the work by Wentz (1962), obtained pressure drop measurements for flow through simple cubic (SC), body-centered cubic (BCC) and face-centered cubic (FCC) packing arrangements. To obtain high porosities spheres were distended by means of thin wires, giving porosities of $0.615<\epsilon <0.882$. Structured close packed beds, one with a BCC arrangement with $\epsilon =0.354$ and the other a SC arrangement with $\epsilon =0.48$, were also investigated. All the packing arrangements had a length of 5 sphere diameters. At the sphere-wall interface the particles were shaped to fit the wall curvature in an attempt to eliminate wall channeling. The modified Reynolds numbers for the measurements were$2550<{Re}_m<64000$, the spheres had a diameter of $d_p=0.0312\text{m}$ and the effective aspect ratio of the beds was $D/d_p=11.38$. A single correlation for the friction factor as function of the modified Reynolds number was derived for all the cases accounting for the pressure drop over the full length of each bed, as well as a single correlation for all cases for the pressure drop over the central layer of each bed. Van der Merwe (2014) set-up numerical models of the close packed SC and BCC beds to validate his computational fluid dynamics (CFD) methodology. He found the agreement between the predicted and the measured pressure drops to be very good. Van Loggerenberg (2020) also simulated the close packed BCC bed using the approach developed by Van der Merwe (2014) and extended the analysis to the fully viscous regime. Allen et al. (2013) measured the pressure drop over an SC packed for modified Reynolds numbers of $308<{Re}_m<7400$. The friction factors obtained from the experimental results were in good agreement with values obtained by Wentz and Thodos, 1963, Van der Merwe, 2014. Allen et al. (2013) also measured the pressure drops over offset simple cubic (rhombohedral-pyramidal) (OSC) and hexagonal closed packed (HCP) arrangements for modified Reynolds numbers of $45<{Re}_m<3500$. The friction factors calculated from the experimental results were in good agreement with friction factors obtained by Wentz and Thodos, 1963, Van der Merwe, 2014, and Van Loggerenberg (2020).

Structured close and “distended” packed beds with a rhombohedral-pyramidal (RP) packing arrangement in a vertical duct with a square cross section were constructed by Susskind and Becker, 1966, Susskind and Becker, 1967. They used spheres with diameters of 0.0032 m, 0.0064 m and 0.0127 m. The porosity was varied between $0.2595<\epsilon <0.3198$ by changing the lateral distance $d_{\mathit{ij}}$ between the centres of adjacent spheres in the first, third, fifth, etc. (A) layers between $1.000<d_{\mathit{ij}}/d_p<1.221$. In the axial direction the spheres of the successive (ABA…) layers were still in contact. No measures were taken to eliminate the wall effect. The modified Reynolds numbers for the tests were $1<{Re}_m<17000$. A correlation for the friction factor as a function of the modified Reynolds number could only be found for the packed beds with $1.000⩽d_{\mathit{ij}}/d_p⩽1.154$. The packed bed with $d_{\mathit{ij}}/d_p=1.154$ is identical to a BCC packing arrangement.

As part of the development of thermal-fluid correlations for pebble bed HTGR cores the KTA made a considerable effort to establish a friction factor correlation for cylindrical randomly packed beds consisting of uniform sized spheres over a large range of Reynolds numbers (KTA, 1980, KTA, 1981). The derivation of the correlation was based on the investigation of various well-tested friction factor correlations and the associated data. It was required that the influence of the walls on the pressure drop must be negligible, the average porosity of the bed known from the original documents, the length of the bed length be larger than 4 particle diameters, the data be for randomly packed beds, and that the sphere diameter be larger than 0.001 m. A Carman-type correlation was proposed that is valid for $10<{Re}_m<{10}^5$, $0.366<\epsilon <0.43$, $L/d_p>5$ and a graph was given depicting the minimum aspect ratio as a function of the modified Reynolds number. Allen et al. (2013) measured amongst other the pressure drop through randomly packed beds of uniformed sized glass spheres with porosities of $0.36<\epsilon <0.40$ for modified Reynolds numbers of $65<{Re}_m<4400$. The friction factors obtained from the experimental results were in good agreement with the values predicted by the KTA correlation.

Eisfeld and Schnitzlein (2001) investigated the influence of the cylinder wall on the pressure drop through cylindrical packed beds with the aim of establishing which existing friction factor correlations are valid when the wall effects are not negligible. They studied 24 correlations with more than 2300 data points. They found the friction factor correlation proposed by Reichelt (1972) to be the most suitable and refined the correlation to fit the data. The proposed correlation is an Ergun-type. For packed beds consisting of uniform sized spheres the correlation is valid for porosities of $0.33<\epsilon <0.882$, modified Reynolds numbers of $0.01⩽{Re}_p⩽17635$ and aspect ratios of $1.624⩽D/d_p⩽250$. Hassan and Kang (2012) measured the pressure drops over random packed beds with aspect ratios of 19.9, 9.5, 6.33 and 3.65 for modified Reynolds numbers of $353<{Re}_m<29936$ and found that for $D/d_p⩽5$ the effect of the wall must be accounted for. Van der Merwe (2014) performed a numerical simulation of the flow through randomly packed beds consisting of uniform sized sphere with aspect ratios of 1.6, 2.01, 2.5, 3.65, 4.0, 5.0 and 6.33 for modified Reynolds numbers of $1<{Re}_m<10000$. The friction factors derived from the numerical results were in very good agreement with the corresponding friction factors predicted by the Eisfeld and Schnitzlein correlation. Guo et al. (2017) measured the pressure drop in packed beds consisting of uniform sized sphere with aspect ratios of $1.025<D/d_p<2.88$ for modified Reynolds numbers of $800<{Re}_m<55400$. An analysis of the results showed the friction factors obtained for the packed beds with $D/d_p<2$ differed markedly from the corresponding values predicted by the Eisfeld and Schnitzlein correlation. For aspect ratios between $2<D/d_p<3$ the packed beds could have a structured, quasi-structured or a random packing arrangement. The friction factors obtained for the randomly packed beds were found to be in good agreement with the corresponding values predicted by the Eisfeld and Schnitzlein correlation.

In support of the PBMR project the High Pressure Test Unit (HPTU) was constructed to measure the pressure drops through random and structured packed beds to check the validity of the KTA friction factor correlation for the PBMR cylindrical and annular configurations, as well as the applicability of the correlation for the porosities typically found in the near wall region (Van der Walt, 2006, Rousseau and van Staden, 2008, Du Toit and Rousseau, 2014). Three pressure drop test sections (PDTS36, PDTS39, PDTS45) with nominal homogeneous porosities of 0.36, 0.39 and 0.45 respectively were manufactured in an effort to isolate the effect of porosity in the near wall region. A structured rhombohedral-hexahedral or cubic close (CCP) packing arrangement (ABC…) was chosen – not a tetrahedral arrangement as reported by Van der Walt (Van der Walt, 2006) or BCC arrangement as reported by Du Toit and Rousseau (Du Toit and Rousseau, 2014). The porosities were obtained by mounting the acrylic spheres on thin cables with spacers to vary the distance between the spheres. The test sections had square cross sections and the wall effect was eliminated by fixing cut spheres to the walls. Van der Walt (Van der Walt, 2006) and Du Toit and Rousseau (Du Toit and Rousseau, 2014) assumed that the effect of the spacers could be neglected, but numerical studies performed by Van Loggerenberg (Van Loggerenberg, 2020) have found that the effect of the spacers must be accounted for. Pressure drops over the test section were measured for modified Reynolds numbers of $1492<{Re}_m<77745$;$1647<{Re}_m<82757$ and $1829<{Re}_m<91335$ for the PDTS36, PDTS39 and PDTS45 test sections respectively. The friction factors for the structured packed beds differed significantly from the corresponding values predicted by the KTA correlation. A cylindrical randomly packed bed (SCPB) and an annular randomly packed bed (SAPB) were also manufactured. The radial dimensions of the packed beds were one tenth of the prototype beds and the SCPB had an aspect ratio of $D/d_p=61.5$ and the SCPB an aspect ratio of $\left(R_o-R_i\right)/d_p=14.1$. $R_o$ and $R_i$ are the inner and outer radii respectively of the annular bed. The modified Reynolds numbers for the pressure drops measured over the packed beds were $1667<{Re}_m<8349$ and $1660<{Re}_m<8318$ for the SCPB and SAPB respectively. The friction factors for the randomly packed beds were in very good agreement with the corresponding values predicted by the KTA correlation.

Erdim et al. (Erdim et al., 2015) performed experiments on packed beds consisting of uniform sized spheres to collect the required data to evaluate 38 different friction factor correlations found in the literature. The sphere diameters were $1.18\text{mm}⩽d_p⩽9.99\text{mm}$ resulting in aspect ratios of $4.0⩽D/d_p⩽34.1$ and porosities of $0.377⩽\epsilon ⩽0.47$. The modified Reynolds numbers for the pressure drops that were measured over the packed beds were $2<{Re}_m<3582$. Based on an analysis of the results they proposed a correlation of the Carman-type which yielded the best fit of their data.

In this paper a brief evaluation is first of all provided of the friction factor as a function of the modified Reynolds number for randomly packed beds. The values predicted by the Ergun and KTA correlations are compared and evaluated at the hand of selected experimental data. The effect of the wall for small aspect ratio randomly packed beds as predicted by the Eisfeld and Schnitzlein correlation is also evaluated and compared with numerical and experimental data. A brief evaluation is secondly provided of experimental and numerical results for the friction as a function of the modified Reynolds number for selected structured packed beds. Where available the experimental results are compared with numerical results. A correlation is proposed to predict the friction factor as function of the modified Reynolds number for OSC, HCP, SC and BCC packed beds.