1 Introduction

In the latter part of the twentieth century, renewable energy sources have been harnessed for electricity generation to mitigate the environmental degradation stemming from the combustion of fossil fuels. The Intergovernmental Panel on Climate Change (IPCC) has ascertained that emissions from fossil fuels are the primary driver of global warming [1]. In 2018, fossil fuels and industry contributed to 89% of global CO2 emissions [1]. This is primarily due to the prevalent use of fossil fuels as the primary energy source in most power plants for electricity generation. As of 2010, the total power generation amounted to 20 TWh, with 81% being derived from fossil fuels, and it is projected to surge by 70% by 2035 [2, 3]. Renewable energy sources are now commercially viable, and concentrated solar energy stands out as a widely utilized option due to its economical operational expenses [4, 5]. Nevertheless, the intermittent availability of sunlight poses a significant obstacle to the widespread adoption of solar energy. To surmount this limitation, a thermal energy storage (TES) system has been proposed to store thermal energy for nocturnal use. The molten salt TES is predominantly employed for storing concentrated solar energy. A packed bed of solid particles has emerged as a promising TES variant and has been suggested for integration into concentrated solar power plants (CSP). For instance, integrating the storage system into a solarized Brayton-Rankine combined cycle would enhance the overall capacity of the power plant [6].

The design of a cost-effective and efficient packed bed TES necessitates the selection of a suitable packing material and the accurate prediction of pressure drop. Crushed rock has been recommended as a promising material for energy storage [7]. The utilization of crushed rock reduces the capital cost of TES by approximately tenfold compared to the use of two molten-salt tanks [8]. However, the flow through a packed bed of crushed rock is intricate and not yet fully comprehended. Extensive research and deliberation have been undertaken to examine the flow behavior of rock-packed beds. Consequently, numerous correlations have been proposed to estimate the pressure drop across a densely packed bed. Ergun [9] formulated a correlation that amalgamates the Blake–Kozeny and Burke-Plummer equations to estimate the pressure drop across a packed bed. The Ergun equation, derived from extensive experimental data, represents the pressure drop along the bed as a function of several parameters and is considered a benchmark, as demonstrated by Eq. 1.

$$\frac{\Delta P}{L}=A \frac{{\left(1-\varepsilon \right)}^{2}}{{\varepsilon }^{3}} \frac{\mu {U}_{s}}{{{D}_{p}}^{2}}+B \frac{\left(1-\varepsilon \right)}{{\varepsilon }^{3}} \frac{\rho {U}_{s}^{2}}{{D}_{p}} ,$$
(1)

where \(\Delta P\) is the pressure drop across the thermal packed bed (Pa), \(L\) is the packed bed’s length (m), \(\varepsilon\) is the bed’s porosity, \({D}_{p}\) is the particle’s diameter (m) that is defined in terms of the particle’s surface area, \({U}_{s}\) is the superficial gas velocity (m/s), \(\mu\) is the fluid viscosity in kilograms per second, \(\rho\) is the fluid density in kilograms per cubic meter, and A and B are Ergun’s constants.

It has become evident that Ergun’s equation does not encompass all conceivable scenarios and is constrained by its simplicity. Consequently, the Ergun equation has been modified to establish a relationship that accurately predicts the pressure drop along a packed bed. According to Du Plessis and Woudberg [10], Ergun’s correlation is only valid for porosities between 0.38 and 0.47. Furthermore, the Ergun equation underestimates the pressure drop through a packed bed of non-spherical particles [11]. Ergun also constructed his model using a column and an isotropic flow. However, when applied to estimate the pressure drop across a packed bed, his equation yields significant errors. According to Zavattoni et al. [12] the pressure drop through a densely packed bed of rock exceeds Ergun’s prediction by 10–30%.

Several modifications have been implemented to enhance Ergun's correlation for predicting pressure drop. Initially, Ergun's constants were substituted with functions dependent on Reynolds number, followed by the inclusion of wall correction. Modifications and extensions of the Ergun equation have been proposed to incorporate the Reynolds number effect for a more accurate prediction of pressure drop, particularly in the transition between laminar and turbulent flow regimes in packed beds. Reichelt [13] integrated wall correction into the constant values by studying irregular particles. Similarly, Eisfeld and Schnitzlein [14] utilized irregular particle shapes to replace constants with wall correction factors linked to particle shape.

Other methods involved the utilization of shape factors such as sphericity (∅), as demonstrated by Singh et al. [15], who correlated pressure drop with sphericity and porosity for regular shapes. Trahan et al. [16] extended this to irregular shapes but encountered challenges in determining sphericity and used a plug flow approach. However, none of these modifications accounted for the flow-directional effect on pressure drop. Notably, most of the newly proposed correlations to estimate the pressure drop through the packed bed compared their results to the Ergun equation. This paper scrutinizes some of the most renowned studies that proposed correlations to estimate the pressure drop through a packed bed of particles. It also examines the major factors influencing the pressure drop measurement. The subsequent section delves into the published correlations used to predict the pressure drop in packed beds of regular and irregular shapes in detail. The third section delineates the parameters influencing the measurement of the pressure drop in a packed bed. Section 4 discusses the study's findings, and Sect. 5 concludes the investigation.

2 Correlations in the literature for predicting packed bed pressure drop

In general, the multitude of correlations proposed in published studies are typically formulated as a function of the most pivotal variables, including Reynolds number, particle shape, porosity, packing density, particle equivalent diameter, and flow direction. These correlations are based on in the Ergun equation and yield a friction factor (f) that is contingent on other parameters, such as particle Reynolds number, porosity, and sphericity. The friction factor is expressed as a set of parameters, as depicted in Eq. 2 [9].

$$\begin{array}{c}f=\frac{\Delta P {D}_{p} {\varepsilon }^{3}}{L \rho {{U}_{s}}^{2}\left(1-\varepsilon \right)}\end{array}$$
(2)

As illustrated in Eq. 3, the Reynolds number \((Re)\) is employed to categorize the flow as laminar, transient, or turbulent. Ergun defined the particle Reynolds number \(({Re}_{p})\) as a function of particle diameter, superficial velocity, porosity, and fluid characteristics, as shown in Eq. 4. Moreover, to facilitate the investigation, the friction factor will be plotted against \({Re}_{p}\) for each correlation scrutinized in this study.

$${R}_{e}=\frac{\rho {U}_{s}{D}_{p}}{\mu }$$
(3)
$${Re}_{p}=\frac{\rho {U}_{s}{D}_{p}}{\mu (1-\epsilon )}$$
(4)

Based on particle shape, packed beds of particles are primarily classified into two categories: spherical particles packed beds and non-spherical particles packed beds. The following is a compilation of the most significant correlations utilized in both types of packed beds.

2.1 Spherical particles packed beds

A packed bed of spherical particles has been utilized in numerous real-world applications, and many studies have developed correlations for predicting the pressure drop and comparing them with the Ergun equation (Eq. 5 in Table 1). Carman [17] formulated a correlation (Eq. 6 in Table 1) to estimate the pressure drop through spherical particles with \({Re}_{p}\) values ranging from 0.1 to 60,000, disregarding the effect of the wall. The model was validated with experimental data from earlier literature. The equation is of the type of Forchheimer’s equation, and he claimed that his equation was more accurate in fitting the data. Hicks [18] evaluated experimental data from earlier studies and found that the Ergun equation fails at \({Re}_{p}\)>500. Subsequently, he proposed a new formula for \({Re}_{p}\) values between 300 and 60,000 (Eq. 7 in Table 1). The findings indicate that Ergun’s constant values are inaccurate, resulting in errors in the pressure reduction prediction across the packed bed. Wilcox and Krier [19] developed a correlation (Eq. 8 in Table 1) valid for \({Re}_{p}\) between 1000 and 100,000 when using spherical glass particles. They determined that when \({Re}_{p}\) > 1000, the Ergun equation is inappropriate. In addition, they observed that the drag coefficient is not only a function of velocity, which influences \(Re\), but also a function of the ratio of the bed to particle diameter and velocity. Similar to Carman’s equation, Brauer [20] developed a formula (Eq. 9 in Table 1) with the same form but distinct constants, highlighting differences from the Ergun equation primarily in their dependence on the Reynolds number for the second term.

Erdim et al. [21] examined 38 correlations from previously published research and constructed an experimental model to evaluate the pressure drop through a packed bed of spherical particles ranging in size from 1.18 mm to 9.99 mm. They argued that their findings could only be applicable to the correlation proposed by Fahien and Schriver [22]. They presented a correlation (Eq. 10 in Table 1) to predict the pressure drop across a packed bed of spherical particles, with constants not dissimilar to those proposed in the literature by other researchers. However, the hypothesized correlation has only been examined for \({Re}_{p}\) between 2 and 3600 and has not been compared to other shapes. Also, Allen et al. [23] experimentally examined the flow across a packed bed of smooth and rough glass spheres and derived an equation as a function of Re (Eq. 11 in Table 1). According to them, the Ergun equation overestimates the pressure drop when \({Re}_{p}\) is greater than 750. They reported that the equation is similar to Carman’s [17] equation when particle Reynolds numbers exceed 300.

Handley and Heggs [24] used a column to study the flow through a packed bed of spherical shapes. They attached a layer of rubber sleeves in order to reduce the wall effect that influences the porosity measurement. They proposed a correlation (Eq. 12 in Table 1) by modifying the Ergun equation to account for spherical shapes. It is appropriate for use when \(8<L/{D}_{P}<50\). Additionally, the proposed equation is valid for \(Re\) greater than 200 where the flow is turbulent. Regar et al. [25] developed a correlation to predict pressure drop across pebble bed reactors (PBRs) with two beds containing 1568 and 1700 pebbles. Simulations were performed utilizing CFD models and the porous media method was employed to incorporate spherical pebbles. The proposed correlation (Eq. 13 in Table 1) is tested for Reynolds numbers ranging from 625 to 10,000. The proposed correlation reduces the error in predicting near-wall velocity from 50 to 5%. Additionally, the correlation is a function of porosity and particle Reynolds number.

Figure 1 compares the above-mentioned correlations used to estimate the pressure drop across a packed bed of spherical particles so far, including the Ergun equation. It also demonstrates that all correlations exhibit the same behaviour when predicting the pressure drop, which is around ± 30% than the Ergun equation. When \({Re}_{p}\) exceeds 500, the Ergun equation behaves differently, whereas the friction factor remains constant when \({Re}_{p}\) is greater than 2000. Equations proposed by Carman [17], Erdim et al. [21], and Jones and Krier [19] behave similarly, and both behave identically to the Ergun equation when \({Re}_{p}\) is less than 700. Furthermore, when \({Re}_{p}\) is greater than 700, Hick's equation [18] captures the flow in the same manner as the previously proposed correlation. It is also worth noting that none of these correlations consider the transition from laminar to turbulent flow. Finally, Table 1 summarizes all the abovementioned correlations in the previous paragraphs and their limitations.

Fig.1
figure 1

Spherical shapes correlations

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Table 1 Common friction factor equations for packed bed of spherical shapes
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2.2 Non-spherical particles packed beds

In many industrial applications, spherical particles are impractical, and instead non-spherical shapes are utilized to store thermal energy. Consequently, the pressure drop correlations are frequently derived using the equivalent particle diameter, defined as the diameter of a sphere (De) that has the same volume as the particle’s volume (\({V}_{p}\)), which can be calculated using Eq. 14. The commonly used pressure drop correlations will be examined in more detail in this section.

$${D}_{e}={\left(\frac{6 {V}_{p}}{\pi }\right)}^{1/3}$$
(14)

Handly and Heggs [24] delved into the analysis of fluid flow through non-spherical particles in a column isolated with a rubber sleeve to mitigate the wall effect. Their findings revealed the applicability of the Ergun equation in predicting pressure drop for non-spherical beds. However, they noted the absence of precise constant values, which vary based on the shape of the particles. Tallmadge [26] utilized data from Wentz and Thodos [27] to establish a correlation (Eq. 14) for \({Re}_{p}\) up to 100,000, highlighting the validity of the Ergun equation up to \({Re}_{p}\) equal 1000. Harrison et al. [28] formulated a correlation by amalgamating the Tallmadge equation (Eq. 15 in Table 2) and a formula proposed by Liu et al. [29], accommodating a broad range of particle Reynolds numbers and bed-to-particle diameter ratios. Their correlation (Eq. 16 in Table 2) was identified as one of the five most accurate equations in predicting pressure drop through non-spherical particles. Nevertheless, the combined correlation can be used for \({0.32<Re}_{P}<7700\) and \(8.3<L/{D}_{P}<50\).

Çarpinlioğlu and Özahi [30] introduced a distinct equation (Eq. 17 in Table 2) based on used the L/Dp ratio and porosity ranging from 0.37 to 0.52, valid for \({Re}_{p}\) values exceeding 700, surpassing the conventional Reynolds number limitation of the Ergun equation. In the same context, Koekemoer and Luckos [31] investigated the pressure drop across packed beds of coal, ash, and char. Different constant values were proposed to be related to Ergun’s constant values. Coal, char, and ash have respective constant values of (77.4 and 2.8), (160.4 and 2.8), and (229.7 and 2.3). As a result, all irregular shapes require different constant values, rendering Ergun’s proposed constant values inapplicable.

Nemec and Levec [11] modified the Ergun equation, considering particle shape (sphericity) when predicting pressure drop across non-spherical shapes with porosities ranging from 0.37 to 0.52 (Eq. 18 in Table 2). Their hypothesized correlation predicted a ± 10% decline in average pressure for non-spherical shapes, while asserting that the Ergun equation accurately predicts the flow over a single phase for spherical shapes but underestimates it for irregular shapes. Additionally, Özahi et al. [32] conducted an investigation into pressure drop, taking into account the shape factor over both spherical and non-spherical shapes. They put forward a correlation (Eq. 19 in Table 2) that mirrors the structure of the Ergun equation but utilizes different constant values. Their belief is that by substituting Ergun constants for 160 and 1.61, the results would be more accurate. Furthermore, Singh et al. [15] conducted experiments to assess the impact of particle shape on the pressure characteristics and packed bed performance. They employed different concrete forms such as T-joint and standard bricks, cubes, and spheres, with a sphericity exceeding 0.55 and Reynolds numbers ranging from 1047 to 2670. Subsequently, they introduced a correlation (Eq. 20 in Table 2) that delineates the friction factor based on sphericity, \(Re\), and porosity.

Hoffmann et al. [33] introduced an alternative method for predicting pressure drop through crushed rocks using the porous media technique, emphasizing that the inertial and viscous terms in the momentum sink are tensors, not scalars, as observed in previous correlations. Their approach involved employing a DEM-CFD method and representing the crushed rocks as ellipsoids with equivalent volumes to random measurements of the average size of crushed rock samples. Subsequently, they proposed a correlation (Eq. 21 in Table 2) and observed its significant sensitivity to elevation and azimuth angles, with dominant diagonal terms and approximately symmetrical tensors (\({C}_{ij}\) ≈ ±\({C}_{ji}\) for non-diagonal terms). The high flow resistance in the y-direction is reflected in \({C}_{yy}\) being about 60% higher than the other diagonal terms. However, the correlation notably underestimated the pressure drop across the crushed rock-packed bed by approximately 50%. In related study, Noël et al. [34] developed closure laws for a macroscopic model of heat and momentum transfer in real rock packings. They used X-ray imaging to examine the packing arrangement of basalt and gravel, covering a Reynolds number range from 5 to 243 with a fixed Prandtl number of 0.7. Their investigation involved a small cubic bed measuring 10 × 10 × 10 cm to enable controlled experimentation of the packing structures. The model showed good agreement with Allen et al. [23] experimental data for crushed rock. They proposed a correlation to estimate the drag force across the bed using the Sauter diameter, emphasizing the significance of the mean Sauter diameter as a key parameter for capturing the similar behaviour of real rock packings in terms of heat and momentum transfer.

Figure 2 presents a comparison of the correlations used to estimate the pressure drop through a packed bed of non-spherical particles and compared with Ergun’s equation. The figure demonstrates that the correlation developed by Özahi et al. [32] is similar to Ergun’s equation, which converges when the \({Re}_{p}\) is greater than 1000. Additionally, correlations developed by Tallmadge [26] and Harrison et al. [28] exhibit similar behaviour in capturing flow characteristics. Table 2 summarizes all the abovementioned correlations for non-spherical shapes and their limitations.

Fig. 2
figure 2

Non-spherical shapes correlations

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Table 2 Common friction factor equations for packed bed of non-spherical shapes
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Based on the prior research, it is obvious that Ergun’s equation is applicable to spherical shapes, single phase flow, and laminar flow with a Reynolds number of less than 1000. Prior studies have utilized Ergun’s equation as a benchmark but have tailored their own correlations by adjusting Ergun’s constant values. In order to gain a deeper comprehension of non-spherical shapes, there was a growing interest in exploring them over time. It is evident that the proposed correlations for non-spherical shapes vary more considerably. It is worth noting that using sphericity for non-spherical shapes can minimize the error in the friction factor when comparing them with Ergun correlation. However, the friction factor increases as the sphericity decreases. Noteworthy, the sphericity lacks any directional effect. Also, modifying Ergun’s correlation by including sphericity does not affect it significantly. This was a result of comparing the correlations that utilized sphericity with Ergun’s correlation. The findings show they behave similarly at high Reynolds numbers.

To conclude, developing an equation to estimate the pressure drop inside a packed bed of particles is challenging, necessitating the consideration of numerous parameters such as particle size and shape, void percentage, aspect ratio, velocity, sphericity, and packing arrangement. The next section investigates these parameters, which have a great effect on the pressure drop through a packed bed of particles.

3 Parameters influencing pressure drop measurement

3.1 Particles size

The size of particles plays a pivotal role in determining the flow characteristics within a packed bed of particles. The pressure drop exhibits an inverse relationship with particle size, increasing as the interparticle void distance decreases and fluid resistance intensifies. Figure 3 depicts the correlation between pressure drop and particle size when Ergun’s equation is applied. The graph showcases the relationship between particle size and pressure drop per unit length as determined by the Ergun equation, keeping all other variables constant. This serves to illustrate the variability in pressure drop in response to a singular parameter. Several studies have scrutinized the impact of particle size on packed bed performance. Furthermore, diverse computational and experimental models [15, 35] have been employed to simulate a densely packed bed of small particles. These models revealed that a densely packed bed of small particles enhances charging and discharging efficiency while concurrently elevating energy consumption and pressure drop. Additionally, particle size influences pressure drop and thermal efficiency, both of which decrease with a reduction in contact surface [15, 36]. Coulson [37] observed that the pressure drop in packed beds containing particles of varying sizes is significantly higher than in beds with uniform particle sizes, attributed to a reduction in bed porosity. Furthermore, Liangxing Li et al. [38] carried out an experimental study to assess how the pressure drop behaves in two-layer packed beds of glass beads arranged both horizontally and vertically, under conditions of single-phase and two-phase flows. Using nine different configurations of the beds, they measured pressure drops and contrasted them with those found in uniform beds. Their results showed that when fine particles were placed over coarser ones in a horizontal layering, the pressure drop increased significantly in the upper layer, whereas vertical layering produced a more balanced pressure drop across the bed. This indicates that complex flow patterns emerge due to the redistribution of flow laterally. The research highlights the need for more comprehensive studies and accurate models to predict flow in layered beds, especially considering the implications for the safety of nuclear reactors in extreme accident scenarios. The size and distribution of particles were investigated using industrial white spruce (Picea glauca) chips, with pressure reduction monitored as a function of chip placement [39]. By employing the Ergun correlation to compare the model with published data, it was discerned that chip sizes and distribution within the column exert a substantial impact on bed porosity and flow behavior [39].

Fig. 3
figure 3

Relationship between the pressure drop and particle’s size [9]

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Owing to the variability in crushed rock sizes, the typical range of crushed rock sizes spans from 30 to 100 mm sieve sizes. Consequently, researchers have delved into the realm of crushed rock sizes to ascertain the optimal rock size for achieving an optimal pressure drop through the rock bed. According to Duffie and Beckman [40], the optimal particle size range for thermal packed beds falls within 10 mm to 30 mm. Danok et al. [41] assessed the impact of heat transfer and pressure loss on the performance of rock-packed beds using two distinct methodologies. The first approach entailed modifying equivalent diameter values while maintaining a constant mass flow, whereas the second approach adopted the opposite strategy. The pressure drop across the rock bed decreased as the equivalent diameter increased, except within the range of 25 mm to 38 mm, where it remained constant. The optimal thermal storage was determined to possess a diameter of 0.01 m. Hoffmann [42] measured the box sizes for over a hundred random samples of a pile of crushed dolerite rock that passed through a 75 mm sieve. He determined the average dimensions for each axis: long axis = 87.9 mm, medium axis = 59.5 mm, and short axis = 37.4 mm. Additionally, he measured the sieve size for the lower and upper sieve sizes, which are 53 mm and 75 mm, respectively.

3.2 Porosity

Porosity (ε) is the ratio of the void volume to the total volume of the packed material, which can be calculated using the following formula.

$$\begin{array}{c}\varepsilon ={V}_{v}/{V}_{t}=1-{V}_{s}/{V}_{t},\end{array}$$
(22)

where \({V}_{v}\) is the volume of voids, \({V}_{s}\) is the volume of solids, and \({V}_{t}\) is the total volume of the bed.

The pressure drop through a packed bed is extremely sensitive to the fraction of space between particles. The pressure drop increases about 3 times for every 10% reduction in porosity, according to the Ergun equation as shown in Fig. 4. The effect of the porosity on the pressure drop and flow characteristics has been investigated as well as the parameters that affect the porosity. According to [43], the diameter of the particles, the shape of the packing, and the voidage along the wall have a significant effect on determining porosity. When applied to a bed of randomly packed spherical particles, the measured value fell between 0.36 and 0.43. In the wall region, however, the porosity value varies significantly, which is dependent on the \(L/Dp\) ratio [44]. The radial porosity changes dramatically near the wall at a distance of up to 1.0 \(Dp\), and then the change decreases [45]. Guo and Dai [46] obtained the same results when constructing a computational fluid dynamics (CFD) model to examine and evaluate the flow and heat transfer for 120 random spheres using the 3D Navier–Stokes equation at low Reynolds numbers ranging from 4.6 to 56. In addition to particle size, the porosity can also be affected by particle form and surface roughness. For instance, the porosity of particles with a regular shape raises as sphericity decreases and declines as the surface becomes softer, resulting in a resistance to pressure drop that is relatively high [11, 14, 23].

Fig. 4
figure 4

Pressure drop porosity dependence [9]

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Enhancing comprehension of the parameters governing void percentage facilitates more accurate predictions of pressure drop. It has been evidenced that particle size exerts the most significant influence on porosity, with bed porosity increasing as the sizes of particles in the packed bed increase [47]. Furthermore, the shape of the particles and the diameter of the bed impact the size distribution of voids, enabling the prediction of pressure decrease by these parameters. Several empirical and semi-empirical equations have been proposed in the literature [48], albeit these correlations hold true solely for large aspect ratios. As stated by Iliuta et al. and Iliuta and Larachi, [49, 50], void size stands as the most pivotal factor for pressure drop estimation. In the pursuit of developing a new pressure drop correlation, the void size distribution of a porous medium was considered instead.

Godbole et al. [51] conducted a comprehensive analysis of 52 porosity equations for upward vertical two-phase flow published in scientific literature. Comparing the most robust porosity equations for 0.5 < \(\varepsilon\) < 0.75 to experimental data revealed a root mean square error of approximately 9.7%. For the range of porosity between 0.75 and 1, the best correlations for porosity exhibited a root mean square error of about 5.3% when compared to experimental data. Utilizing a porosity equation with 5% precision to estimate both porosity and frictional pressure drop could yield a substantial error. However, these porosity ranges are considered to be beyond the typical porosity range of the crushed rock bed.

Porosity can be predicted for spherical particles when the particle sizes are known. Numerous correlations have been developed in the literature to estimate the bulk porosity of spherical particles in a cylindrical container [45, 52, 53]. However, predicting the porosity of a packed bed of non-spherical particles, such as crushed rocks, is challenging as it depends on the shape factor.

Conversely, in numerical models, particles sizes are typically altered to eliminate interaction regions. The two predominant approaches employed in the literature involve reducing or increasing the initial particle volume. Equation 7 can be utilized to compute the proportional relationship between porosity and percentage volume change. Porosity increases by 3% for every 1% drop in volume. Owing to the correlation between pressure drop and packing porosity, a 10% change in packing porosity could result in a 30% pressure drop error during storage. To mitigate the escalation in pressure drop throughout the packed bed, porosity variation should not exceed 3%.

$$\begin{array}{c}\varepsilon =1-\frac{\sum_{i=1}^{N} \frac{\pi {\left({{D}_{e}}_{i}-\Delta d\right)}^{3}}{6} }{{V}_{t}}\end{array}$$
(23)

In summary, managing the porosity of a packed bed of particles is exceedingly challenging but can be confined within a 1% to 2% margin for particles of the same size. In a packed bed of crushed rock, the average porosity for medium-sized particles can be estimated to range between 39 and 55%.

3.3 Particle surface

The surface roughness of a particle denotes the degree of smoothness that hinders fluid flow through a packed bed of particles, thereby influencing fluid velocity and subsequent fluid characteristics. When the Ergun correlation is applied, the friction coefficient for flow in pipes can be regarded as a function of surface roughness. However, the Moody chart indicates that surface roughness exerts no impact on pressure drop in laminar flow. In an exploration of the effect of roughness on flow characteristics, Meier et al. [54] devised an experimental model and compared it with a numerical model. They found that particle surface roughness influences fluid flow in a packed bed with low Reynolds numbers and porosity, both of which are affected by particle surface roughness. Similar to sphericity, particle roughness influences the surface’s form and quality. Consequently, as particle roughness escalates, pressure drop increases in both laminar and turbulent flows. This aligns with the assertion of Nemec and Levec [11] that pressure drop rises with increasing surface roughness. However, Eisfeld and Schnitzlein [14] contend that estimating the percentage increase in pressure drop due to particle surface roughness is unfeasible. On the other hand, it is possible to estimate the effect of particle surface on pressure drop, although it may be costly from a business perspective.

In summary, determining the effect of particle roughness alone in crushed rock beds can be difficult. This is because the process of creating smooth rocks is expensive, and the effect of roughness may be insignificant enough to be ignored.

3.4 Particle arrangement

Particles are often packed into a packed bed in one of two ways: systematic packing or random packing. When particles are placed systematically, structured packing is employed, whilst unstructured packing, which is often used in industrial applications, is used when the bed is packed randomly [46]. The structured packed bed has substantially better flow and heat transfer performance than a randomly packed bed, which has significantly lower flow and significantly higher heat transfer performance. The unstructured bed is widely used due to its low cost and practicality [55,56,57]. The structure method, however, can only be applied to a packed bed of regular particles and, consequently, it cannot be utilized in crushed rock-packed beds.

In a real crushed rock packed bed, the rocks are likely poured at random, based on pouring rates, rock sizes and shapes, and pouring angle. As proposed by Hoffmann and Lindeque [58], the rocks may be horizontally aligned along their long axis. Due to the fact that every packing configuration is different, establishing a correlation between pressure drop and particle dispersion can be difficult and expensive.

3.5 Wall effect

The flow behavior of a fluid near the wall of a packed bed exhibits distinct characteristics compared to the flow far from the wall. The flow along the wall is influenced by various factors such as particle morphology and dimensions, distance from the flow center, flow velocity, and the ratio of bed-to-particle diameter. The impact of the wall effect on pressure drop is contingent upon the bed-to-particle diameter ratio. It increases as the magnitude of wall effects in the viscous flow regime increases, whereas it decreases as the magnitude of wall effects increases in the inertial flow regime [59]. The bed-to-particle diameter ratio is deemed the most pivotal parameter affecting the wall effect, consequently garnering the focus of the majority of studies. The study [60] reveals that the wall effect becomes significant as the aspect ratio decreases, resulting in an increase in the porosity. In contrast, the wall effect reduces as the bed-to-particle diameter ratio increases and becomes negligible above 40 [54]. Similarly, in accordance with Bruch et al. [61], the wall effect can be disregarded when the bed-to-particle diameter ratio exceeds 30. However, a significant escalation in the wall effect is observed when the bed-to-particle diameter ratio falls below 10. [14] and [62] observed that the wall effect is contingent on the Reynolds number, with the pressure drop induced by the wall effect escalating in laminar flow regimes and diminishing in turbulent flow regimes. Montillet [63] proposed a correlation to estimate the wall effect on pressure drop through a densely packed bed, utilizing spherical particles with aspect ratios lower than 20 and Reynolds numbers as high as 1500. Employing a similar approach, Montillit et al. [64] introduced a comparable equation with aspect ratios ranging from 3.5 to 40 and Reynolds numbers spanning from 10 to 2500. However, both were utilized and validated for small spherical particles and reported insignificant changes when the bed-to-particle diameter ratio exceeded 40. Figure 5 compares the aforementioned relationships and demonstrates that, for bed-to-particle diameter ratios greater than 30, the alteration in the friction factor is negligible.

Fig. 5
figure 5

Influence of bed-to-particle diameter ratio on the friction factor [63, 64]

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The wall effect on the pressure drop through a stationary packed bed of spheres is stated to be negligible when the bed-to-particle diameter ratio is above 20 [59]. In summary, based on Allen's calculation, the length of a packed bed of crushed rock for a 50 MWe solar is projected to be approximately 7 m [8]. Consequently, the ratio of bed diameter to average rock diameter would unquestionably exceed 40. Hence, it is unnecessary to account for the wall effect for a large bed.

3.6 Particle shape

The morphology of a particle is a critical parameter in simulating a packed particle bed as it profoundly influences the flow dynamics within the bed. It is used in computing various metrics, including the equivalent diameter, particle volume, surface area, and sphericity. Typically, the shape of crushed rock is characterized by three parameters: sphericity, roundness, and surface texture [65]. Extensive research has been conducted on the impact of particle shape on pressure drop, and the findings have been consolidated into correlations based on sphericity. Sphericity denotes the ratio of the surface area of a spherical particle to the surface area of an actual particle of the same volume, ranging from 0 to 1 and can be predicted using 1D and 2D components [66]. Diverse methodologies have been utilized to estimate the sphericity of particles by measuring their volume and surface area. The lengths of particles in 1D can be measured using a ruler or a calliper, while the sizes of particles in 2D can be determined through image analysis of particle projections. A novel technique known as laser scanning [67] is used to estimate the particle's 3D shape. Additionally, Blott and Pye [68] suggested determining the sphericity of particles using a combination of \(I/L\) (elongation ratio) and \(S/I\) (flatness ratio) techniques. Where \(L\) is the longest dimension of a particle, I is the longest dimension perpendicular to \(L\), and S is perpendicular to both \(I\) and S [69]. Typically, the method and methodology used to assess \(L, I,\) and \(S\) values are applied to determine sphericity. Consequently, most contemporary procedures entail some degree of inaccuracy, particularly when dealing with irregular shapes and very small particles. It would be more precise to measure the dimensions of the particle's box surfaces, which could reduce the margin of error. Therefore, measuring the distance between surfaces simplifies the process. Furthermore, employing 3D scanning enables an accurate determination of particle sizes. Scanning the surfaces of crushed rock and measuring the distances between the surfaces facilitates the proper depiction of the correct shape in the representation of crushed rock.

Experiments involving diverse particle geometries were conducted to assess the influence of the shape factor on the drag coefficient. It was observed that as sphericity decreases, the drag coefficient increases in one direction and decreases in another [69]. The impact of irregular particle dimensions (size and shape) on packed bed behaviour has been thoroughly investigated, encompassing both regular and irregular shapes. Standish and Drinkwater [70] revealed that particle shape is one of the key criteria that directly affects the overall performance of the packed bed. Furthermore, the pressure drop of the packed bed has a direct relationship with the operation cost and capital cost [71]. To accurately anticipate pumping energy, Chandra and Willits [72] recommended considering the influence of bed porosity. Yow et al. [73] also investigated and analyzed experimental data to ascertain the effect of sphericity on the drag coefficient for various particle shapes, including cubes, discs, cylinders, and bricks. They noticed that the drag coefficient increases significantly as sphericity decreases.

Also, the particle shape influences packing density, thereby altering the pressure drop across the packed bed of particles. Aijun and Zhihong [74] developed a DEM model and evaluated the impact of particle shape on packing density using a 2D imaging technique. To examine the influence of particle shape on porosity, three shapes were used to simulate a change in one shape while maintaining the other shapes relatively constant. The studies demonstrated that elongation influences porosity, which increases as the elongation rate increases (elongation increased from 0.310 to 0.912, and the porosity raised from 17% to 20.2%). Additionally, when the roundness of the particle is altered from 0.257 to 0.849, the porosity increases from 16.98% to 24.20%. However, this method may lake accuracy when applied to crushed rock in general due to the varying sizes and shapes of crushed rocks. Utilizing experimental data, Hartman et al. [75] also established a correlation between sphericity and porosity, albeit valid for porosities greater than 0.42. He et al. [67] studied the effects of particle shape and Reynolds number on the drag force by simulating structure packing with a solid volume fraction ranging from 0.1 to 0.35 at low moderate Reynolds numbers. Ellipsoidal assemblies have a drag force that is 15% to 35% greater than spherical particles at the same solid fraction and Reynolds number. He et al. [67] compared their findings to a correlation established using a mixture of isolated non-spherical particles and a correction factor established by Di Felice [76]. According to their findings, at low Reynolds numbers, the total correlation underpredicts the ellipsoid particle's drag force. Knowing that Di Felice used a different approach known as the immersed particle approach. The approach method is based on the correction of drag across a single particle in a free stream by the presence of nearby particles through porosity. However, this method is not deemed effective for predicting pressure drop as the drag coefficient of irregular shapes is dependent on flow direction.

In the same context, [55] examined various particle shapes, such as spherical, flat ellipsoidal, and long ellipsoidal forms, utilizing Reynolds numbers between 1 and 5000 and temperatures between 293 and 333 K. The pressure drop across an irregular particle bed is greater than that across a spherical particle bed when a constant Reynolds number is employed. The results demonstrate that, for the same packing structure, particle shape significantly influences pressure drop and tortuosity. Furthermore, a suitable selection of particle shape can mitigate pressure drop and enhance the overall efficiency of the packed bed. Hoffmann and Lindeque [58] proposed an ellipsoidal shape with dimensions of 87.9 mm, 59.5 mm, and 37.4 mm to represent crushed rocks. They analysed the pressure drop using the momentum source equation, which is based on viscous and inertial factors. Notably, these terms are tensors, not constants, as stated in all previous pressure drop correlations. In a separate study [42], a correlation was proposed for predicting the pressure drop through a bed of ellipsoidal particles and reported that it was underestimated for crushed rocks.

While the optimal shape for representing crushed rocks in simulations has not been definitively identified, the ellipsoidal shape stands as one of the most suitable options for simulating and predicting flow behaviour and pressure drop with reasonable accuracy. Moreover, extensive research should be dedicated to exploring the shape of particles that best represent crushed rocks. A shape with the same volume box as an ellipsoidal shape but with distinct sharp edges may offer some success in representing crushed rock. Given that crushed rock exhibits an irregular shape with numerous edges, it is typically characterized by an uneven surface.

3.7 Particle orientation

The flow behaviour through a packed bed of particles depends on the particles' distribution inside the bed, which is affected by the particles’ orientation to the flow, and it is not captured by sphericity. The angle between the flow direction and the particle's major axis has a considerable effect on the pressure drop inside a densely packed bed of crushed rocks or irregularly shaped particles. The force resistance increases as the particle orientation increases against the flow direction. In a study by [77], a novel relationship between the attack angle and the drag force of an ellipsoid particle was established. The results indicated that the drag force is a strong function of the angle of attack, which has a major impact on it (about 80% for an ellipsoid). Haider and Levenspiel [78] developed correlations for experimentally determining the drag coefficient of spherical and non-spherical shapes. They validated the results with recently proposed equations from the literature. They proposed a correlation that is a function of both Reynolds number and sphericity with a 3% deviation from the literature. The authors in [79] established and experimentally validated a correlation to determine the drag coefficient as a function of Reynolds number, sphericity, and particle orientation. Using longitudinal and transverse sphericity, particle orientation was determined. When comparing the relationship to recently proposed correlations in the scientific literature, the authors found that it differed by 14.1%.

Previous studies have investigated the impact of drag force on a single particle. However, the research also focused on the effect of drag force on a collection of particles. Rong et al. [80] developed a correlation for calculating the average drag force of ellipsoid particles in a bed that flows uniformly. Results indicate that the correlation provides more precise results than correlations found in the literature. In addition to that, the aspect ratio and sphericity have a substantial effect on the mean drag force. Hoffmann et al. [33] examined the effect of particle orientation on pressure drop and reported that the pressure drop responds more to the elevation angle than to the azimuth angle. They developed a correlation for calculating the pressure drop through a packed bed of ellipsoidal particles and reported that it underestimates the pressure drop over crushed rocks by 50%.

4 Discussion

Predicting the pressure drop across a packed bed of crushed rock is a pivotal factor influencing the energy consumption and uneven flow distribution within the bed.To construct an economical model of a rock-packed bed, it is necessary to precisely predict the pressure drop across the bed. Drawing from a comprehensive analysis of the existing literature on pressure drop through packed beds, a thorough knowledge of the complexities inherent in flow behaviour has emerged. Central to the discussion are the fundamental questions surrounding the appropriate nature of coefficients employed in pressure drop predictive models. The literature gives a range of perspectives, necessitating a careful analysis of whether coefficients should be constant, fluctuate as functions of relevant general factors such as Reynolds number, porosity, and sphericity, or transform into tensors. This critical argument incorporates both the pragmatic appeal and limitations of each technique, influencing the course of improvements in packed bed pressure drop modelling. Additionally, the parameters that influence the pressure measurement should be understood. According to the literature, the parameters that affect the pressure drop have been thoroughly discussed and examined. In this section, the summary of the findings in the literature is presented and criticized in detail.

In general, the proposed correlations for spherical and non-spherical shapes in the literature can be employed to estimate the pressure drop through a packed bed of rock with some margin of error. To apply these correlations, the equivalent diameter of crushed rock must be computed. Alternatively, the porous media approach introduced by Hoffmann et al. [33], utilizing tensors, was compared to correlations employing the common approach, which utilizes constant values, such as the Harrison et al. [28] equation (Eq. 16 in Table 2), as illustrated in Fig. 6. The graph demonstrates a substantial concurrence between the two equations in terms of rate of change, exhibiting similar behaviour when the Reynolds number exceeds 2000. Nevertheless, disparities in the friction factor values are evident. Consequently, unfortunately, none of the proposed correlations adequately anticipates the pressure drop across the bed of crushed rock.

Fig. 6
figure 6

Comparison of non-spherical and porous media approach equations

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Porosity is one of the most challenging parameters to control and has a significant impact on pressure reduction. Attaining effective control over porosity necessitates the consideration of numerous parameters, encompassing packing method, particle dimensions and morphology, container size, as well as the velocity and flow rate employed during container filling. Regrettably, the existing literature lacks a definitive correlation capable of accurately predicting the porosity of a packed bed where the pressure drop is highly sensitive to the interparticle spaces. The primary controllable factors for a crushed rock packed bed encompass the crushed rock size and bed dimensions, which can be specified within a certain range.

Particle size has a direct effect on the flow characteristics, whereas the pressure drop increases with a decrease in particle size. On the other hand, smaller sizes enhance energy storage capacity due to increased contact surfaces. Given the negligible pressure variation beyond this range, the optimal crushed rock size may fall within the 60 mm to 80 mm range, as illustrated in Fig. 7. Additionally, the size of the crushed rock is dependent on the recommended storage method, the required capacity, and the available volume for construction. Furthermore, it is evident from the graph that it makes no difference whether the flow is laminar, transitional, or turbulent.

Fig. 7
figure 7

Impact of the particle size on the pressure dropper unit length at different \({Re}_{p}\) [9]

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The wall effect is determined by the bed-to-particle diameter ratio, which can be disregarded when this ratio exceeds 20 for large beds. The wall effect can be estimated numerically because it is challenging to estimate it experimentally. For a real packed bed, the wall effect must be considered, and a perfect design must be developed to minimize the wall effect on pressure drop. Notably, large packed beds are almost always free-standing piles, resulting in minimal wall contact.

The characterization of crushed rock particles involves assessing their size, shape, roundness, irregularity, and roughness, among other properties. Among these parameters, particle size holds the utmost significance due to its influence on packing, permeability, and mechanical behaviour. Roundedness and shape influence the interlocking and stability of particle arrangements, followed by roughness, which impacts friction and particle interactions. Overall shape and irregularity play a smaller but still noticeable role. Size distributions that are monodisperse simplify experiments and modelling. Measuring irregularity and texture remains difficult despite technological advancements that allow for precise measurements. Depending on the application, size, sphericity, and shape typically carry the highest significance.

Many shapes, including bricks, a compound shape composed of spherical shapes, and an elliptical shape, have been suggested to simulate flow through a packed bed of crushed rocks. Despite this, none of these shapes adequately represent the crushed rock, resulting in an underestimation of the pressure drop through a packed bed of crushed rock. However, the ellipsoidal shape had some success in representing the rocks. This is because crushed rocks have irregular shapes due to the crushing process. In contrast to spheres, which possess uniform and round surfaces, ellipsoidal shapes can more effectively imitate these irregular shapes. When irregular shapes are utilized, the packed bed’s porosity can be minimized, resulting in a higher packing density, which better represents the actual conditions in a real-world packed bed. Additionally, the flow over irregular particles improves the flow’s similarity to that through crushed rocks. The ellipsoidal particle has a high degree of similarity to crushed rock in terms of sphericity, angularity, aspect ratio, bounding box, and roundness [81]. Moreover, they generate a higher packing density and a porosity comparable to that of crushed rock beds. Sphericity, angularity, and roundness lack direction information; however, the aspect ratio is the only parameter containing direction information.

The ellipsoidal shape proposed by Hoffmann [42] has achieved some success in capturing the flow characteristics. However, the crushed rock has an irregular shape and contains various sharp edges. Therefore, a compound shape including sharp edges may give some accuracy to capture the flow behaviour. To create a compound shape, simply, a brick of the same size should be developed to fill the box. All the three sizes should be divided into equal three subsequent sizes, and then remove the triangle shapes from the brick's corners. This leads to a compound shape having more than 20 edges.

Describing the flow through crushed rocks is challenging as it results in a three-dimensional flow in which the orientation and distribution of the particles impact the pressure drop. A number of studies have been conducted to investigate the relationship between particle orientation and pressure drop. However, it is not yet fully understood whereas additional research needs to be carried out in this field.

5 Conclusion

The pressure drop across a packed bed of crushed rock plays a pivotal role in the design and efficiency of thermal storage systems, highlighting the critical need for accurate prediction methods. This comprehensive review delves into various aspects of pressure drop across gravel packed beds and elucidates its practical implications. Through the evaluation of existing prediction correlations, it becomes evident that precise and reliable methods are essential for optimizing the design and operation of thermal energy storage systems. The significance of pressure drop prediction in industrial applications cannot be overstated, underscoring the urgency for further research and development in this domain. Real-world applications of gravel packed beds, particularly in thermal energy storage and heat transfer processes, demonstrate considerable potential. The economic viability and cost-effectiveness associated with crushed rock as a packing material make it an attractive option for a range of industrial applications, including concentrated solar power plants and thermal energy storage facilities. Advancements in crushed rock packed bed design and operation focus on optimizing particle shape, size distribution, and packing arrangement to enhance flow behavior and pressure drop characteristics. Moreover, advancements in porous media methods show promise in accurately predicting pressure drop, paving the way for future research and development. While various particle shapes, such as spheres, bricks, compound shapes, and ellipsoids, have been proposed to represent crushed rocks, the understanding of flow through a packed bed of ellipsoidal particles remains incomplete. The influence of particle shape and orientation on pressure drop underscores the complexity of flow behavior, necessitating further investigation to develop precise predictive models. The effect of particle orientation on pressure drop and fluid properties underscores the intricate nature of flow dynamics in crushed rock packed beds, warranting thorough research efforts. Utilizing a spherical container for examining particle orientation is recommended, as it allows for 360-degree rotation to explore all directions comprehensively. In conclusion, this review emphasizes the importance of accurate prediction methods for pressure drop across crushed rock packed beds and highlights the need for further research to enhance the understanding and predictive capabilities in this vital area of industrial application.