Mean porosity variations in packed bed of monosized spheres with small tube-to-particle diameter ratios

Powder Technology 354 (2019) 842–853

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Mean porosity variations in packed bed of monosized spheres with small tube-to-particle diameter ratios Zehua Guo a,b, Zhongning Sun a,b, Nan Zhang a,b,⁎, Xiaxin Cao a,b, Ming Ding a,b a b

Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, Heilongjiang 150001, PR China College of Nuclear Science and Technology, Harbin Engineering University, Heilongjiang 150001, PR China

a r t i c l e

i n f o

Article history: Received 18 January 2019 Received in revised form 26 March 2019 Accepted 1 July 2019 Available online 02 July 2019 Keywords: Packed bed Mean porosity Layer stacking model Small tube-to-particle diameter ratio

a b s t r a c t The transport phenomena in packed beds are sensitive to porosity variations. However, empirical correlations in the literature are not in agreement regarding the prediction of the mean porosity for packed beds, resulting in challenges for reactor design. In this study, the mean porosity variations in a densely packed bed with 1.2 b D/ d b 7.1 are investigated experimentally. Moreover, a layer-stacking model is developed to explore the mean porosity variations for different D/d ratios. It is observed that analytical expressions obtained by the model can effectively track the experimental data for D/d b 3. When the D/d ratio increases further, changes in the mean porosity values can be interpreted by the model. Furthermore, it is confirmed that the mean porosity variations exhibit several inherent fluctuations as the D/d ratios vary, which may contribute to the significant differences between values predicted by the correlations in the literature. This significantly reduces the applicability of empirical correlations in the rigorous analysis of packed beds. Finally, the significance of the model is discussed with respect to the prediction of the pressure drop in the packed bed. This study can provide new perspectives to model mean porosity variations in packed beds, which is also useful for the evaluation of the hydraulic performance. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Packed beds can be utilized for several applications in chemical industries and other related industries, including chemical reactors, grain dryers, nuclear reactors, and heat storage. In recent years, packed beds with small tube-to-particle diameter ratios have been discussed frequently in the literature, exhibiting advantages in pressure drop reduction and rapid heat removal [1–4]. The design of new packed bed reactors is based on the mechanisms of heat and mass transfer, as well as the flow and pressure drop of the fluid through the bed of solids. These mechanisms are affected significantly by porosity variations in the packed bed [5,6]. In a packing system, the packing geometry is disturbed by the confining wall. When decreasing the tube-to-particle diameter ratio, the wall exerts a significant influence on the packing structure in a packed bed: it not only changes the local porosity, but also the mean porosity

⁎ Corresponding author at: Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, Heilongjiang 150001, PR China. E-mail addresses: [email protected] (Z. Guo), [email protected] (N. Zhang).

https://doi.org/10.1016/j.powtec.2019.07.001 0032-5910/© 2019 Elsevier B.V. All rights reserved.

of the packed beds. Therefore, various approaches have been presented to describe the porosity variations in packed beds with small tube-toparticle diameter ratios. Certain experiments and simulations have been aimed specifically at capturing the oscillatory profile with radial distance from 0 to R [7–9]. Moreover, correlations have been developed to characterize the damping oscillations [10–13]. Studies concerned with the global packing structure have also been widely conducted. The relevant correlations are summarized in Table 1. Moreover, attempts have been conducted to reconcile the radial porosity variations with the integrated bulk porosity as a function of D/d [14]. However, to achieve concrete results, further investigations are necessary for generalized expressions. All of these investigations provide essential foundations for the design and analysis of systems using packed beds with small tube-to-particle diameter ratios. The prediction of pressure drop when using packed beds is an important engineering issue, as it directly influences convection heat transfer, chemical reaction rates, filtration effectiveness, and required pumping power. A variety of correlations exist for the prediction of pressure drop in packed beds, and a summary was made by Erdim et al. [24]. According to these correlations, the pressure drop in a packed bed is primarily influenced by the characteristic particle diameter, superficial velocity, working fluid properties, and mean porosity of the packed bed. For example, the most widely used empirical correlation

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Table 1 Summary of empirical correlations for mean porosity predictions in packed bed. Author

Correlation

Carman [15] ε ¼ 1− Sato et al. [16] Dixon [17]

Fand and Thinakaran [18]

Foumeny et al. [19]

Zou and Yu [14]

de Klerk [20] Benyahia and O'Neill [21]

2 d 3 D

Range of applicability !3

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! d 1= 2 −1 D

D ≥2:5 d D ≤1:866 d D 1:866≤ ≤2 d D ≥2 d

d ε ¼ 0:3494 þ 0:4381 D !3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 d d ε ¼ 1− 1= 2 −1 3 D D  d ε ¼ 0:528 þ 2:464 −0:5 D !2 d d ε ¼ 0:4 þ 0:05 þ 0:412 D D !3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 d d 1= 2 −1 ε ¼ 1− 3 D D D ε ¼ −0:6649 þ 1:8578 d 0:151 ! þ 0:36 ε¼ D −1 d !3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 d d ε ¼ 1− 1= 2 −1 3 D D rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! D−0:923 D ε ¼ 0:383 þ 0:254 1= 0:723 −1 d d !3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 d d 1= 2 −1 ε ¼ 1− 3 D D !2 d d ε ¼ 0:681−1:363 þ 2:241 D D " ! # d ε ¼ 0:372 þ 0:002 exp 15:306 −1 D ! D ε ¼ ε b þ 0:35 exp −0:39 d ε ¼ 0:39 þ 1

1:74 D þ 1:14 d

Ribeiro et al. [22]

D ≤1:866 d



Cheng [23] ε¼

D ≤1:866 d D 1:866≤ d D ≤1:866 d D 1:866≤ ≤3:95 d D ≥3:95 d

N/A

1:5 ≤

!2

ε ¼ 0:373 þ 0:917 exp −0:824

D ≤1:866 d D 1:866≤ ≤2:033 d D ≥2:033 d

D d

!

   0:27 −3  1:9 −3 −1=3 D−d d 0:8 þ 0:38 1 þ d D−d

2≤

D ≤50 d

D ≤19 d

D ≤100 d

for pressure drop is the Ergun equation [25]: ð1−ε Þ2 V ð1−εÞ V 2 ΔP ¼ 150μ þ 1:75ρ 3 − L d ε3 d2 ε

ð1Þ

In Eq. (1), except for the mean porosity, all other parameters can be determined accurately according to the designed operating conditions. The porosity value is typically predicted using empirical correlations published in the literature [16,23]. Still, correlations continue to be proposed. For comparison, average porosities predicted by the correlations in Table 1 are shown in Fig. 1 for packed beds with D/d b 30. As shown in Fig. 1, only correlations for D/d b 2 predict values for the mean porosity that are in agreement. This is because configurations with a regular structure are obtained within this range of tube-toparticle diameter ratios. Therefore, in a landmark paper, Carman [15] calculated the mean porosity variations in a theoretical manner, which is widely recommended by researchers [14,17–19]. By varying the tube-to-particle diameter ratio, experiments have been performed by numerous researchers, and various empirical correlations have been proposed (see Table 1). However, although a generally satisfactory goodness of fit was reported in every study, their predicted porosity values show poor agreement. In Fig. 1, certain predicted values are almost 25% higher than the lowest value.

Fig. 1. Predictions of mean porosity values for packed beds with D/d b 30 by correlations in Table 1.

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From Eq. (1), it is clear that the mean porosity value is a critical parameter. The error in pressure drop resulting solely from an error in porosity can be estimated according to the error propagation, as per Eq. (2) [26]:   ∂ðΔP Þ    ∂ε  3−mε Δε   ¼ ; fm ¼ 1 ðLaminar flowÞ; m ¼ 2 ðTurbulent flowÞ ΔP 1−ε ε Δε ð2Þ It can be observed that the pressure drop prediction is highly sensitive for variations in mean porosity. A small deviation in the predicted mean porosity value can translate into a significant error in the pressure drop evaluation. A 2.5% deviation in the mean porosity predictions may lead to a ~10% error in the predicted pressure drop in laminar flow conditions. Although the mean porosity is a crucial parameter in reactor design, unfortunately, predictions of the mean porosity value for packed beds with small tube-to-particle diameter ratios (typically D/d b 10, see Fig. 1) could vary significantly when using different empirical correlations. Therefore, tubular packed bed reactors must be oversized using a capacity safety factor in the design owing to this inherent error in pressure drop calculations. Despite the significance of this problem, the differences in empirical correlations for the prediction of the mean porosity have not been investigated extensively. The primary reason is that the randomness of the particle distribution in a packed bed results in significant difficulty in the modeling of the packing. Furthermore, the effects of other complex interactions such as the geometrical parameters, particle materials, and filling methods, as well as the vibration mode on the packed bed structures render it substantially more difficult to provide a generally satisfactory solution to this problem. Therefore, to the best of the authors' knowledge, only a few models or analytical expressions have been developed to characterize the mean porosity variations in a packed bed with D/d N 2. It has been reported that densely packed beds are always associated with the occurrence of regular packing structures. Yu et al. [27] and An et al. [28] demonstrated that, by controlling the vibrational and feeding conditions properly, the transition from disordered to ordered and the densest packing of particles can be obtained consistently. Reimann et al. [29] investigated the effects of the container geometry and filling and vibration procedures on the packing structures of a packed bed using X-ray computed tomography. They found that the zones with regular structures, which were in a dense hexagonal array with horizontal sphere chains, could gradually dominate the packed bed with appropriate vibrations. Mcgeary [30] studied the structure of a packed bed of uniform or composite spheres. He found that, in a packed bed of uniform spheres, the vertical sphere positions in the inner layers were the same as those of the layers adjacent to the container wall, and that the packing consisted almost entirely of an orthorhombic arrangement owing to the gravitational stability. The dense packing generally represents a limiting reference for packed beds at a specific D/d. Therefore, although packed beds used for industrial applications are not necessarily a dense packing and the mean porosity could vary with different loading strategies, explorations into the packing of densely packed beds can lay the basic foundation for a better understanding of the packed bed structure. And, fortunately, the regular structures of the dense packing can provide significant benefits in packed beds modeling. Therefore, in this study, experiments on mean porosity variations are conducted for densely packed beds with D/d b 7.1. A layerstacking model is developed to describe the packing configuration of a packed bed with the appropriate simplifications. It can be observed that simple equations and factors can effectively capture the packing state of a packed bed, and is in good agreement with the experimental results of the mean porosity variations. Furthermore, the

general trend of the mean porosity variations with increasing D/d ratios can be interpreted. Rapid oscillations of the mean porosity value are confirmed by the theoretical analysis of the packed beds with small tube-to-particle diameter ratios. This could contribute greatly to the understanding of the significant differences between the values predicted by the empirical correlations in Table 1. Finally, the significance of the layer stacking model for the pressure drop characteristics of the packed beds is discussed, which can provide new insights into the evaluation of the hydraulic performance of packed beds. 2. Development of layer stacking model According to previous studies [29–31], a densely packed bed exhibits a distinct layered configuration. Therefore, it is assumed that a densely packed bed consists of a number of similar layers, as illustrated in Fig. 2. Each layer contains a maximum of n spheres with an equal height d. After assembly, a certain amount of overlap occurs between each layer. Hence, if the packing state of the layer and the overlap can be quantified properly, the mean porosity of the packed bed can be calculated. Following these assumptions, the definitions of two factors are presented as Eqs. (3) and (4) for quantification. k1 ¼

k2 ¼

nπr 2

ð3Þ

πR2 Nd L

ð4Þ

The factor k1 evaluates the packing fraction of the spheres in the layer projected onto a cross sectional plane, while the factor k 2 is used to evaluate the ratio between the estimated height and the actual height of the packed bed consisting of N layers which gives an indication of the overlap between the layers. The mean porosity value for a packed bed of monosized spheres, which is a function of the ratio between the volume of solid particles and volume of empty column, is expressed by ε ¼ 1−

Vs Vc

4 3 πr ¼ 1−M 3 2 : LπR

ð5Þ

where M is the total number of particles in a packed bed (nN). Combining Eqs. (3)–(5), the formula expression for the mean porosity can be written with k1 and k2, as follows: 4 3 πr 2nπr 2 Nð2r Þ 2 : ¼ 1− k1 k2 ε ¼ 1−M 3 2 ¼ 1− 3 πR2 L 3 LπR

ð6Þ

According to Eq. (6), the mean porosity of a packed bed is directly related to the particle packing fraction in the layer and the overlaps between layers. Carman's equation [15] is entirely based on geometry and widely recommended in the literature. Therefore, the layer stacking model is first developed for packed beds for D/d b 1.866 to validate its assumptions. The packing state of a packed bed with D/d b 1.866 is illustrated in Fig. 3. According to the model, the layer can accommodate one particle only (n = 1) within this D/d range, such that the factor k1 can be expressed as follows: k1 ¼

1  πr2 πR

2

πr 2 1 ¼  2 ¼  2 : D D r π d d

ð7Þ

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Fig. 2. Assembly procedures of densely packed bed according to layer stacking model.

Each additional layer corresponds to the increase in height l of the bed: l¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 d −ðD−dÞ :

ð8Þ

Therefore, the packed bed consisting of N spheres shown in Fig. 3 thus has a length of: L¼

d d þ ðN−1Þl þ 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

2

¼ d þ ðN−1Þ d −ðD−dÞ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3  2 D 4 −1 5 ¼ d 1 þ ðN−1Þ 1− d

ð9Þ

Carman's equation for packed beds with three typical D/d ratios in the range D/d b 1.866 respectively. The new analytical results vary as asymptotic curves to the values predicted by Carman's equation: when a packed bed has a low N value, the new analytical result is higher; with increasing N, the new expression gradually approaches to the value given by Carman's equation. This variation is caused by the end effect or the thickness effect that is the result of particles close to the top and bottom end tending to conform to the surfaces with high void fraction [14]. Therefore, since Carman ignored the end effect and the effect can be notable in a packed bed with low N, the new expression presents 1 a higher result. As N → ∞, the term in Eq. (11) becomes an infinitesimal N

According to the definition of k2, it can be expressed as Nd L ¼ 2

k2 ¼

Nd 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 D −1 5 d41 þ ðN−1Þ 1− d

ð10Þ

1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 ¼2    2 D 4 1 þ 1− 1 1− −1 5 N N d Following Eq. (6), the mean porosity of the packed beds at 1 b D/d b 1.866 can be expressed as 2 ε ¼ 1− k1 k2 3 2 1 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 ¼ 1−  2 2    2 3 D 1 1 D 4 þ 1− 5 1− −1 d N N d

ð11Þ

Fig. 4(a)-(c) displays the comparison between the analytical results according to Eq. (11) when increasing N and the values predicted by

Fig. 3. Packing state of a packed bed for D/d b 1.866.

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Fig. 4. Comparison of mean porosity values between packed beds with/without end effect according to analytical expressions: (a) D/d = 1.2; (b) D/d = 1.5; (c) D/d = 1.8; (d) D/d = 2.1; (e) D/d = 2.2; (f) D/d = 2.6.

value, thus achieving an equation (Eq. (12)) the same as that of Carman's.

2 1 1 ε inf ¼ 1−  2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 3 D D −1 1− d d  2 2 d 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1− 3 D 2d −1 D

successive layers is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l ¼sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d −x2 ffi  2 ðD=d−1Þd 2 ¼ d − 2 sin45 ° 2

ð12Þ

In a similar manner a general expression can be derived for the average porosity for layered packed beds with 2 ≤ D/d b 3. When increasing the tube-to-particle diameter ratios further to 2 b D/d b 3, the factor k1 for a given n can be expressed as follows:

k1 ¼ n 

 2 d D

ð13Þ

The overlap between layers can be obtained by using the geometrical similarity of the layers. For example, in a packed bed with D/d = 2.1, two particles are present in each layer, similar to the condition shown in Fig. 5. Because the particles are placed against the container wall in a dense packed bed, the distance between the particle centers is 1.1d in a layer. Thus, mutually orthogonal lines of length 1.1d can be obtained connecting the particle centers in each layer, such as the black and red lines in the top view in Fig. 5. As the upper layer is built on the lower neighbor, the distance between the spheres in contact is d (the green line in the side view of Fig. 5). Hence, the height increase between

Fig. 5. Packing state of two successive layers in packed bed with n = 2.

ð14Þ

Z. Guo et al. / Powder Technology 354 (2019) 842–853

where x is an auxiliary line, as shown by the red dashes in Fig. 5. Therefore, the packed bed consisting of N layers thus has a length of: d d þ ðN−1Þl þ 2 2 ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 2 ∘ ðD=d−1Þd ¼ d þ ðN−1Þ d − 2 sin45 2 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9  2 < ðD=d−1Þd = ¼ d 1 þ ðN−1Þ 1− 2 sin45∘ : ; 2

the calculations within this range of D/d, the work by Govindarao et al. [32] is recommended. 3. Experimental method

L¼

ð15Þ

In this study, all experiments were conducted with glass spheres and cylindrical columns constructed from polymethyl methacrylate. The diameters of the spheres and columns, presented in Table 3, were measured using a caliper with a precision of ±0.05 mm. To determine the average size of each particle type, a sample of 100 particles was analyzed, as per Eq. (20).

And, the factor k2 can be expressed as follows: Nd k2 ¼ L Nd 9 ¼ 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 = < ∘ ðD=d−1Þd d 1 þ ðN−1Þ 1− 2 sin45 : ; 2 ¼

P100 d¼

ð16Þ

1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2 1 1 ðD=d−1Þd þ 1− 1− 2 sin45∘ N N 2

According to Eq. (6), the mean porosity variation in the packed bed with 2 b D/d b 2.1547 can be expressed as follows (with n = 2): 2 ε ¼ 1− k1 k2 3   2 2 d ¼ 1−  2  3 D

1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 ðD=d−1Þd 1 1 þ 1− 1− 2 sin45∘ N N 2 

ð17Þ

1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2 1 1 ðD=d−1Þd þ 1− 1− 2 sinα N N 2

i¼1 di 100

ð18Þ

ð20Þ

To obtain densely packed beds, alternations in feeding and vibrations were used in the assembly stage [28]. The batch-wise addition of spheres can effectively prevent jamming during the packing [27]; therefore, at each time, a small batch of beads was poured slowly into the column. The number was dependent on the tube-to-particle diameter ratios: approximately 20 to 30 particles per batch for D/d N 4 and ~10 particles per batch for D/d b 4. Following each pouring, the packed bed was vibrated vertically by hand for 3 to 5 min. The vibration amplitude should cause the particles to move a distance that is approximately equal to the radius r, such that the newly added spheres have sufficient time and space to organize themselves into a stable and dense packing state during the vibrations. Moreover, an excessively large amplitude Table 2 Maximum spheres n in a layer and deflection angle α between successive layers. D/d range

Similarly, for other packed beds with 2.1547 b D/d b 3, the lines between the centers in a single layer with a certain n are the same as those in the plane figures, differing only in the deflection angles α between successive layers. Therefore, the factors can be determined in the same manner as above. And a general expression can be derived for the average porosity for layered packed beds consisting of N layers with 2 ≤ D/d b 3:  2 2 d ε ¼ 1−  n 3 D

847

Graphic illustration

n and α

D 2 ≤ b2:1547 d

n=2 α = 45°

D 2:1547 ≤ b2:4142 d

n=3 α = 30°

D 2:4142 ≤ b2:7013 d

n=4 α = 22.5°

D 2:7013 ≤ b3 d

n=5 α = 18°

Table 2 summarizes the growth of n for 2 b D/d b 3, and the corresponding variations of α. And, for a packed bed with an infinite length (N → ∞), a simplified expression can be derived:  2 2 d 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εinf ¼ 1−  n  2 3 D ðD=d−1Þd 1− 2 sinα 2

ð19Þ

The comparison of the mean porosity for a packed bed at 2 b D/d b 3 is presented in Fig. 4 (d)-(e) between a “real” one consisting of N layers (according to Eq. (18)) and ideal one with infinite length (according to Eq. (19)). With the layer-stacking model, the packing configuration of the packed bed is decomposed into the cross sectional and axial-packing states. Section 4 reveals that the mean porosity variations can be interpreted effectively with a simple two-dimensional illustration, and certain useful insights into the variations can be obtained. However, it is noteworthy that the variation for 1.866 b D/d b 2 is not included in this study. For this D/d ratio, although the packing is also effectively structured, a particle may be in contact with particles in multiple layers, which contradicts the layer-stacking model assumption. Therefore, for

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should be avoided, as it could cause the packed bed to exhibit a substantially looser and more random structure. During the experiments, it was observed that the particles exhibited a tendency to form a significantly ordered structure as a result of the vibration. Eventually, the packed beds exhibited a layered configuration, as illustrated in Fig. 6. A similar structure was reported in the literature [29,30,33]. Therefore, by the visual inspection, densely packed beds with an ordered structure could be obtained using the current packing method; on the other hand, a full confirmation of the degree of regularity inside packed beds needs internal packing structure analysis, such as the porosity variations in the radial and the axial directions, which can be accounted for in further studies of dense packed beds. Following assembly, the packed bed height was measured using a long ruler with a precision of ±0.5 mm. As the influence of the end effect on the mean porosity variations is not the primary purpose in the present work, the measurements are suggested to be performed in a packed bed with large L/d ratios to eliminate the influence; after a tradeoff between manual work and sample representativeness, experiments were conducted with packings of at least L/d N 15, referring to the comparison presented in Fig. 4 and the work by Foumeny et al. [19]. The mean porosity value of the packed bed was determined by the particle counting method. This method consists of counting the total number of beads inside a column manually. For each packed bed, the counting was repeated several times until a constant value was obtained. The entire volume occupied by the spheres was calculated by multiplying the number of spheres by the volume of an ideal sphere; thereafter, the mean porosity value was determined by Eq. (5). Experiments were performed thrice for each tube-to-particle diameter ratio. The measurements of the different packings are presented in Table 4, and the average value is used in the following sections. 4. Results and discussion 4.1. Experimental data and comparison with empirical correlations Fig. 7 displays a comparison between the experimental data from the present work and the values predicted by the correlations in Table 1. As shown, the data from the experimental measurements for packed beds with D/d b 1.775 are in good agreement with the correlations that employ Carman's equation for the mean porosity prediction in the range D/d b 1.866. When increasing the ratio to D/d N 2, a discrepancy can be observed between the experimental data and the prediction by the correlations: the data fail to follow the smooth decreasing trend as most empirical correlations present. Instead, an inverse trend where the mean porosity value increases with increasing the D/d ratio is found in several D/d ranges, such as 2.727–2.88, 3.117–3.367, 3.556–3.752, etc. Moreover, it can be observed that small changes in the D/d ratio can result in a considerable impact on the mean porosity value, which was seldom pointed out in previous studies. For example, an abrupt decrease occurs from 0.497 to 0.428 as the D/d ratio merely increases from 2.88 to 3.117. As a result, effected by the increases and the abrupt drops, the experimental data result in a band of values in a narrow range of the tube-to-particle diameter ratios. Scrutinizing previous studies, it is suggested that the “abnormal” mean porosity variations does not occur only in the dense packed beds constructed using the rigorous method of this study. The band with a wide data spread of mean porosity values can also present in packed beds obtained by most other packing methods regardless of Table 3 Cylindrical column diameter (D), particle diameter (d), and corresponding D/d ratio range. D (mm)

d (mm)

D/d ratio range

71.0 50.5 40.9

60.0, 40.0, 16.2, 15.0, 14.2, 12.0, 10.9, 10.0 40.0, 29.4, 16.2, 15.0, 14.2, 12.0, 10.9, 10.0, 8.0 29.4, 24.3, 16.2, 15.0, 14.2, 12.0, 10.9, 10.0, 8.0

1.18 to 7.1 1.26 to 6.31 1.39 to 5.11

whether the bed is packed dense or loose. Dixon [17] and Benyahia and O'Neill [21] studied the mean porosity variations in loose packed beds. In both of the studies, the packed beds were obtained by slow pouring of particles into columns and the measured mean porosity data exhibited a rapid oscillation pattern for D/d N 2 as the data presented in Fig. 7. In the study of Zou and Yu [14], the experiments were performed for both the dense and loose packing conditions. Since the D/d combinations were well-distributed with a relative large spacing over a wide D/d range, the oscillations of the data were not notable; but, an obvious increasing trend was identified for both of the packing modes as the D/d ratio varies from 2.86 to 4. The characteristics of the experimental data consequently influenced the empirical correlations derived in their studies. Although obvious oscillations were observed, Dixon [17] and Benyahia and O'Neill [21] ignored these and fitted quadratic functions to the data to characterize the decreasing trend. In the work of Zou and Yu [14], they accounted for the porosity value increase for 2.86 b D/d b 4 and carefully fitted the data with a separate parabolic equation for 1.866 b D/d b 3.95 to reflect the non-monotonic variation (see Table 1). Nevertheless, although Ribeiro et al. [22] followed the study of Zou and Yu [14], they removed the parabolic equation and fitted the data using a single exponential function because only six data points were obtained by their experiments over a wide range of D/d ratios, nicely presenting a general decreasing trend. Therefore, it is clear that different correlations can be fitted to the sets of experimental data with various distributions of D/d ratios, since a small change in the D/d ratios could result in a notable variation on the mean porosity value. This could be an important contribution to clarify the significant disparity between the values predicted by empirical correlations as shown in Figs. 1 and 7. However, although the oscillatory nature of the mean porosity value as a function of the D/d ratio is a real effect and has a considerable impact on the derived empirical correlations, very few investigations have given attention to these “abnormal” variations. Meanwhile, a common deficiency can be found for most of the correlations in Table 1: the correlations fail to account for the oscillatory nature. Therefore, special attention is focussed on the “abnormal” variations in the following discussion, and the layer stacking model is employed to interpret the corresponding packing state changes as the D/d ratio varies. 4.2. Packed beds at D/d b 3 Fig. 8 displays the variations in k1 and k2 according to the derived equations for packed beds with infinite length in Section 2. Moreover, a comparison between the experimental results and analytical expressions is plotted in Fig. 9. It can be observed that the expressions are in agreement with the variations in the experimental data for D/d b 3. For a packed bed with D/d b 1.866, the mean porosity value first increases, and subsequently decreases. According to the model, the variation depends on two factors—k1 decreases gradually and k2 increases with D/d ratio, as illustrated in Fig. 8. Within this range of tube-toparticle diameter ratios, the initial condition of the packed bed is a sphere completely fitting into a tube with the same diameter. As the diameter of the spheres decreases gradually or the container diameter increases, the packing fraction in a layer becomes less, leading to a rapid decrease in k1. However, the overlap between successive layers becomes larger. This enables the column to accommodate more layers for the same container height, and leads to an increase in k2. Clearly, the factors play different roles in the mean porosity variations. With the combination of increasing and decreasing values for k2 and k1 respectively, the mean porosity value is non-monotonic in this range, as per Eq. (12). The structure transformation that results in nonmonotone variations in the mean porosity can be described effectively by the model. As shown in Fig. 9, the experimental mean porosity values decrease from 0.492 to 0.479 as the D/d ratio varies from 2.525 to 2.727. This is consistent with the general decreasing trend predicted in the literature.

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Fig. 6. Layered configuration of packed beds at small tube-to-particle diameter ratios: (a) D/d = 2.88; (b) D/d = 5.05.

According to the layer-stacking model, the decrease is primarily caused by the additional particles in the layer. For a packed bed with D/d = 2.525, a maximum of four particles can be accommodated in the layer with k1 = 0.628. When the D/d ratio increases to 2.727, the value of n becomes 5. This leads to a significant increase in k1 (= 0.67). Although the value of k2 decreases simultaneously from 1.23 to 1.18, the increase in the accommodation capacity of the layer is dominant in the mean porosity variations within this range of D/d. Consequently, the mean porosity value decreases. However, for specific ranges of the D/d ratio for a given n, the mean porosity variations exhibit an increasing tendency, which is rarely

Table 4 Detailed experimental results of mean porosity variations for packed beds with different D/d. D/d

1.183 1.263 1.391 1.667 1.718 1.775 2.525 2.727 2.880 3.117 3.367 3.408 3.556 3.752 4.090 4.208 4.383 4.633 4.733 5.000 5.050 5.113 5.917 6.313 6.514 7.100

Experimental results

Average value

1

2

3

(mean value ± one standard deviation)

0.5120 0.5632 0.6385 0.6735 0.6740 0.6422 0.4971 0.4750 0.4920 0.4198 0.4640 0.4601 0.4467 0.4613 0.4285 0.4309 0.4256 0.4267 0.4153 0.4047 0.4079 0.4305 0.3935 0.4073 0.3896 0.4009

0.5039 0.5650 0.6310 0.6720 0.6750 0.6363 0.4927 0.4811 0.5007 0.4235 0.4691 0.4592 0.4415 0.4650 0.4290 0.4321 0.4271 0.4251 0.4140 0.4056 0.4073 0.4319 0.3907 0.4041 0.3855 0.4052

0.4961 0.5574 0.6195 0.6885 0.6891 0.6490 0.4861 0.4814 0.4989 0.4280 0.4721 0.4555 0.4405 0.4669 0.4307 0.4351 0.4214 0.4211 0.4122 0.4014 0.4067 0.4325 0.3924 0.4051 0.3862 0.4038

0.5040 ± 0.0080 0.5619 ± 0.0040 0.6297 ± 0.0096 0.6780 ± 0.0091 0.6794 ± 0.0084 0.6425 ± 0.0064 0.4920 ± 0.0055 0.4792 ± 0.0036 0.4972 ± 0.0046 0.4238 ± 0.0041 0.4684 ± 0.0041 0.4583 ± 0.0024 0.4429 ± 0.0033 0.4644 ± 0.0028 0.4294 ± 0.0012 0.4327 ± 0.0022 0.4247 ± 0.0030 0.4243 ± 0.0029 0.4138 ± 0.0016 0.4039 ± 0.0022 0.4073 ± 0.0006 0.4316 ± 0.0010 0.3922 ± 0.0014 0.4055 ± 0.0016 0.3871 ± 0.0022 0.4033 ± 0.0022

mentioned in the literature. For example, the experimental results increase from 0.479 to 0.497 as D/d varies from 2.727 to 2.88 in Fig. 9. It can be observed that, within this range of tube-to-particle diameter ratios, the layer can accommodate the same number of particles (n = 5). With an increasing D/d value, k1 decreases from 0.685 to 0.558, while k2 increases from 1.175 to 1.269. Clearly, k1 exhibits a substantially larger change in value (~22.8%) than k2 (~8%) in this range. Thus, with an increase in the D/d ratios, the decrease in k1 leads to an increase in the mean porosity value. The overlap ratio (k2) gradually becomes less sensitive to the diameter ratio variations. The variations in k1 gradually becomes more important in the overall decreasing trend of the mean porosity variations. Meanwhile, it is noteworthy that the increased accommodation capacity in the layer is not continuous as the D/d ratio varies. The maximum accommodation capacity of the layer is a step function of the tube-to-particle diameter ratios (see Fig. 8). Once n increases, according to the definition, abrupt changes occur in the factors, which lead to a significant reduction in the value of the mean porosity. Thus, combining

Fig. 7. Comparison between current experimental results and correlations in literature.

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Fig. 8. Variations in factors k1 and k2 by derived equations for packed beds with infinite length.

the increases in each interval and the significant decreases at the transition points, the mean porosity values vary with discernibly fluctuations. 4.3. Packed beds at D/d N 3 With further increases in the D/d ratios, additional particles can be accommodated in the layer. From Section 2, it is clear that the mean porosity value is a function of the tube-to-particle diameter ratios in each specified range with different n values. Therefore, the maximum accommodation capacity of the layer is crucial. This is related to an analogous problem regarding the distribution of non-overlapping circles in a large

containing circle, which has been studied by mathematicians for several decades [34]. According to the results published on the website [35], variations in the factor k1 when n is b100 are displayed in Fig. 8. It was assumed that additional particles in the layer should result in a larger k1. However, although an increased number of particles are accommodated in the layer, decreases in the value of factor k1 can be identified for each additional particle as the tube-to-particle diameter ratio increases from 3 to 3.81 (see Fig. 8). To understand the decrease, the layer configurations are illustrated in Fig. 10. In this D/d range, most particles are positioned in an annular section within a particle diameter d from the wall and only one sphere in the center. Consequently, the majority of the empty area is concentrated in the center part when the containing circle is enlarged. The distribution leads to a difference in the packing fractions for the near-wall and central regions. Therefore, to explore the effect with this uneven distribution further, the bed is subdivided into a near-wall and central region. According to the definition of k1, it can be expressed as Eqs. (21) and (22) to represent the packing fractions in the two zones. n1 πr2

A

k1 ¼

B

k1 ¼

Fig. 9. Experimental measurements and analytical expressions from current work.

2

2

πR −πðR−dÞ n2 πr 2 2

πðR−dÞ

ð21Þ

ð22Þ

Fig. 11 depicts the variations in factors kA1 and kB1 with each additional sphere obtained from experiments for a single layer. It is clear that the variations are asynchronous. According to the layer-stacking model, the layer structure affects the development of the overlap ratio. Therefore, it is difficult to determine the exact values of k2 with the uneven distribution. However, certain insights can be obtained by analysis using the layer-stacking model. For the bed composed of the layers illustrated in Fig. 10, the particles adjacent to the wall are likely to form a ordered ring structure in the annular zone. As the particles in the center

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Fig. 10. Configuration of layers with n = 7, 8, 9.

part become in contact with the particles at the container wall, a channel will be formed along the centerline. There is no channel along the center line for D/d b 2. Because the mean porosity of packed beds with D/d b 2 is a typical non-monotonic function of the ratios (see Fig. 1), the mean porosity in the central zone of the packed beds varies similarly, thereby contributing to rapid oscillations. Fig. 12 displays the proportions occupied by the annular and central region in a layer with increasing D/d ratio. It is clear that the dominant status changes gradually in the layer with increasing D/d ratio: when D/d b 2, the annular region adjacent to the container wall occupies the layer entirely; with the ratio increasing to 2 b D/d b 3, the central region appears, thus causing the contribution by the annular region to start declining, but it is still unable to accommodate any particle in the center; when increasing the ratio further, the contribution by the center increases continuously and, consequently, packings with sufficient large D/d ratios would primarily be determined by the particle distribution in the central part. Therefore, the evolution of the particle distribution in the layer affects the packing as well as the mean porosity value significantly as the D/d value increases. According to the layer stacking model, illustrations of several typical particle distributions in a layer with D/d = 4.86, 6.96, and 10.57 are presented in Fig. 13. When the layer contains a small number of large particles, such as the layer with D/d = 4.86 or the layers shown in Fig. 10, the particle size must be effectively reduced (or the D/d ratio increased) to accommodate an additional particle and, consequently, a notable change will occur in the packing fraction (as well as the value of k1) by a particle being added or removed. Therefore, in Fig. 8, within the

range D/d b 6–7, stepwise variations of k1 can be observed, which correspond to the D/d ratio ranges of the layers, each associated with a given n, and the k1 value in each interval shows a relatively wide range. Consequently, the influences presented in Section 4.2 by the step function of n and the non-monotone nature of k1·k2 are still present, which can result in clear fluctuations in the mean porosity value. Additionally, as the uniformity of the center and annulus in a layer with D/d N 3 is affected by the position of the additional particle, further deterioration is expected in the fluctuation by the expected uneven particle distribution. Correspondingly, in Fig. 9, it can be observed that the measured porosity value indicates clear fluctuations for 3 b D/d b 7.1 with several “abnormal” increases. By increasing the D/d ratio, more smaller particles appear in the layer resulting in higher packing fractions (k1), as shown in Fig. 13. Correspondingly, a rapid increasing trend in k1 can be seen in Fig. 8 in the range of 2 b D/d b 10. This is the major contribution to the decrease in the mean porosity value for 2 b D/d b 10 (see Fig. 1). When the D/d ratio increases further, such as the layer with D/d = 10.57 in Fig. 13, two important features of the particle distribution in the layer emerge: first, the number of particles in the layer becomes sensitive to the D/d ratio variations (in other words, the changes in D/d ratio will result in an almost continuous variation of n), and the influence on k1 by a single particle becomes weaker; next, a significant homogeneous particle distribution in the layer can be achieved. Both features contribute to a smooth variation in the packing density versus the D/d ratio: with the continuous n variations, smoother k1 increases are observed for D/d N 7 in Fig. 8; further, the homogeneity of the particle layer is beneficial

Fig. 11. Variations in factors kA1 and kB1 corresponding to each additional sphere in the layer.

Fig. 12. Proportions of annuli and central regions occupied in layer.

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Fig. 13. Typical particle distributions in layer with D/d = 4.86, 6.96, and 10.57 (n = 19, 38, and 91) by Wu [36].

for more regular overlaps between the layers, thus resulting in an approximately constant value of k2. Consequently, the fluctuations are expected to be reduced for D/d N ~10 and a gentle decreasing slope is presented in Fig. 1 exhibited by the different empirical correlations. 4.4. Further discussions A further key value is the mean porosity of densely packed beds with large tube-to-particle diameter ratios, which is important in numerous practical applications. According to the variations in Fig. 8, it is clear that the oscillations of k1 weaken gradually as D/d increases. This is because a small change in the D/d ratio can lead to an increase in n with several particles for D/d N 10. As N90 spheres are accommodated in the layer, an additional sphere renders a small contribution to the overall layer packing fraction; consequently, the curve of k1 increases almost smoothly. Therefore, a correlation can be fitted according to the variations in n for 10 b D/d b 50, as presented on the website [35].  2 d d k1 ¼ 3:0707 −1:2814 þ 0:9036 D D

ð23Þ

Fig. 14 illustrates the packing arrangement in an infinite packed bed. According to the layer-stacking model, each layer corresponds to the inpffiffiffi 3 d for the bed. Moreover, the mean porosity varcrease in height l ¼ 2 iations in the packed bed with infinite packing length at 10 b D/d b 50 can be expressed as follows: pffiffiffi 2 4 3 ε ¼ 1− k1 k2 ¼ 1− k1 : 3 9

ð24Þ

The mean porosity is an overall evaluation of the packed bed geometry. For packed beds with small tube-to-particle diameter ratios, rapid oscillations in the mean porosity value demonstrate that multiple combinations of the cross-sectional and axial packing fractions are possible to obtain different packed beds with a given mean porosity value. The mean porosity value cannot differentiate between the packing states of packed beds with the same mean porosity. In terms of the classical study of pressure drop in a packed bed, although numerous studies

Fig. 14. Orthorhombic arrangement in infinite packed bed.

have been devoted to packed beds with small tube-to-particle diameter ratios, a lack of consensus still remains on this issue. Moreover, previous studies have demonstrated that certain packed beds exhibit a larger pressure drop than those with lower porosity values [37,38]. Clearly, this contradicts the general understanding that increasing the porosity of a packed bed can reduce the pressure drop according to the empirical correlations, such as that of Ergun's (Eq. (1)). To address this problem, compared to the coefficient modifications in Ergun's equation, fundamental models on the description of hydraulic performance in packed beds are more desirable. In this respect, the layer-stacking model is more suitable for the description of the packing state than the mean porosity value. Therefore, it can provide a new perspective for investigating the interactions between pressure drop and packed bed geometry. 5. Conclusion In this study, experimental and theoretical work was performed to investigate the mean porosity variations in packed beds with small tube-to-particle diameter ratios. The layer-stacking model developed in this work could effectively track the experimental results for D/d b 3. Moreover, it could describe the cross-sectional and axial packing state of the packed bed, as well as interpret the mean porosity variations. From the experimental results and model analysis, rapid oscillations of the mean porosity values were confirmed to be prevalent in densely packed beds with small tube-to-particle diameter ratios, which were primarily caused by the non-monotonic nature of k1·k2, which are step functions of n, and the non-uniform particle distribution in the layer. This may contribute to the differences between the empirical correlations in the literature. For the design of new reactors using packed beds with small tube-toparticle diameter ratios, multiple combinations of the cross-sectional and axial packing factors could be used to achieve the same packing density. Thus, empirical correlations cannot distinguish the difference, as they use only the mean porosity value to characterize the influence of the packing structure; different pressure drop performances could be expected in these packed beds with the same mean porosity value as well as the same particle size. Therefore, the interaction between the packing state and fluid flow in packed beds should be reevaluated. And, the layer stacking model can provide a new perspective for this issue, as well as reactor design and optimization. Nomenclature d particle diameter [mm] di particle diameter of i-th sphere [mm] k1 coverage of layer k2 axial packing density kA1 coverage of annuli region in layer kB2 coverage of central region in layer l height growth by a layer to packing [mm] n maximum number of particles in layer n1 number of particles in annuli region of layer n2 number of particles in central region of layer r particle radius [mm]

Z. Guo et al. / Powder Technology 354 (2019) 842–853

x D L M N R V Vs Vc

auxiliary line for Eq. (11) [mm] tube diameter [mm] bed height [mm] number of particles in packed bed number of layers in packed bed tube radius [mm] fluid superficial velocity [m/s] total volume of solid particles [m3] volume of empty column [m3]

Greek letters △P pressure drop of packed bed [Pa] α deflection angle between successive layers ε mean porosity εinf mean porosity in packed bed of infinite length μ kinematic viscosity coefficient [Pa·s] ρ density [kg/m3] Acknowledgements The authors acknowledge the financial support provided by the Ph. D. Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (HEUGIP2018021), the Fundamental Research Funds for the Central Universities (HEUCF181503) and the National Natural Science Foundation of China (No. 11705036). References [1] X. Guo, Y. Sun, R. Li, F. Yang, Experimental investigations on temperature variation and inhomogeneity in a packed bed CLC reactor of large particles and low aspect ratio, Chem. Eng. Sci. 107 (2014) 266–276. [2] Y. Dong, B. Sosna, O. Korup, F. Rosowski, R. Horn, Investigation of radial heat transfer in a fixed-bed reactor: CFD simulations and profile measurements, Chem. Eng. J. 317 (2017) 204–214. [3] S. Das, N.G. Deen, J.A.M. Kuipers, A DNS study of flow and heat transfer through slender fixed-bed reactors randomly packed with spherical particles, Chem. Eng. Sci. 160 (2017) 1–19. [4] A.G. Dixon, Local transport and reaction rates in a fixed bed reactor tube: endothermic steam methane reforming, Chem. Eng. Sci. 168 (2017) 156–177. [5] Z. Guo, Z. Sun, N. Zhang, M. Ding, S. Shi, CFD analysis of fluid flow and particle-tofluid heat transfer in packed bed with radial layered configuration, Chem. Eng. Sci. 197 (2019) 357–370. [6] Z. Guo, Z. Sun, N. Zhang, M. Ding, H. Bian, Z. Meng, Computational study on fluid flow and heat transfer characteristic of hollow structured packed bed, Powder Technol. 344 (2019) 463–474. [7] G.E. Mueller, Radial porosity in packed beds of spheres, Powder Technol. 203 (2010) 626–633. [8] M. Suzuki, T. Shinmura, K. Iimura, M. Hirota, Study of the wall effect on particle packing structure using X-ray micro computed tomography, Adv. Powder Technol. 19 (2008) 183–195. [9] J. Theuerkauf, P. Witt, D. Schwesig, Analysis of particle porosity distribution in fixed beds using the discrete element method, Powder Technol. 165 (2006) 92–99. [10] C.G. du Toit, Radial variation in porosity in annular packed beds, Nucl. Eng. Des. 238 (2008) 3073–3079. [11] R.F. Benenati, C.B. Brosilow, Void fraction distribution in beds of spheres. AIChE J 8: 359-361, AICHE J. 8 (1962) 359–361.

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