Experimental and statistical analysis of the void size distribution and pressure drop validations in packed beds

chemical engineering research and design 1 0 6 ( 2 0 1 6 ) 115–125

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Experimental and statistical analysis of the void size distribution and pressure drop validations in packed beds Wei Du a,∗ , Ni Quan a , Panpan Lu a , Jian Xu a , Weisheng Wei b,∗ , Lifeng Zhang c a b c

State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Beijing, PR China School of Chemical Engineering, Fuzhou University, Fuzhou, PR China Department of Chemical and Biological Engineering, University of Saskatchewan, Saskatoon, SK, Canada

a r t i c l e

i n f o

a b s t r a c t

Article history:

The void size distribution inside a packed bed is one of the most important parameters

Received 10 July 2015

affecting the reactor performance. The non-uniform distribution of material and energy

Received in revised form 24

flows can be avoided by a reasonable arrangement of the void size distribution. In this work,

November 2015

the void size distributions in a packed bed with structured and random packing were inves-

Accepted 29 November 2015

tigated through both theoretical calculations and statistical analysis of experimental results

Available online 11 December 2015

to characterize the bed voids between the particles. In theoretical analysis, the equivalent

Keywords:

structured packing were first calculated, and then the wax replacement method was adopted

Packed bed

to measure the voids inside a randomly packed bed. Different two-dimensional layers of the

diameters of voids at different two-dimensional layers in a packed bed with three types of

Void distribution

bed were sliced and photographed for further analysis. The effects of particle size, shape,

Particle size

and packing method were investigated, and the results showed that the relative void size

Particle shape

(void size/particle size) was not affected by the particle size. In contrast, the particle shape,

Pressure drop

which can be represented by particle sphericity, significantly affected the void size distribution. A more flat void size distribution curve was obtained for particles with sphericity close to unity. A new correlation was established based on statistical analysis of the experimental data to account for the void size distribution. This new correlation was validated by predicting the pressure drop of a packed bed and the results of comparison showed good agreement with experimental results. © 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

Introduction

Column reactors randomly packed with particulate solids are extensively employed in various chemical processes, particularly for reactions requiring effective interphase contact, such as heterogeneous catalytic processes. Typical applications in chemical process industries, including trickle-bed reactors (TBRs) (Shah et al., 1986) and packed-bed reactors (Wakao and Kaguei, 1982), involve contacting gas–liquid mixtures in a

∗

countercurrent manner to selectively transfer a species from one phase to another. Both spherical and non-spherical particles are commonly employed in packed bed reactors. In addition, the particles may be mono-sized or vary in shapes and sizes. The characterization of the structure of packed beds is essential for designing and modeling packed bed reactors. The microscopic structures of packed beds determines local fluid flow, as well as overall heat and mass transfer, which in turn influences macroscopic performance of reactors when

Corresponding authors. Tel.: +86 1089734981. E-mail addresses: [email protected] (W. Du), [email protected] (W. Wei). http://dx.doi.org/10.1016/j.cherd.2015.11.023 0263-8762/© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

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chemical engineering research and design 1 0 6 ( 2 0 1 6 ) 115–125

Nomenclature Ar d dp de D E1 ,E2 f Fr g Ga H hd he hst k k˛ L m Ni P Re S SP u U V x xm

Archimedes number void equivalent diameter (mm) particle equivalent diameter (mm) equivalent diameter average porosity size (mm) Ergun’s constants void distribution of function Froude number gravitational acceleration (m/s2 ) Galileo number packing height (m) liquid hold-up dynamic liquid hold-up, dimensionless the static liquid hold-up, dimensionless total pressure drop coefficient relative permeability of the ␣ phase length of the reactor (m) quality of particle (kg/m3 ) number of the void with the diameter of the void is di pressure drop (Pa) Reynolds number equivalent volume sphere surface area (mm2 ) surface area of a particle (mm2 ) superficial velocity (m/s) flow velocity (m/s) particle volume (m3) ratio of void equivalent diameter and particle equivalent diameter maximum of the ratio of void equivalent diameter and particle equivalent diameter for non-spherical particle

Greek letters density of the gas phase (kg/m3 )  ϕ particle shape coefficient ε average porosity dynamic viscosity of the gas phase (Pa s)  resistance coefficient  εb bed porosity specific surface shape factor ϕc volume shape factor ϕv ϕ Carmen coefficient lurface area shape factor ϕs liquid holdup ˇd  interfacial tension (N/m) standard deviation ω Ф sphericity Subscripts ˛ gas/liquid/solid phase gas phase g l liquid phase solid phase s

applied in many chemical engineering operations (e.g., separation and reaction processes). The packing materials are dumped randomly into the bed, and the macroscopic parameters, such as the packing density and bulk porosity, of the packed bed are measured. These properties are usually used

as inputs to transport models, such as the Ergun equation, Darcy’s law, and their variants. Spherical particles are commonly used packing materials as they are the most well studied in packed beds. Ordered packing methods include rhombohedral and cubic packings, embracing a range of porosity between 0.26 and 0.48, respectively (Zhang et al., 2006). In contrast, random packings exhibit a much narrower range of porosity. The minimum bulk porosity of sphere packings is 0.36 (for uniform-sized spheres), and the values for actual packings typically range from 0.36 to 0.42. Cylindrical particles are also frequently used in chemical processes. However, the packing behaviour of cylindrical particles is fundamentally different from spheres because cylinders exhibit more orientation freedoms, and their geometry contains various surface elements (i.e., flat surfaces, curved surfaces, and corners). These differences presents more diverse packing structures as shown by the range of porosity values reported in the literature; for equilateral cylinders, the porosity of a packed bed ranges from 0.25 (Roblee et al., 1958) to 0.445 (Coelho et al., 1997), a much broader range than that for spherical particles. In previous studies, the void in packed beds are usually characterized by an averaged parameter, i.e. the mean porosity (e.g. Nguyen et al., 2005), or the radial and axial porosity distribution in cylinder (Du Toit, 2008; Mueller, 1991; Theuerkauf et al., 2006). However, such descriptions cannot represent detailed information pertaining to the local structural properties in a packed bed, where considerable flow mal-distribution could occur due to larger voids and flow channels existing in the bed. Therefore, the characteristics of the packing structure in the packed bed are important parameters for flow and reactor modelling. The local porosity distribution, in both radial and axial directions, is often utilized to provide a more detailed insight of the packing structure. Different techniques, such as X-ray computed tomography (Toye et al., 1998) and nuclear magnetic resonance (NMR) imaging (Nguyen et al., 2005), have been used to measure the void size distribution in the bed. Experimental porosity profiles, particularly the radial ones, can be found in the literature for different particle-to-tube ratios, as well as for mono-sized and binary or poly-dispersed sphere mixtures (Du Toit, 2008; Martin, 1978; Gotoh et al., 1986; Alonso et al., 1992, 1995). Especially, the radial porosity distribution for mono-sized spheres in cylindrical columns are commonly used, and they have been investigated by various experimental and systematic methods (Roblee et al., 1958; Benenati and Brosilow, 1962; Ridgway and Tarbuck, 1968; Martin, 1978; Cohen and Metzner, 1981; Kufner and Hofmann, 1990; Mueller, 1992; Govindarao et al., 1992; Sederman et al., 2001; Wang et al., 2001; Mariani et al., 2009). In packed beds with random packing, the void fraction in the vicinity of the tube wall approaches unity, leading to a maximum fluid velocity near the wall (Bey and Eigenberger, 2001). Previous studies usually employed porosity to characterize a packed bed, and the interstitial spaces (termed voids in this study) between particles are considered to be interconnected with each other. However, this study showed that part of the void can be regarded as independent, the void size is not uniform, and the shape and size of the void vary through the whole bed. Therefore, the radial and axial porosity distributions still cannot represent the complexity of a packed bed. The void size distribution determines the flow of reactants and consequently affects the mass and heat transfer, hydrodynamics, and reaction behaviour. Therefore, the void size and its distribution require further investigations. Although the size

chemical engineering research and design 1 0 6 ( 2 0 1 6 ) 115–125

2.2.

Fig. 1 – Flow sheet of the experimental setup. 1. Air compressor; 2. air buffer; 3. air flow meter; 4. valve; 5. liquid flow meter; 6. liquid controlling valve; 7. pump; 8. liquid storage tank; 9. distributor; 10. column; 11. differential pressure drop transducer; 12. A/D converting card; and 13. PC.

and shape of voids are non-uniform and randomly distributed inside a bed, the void size distribution can be assumed to obey a certain distribution function from a statistics point of view. Thus, in this study, the void distribution in packed beds with different particles was experimentally analyzed, and then a suitable statistical distribution function was derived. Based on the derived distribution function in this work, the prediction of the pressure drop in packed beds was considerably improved.

2.

Experimental setup and measurements

2.1.

Experimental setup

The experimental set-up is shown in Fig. 1. The two columns (10 cm and 30 cm in diameters) adopted in the experiment were composed of three sections, namely, the upper gas–liquid distribution, middle packing, and lower gas–liquid separation sections. The 10 cm diameter column had a height of 1m while the 30 cm diameter column had a height of 40 cm. Air and water were used as the gas and liquid phases, respectively, and the flow rates were controlled by a set of flow meters. Water was pumped into a distributor installed on the top of the bed and then flowed down through the packed bed. The superficial liquid velocity through the bed was controlled within 1.77–14.15 mm/s. Air was supplied by an air compressor and introduced into the column through the distributor. The air and liquid flows were introduced in a co-current manner. The bed was packed by random packing and fully wetted before the experiments. The packing materials were ceramic balls (3.5 mm and 5.7 mm in diameter), and ceramic cylinders (2.5 mm in diameter and 4.6 mm in height). The experiments were conducted at ambient conditions. The water had a density of 999.4 kg/m3 and a viscosity of 1.3 mPa s. The air had a density of 1.239 kg/m3 and a viscosity of 0.0178 mPa s. The bed pressure drop was measured by four pressure transducers (Omega, PX164-010D5V) installed on the column wall at heights of 50, 450, and 850 mm, as well as on the top of the column. The signals were collected by an ART A/D converting card and then transferred to a PC. The data sampling frequency was 500 Hz. The pressure transducers were calibrated by a U-tube manometer.

117

Void size measurement in random packed beds

Previous studies have investigated the local void characteristics in packed beds with spherical and cylindrical particles using experimental methods (Di Felice and Gibilaro, 2004; Ismail et al., 2002; Zou and Yu, 1996a,b), analytical techniques (Wang et al., 2001), and empirical correlations (Dixon, 1988; Mueller, 1992). The void size between particles is usually defined as the gap size between two adjacent particles, but this definition can hardly be applied to the voids formed by three or more particles. Therefore, a revised definition for such voids is based on first connecting the centers of these particles (as shown in Fig. 2). Since our study is applied to gas–liquid flow in packed beds, thus from view of liquid flow hydrodynamics (especially liquid film thickness), those gaps between two particles (alone the center connection line) larger than dp /2 cannot be considered as independent (as can be seen from our previous results, Du et al., 2013), therefore, we first paved all section with triangles and then omitted these lines (red lines in Fig. 2a), the left lines (black lines in Fig. 2a) are boundaries of voids. As seen from Fig. 2a, the surrounding particles can be 3, 4 or even more. By this definition, the area not occupied by the particles inside this region is the void size. The equivalent diameter of this void (de ) is defined by:

 de = 2

S

(1)

where S is the area not occupied by particles, m2 . The void size inside the packed beds can be measured by two ways. One technique is the direct analysis method, in which the whole bed is treated with particles uniformly packed inside. In this technique, the shape and size of void can be analyzed and calculated mathematically. The method was used for an ideal case analysis in this study. However, such packing manner is quite different from that in a real bed, where the particles are typically randomly packed. Therefore, a statistical analysis is required to account for non-ideality. The experimental measurement was based on the wax replacement method (Roblee et al., 1958). The spherical particles used in this study were soybeans and green beans, whereas the cylindrical and rectangular particles were cut from carrots and garlic sprouts. Different types of particles were randomly packed into beakers, and melted wax was poured into the beakers with these particles. After cooling down, the packed bed was cut into different horizontal layers, which were pictured and then processed by commercial software Photoshop. The void between particles, as previously defined, was measured by counting the pixel numbers, which were proportional to the actual area. The overall porosity of a packed bed (ε) was measured by the water displacement method and calculated by: ε=1−

Vparticle Vbed

(2)

The particle density can be calculated from =

m V

(3)

The sphericity of a particle is determined as follows: ϕ=

d2ev d2es

(4)

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Fig. 2 – Soybean section. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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The particles used in this work are summarized in Tables 1 and 2.

3.

Void size distribution in packed beds

The local density of a packed bed only defines the porosity; the void size distribution is more important and therefore affects the gas–liquid flow in the bed. The previous studies have reported many results about radial and axial void distributions (Nguyen et al., 2005; Du Toit, 2008; Mueller, 1991; Theuerkauf et al., 2006). These representations are not enough for revealing the void size distributions in a packed bed. In this work, we adopted a new concept to describe the distribution. The void in the packed bed is considered as independent at different bed layers. The total number is then uncountable, but the total bed void can be integrated or statistically calculated. Statistical analysis of void can be considered to follow a certain distribution function. With the initial assumption that the voids are independent of each other, the void distribution function is defined as



the particles were first arranged three-dimensionally in commercial software Solidworks. The whole packed bed was then sliced into a sufficient number of horizontal layers for statistical analysis. The void (nonparticle) area for each layer then can be easily derived by particle cross-section areas at different heights. The void area in Fig. 3a can then be calculated from:

 S = (4 − ) ∗

 √  S= 2 3− ∗

 S

= 2dp





dp 2

(5)





Direct analysis of structured packing

Direct analysis of the packed bed has been widely performed to evaluate properties, such as porosity. Mueller (2005) developed simple analytical and semi-analytical equations based on fundamental principles to calculate the local radial porosity and porosity distribution. The predicted results are benchmarked with an existing analytical equation and available experimental data. Nandakumar et al. (1999) developed an algorithm to analyze and predict the geometrical properties of a randomly packed structure. Once the packing manner is well defined, both macroscopic quantities, such as the overall porosity, specific surface area, and the number of packing particles per unit volume, as well as microscopic properties, which include the porosity variation in any direction, can be determined. Abreu et al. (2003) investigated the influence of the particle shape on the packing and segregation of spherocylinders by Monte Carlo simulations. Montillet and Le Coq (2001) used X-ray image analysis to investigate the characteristics of fixed beds of anisotropic particles. This method is reliable in analyzing the structure of packed beds. The most commonly observed regular packing forms are shown in Fig. 3. In Fig. 3a, all particle centers are arranged in a cubic mode; in Fig. 3b, from left or right view, all particle centers are in a square mode, however in a regular triangle mode from a top view; in Fig. 3c, particle centers on each layer are in a regular triangle mode from both left and top view. Given that the porosity is fixed for each packing manner, the void size distribution can then be analyzed by calculations. In this study,

dp 2

(6)

2 + ∗

h 2

(7)

2



2 − h2 √

for 0 < h <

number of void with diameter less than d total number of void

+ ∗ h2

The void area in Fig. 3c can be calculated by:



where d is the equivalent diameter of the void (m), and P < d is the distribution probability of those voids with diameters smaller than d. The void distribution is mainly determined by the particle size, the particle shape, and the packing manner. To clearly reveal the influences of each parameter on the void size distribution in a packed bed, a theoretical method, in which particles are packed in certain manner, was employed for the direct analysis.

3.1.

2

The void area in Fig. 3b can be calculated by:

f (d) = P < d =

dp 2



3 dp 4

−



dp 2

2  √ 2 dp 3 S =

− dp − h 2 2   √ dp 3 for dp < h < 4 2 S

 √





2 − h2

(8)

2

dp = 2 3− ∗ + ∗ h2 2   √ dp 3
where h is the height from the particle center (m).

3.2.

Influence of particle size

The granularity, as a measurement of particle size, is one of the basic characteristics of particles. The particle size has been proven to have the most significant impact on the porosity. Suzuki et al. (2001) found that when dp > 15 ␮m, the bed porosity remains almost constant, but when dp < 15 ␮m, the bed porosity increases with increasing particle sizes in the packed bed. Abe and Hirosue (1982), as well as Mota et al. (1999), found that the percentage of larger particles in a packed bed of binary mixtures affects the void distribution. Latham et al. (2002) showed that the minimum bed porosity occurs when the content of the larger particles is 75%. Suzuki et al. (1999) also showed that mixed packing of particles exhibits a minimum porosity. The size of the catalysts in TBRs usually ranges from 0.5 to 10 mm. In this study, the influence of granularity on the void size distribution was investigated in packed beds of spherical particles (sphericity = 1). The particles had diameters of 2, 4, and 8 mm with the calculated porosity of 0.395 for all beds. The narrowest distribution was found to be for the 2 mm particle, the void size of which ranged from 0.7 to 2 mm (as shown in Fig. 4a). The void size distribution was significantly broadened by increasing the particle size. The void distributions from 1.3 to 4 mm and 2.6 to 8 mm were observed for the 4 mm and 8 mm particle beds, respectively (Fig. 4a). In an ideal packing, the void size increases linearly with an increase in the particle size. Therefore, a dimensionless void size x, defined

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Table 1 – Parameters of the properties of ceramic materials. Particles

 10.0 mm

 8.0 mm

 5.7 mm

 3.5 mm

2.5 × 4.6 mm

Static liquid holdup Volume equivalent diameter dev Superficial area equivalent diameter des Sphericity ϕ

– – – 1.00

– – – 1.00

0.041 – – 1.00

0.03 – – 1.00

0.045 3.5 3.61 0.64

Table 2 – Physical properties of packed materials. Material

Diameter (mm)

Soybean Green beans Green beans Garlic sprouts Carrots

5.0 7.0 9.0 5.0 × 5.0 × 5.0 4.5 × 4.5 × 4.5

Shape Spherical Spherical Spherical Cylindrical Rectangular

Porosity

Density (kg/m3 )

Bulk density (kg/m3 )

0.56 0.53 0.63 0.56 0.56

1239 1201 1179 1055 1032

718 712 694 785 765

Fig. 3 – Different packing manners. as the ratio of the void size to the particle size (the equivalent diameter is used in this study) (x = de /dp ) is introduced for further analysis. With this new parameter, the dimensionless void distribution function can be expressed by f (x) = P {< x} =

number of void with size less than x total number of void (9)

The void distribution is plotted against the dimensionless void size as shown in Fig. 4b. It is seen in this figure that the dimensionless void size is independent of the particle size, and the distribution curves for the three particles overlapped with each other within the range of 0.3–1.

3.3.

and 0.395 for the two methods, respectively. It can be seen in this figure that the packing manner with a higher bed porosity resulted in a wider bed void size distribution range. The experimental studies were carried out to further verify the effects of packing method. The porosities of different particles with three packing methods, (1) loading particles without vibration; (2) loading particles with vibration; and (3) batch loading particles with vibration, were measured. The results are shown in Table 3. From this table, it can be seen that the packing method clearly influences the porosities, and the lowest value is obtained with the packing method (3). The bed packed by this method was found to have a wider range of the void size distribution and larger pores, resulting in the total bed void, and smaller channels, where the capillary effect was dominant. A higher liquid holdup and a bigger pressure drop was expected.

Influence of packing manner 3.4.

The packing manner in a trickle-bed reactor can influence the void shape and its size distribution in the bed, which in turn influences the liquid holdup and the bed pressure drop. As a result, the reactor performance will be influenced by the packing method. Zhang et al. (2006) investigated the porosity of a TBR with cylindrical particles and found that the overall packing density was found to be the major factor affecting the packing structure. Different particle arrangements in structured beds can provide a simple and direct analysis of its influences, and thus were investigated in this study. Fig. 5 shows the influence of the packing method on the void size distribution. To study this effect, the bed porosity was measured in beds packed with the same particles but different packings (Fig. 3a and b). The resulting bed porosity was 0.476

Influence of particle shape

In regard to the effect of the particle shape on the bed porosity, only very few studies are available in the literature. Zou and Yu (1996a,b) reported that the bed porosity increases with an increase in sphericity. However, their studies were only focusing the overall bed porosity not the void size distribution. Therefore, three types of particle shapes, i.e., spherical, cylindrical, and cuboid were employed to investigate this effect. All particles had equivalent diameters of 10 mm, and the resulting bed porosities for three particles were quite similar at approximately 0.42. The corresponding spherical factors for the different particles were 0.94, 0.72, and 0.67, respectively. The narrowest void size distribution was observed in the bed packed with cuboid particles, which had the lowest

chemical engineering research and design 1 0 6 ( 2 0 1 6 ) 115–125

121

Fig. 6 – Void distributions for packed beds with different particle shapes.

Fig. 4 – (a) Void distributions for packed beds with different particle sizes, (b) dimensionless void distribution for packed beds with different particle sizes.

Fig. 5 – Void distributions for packed beds with different packing manners. Table 3 – Porosities of three different loading types for spherical and cylindrical particles. Particles

5.7 mm spherical particle 3.5 mm spherical particle Cylindrical particle

Packing method 1

2

3

0.436 0.427 0.471

0.423 0.415 0.465

0.407 0.397 0.43

spherical factor (as shown in Fig. 6). The distribution function f approaches 1 at x = 0.35, indicating relatively smaller voids inside this bed. In contrast, the void size distribution for the bed with cylindrical particles is wider as f approaches 1 at x = 0.65 while the largest void distribution was found in

the bed with spherical particles with x ranging from 0 to 0.9. These results show that the particle shape has a detrimental impact on the void size distribution; thus, the higher the sphericity of the particles, the broader the void size distribution in the bed. When packed with spherical particles, the void in the bed may be considered uniformly distributed in all directions. By contrast, the void in a non-spherical particle bed is anisotropic, and thereby its distribution is determined by the packing method. The capillary effect cannot be ignored in smaller channels formed between the particles in such beds. The liquid distributions on the top of both spherical and cylindrical particle beds were uniform, but the liquid distribution variations were observed when varying the bed depth (Wang, 2012). Stagnant liquid was found near the particle–particle contact surface in the packed bed with cylindrical particles, suggesting occurrence of gas channeling and a non-uniform void distribution in the bed. Similar results were reported by Zou and Yu (1996a,b). This phenomenon was caused by collective effects of the much higher tortuosity of the void in a cylindrical particle bed than that in a spherical particle bed and the obvious difference in the void shape and distribution between the two beds. The spherical particles contact each other by points, but contacting lines are usually found between cylindrical particles. Therefore, the liquid flow inside a cylindrical particle bed experiences more resistance, and the liquid holdup and the pressure drop are higher because of the complexity of the void distribution (Ikegami, 2005).

4.

Void distribution in packed beds

4.1.

Derivation of void distribution model

The derivation of the void distribution model can be illustrated by Fig. 7. According to Fig. 5 and calculation of the structural packing method, for cases 1 (porosity = 0.476) and 2 (porosity = 0.395), the following equations are obtained:

  f =



x2 + 1 − (4/ ), ε = 0.476

√ x2 + 1 − (2 3/ ), ε = 0.395

(10)

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Fig. 8 – Variation in the pressure drop with gas velocities.

4.2. Prediction of pressure drop by the void distribution model Fig. 7 – Comparison of predicted and experimental void distributions.

Therefore, the general form for the void distribution can then be expressed by the following equation

f =



x2 + 1 −

2 3(1−ε)

(11)

Based on Eq. (11), the maximum void size in the bed with 2/ (3(1 − ε)), and it increases with structured packing is increasing the bed porosity, which is consist with experimental findings. As shown in Fig. 6, the void size is smaller in non-spherical particle beds. Thus, the maximum void size for a non-spherical particle bed is assumed to be as follows:

xm =



2 ϕ 3(1−ε)

(12)

The void distribution function for structured packing can then be expressed by:

f =



x2 + 1 −

2 ϕ2 3(1−ε)

(13)

The void distribution for random packing is considered to have a relatively large deviation from that for structured packing. Yu (1988) stated that the packing manner can be predicted by the Gauss and normal distribution functions. Thus, a new expression is derived based on the experimental results:

f =

1 − exp (− (c · x/ϕ · xm )) 1 − exp (− (c/ϕ))

(14)

The constant c in the above equation is equal to 6, as regressed from the experimental results. Fig. 8 shows a comparison of the experimental data with the predicted values. In this figure, w is the residue of the standard deviation, and all calculated f values fall within the region between y = x ± 2w, indicating good agreement of the predictions from the proposed model with the experimental data.

The void between particles provides flow pathways for reactants, resulting in the pressure drop. The void size distribution is a function of the particle shape, the particle size, and the bed diameter. Therefore, the pressure drop can also be correlated to these parameters. A few empirical or semi-empirical correlations have been proposed in the literature (Holub et al., 2004). However, these correlations are only valid at large bedto-particle-diameter ratios. Iliuta and Larachi (2009), Larachi et al. (2000), and Iliuta et al. (1996) showed that the void size is the most important parameter in predicting the pressure drop. Hence, an attempt was made to develop a new pressure drop function by taking into account the void size distribution in the bed. The pressure drop, one of most important parameters in TBRs, cannot be accurately predicted only by the overall porosity of the packed bed without taking into account the actual void size distribution. Considering that the pressure drop in a packed bed is related to the void size, the introduction of a new void distribution model is to improve the prediction of the pressure drop in TBRs. Assuming that the interstitial spaces inside a packed bed can be considered as many parallel serpentine channels of different sizes, and only the gas phase exists in the bed under operating conditions related to industry practice, such as TBRs (high temperature and pressure), the pressure drop in the channel is given by (Feng, 1989):

P =

1 L k · Re−0.25 ·  · 2g D

u 2 ε

(15)

where  is the gas density (kg/m3 ), P is the pressure drop (Pa), u is the superficial velocity (m/s), k is the total pressure drop coefficient (According to literature, for each serpentine channel, pressure drop coefficients are 0.1 and 0.5 for turbulent and laminar flows, respectively, and the total pressure drop coefficient k is a summary of these individual coefficients. In this study, only the total pressure drop coefficient was obtained through regression based on the experimental results), Re represents the Reynolds number, and D is the diameter of the void. In this work, D is estimated by: D = dp

∞ (d /d ) Ni i=0 i p ∞ i=0

Ni

(16)

chemical engineering research and design 1 0 6 ( 2 0 1 6 ) 115–125

∞

∞  d

i

dp

i=0

N=

x

df dx dx

123

(17)

0

D = dp

∞ x · df

(18)

0

By incorporating Eqs. (13) and (14), the following equations are derived for structured and random packings, respectively.

 D ≈ dp



 1  xm 2 − 1 xm ln (xm + 1) − ln xm 2 − 1 + 6 2 2

 ,

structural packing

ϕ ϕ2 D ≈ dp xm = dp 6 6



(19)

2 , random packing 3 (1 − ε)

(20)

In the pressure drop measurements, the particles used were spherical and cylindrical with equivalent diameters between 1.7 and 10 mm, and the porosities of the packed beds are between 0.27 and 0.59. The pressure drop correlation can then be regressed from the experimental results as follows:

2

0.106  P u = · Re−0.25 L 2g D ε

kPa/m

(21)

Fig. 9a indicates the pressure drop gradient varies with the superficial gas velocity in a bed with 10 mm spherical particles. The porosities of structural and randomly packed beds were 0.395 and 0.338, respectively. The pressure drop gradient in the structured packed bed increases almost linearly with an increase in the superficial gas velocity. However, it increases in an exponential manner in a randomly packed bed. The pressure drops for both beds were well predicted by the new correlation. Fig. 9b shows good agreement between the model prediction and the experimental data.

4.3.

Consideration on gas–liquid two-phase flow

The analysis above is based on a single-phase (gas) flow through a packed bed to validate the newly derived void distribution model. However, in most industrial applications of packed beds (e.g., trickling bed reactor), both gas and liquid flows are involved, flowing either co-currently or counter-currently through a bed packed with catalyst particles. Therefore, the effect of the liquid flow on the pressure drop should be considered. In the literature, there are many correlations proposed to predict the pressure drop. In this work, two most commonly used ones (Attou et al., 1999; Nemec and Levec, 2005) were chosen to compare with the experimental results. The correlations are given by P = −L g L



1+

(1 − ˇ)



 

AGL G Ur + BGL G Ur2 − ALS L UL +BLS L UL2 ˇL g

 . (22)

−P/L 1 = G g kG

 A

Re∗G

Ga∗G

+B

Re∗2 G Ga∗G

 (23)

Fig. 9 – Comparison of predicted and experimental pressure drops.

Fig. 10 shows that predictions of the Attou et al. (1999) correlation agree with the experimental data in the large spherical particle (5.7 mm) bed, but the model overpredicts the pressure drop in the smaller spherical particle (3.5 mm) bed at high gas superficial velocities and underestimates the pressure drop in the cylindrical particle bed. For the Nemec and Levec (2005) correlation, the pressure drop in the large spherical particle bed is underestimated by this correlation, but those in the smaller spherical and cylindrical particle beds are overpredicted. Deviations are due to the fact that all correlations (Eqs. 22 and 23) were derived under assumptions that the flow channels were straight, and the liquid formed a uniformly distributed film on particle surfaces, which were contradicting the real flow behavior. Moreover, the void distribution in the bed was not considered in these correlations, and only an average porosity was used. Fig. 9 also shows that the three correlations are sensitive to the void size and the liquid holdup. Furthermore, the liquid holdup is influenced by the liquid properties such as contact angle and surface tension. The correlations between liquid properties and the bed structure are important factors to estimate the holdup and the pressure drop. Our previous studies have discussed the influence of liquid properties on gas–liquid flow behaviour (Du et al., 2013, 2015). However, it still requires a detailed liquid distribution for gas–liquid flow in paced beds. Thus, taking into account the void size distribution and the liquid film distribution is essential for the gas–liquid two-phase flow analysis in TBRs, and which requires further investigations.

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Fig. 10 – Comparison of predicted and experimental pressure drops for packed beds with different particle shapes at different gas and liquid velocities.

5.

Conclusions

In this work, the void size distribution in packed beds was experimentally measured, and a statistical distribution function was developed for beds with structured and random packings. The effect of the void size distribution on the pressure drop was also investigated. The following conclusions were drawn: 1. The void size distribution range in a packed bed was wide. Both large pores and small channels were present in such a bed. 2. The particle shape significantly affected the void size distribution, and the spherical particle bed exhibited a wider void size distribution. 3. The void distribution function for structured packed beds can be expressed by f =



x2 + 1 −

2 ϕ2 3(1−ε)

The void size distribution for randomly packed beds can be predicted by the Gauss distribution function: f =

1 − exp (− (c · x/ϕ · xm )) 1 − exp (− (c/ϕ))

4. The prediction of pressure drop gradient, one of the most important parameters of TBRs, can be improved by taking into account both the void size and liquid film distribution.

Acknowledgements Authors should thank the National Natural Science Foundation of China under Grant No. 21176256 and No. 21476255 and

Science Foundation of China University of Petroleum, Beijing (No. KYJJ2012-03-01) for funding this project.

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