Packed bed pressure drop dependence on particle shape, size distribution, packing arrangement and roughness

Powder Technology 246 (2013) 590–600

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Packed bed pressure drop dependence on particle shape, size distribution, packing arrangement and roughness K.G. Allen ⁎, T.W. von Backström, D.G. Kröger Department of Mechanical and Mechatronic Engineering, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa

a r t i c l e

i n f o

Article history: Received 18 December 2012 Received in revised form 10 June 2013 Accepted 15 June 2013 Available online 24 June 2013 Keywords: Packed bed Granular materials Fluid mechanics Roughness Shape Packing arrangement

a b s t r a c t Packed beds have been used or proposed for many different applications, including thermal storage in buildings and in solar thermal power plants. In order to size the blowers and predict the operating and capital cost, the packed bed pressure drop must be known. The Ergun equation, commonly used for predicting packed bed pressure drop, over-predicts the pressure drop through randomly packed or structured beds of smooth spheres at Ergun Reynolds numbers in excess of ≈700, and previous work has found it to under-predict the pressure drop through beds of rock by a factor as high as 5. Present measurements of the pressure drop for air flow through beds of rough spheres, smooth cylinders, cubes and crushed rock are significantly higher than those for smooth spheres, and all differ from the Ergun equation. Particle shape, arrangement (including packing method) and surface roughness are shown to influence the pressure drop. Recent correlations for nonspherical particles are shown to differ significantly from present measurements. Different pressure drop measurements obtained for irregularly shaped rock packed into the test section in two different directions relative to the flow direction show that random packing is not necessarily isotropic. In order to predict the pressure drop over a packed bed of irregular particles such as crushed rock with any degree of accuracy, an empirical equation must be obtained from a sample of the particles for a given packing arrangement. © 2013 Elsevier B.V. All rights reserved.

The Ergun Reynolds number ReErg is defined as

1. Introduction Packed beds of particles have been used in a number of different applications such as building heating and cooling (for example Hughes et al. [1]), absorption or ion-exchange resin beds [2]. Packed beds of rock have been proposed for use as thermal storage in solar thermal power generation, an application for which there is “great potential”, although further work is needed on understanding pressure drop [3]. The pressure drop through a packed bed must be known in order to estimate the capital and operating costs and to size the blowers or pumps required to force fluid through it. The present work looks at pressure drop through packed beds of regular, irregular, rough and smooth particles in order to show which are the most important parameters that influence the pressure drop, and to better understand the complexity of pressure drop through packed beds, particularly rock beds.

ReErg ¼

Rep ρvs D : ¼ μ ð1−ε Þ 1−ε

ð2Þ

The superficial speed vs is defined as vs ¼

_ m : ρAcs

ð3Þ

D is the particle size, defined by Ergun in terms of particle volume and surface area: D¼

6ΣV p : ΣAp

ð4Þ

1.1. Correlations for packed bed pressure drop prediction in the literature The Ergun friction factor fErg of a packed bed is [4] fErg ¼

Δp ε3 150 ¼ D þ 1:75: 2 Lρvs ð1−ε Þ ReErg

⁎ Corresponding author. Tel.: +27 21 808 4272. E-mail addresses: [email protected] (K.G. Allen), [email protected] (T.W. von Backström), [email protected] (D.G. Kröger). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.06.022

ð1Þ

Ergun proposed estimating this ratio indirectly by means of pressure drop measurements over a packed bed of the material at low flow rates in the viscous regime. For a sphere, Eq. (4) reduces to the sphere diameter. Ergun does not explicitly state the range of validity of Eq. (1). The measurements on which his correlation is based, or with which he compared it, are in the range 1 b ReErg b 2400. He does not specifically state what particles he used; tables and graph legends in Ergun [4] or Ergun and Orning [5] suggest spheres, pulverized coke/coal, sand, cylinders and tablets were used. The correlation

K.G. Allen et al. / Powder Technology 246 (2013) 590–600

is not based solely on pressure drop measurements through spheres or smooth particles. Montillet [6] states that equations of the form Δp/L = avs + bvs2 – such as the Ergun equation – should not be used for Rep N 500–600, because, in the turbulent regime, the pressure drop in a finite packed bed is not proportional to the square of the flow speed. He attributes this to the combined effect of transition to a new flow regime and the finite nature of the packed bed. Niven [7] argues that the vs2 pressure loss term is strongly dependent on local losses — expansion, contraction and change in flow direction, which occur even in laminar flow. He states that in fully turbulent flow, the local losses should be dominant, rather than turbulent losses, which results in “a second transition, from laminar to turbulent flow,” which occurs at Reynolds numbers higher than the transition within the Ergun equation. The Ergun equation, for ReErg N 700, over-predicts the pressure drop for randomly packed beds of smooth spheres. The over-prediction can be seen in Ergun [4] from some of the graphical data he shows. On the other hand, the measurements of Zavattoni et al. [8] for rock beds were 10–30% higher than the Ergun equation, and Shitzer and Levy [9] measured rock bed pressure drops a factor of 1.5–5 times higher than the Ergun equation. Tobiś [10] has shown, by inserting obstacles into the flow passages between spheres in a simple cubic packing arrangement, that the constant 1.75 in the Ergun equation can vary by a factor up to almost five, depending on the alignment and shape of the obstacle. Mayerhofer et al. [11] have shown, for irregularly shaped wood chips, that the packing alignment of the wood chips relative to the air flow direction influences the pressure drop. Hicks [12] warns that the constants in the Ergun equation may be dependent on the Reynolds number. He notes that, although the Ergun equation is “advanced by several textbooks without restriction to flow range,” it may not be applicable for spheres when ReErg N 500. A range of correlations for spherical and non-spherical particles is given below. The selection includes recent correlations, two of which include parameters to estimate the influence of wall effects. Carman [13] gives the following correlation for spheres in a test section with negligible wall effects, for 0.1 b ReErg b 60 000: fC ¼

Δp ε 180 2:87 ¼ D þ : Lρv2s ð1−ε Þ ReErg Re0:1 Erg

Another correlation for pressure drop in a bed of spheres of uniform diameter is found in Idelchik [17], which presents a correlation by Bernshtein for void fractions between 0.3 and 0.8: 3

fI ¼

ð5Þ

3

Δp ε ε 0:765 30 3 ¼ D þ þ 0:3 ReI Re0:7 Lρv2s ð1−ε Þ ð1−ε Þ ε4:2 I

! ð9Þ

where ReI = (0.45/ε0.5) ReErg. The range of applicability of the equation is not specifically stated; Idelchik uses it in graphs over the range 0.001 b ReI b 1 000. For beds of spheres, Montillet et al. [18] propose fM

!  0:20 Δp ε3 Dc 1000 60 ¼a ¼ D þ 0:5 þ 12 D Rep Lρv2s ð1−ε Þ Rep

ð10Þ

where a is 0.061 for dense packings (ε b 0.4) and 0.050 for loose packings (ε N 0.4). The equation is valid for 3.8 b Dc/D b 50 and 10 b Rep b 2500. For Dc/D N 50, (Dc/D)0.2 is set to 2.2. This equation was obtained from measurements with water or aqueous solutions of glycerol in a cylindrical column. Montillet et al. do not state how the value Dc should be calculated for test sections of a non-circular cross-section. Singh et al. [19] present a correlation for pressure drop through beds of differently shaped particles, with the particle shape taken into account by means of a sphericity factor ψ: ψ¼

" #1=3 36πV 2p As ¼ : Ap A3p

ð11Þ

Here As is the surface area of a sphere that has the same volume as the particle. The correlation is based on data for pressure drop through spherical and other non-spherical objects from measurements in the range 1000 b Rep b 2700 (approx. 1500 b ReErg b 5000), and the particle diameter Dve is defined as the diameter of a sphere that has the same volume as the particle volume Vp: fS ¼

3

591

Δp ε3 ε3 −0:2 0:696 −2:945 11:85ðlogψÞ2 ¼ 4:466Rep ψ D ε e ð12Þ 2 ve ð1−ε Þ ð1−ε Þ Lρvs

where Hicks [12] proposes a relation for the range 300 b ReErg b 60 000 for spheres: fH ¼

Δp ε3 6:8 ¼ D : 2 ð 1−ε Þ Re0:2 Lρvs Erg

ð6Þ

Brauer [14] gives an equation for packed beds of spheres, plotted against measured data for the range 0.01 b ReErg b 40 000, which may be written as ð7Þ

This equation is almost identical to the equation used by the KTA 3102.3 standard for pebble bed nuclear reactors [15]; the only difference is that the constant 3.1 is changed to 3. Jones and Krier [16] give a correlation for spherical glass beads in the range 1000 b Rep b 100 000, 8 b Dc/D b 52, which can be written as 3

f JK ¼

Δp ε 150 3:89 ¼ D þ : Lρv2s ð1−ε Þ ReErg Re0:13 Erg

6 V π p

1=3

:

ð13Þ

An equation for spherical or non-spherical particles with wall correction terms is found in Eisfeld and Schnitzlein [20]: f ES ¼

Δp ε3 K A2 A ¼ 1 wþ w: D 2 Lρvs ð1−ε Þ ReErg Bw

ð14Þ

Aw and Bw are the wall correction terms, defined as

3

Δp ε 160 3:1 ¼ D þ : fB ¼ ReErg Re0:1 Lρv2s ð1−ε Þ Erg

 Dve ¼

ð8Þ

Aw ¼ 1 þ

2 3ðDc =DÞð1−ε Þ

h i2 2 Bw ¼ k1 ðD=Dc Þ þ k2 :

ð15Þ ð16Þ

The values of K1, k1 and k2 presented by Eisfeld and Schnitzlein are shown in Table 1. They are based on experimental data largely with spheres and cylinders for 0.33 b ε b 0.88, 0.01 b Rep b 17 700 and 2 b Dc/D b 250. Eisfeld and Schnitzlein do not specify how Dc should be calculated for non-circular bed cross-sections. For non-spherical particles, Nemec and Levec [21] propose altering the constants in the Ergun equation by means of the particle

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K.G. Allen et al. / Powder Technology 246 (2013) 590–600

Table 1 Values of K1, k1 and k2 for Eq. (14) from Eisfeld and Schnitzlein [20]. Particle shape

K1

k1

k2

Spheres Cylinders All particles

154 190 155

1.15 2.00 1.42

0.87 0.77 0.83

sphericity ψ used in Eq. (11). For cylindrical particles, the friction factor is altered to

3

f NL ¼

Δp ε 150 1:75 ¼ D þ : Lρv2s ð1−ε Þ ψ3 =2 ReErg ψ4 =3

ð17Þ Fig. 2. Influence of different packing arrangements of smooth spheres on friction factor.

This relation is based on experimental data for ReErg b 400. 1.4. Influence of particle roughness 1.2. Comparison of friction factor correlations for beds of randomly packed spheres The different predictions for randomly packed beds of spheres are compared in Fig. 1. The equations of Eisfeld and Schnitzlein [20] and Montillet et al. [18] are plotted for a container to particle ratio of Dc/D = 25, where wall effects should be negligible. As the Reynolds number decreases below 500, the different correlations tend to converge; however, there is an increasing difference between them for values of ReErg above 500. 1.3. Influence of packing arrangement of smooth spheres Measurements from literature for different packing arrangements of spheres are plotted in Fig. 2. Wentz and Thodos [22] tested five-layer thick arrangements of 31 mm spheres at a container to diameter ratio of 11. They do not specify if they took measures to reduce edge effects. The other data shown was read off the graph of Martin et al. [23], who used fractional spheres at the walls to eliminate excess porosity. Their tests were for an estimated test section to particle size ratio Dc/D ≈ 8 or 9, based on a picture of their test section. They found that rotating the packing arrangement so that the flow passed through it in a different direction altered the apparent friction factor. They record that the simple cubic packing was extremely unstable and shifted very easily, which may explain why the measured friction factors for the simple cubic packing reach a constant value and increase for ReErg N 1000.

According to Eisfeld and Schnitzlein [20], “an observable influence of the particle surface roughness on the pressure drop is physically hardly plausible…”. They state that those who do find an increase in pressure drop due to surface roughness probably misinterpret the data. On the other hand, Nemec and Levec [21] state that pressure drop is increased by surface roughness in both the viscous and inertial flow regimes, and Crawford and Plumb [24] present measurements showing a “substantially increased” pressure drop as a consequence of surface roughness on spheres for 10 b ReErg b 2000. In light of the divergence between existing correlations, and differences between the correlations and measurements over rock beds, further measurements were made on particles of different shape, roughness, packing arrangement, and size. A brief summary of packed bed pressure drop analysis based on the assumption of duct flow – the assumption underlying the Ergun equation [25] – is first presented and applied to different packing arrangements. 2. Analysis of packed bed pressure drop assuming duct flow The pressure drop factor or apparent friction factor fda of a tortuous duct, which takes into consideration both frictional and form drag, is ΔpD f da ¼  2 h  : L ρvd =2

ð18Þ

To illustrate how this is applied to a packed bed of smooth spheres, consider a simple cubic packing, divided into repeatable control volumes of total volume Vt = Pt1Pt2Pl, pitched at Pl = D in the flow direction and Pt1 = D and Pt2 = D in the transverse direction, with fluid flowing upstream from the packing at a speed vs in the direction shown in Fig. 3.

Fig. 1. Comparison of friction factor correlations for a randomly packed bed of smooth spheres.

Fig. 3. Plan view and control volume in structured simple cubic packing indicating pitch between volumes.

K.G. Allen et al. / Powder Technology 246 (2013) 590–600

593

The total control volume is Vt = D3; the particle volume Vp = πD3/6, and the particle surface area Ap = πD2. The average area available for the fluid flow is Aflow ¼

Vv Pl

¼

V t −V p ¼ Pl

2

D −

! πD2 : 6

ð19Þ

The mass conservation equation for the control volume may be expressed in terms of the average duct speed vd in the packed bed, and the upstream superficial speed vs in the cross-sectional area Acs = Pt1Pt2: vd ¼

Acs vs vs : ¼ Aflow 1−π=6

Fig. 4. Plan view and control volume in offset simple cubic (rhombohedral) packing indicating pitch between volumes.

ð20Þ The apparent friction factor for the shown control volume is

The volumetric void fraction ε is ε SC ¼

V v V t −V p D3 −πD3 =6 π ¼ ¼ ¼ 1− ¼ 0:476: Vt Vt 6 D3

ð21Þ

3 3 4V p Δp εOSC Δp εOSC 4D   2  ¼ 2 ð 1−ε Þ A ð 1−ε P l ρvs =2 ðD=2Þ ρvs =2 OSC p OSC Þ 6 ¼ f ðReOSC Þ



fOSC ¼

ð29Þ

From Eqs. (20) and (21), vd ¼

where

Acs vs v ¼ s : Aflow εSC

ð22Þ ReOSC ¼

The hydraulic diameter of the duct is defined as Dh ¼

4Aflow P wet

SC

2.2. A mixture of two sizes of spheres

  3 3 4Aflow P l 4V v 4 D −πD =6 4Dð1−π=6Þ ¼ ¼ ¼ ¼ : Ap Ap π πD2

ð24Þ

The friction factor fda is a function of Reynolds number. The latter is defined as Re ¼

ρvd Dh : μ

ρvs 4V v ρv V ¼4 s t μεSC Ap μ Ap

¼4

V p 2 ρvs D ρvs : ¼ μ ð1−ε SC Þ Ap 3 μ ð1−ε SC Þ

  3 3 V p1 þ V p2 π D1 þ D2 ¼ : ΣV p ¼ 2 12

ð31Þ

The void fraction for this arrangement of mixed spheres (MS) is

ε MS

  8π D31 þ D32 ΣV p Vv ¼ ¼ 1− ¼ 1– : Vt Vt 12ðD1 þ D2 Þ3

ð32Þ

The hydraulic diameter is

ð26Þ

Substitute Eqs. (21), (22), and (24) into Eq. (18) to obtain the apparent friction factor for simple cubic packing, for the shown control volume: fSC ¼

Eq. (18) is now applied to a control volume which contains smooth spheres of two different sizes, arranged as shown in Fig. 5. The sphere volume

ð25Þ

Substitute Eqs. (21), (22) and (24) into Eq. (25) to obtain the Reynolds number for simple cubic packing: ReSC ¼

ð30Þ

ð23Þ

where Pwet is Ap/Pl, the wetted perimeter of the duct. The hydraulic diameter is therefore

Dh

V p 2 ρvs D ρvs 4V v ρvs : ¼4 ¼ μεOSC Ap μ ð1−ε OSC Þ Ap 3 μ ð1−ε OSC Þ

Dh

MS

    3 3 3 4V v 4 ðD1 þ D2 Þ =8 − π D1 þ D2 =12   ¼ ¼ : ΣAp π D21 þ D22 =2

ð33Þ

3 3 4V p Δp εSC Δp εSC 4D  2  ¼  2  ¼ f ðReSC Þ: P l ρvs =2 ð1−ε SC Þ Ap D ρvs =2 ð1−ε SC Þ 6

ð27Þ 2.1. Offset simple cubic layers An offset simple cubic layer arrangement (rhombohedral packing — Martin et al. [23]) is shown in Fig. 4. pffiffiffi For this packing arrangement, Pl = D/2, Pt1 = 2D and Pt2 = D. p ffiffiffi The control volume Vt = D3 = 2; the particle volume Vp = πD3/6, and the particle surface area Ap = πD2. The void fraction pffiffiffi V 2π εOSC ¼ v ¼ 1− ¼ 0:2595: ð28Þ Vt 6

Fig. 5. Plan view and control volume of two different sphere sizes alternated in packing position.

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K.G. Allen et al. / Powder Technology 246 (2013) 590–600

For this packing arrangement, Eq. (18) can be written as an apparent friction factor fMS: 4ΣV p Δp ε3MS  2 P l ρvs =2 ð1−ε MS Þ ΣA  p  4 D31 þ D32 Δp ε3MS   ¼ f ðReMS Þ ¼  2  P l ρvs =2 ð1−ε MS Þ 6 D21 þ D22 

fMS ¼

ð34Þ

where 

ReMS ¼ 4



D31 þ D32 ΣV p 2 ρvs ρvs  2 : ¼ μ ð1−ε MS Þ ΣAp 3 μ ð1−ε MS Þ D1 þ D22

ð35Þ

In the case where D1 = D2, Eqs. (34) and (35) reduce to the same form as those obtained for single-sized spheres. 2.3. Generalized equations for fda and Re In general, for different particle sizes and shapes, the apparent friction factor can be written in terms of the sum of the particle volumes ΣVp and surface areas ΣAp as fda ¼

3 4ΣV p Δp ε  2  ¼ f ðReÞ P l ρvs =2 ð1−ε Þ ΣAp

ð36Þ

where Re ¼ 4

ρvs ΣV p : μ ð1−ε Þ ΣAp

ð37Þ

This applies to spherical or non-spherical particles. The packing dimension ΣVp/ΣAp, the ratio of the total particle volume to surface area – used as early as Blake [26] and Carman [13] – should be understood to refer to the duct size rather than an equivalent particle size 6ΣVp/ΣAp, as used by Ergun. The apparent friction factor fda may depend on the particle shape, surface roughness, packing arrangement and void fraction; consequently, every packing arrangement should have a particular apparent friction factor associated with it. 3. Experimental apparatus, particle characteristics and test method 3.1. Apparatus The wind tunnel and test section shown in Fig. 6 were used for measuring the pressure drop over a packed bed test section. Air is drawn through a bell mouth at the tunnel inlet and then through the test section, where the pressure drop is measured. Subsequently, the air passes into a settling chamber upstream of a 50 mm elliptical nozzle. At flow rates above 0.02 kg/s, the pressure drop over this nozzle is measured and used to calculate the air mass flow rate as described in Kröger [27]. The air then passes into a fan controlled by a variable speed drive, and out of the fan exhaust. At flow rates below

0.02 kg/s, a DISA 55D41 calibration unit with nozzle was attached to the wind tunnel exit to measure the flow rate. The pressure drop is measured by means of two Endress-Hauser 0–2500 Pa pressure transducers and one Foxboro 0–25 000 Pa transducer, which were calibrated by means of two different Betz 5000 manometers. 3.2. Particle characteristics and test method For random packing, the test section was packed by pouring the particles into it from one of two different directions: either through the lid indicated in Fig. 6 in a cross-current direction relative to the air flow, or through the flow inlet in a co-current direction relative to the air flow by temporarily removing the metal mesh and standing the test section on the flow outlet. The packing was not vibrated after it was poured in. The repeatability of the packing was tested for every type of packing material by emptying the test section, re-filling it, and taking a second set of measurements. The pressure drop over the empty test section was measured and found to be negligible (b 0.1– 1%) in comparison to that measured with packing. For structured packing, the particles at the wall were embedded in a dough mixture to reduce channelling of air through low resistance regions. The mesh holding the particles at the inlet and outlet was moved to fit each packing arrangement, so the length of the packed section varied between 270 and 300 mm. For random packing the wall was lined with a sponge layer 10–15 mm thick or a soft cloth 3–5 mm thick, which reduced the effective packing cross-section from the nominal test section dimensions shown. Tests were also done without the lining to determine its influence. Pressure drop measurements were made for randomly packed beds of smooth and rough glass spheres, wooden cubes, wooden cylinders, acorns (ellipsoids), mixed smooth spheres of different sizes, naturally weathered (rounded) rock and crushed rock. Data was also obtained for smooth spheres in different structured packing arrangements, and partially aligned ellipsoids. The uniformly shaped particle dimensions were measured by means of a vernier calliper. The wooden cylinders and cubes were cut from lengths of the required cross-section with a circular saw, and the splinters removed. The measured average surface roughness of the cubes was 7 μm. Two different sets of mixed spheres were used — a mixture of 8 mm plastic beads and 16 mm glass marbles, and a mixture of marbles sized between limits of 16 mm–44 mm. The acorns were approximated as ellipsoids and the length of the major and minor axes was measured. Sand from a vibrating sieve, sized between 0.18–0.25 mm and 0.425–1 mm, was stuck onto glass marbles by means of gelatine or wood glue to make the roughened spheres. Rock samples were scanned with a NextEngine 3D scanner in order to calculate the average volume and surface area of the rock. These scans are time-consuming, so only 30–50 were done for each rock type. The average volume per rock from the scans was checked against that measured by a volume displacement test on a larger number of samples (≈2% of the test section mass) to confirm that it was a representative average volume. The spheres were packed in the test section in 3 different structured arrangements: Hexagonal close (HC) packing, a layered packing

Fig. 6. Horizontal wind tunnel layout and nominal test section dimensions in mm.

K.G. Allen et al. / Powder Technology 246 (2013) 590–600

with offset simple cubic (OSC) layers so that the layers packed closer together than they would for normal simple cubic packing (Martin et al. [23] refer to this as a rhombohedral arrangement), and simple cubic (SC) packing. A plan view of the simple cubic packing layers in the horizontal test section is shown in Fig. 7a. These layers were repetitively packed directly on top of each other until the test section was filled. The offset simple cubic layers are shown in Fig. 7b. The HC packing was orientated to the flow direction as shown in c). The measured void fractions of b) and c) in Table 2 were higher (0.28– 0.29) than the theoretical value of 0.26. This is a consequence of small gaps between some of the marbles during packing, which led to extra space between the marbles of up to 2–3 mm per row. The rounded rock had no specified size range. The sizes of the crushed rock were 13.2 mm (predominantly hornfels and schist) and 26.5 mm, as defined by South African Bureau of Standards (SABS) specification 1083 for aggregates. For example, 26 mm aggregate is specified to be in a size range such that 100% of the rock mass falls through a grid with square holes of side length 37.5 mm; 85– 100% through 26.5 mm, 0–50% through 19 mm, 0–25% through 13.2 mm, and 0–5% through 9.5 mm [28]. The 26 mm rock used in the test section was hand-collected, which may have reduced the percentage of the smallest particle sizes and dust. The tested particle dimensions and packing void fractions are summarized in Table 2. The sphericities shown for the rock were calculated as an average of the measured sphericities of each of the sample rocks that were scanned. The density of irregular particles was measured by volume displacement of liquid for particles of known mass. The void fraction of the packing was obtained by one or more of three methods. (I) The total mass of particles in the test section was measured or (II) the total number of regular particles of known dimension in the test section was counted. The total particle volume was obtained from I or II, and subtracted from the total volume occupied by the packing to give the void volume. Alternatively, (III) the void fraction was measured by pouring the particles into a waterproof container of similar volume and shape, and measuring the quantity of liquid required to fill the void volume in a known total packing volume. The disadvantage of (I) and (II) is that, if a wall lining is used, the uncertainty of the volume in which the particles are packed increases. The disadvantages of method (III) are twofold: the particles are poured into a different container, so it is not a direct measurement of the void fraction in the test section; and, if the particles absorb the liquid, it causes an erroneous measurement. To reduce the potential error, the particles must be packed according to the same method in a container of similar shape and size. The particles were pre-soaked in the liquid to avoid absorption during void measurement, and excess liquid was drained off the particles before the measurement. The different void fractions for the randomly packed smooth marbles and 13.2 mm rock were obtained by varying the height from which the particles were poured into the test section, from the level of the test section opening up to 1 m above the opening. The lower void fraction (0.37) of the wooden cubes was obtained by tamping during the packing process.

Fig. 7. Plan view of different horizontal sphere packing arrangements with the flow direction shown: a) simple cubic layer; b) offset simple cubic layers; c) hexagonal close layers.

595

Table 2 Summary of packing characteristics. Particle type

6ΣVp/ΣAp, m ΣVp/ΣAp, m ε

Random packing, single size spheres (smooth glass) a Structured single size spheres (smooth glass) Random packing, single size rough spheres (0.18–0.25 mm sand) a Random packing, single size rough spheres (0.43–1 mm sand) Random packing, mixed spheres (8 mm & 16 mm) Random packing, mixed spheres (16–44 mm) Random packing, cubes (wood) a Random packing, cylinders (wood) Random packing, acorns (approx. ellipsoids) Aligned acorns (approx. ellipsoids)

15.8

2.6

15.8

2.6

0.36; 0.38; 1 0.39; 0.40 0.28–0.48 1

16.2–16.3

2.7

0.43–0.44

1

16.3–16.4

2.7

0.44–0.45

1

10.5

1.7

0.37

1

16.3

2.7

0.37

1

16.2 2.7 20.6; 17.9

2.7 2.7 3.4; 3

0.37; 0.43 0.40 0.34–0.35

0.81 0.87 0.95

20.6; 17.9

3.4

21.3

3.5

0.33; 0.34; 0.95 0.35 0.38 0.85

7.8

1.3

24.4

4.1

Random packing, rounded smooth rock Random packing, crushed “13.2 mm” rock a Random packing, crushed “26.5 mm” rock a

ψ

0.44; 0.45; 0.76 0.46; 0.47 0.42 0.80

a Experimental results presented in part or in whole at the SASEC 2012 conference [29].

Pictures of some of the particles tested are shown in Fig. 8. The diameter of the sand-roughened spheres was measured with vernier callipers and by volume displacement in water. The reason that there is not much difference in the diameter of the spheres covered in two different grain sizes of sand is that the larger grains did not bond as well and only about 30% of the marble surface was covered. The surface area was calculated based on the measured diameter — the micro-surface area of the sand particles was ignored. The reduction of the void volume by the sand particle volume is taken into account in the void fraction, which was measured with methods I and III. The increased wetted surface area from the sand roughness was not taken into account. This would reduce the hydraulic diameter and lower Re and fda (see Eqs. (36) and (37)). Any reduction in wetted surface area that occurs for the non-spherical particles as a consequence of overlapping particles will increase the hydraulic diameter and therefore increase Re and fda. No attempt was made to quantify these effects. 3.3. Uncertainty and error of measurement The minor Betz manometer divisions represent 2 Pa, which allows readings to an accuracy of ± 0.5 Pa. Comparison between repeated calibrations of the Endress Hauser pressure transducers, which give a voltage output sensitive to ambient temperature, shows a standard deviation of approximately 0.5 Pa at a pressure of 10 Pa increasing linearly to 6 Pa at the upper limit, 2500 Pa. The Foxboro transducer was more inaccurate, with a standard deviation of 4 Pa at 10 Pa, increasing roughly linearly to 21 Pa at 5000 Pa, the maximum pressure at which measurements were taken. The standard deviation of the test section dimension measurements varies between 0–3 mm for transverse lengths, depending on whether or not a wall lining is used, and 0–1 mm for the bed flow length. The standard deviation of the cylinder and cube dimensions is less than 0.2 mm. That of the spheres is less than 0.3 mm. The value of ΣVp/ΣAp calculated from the ellipsoid approximation of the acorns was about 3% lower than that obtained from a sample 3D

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K.G. Allen et al. / Powder Technology 246 (2013) 590–600

Fig. 8. Pictures of test section fill (minor ruler divisions in mm). a) Smooth glass spheres (marbles); b) 0.18–0.25 mm sand-coated marbles; c) wooden cubes; d) wooden cylinders; e) ellipsoids (acorns); f) crushed “13 mm” rock; g) crushed “26 mm rock”; h) rounded rock.

scan. The total rock volume from the 3D scans was on average 3% less than that measured by volume displacement in water. The scanner is specified to measure points to within 0.2–0.3 mm for the range in which the particles were measured. However, the post-scan processing is probably only accurate to within about 1–2 mm as a consequence of trimming edges or aligning multiple scans. 4. Experimental results and discussion Present measurements for randomly packed smooth spheres (16 mm marbles; cross-current packing) are compared with the data of Kays and London [30] in Fig. 9. The data in Fig. 9 can be adequately represented by 4ΣV p 172 4:36 Δp ε3 ¼ þ 0:12 : fda ¼  2  ð 1−ε Þ Re ΣAp L ρvs =2 Re

ð38Þ

Fig. 9. Comparison of friction factor measurements through randomly packed beds of smooth spheres with the Ergun equation.

K.G. Allen et al. / Powder Technology 246 (2013) 590–600

597

Fig. 10. Comparison of Eq. (38) with other correlations for randomly packed smooth spheres.

Fig. 12. Measured friction factor for randomly packed rough spheres and mixed sphere sizes.

Eq. (38) is compared with some of the other correlations in Fig. 10. For Re b 500, Eq. (38) is almost indistinguishable from Eq. (10) for Dc/D = 16, the present value if Dc is taken as the hydraulic diameter of the test section. For Re N 300, Eq. (38) is similar to Eqs. (8) and (5). Eq. (38) provides a common basis of comparison for all further measurements presented. Measured values of fda for different structured packing arrangements of smooth spheres are compared with those for random packing in Fig. 11. The simple cubic packing differs less (up to 30%) from the randomly packed sphere data than the data of Martin et al. [23] shown in Fig. 2. For Re N 600, the slope of fda steepens as the void fraction decreases from 0.48 to 0.28. It is possible that the steeper slope of fda for the lowest void fractions may be linked to the structure of the packing: for low void fraction, the flow is forced to follow the surfaces of the particles more than for high void fraction. If so, in dense packings, form drag is reduced and surface friction losses are higher, whereas, in low density packings, the reverse is true. The influence of mixed particle sizes and roughness is shown in Fig. 12 for random packing. The first mixture was obtained by adding 8 mm beads to the 16 mm marbles, until 6ΣVp/ΣAp was reduced from 15.8 mm to 10.5 mm. The second mixture was obtained by using marble sizes between 16 and 44 mm, although there were only sufficient large marbles to increase the value of 6ΣVp/ΣAp from 15.8 mm to 16.3 mm. Neither mixture significantly altered fda from that of single-sized spheres (b9%). However, the surface roughness, for both sand grain sizes, increased fda by up to 20–45% of the value

for smooth spheres. At low Re (b100), the measurements appear to tend towards those obtained for smooth spheres. This trend is consistent with pipe flow under low Reynolds number laminar flow conditions, where surface roughness has no effect on the apparent friction factor. If the additional surface area due to particle roughness was included in the calculation of fda and Re, the values of both would reduce and the data would move towards the origin on both axes. The influence of particle shape and alignment for smooth ellipsoids (acorns) is shown in Fig. 13. The alignment of the acorns with the major axis approximately parallel to the flow direction resulted in values of fda as much as 30–40% lower than the other arrangements. The relatively poor repeatability for the parallel alignment is because the acorns do not all align parallel to the flow as a consequence of different acorn sizes, which disturbs the packing arrangement. The smooth acorns in random packing give fda values similar to those for randomly packed smooth spheres. The influence of particle shape is seen in Fig. 14 for randomly packed beds of cubes and cylinders. The fda values for the cubes and cylinders are respectively up to almost 200% and 100% higher than those for smooth spheres. To ensure that there were no entrance or exit effects influencing the measurements, an additional test was conducted with the wooden cubes in a longer test section with L/D ≈ 30. This data is also shown in Fig. 14. If the reduction in particle surface area exposed to the flow (caused by particle overlap) was included in the calculation of fda and Re, the values of both would increase and the data points would move further away from the origin and the smooth sphere data.

Fig. 11. Friction factors for different structured packing arrangements of smooth spheres.

Fig. 13. Measured friction factor for randomly packed and aligned ellipsoids (acorns).

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Fig. 14. Measured friction factor for randomly packed cubes and cylinders.

Fig. 15 plots the measured friction factors for the different sets of rocks, packed through the test section lid in a cross-current direction relative to the air flow. The values of fda are up to 35% more than the smooth sphere curve. Even though the one rock set had rounded edges, its measurements lie between those of the two different sizes of crushed rock, which had sharp edges. Fig. 16 plots the measured friction factors for the two sets of crushed rocks, packed in co-current and cross-current directions relative to the air flow. Friction factors for the co-current packing are up to 35–55% more than those for cross-current packing for the same set of rock, and as much as 80% more than those for smooth spheres. Particle alignment has been shown to influence the apparent friction factor, as shown by the measurements for acorns (Fig. 13) and the results of Tobiś [10] and Mayerhofer et al. [11] which show strong dependence on the alignment of particles. It is suggested that, because the crushed rock is asymmetric, it tends to align during packing so that the larger areas are parallel to the bottom of the test section; that is, irregularly shaped particles poured in the presence of gravity will not form an isotropic packing arrangement. This would result in the larger area of the rock particles obstructing flow when the section is packed in a co-current direction relative to the air flow, which could cause a higher form drag and value of fda. At low Re in the viscous flow region, particle form drag is negligible relative to surface friction, so the difference in fda would reduce with reducing Re, until the measurements converge, provided the duct shape is not particularly different. This appears to be the case for the ΣVp/ΣAp = 1.3 mm crushed rock, which could be tested to lower values of Re than the larger rock. The packing alignment effect

Fig. 16. Friction factor for two different sizes of randomly packed crushed rock (comparison of co-current and cross-current measurements).

is visibly discernible for a single layer of rock on a flat surface (for example, Fig. 8f), but not for further layers when the surface becomes more uneven. Discrete element modelling (DEM) by Nel [31] found that the particles tended to align with the major axis at an average of about 25° to the container floor. Curve fits to a selection of the experimental data shown above are given in Table 4.

5. Conclusion The Ergun equation should not be used for pressure drop prediction in packed beds of smooth spheres where ReErg N 700, as it over-predicts the pressure drop. It is not suitable for predicting pressure drop over packed beds of rock or other non-spherical particles like cylinders or cubes, as the measured values can differ from the Ergun equation by as much as 100% or more, which is shown by present measurements and previous work. The percentage change in fda for the different particles, relative to the value for smooth spheres, is summarized in Table 3. The apparent friction factor of a packed bed has been shown to change significantly (of the order of 10–100%) from that of randomly packed smooth spheres, depending on particle shape, packing arrangement and surface roughness. A variation in the particle size ratio of a bed of randomly packed spheres of almost 40% by the addition of spheres of half the diameter did not result in much deviation of fda from that of randomly packed spheres of a single size. The apparent friction factor (for irregularly shaped particles) is dependent on the packing method, which presumably alters the packing arrangement. Random packing of irregularly shaped particles is not necessarily isotropic, as shown by the crushed rock packed in a co-current direction relative to the air flow, which gave rise to

Table 3 Alteration in fda caused by particle shape, roughness, arrangement and size distribution.

Fig. 15. Friction factor for randomly packed crushed rock.

Particle type

% Increase in fda relative to smooth glass spheres 50 b Re b 100

1900 b Re b 2300

Cubes Cylinders Acorns, parallel Acorns, perpendicular or random Mixed spheres Rough spheres Cross-current rock Co-current rock

85–90 50–55 −25 to −30 −10 to −15 5–10 25 0–15 15–50

180–185 95–100 −35 to −45 −10 to −15 5–10 40–45 0–35 40–60

K.G. Allen et al. / Powder Technology 246 (2013) 590–600 Table 4 Friction factor curve fit constants for an equation of the form fda = (a/Re) + (b/Rec). Particle type and packing description

a⁎

b⁎

c⁎

Wooden cubes (random) Wooden cylinders (random) Rough spheres (random) Acorns (parallel to flow) “26.5 mm” crushed rock (co-current, random)

240 216 185 150 200

10.8 8.8 6.35 3.25 8

0.1 0.12 0.12 0.15 0.12

⁎ Only valid over the range of Re for which the measurements were done.

apparent friction factors up to 60% higher than those obtained from the same rock packed in a cross-flow direction. It is important that the packing direction of the test section relative to the flow be recorded and stated. For crushed rock, which may vary from quarry to quarry, or even within a quarry where the rock is not homogeneous, it is not possible to take all these factors into account. In order to predict the pressure drop over a packed bed, a sample test must be done to obtain an empirical correlation for the particular material and packing direction relative to the air flow. As shown by the duct analysis, the appropriate dimension for the flow channel is obtained from the ratio of the total particle volume to surface area in the packed bed, not the diameter of a volume-equivalent sphere. For rough particles, this raises the question as to which scale and degree of accuracy the surface area should be based on. The results in this work are based on surface dimensions to the order of approximately 1 mm, as limited by the measurement equipment accuracy. Nomenclature Total packed bed cross-sectional area, m2 Acs Aflow Open flow area in packed bed/control volume, m2 Ap Particle surface area, m2 As Surface area of volume equivalent sphere, m2 D Particle diameter, 6ΣVp/ΣAp m Test section container diameter, m Dc Hydraulic diameter of a flow duct, m Dh Volume equivalent sphere diameter, m Dve Duct apparent friction factor fda Brauer friction factor fB Carman friction factor fC Ergun friction factor fErg Eisfeld, Schnitzlein friction factor fES Hicks friction factor fH Jones, Krier friction factor fJK Idelchik friction factor fI Montillet friction factor fM Nemec, Levec friction factor fNL Singh friction factor fS L Length of packed bed, m _ m Air mass flow rate through packed bed, kg/s Longitudinal pitch of control volume, m Pl Transverse pitches of control volume, m Pt1,t2 Wetted perimeter of a flow duct, m Pwet Re Duct Reynolds number, 2ρvD/[3μ(1 − ε)] Ergun Reynolds number, ρvD/[μ(1 − ε)] ReErg Particle Reynolds number, ρvD/μ Rep Particle volume, m3 Vp Vt Combined particle and void volume, m3 Vv Void volume, m3 vs Superficial flow speed of air, mf/ρAcs, m/s Duct flow speed, m/s vd Greek alphabet Δp Difference in pressure, Pa ε Void volume fraction of packing, Vv/Vt

μ ψ ρ Σ

599

Fluid viscosity, kg/ms Sphericity of a particle Fluid density, kg/m3 Sum of

Abbreviations/subscripts HC Hexagonal close MS Mixed spheres SC Simple cubic OSC Offset simple cubic

Role of the funding source Funding for this research was from the NRF, the Department of Science and Technology Solar Thermal Spoke and the Stellenbosch University Hope Project, via STERG and the CRSES. None of these had any role in the study design, the collection and interpretation of the data, the writing of this paper or the decision to submit. Acknowledgments Thanks to the Mechanical and Mechatronic laboratory staff, the Centre for Renewable and Sustainable Energy Studies (CRSES), and the Solar Thermal Energy Research Group (STERG) for their technical or financial support. Thanks to Lisa Conradie for logging a few of the datasets. The financial assistance of the National Research Foundation (NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the NRF. References [1] P.J. Hughes, S.A. Klein, D.J. Close, Packed bed thermal storage models for solar air heating and cooling systems, ASME Journal of Heat Transfer 98 (1976) 336–338. [2] N. De Nevers, Fluid Mechanics for Chemical Engineers, 2nd ed. McGraw-Hill, New York, 1991. [3] H.W. Fricker, High temperature heat storage using natural rock, Solar Energy Materials 24 (1991) 249–254. [4] S. Ergun, Fluid flow through packed columns, Chemical Engineering Progress 48 (1952) 89–94. [5] S. Ergun, A.A. Orning, Fluid flow through randomly packed columns and fluidized beds, Industrial and Engineering Chemistry 41 (1949) 1179–1184. [6] A. Montillet, Flow through a finite packed bed of spheres: a note on the limit of applicability of the Forchheimer-type equation, ASME Journal of Fluids Engineering 126 (2004) 139–143. [7] R.K. Niven, Physical insight into the Ergun and Wen & Yu equations for fluid flow in packed and fluidized beds, Chemical Engineering Science 57 (2002) 527–534. [8] S.A. Zavattoni, M.C. Barbato, A. Pedretti, G. Zanganeh, CFD simulations of a pebble bed thermal energy storage system accounting for porosity variations effects, SolarPACES 2011, Granada, 2011. [9] A. Shitzer, M. Levy, Transient behaviour of a rock-bed thermal storage system subjected to variable inlet air temperature: analysis and experimentation, ASME Journal of Solar Energy Engineering 105 (1983) 200–206. [10] J. Tobiś, Influence of bed geometry on its frictional resistance under turbulent flow conditions, Chemical Engineering Science 55 (2000) 5359–5366. [11] M. Mayerhofer, J. Govaerts, N. Parmentier, H. Jeanmart, L. Helsen, Experimental investigation of pressure drop in packed beds of irregular shaped wood particles, Powder Technology 205 (2011) 30–35. [12] R.E. Hicks, Pressure drop in packed beds of spheres, Industrial and Engineering Chemistry Fundamentals 9 (1970) 500–502. [13] P.C. Carman, Fluid flow through granular beds, Transactions of the London Institute of Chemical Engineers 15 (1937) 150–166. [14] H. Brauer, Grundlagen der Einphasen –und Mehrphasenströmungen, Sauerländer AG, Aarau, 1971. [15] Geschäftsstelle des Kerntechnischen Ausschusses, KTA 3102.3: Reactor Core Design of High Temperature Gas-cooled Reactors. Part 3: Loss of Pressure through Friction in Pebble Bed Cores, 1981. [16] D.P. Jones, H. Krier, Gas flow resistance measurements through packed beds at high Reynolds numbers, ASME Journal of Fluids Engineering 105 (1983) 168–173. [17] I.E. Idelchik, Flow Resistance: A Design Guide for Engineers, (English translation from Russian 2nd edition) Hemisphere Publishing Corporation, New York, 1989. [18] A. Montillet, E. Akkari, J. Comiti, About a correlating equation for predicting pressure drops through a packed bed of spheres in a large range of Reynolds numbers, Chemical Engineering and Processing 46 (2007) 329–333.

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