A revisit of pressure drop-flow rate correlations for packed beds of spheres PDF

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Powder Technology 283 (2015) 488–504 Contents lists available at ScienceDirect Powder Technology journal homepage: www.elsevier.com/locate/powtec A revisit of pressure drop-flow rate correlations for packed beds of spheres Esra Erdim a,1, Ömer Akgiray b,⁎, İbrahim Demir a a Istanbul Technical Univers...

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Powder Technology 283 (2015) 488–504

Contents lists available at ScienceDirect

Powder Technology journal homepage: www.elsevier.com/locate/powtec

A revisit of pressure drop-flow rate correlations for packed beds of spheres Esra Erdim a,1, Ömer Akgiray b,⁎, İbrahim Demir a a b

Istanbul Technical University, Faculty of Civil Engineering, Department of Environmental Engineering, Istanbul, Turkey Marmara University, Faculty of Engineering, Department of Environmental Engineering, Istanbul, Turkey

a r t i c l e

i n f o

Article history: Received 13 January 2015 Received in revised form 1 June 2015 Accepted 4 June 2015 Available online 14 June 2015 Keywords: Head loss Packed bed Porous media Pressure drop

a b s t r a c t A large number of correlations can be found in the literature for the calculation of pressure drop caused by fluid flow through packed beds. New correlations continue to be proposed and there appears to be no general agreement regarding which correlation is the most accurate. In this work, experiments have been carried out with water using glass spheres of nine different sizes, varying from 1.18 mm to 9.99 mm. For each size, experiments were repeated with at least two different porosities. A total of 38 correlations from the literature have been evaluated and a uniform notation was established to facilitate the comparison of the correlations. While the Ergun equation remains the most widely-used correlation, the data collected in this work shows that it should not be ) is proposed to represent the data collectused above Rem ≈ 500. A simple new equation (fV = 160 + 2.81Re0.904 m ed in this work. The new equation yields the smallest mean error among all the correlations considered here. While a substantial amount of the data collected in this work involved D/dP ratios less than 10, the correlations that fit the current data best do not have any wall effect correction terms. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Flow of fluids through packed beds of solid particles occurs in a variety of important applications in several engineering fields. A quantity of primary interest is the pressure drop (or the head loss) generated as a result of fluid flow through the porous medium. The equation published by Ergun [1] about sixty years ago remains the most popular and the most widely-quoted pressure drop–flow rate relation for fluid flow through packed beds. For incompressible flow through a bed of spherical particles of identical size, the Ergun equation can be written as follows: −ΔP ð1−ε Þ2 V ð1−ε Þ V 2 þ 1:75ρ ¼ 150μ 2 3 L ε ε 3 dp dp

ð1Þ

Abbreviations: −ΔP, piezometric pressure drop −(P2 − P1); A, a characteristic area for the bed; Ax, the cross-sectional area of the empty bed; D, column diameter; dP, particle diameter; ε, porosity of the bed; εb, bulk zone porosity; f, friction factor; fk, friction factor ( fk = 3f);
 Þ2 Re fV, friction factor (f v ¼ f k 1−ε ); fP, modified particle friction factor f p ¼ f v ð1−ε ; Fk, kinetic ε 3 Re

force exerted by the fluid on the solids; h, head loss in the bed; K, a characteristic kinetic energy per unit volume; L, depth of the bed; V, velocity based on the empty cross-section of the bed; Vb, bulk zone velocity; ρ, fluid density; μ, fluid viscosity; Re, Reynolds number ( Re Re ); Re1, modified Re number (6ð1−ε Rem, modified Re number (1−ε Þ); T, tortuosity.

ρdp V μ );

⁎ Corresponding author. Tel.: +90 216 348 02 92/261; fax: +90 216 348 02 93. E-mail address: [email protected] (Ö. Akgiray). 1 Present address: Marmara University, Faculty of Engineering, Department of Environmental Engineering, Istanbul, Turkey.

http://dx.doi.org/10.1016/j.powtec.2015.06.017 0032-5910/© 2015 Elsevier B.V. All rights reserved.

where −ΔP = piezometric pressure drop in the bed, L = depth of the bed, V = velocity based on the empty cross-section of the bed, ε = porosity of the bed, dP = particle diameter, and ρ = fluid density, μ = fluid viscosity. It may be noted that −ΔP = ρgh, where h = head loss in the bed. In addition to a number of historically significant equations that predate it, many additional equations have been proposed since the publication of the Ergun equation and new correlations continue to be developed and published. Each new correlation is claimed to be more accurate than, or in some other way superior to the previously proposed correlations. However, there aren't many independent comparisons of these correlations. The few studies evaluating and comparing various correlations ignore most of the correlations in the literature and focus only on a few correlations. The purposes of this work can be outlined as follows: (1) Review all the widely-used and/or well-known correlations that exist in the relevant literature. (2) Present all the mentioned correlations using a uniform notation so that their application and comparison with each other are facilitated. (3) Test and compare the accuracy and the applicability of the correlations. Some recently proposed correlations have also been included in this evaluation. In what follows, the definitions of various quantities of interest and the notation used in this work are first described. Correlations considered in this work are then rewritten using the described notation instead of the notation used by their original authors. Expressing correlations from different sources using a uniform notation will facilitate the comparison of the mentioned correlations and the observation of differences as well as similarities between the different equations.

E. Erdim et al. / Powder Technology 283 (2015) 488–504

489

Following the presentation by Bird, Stewart and Lightfoot [2], a friction factor f for flow through a packed bed of particles can be defined as follows:

This quantity is also referred to as the “dimensionless pressure drop” (Eisfeld and Schnitzlein [6]). The Ergun equation can be expressed as follows in terms of these friction factors:

F k ¼ f AK

f v ¼ 150 þ 1:75

ð2Þ

where Fk = kinetic force (i.e. force due to motion of the fluid) exerted by the fluid on the solids, A = a characteristic area, K = a characteristic kinetic energy per unit volume, and f = friction factor. It may be noted that f is dimensionless. Choosing A to be the total external surface area of the particles that make up the packed bed and noting that the specific surface (particle surface area per unit particle volume) is given by 6/dp for spheres, one can write: 6 ð1−ε ÞAx L: dp

fp ¼

Re ð1−ε Þ

150ð1−ε Þ2 ð1−ε Þ þ 1:75 : Reε 3 ε3

ð12Þ ð13Þ

The Reynolds number Re appearing in these equations is defined by (Eq. (7)). A number of “modified Reynolds number” definitions are also used, the two most common being the following:

ð3Þ

Rem ¼

dp ρV Re ¼ μ ð1−ε Þ ð1−ε Þ

ð14Þ

Here Ax represents the cross-sectional area of the empty bed. Using the Dupuit assumption, i.e. interstitial average velocity in the direction of the column axis is given by Vε = V/ε, the characteristic kinetic energy is written as

Re1 ¼

dp ρV Re ¼ : 6μ ð1−ε Þ 6ð1−ε Þ

ð15Þ

A¼

K¼

  1 V 2 : ρ 2 ε

ð4Þ

A force balance on the fluid yields: F k ¼ ð−ΔP ÞAx ε:

ð5Þ

Combining (Eqs. (2–5)) then gives: f ¼

1 ð−ΔP Þ dp ε3 : 2 3 L ρV ð1−ε Þ

ð6Þ

This expression can be taken as the definition of the friction factor f. The numerical factor 1/3 is not widely used, however, and the commonly used friction factor definitions are presented below. Ergun [1] defined the Reynolds number Re and a friction factor fk as follows: Re ¼ fk ¼

dp ρV μ

ð7Þ

−ΔPdp LρV

2

ε3 : ð1−ε Þ

ð8Þ

It may be noted that fk = 3f, where f is given by (Eq. (6)). It may also be useful here to note that the friction factor employed by Blake [3] and Richardson, Harker and Backhurst [4] is equal to fk/6 = f/2. With the above definitions of fk and Re, the Ergun equation (Eq. (1)) can be expressed in the following simpler form: fk ¼

150ð1−ε Þ þ 1:75: Re

ð9Þ

Ergun [1] defined another friction factor, denoted by fV, which can be expressed as 2

fv ¼

ε3 Re ¼ fk : μVL ð1−ε Þ2 1−ε

−ΔPdp

ð10Þ

A “modified particle friction factor” fP is sometimes employed (Montillet, Akkari and Comiti [5]): fp ¼

−ΔPdp 2

ρV L

¼ fk

ð1−ε Þ ð1−3Þ2 ¼ fv 3 : 3 ε Re ε

ð11Þ

The latter definition can be traced back to Blake [3], whereas the definition in (Eq. (14)) differs from this definition by only the numerical factor 6. The notation Re1 used in (Eq. (15)) is adopted from Richardson, Harker and Backhurst [4] who used it for packed beds and from Dharmarajah and Cleasby [7], Akgiray and Soyer [8] and Soyer and Akgiray [9] who employed it to correlate liquid–solid fluidization and sedimentation data. The pressure drop–flow rate correlations considered and tested in this work are listed in Table 1 in terms of fP and Re defined as above. It may be noted that some of these equations can be written in more compact forms using alternative definitions of the Reynolds number and/or the friction factor. The Carman equation, for example, can be expressed in terms of fk and Re1 or Rem in forms simpler than the rather bulky expression (Eq. (17)) in Table 1: fk ¼

30 180 þ 2:4Re−0:1 ¼ þ 2:87Re−0:1 : 1 m Re1 Rem

ð16Þ

The equations in Table 1, however, are all expressed in terms of fP and Re so that they can be compared with each other directly and their differences as well as similarities can be observed more easily. Carman [10], in a landmark paper, presented a review of earlier studies including those of Darcy [11], Blake [3], Kozeny [12], Burke and Plummer [13], and Fair and Hatch [14]. Considering the data collected by earlier researchers, Carman [10] presented an equation of the Forchheimer type (Eq. (17) in Table 1) and stated that this equation fitted the data best. Rose [15] carried out a comprehensive analysis of data, including his own and those of other workers between 1922 and 1945. He proposed an empirical equation containing three terms (Eq. 18). It may be noted that, in this particular paper, Rose [15] did not discuss the effect of porosity on pressure drop nor did he explicitly mention the fact that his equation is applicable only at ε = 0.4. Furthermore, the Rose equation has been occasionally quoted (e.g. Montillet, Akkari and Comiti [5]; Özahi, Gundoğdu and Çarpinlioğlu [16]) without the porosity correction function h(ε). This porosity correction function was presented in graphical form in another paper by Rose [17] and then given again in graphical form by Rose and Rizk [18] by extending it to higher porosities. Since the h(ε) curve given by Rose and Rizk [18] is very inconvenient to use when a large number of calculations are needed (as is the case in this work), the following curve-fitting polynomial has been developed here to be used (within the range 0.32 b ε b 0.90) in conjunction with the Rose equation: hðε Þ ¼ 54:3218ε 4 −156:3496ε 3 þ 169:7978ε 2 −83:0717ε þ 15:6676: ð19Þ

It may be prudent here to emphasize that this rather bulky equation is intended to be neither a new correlation nor a suitable expression for

490

E. Erdim et al. / Powder Technology 283 (2015) 488–504

Table 1 The pressure drop-flow rate relations considered in this work. Author(s) Carman [10] Rose [15] Morcom [19] Rose and Rizk [18]

Relation
 Re 0:9  ð1−εÞ2 f P ¼ 180 þ 2:871 1−ε ε3 Re
 ffiffiffiffi p60 þ þ 12 h ð ε Þ f P ¼ 1000 Re Re

h(ε) given in graphical form, h(0.4) = 1 3   þ 13:73 0:405 f P ¼ 784:8
Re ε 1000 125 ffiffiffiffi þ 14 hðε Þ f P ¼ Re þ pRe

Equation

Range of applicability

17

0.01 b Re1 b 10,000

18

0.01 b Re b 10,000

20

Re b 750

21

0.01 b Re b 10,000

23

Re b 10,000

26

0.2 b Re1 b 700

27

2 b Rem b 20,000

28

N/A

h(ε) given in graphical form, h(0.4) = 1 Leva [20] Ergun [1] Brauer [23] Fahien and Schriver [25]

Wentz and Thodos [26] Handley and Heggs [27] Mehta and Hawley [28] Hicks [29] Tallmadge [30] Kuo and Nydegger [31] Macdonald, El-Sayed, Mow and Dullien [32] KTA [33] Jones and Krier [34] Foscolo, Gibilaro and Waldram [35] Meyer and Smith [36] Paterson, Burns, Griffiths, Kesterton and Paveley [37]

Watanabe [38]; Kurten, Raasch and Rumpf [39]; Steinour [40] Fand and Thinakaran [42]

ð3−nÞ

f P ¼ 2 f m ð1−εεÞ3 fm and n given in graphical form as functions of Re   Re  ð1−εÞ2 f P ¼ 150 þ 1:75 1−ε ε 3 Re
 Re 0:9  ð1−εÞ2 f P ¼ 160 þ 3:1 1−ε ε3 Re

 ð1−εÞ f 1L f 1T þ ð 1−q Þ f þ f p ¼ q Re 2 ε3
m 2  Rem ε ð1−ε Þ 29 q ¼ exp − 12:6 Rem f 1L ¼ 1360:38 f 1T ¼ ð1−εÞ ð1−εÞ1:45 ε 2
 ð1−εÞ 0:351 fP ¼ ε3 ðRe=ð1−εÞÞ0:05 −1:2   Re  ð1−εÞ2 f P ¼ 368 þ 1:24 1−ε ε 3 Re   Re  ð1−εÞ2 M ε3 Re f P ¼ 150 M þ 1:75 1−ε 4dP M ¼ 1 þ 6ð1−ε ÞD 1:2

1−ε Þ f P ¼ 6:8 ðRe 0:2 ε 3
 Re 5=6  ð1−εÞ2 f P ¼ 150 þ 4:2 1−ε ε 3 Re
 Re 0:87  ð1−εÞ2 f P ¼ 276:23 þ 5:05 1−ε ε3 Re   Re  ð1−εÞ2 f P ¼ 180 þ 1:8 1−ε ε3 Re
 Re 0:9  ð1−εÞ2 f P ¼ 160 þ 3 1−ε ε 3 Re
 Re 0:87  ð1−εÞ2 f P ¼ 150 þ 3:89 1−ε ε3 Re

Þ f p ¼ ð17:3 þ 0:336ReÞ ðε1−ε 4:8 Re   Re  ð1−εÞ2 f P ¼ 90 þ 0:462 1−ε ε4:1 Re   Re  ð1−εÞ2 f P ¼ 150 A þ 1:75 B 1−ε 3
ε Re
 
 2 dp A ¼ 1 þ 1:22 D B ¼ exp 1:66 1−dp =D −1


 ð1−εÞ2 p6ffiffiffiffi f P ¼ 6:25 21 Re þ Re þ 0:28 ε3   Re  ð1−εÞ2 M ε3 Re f P ¼ AW M þ BW 1−ε

4dP M ¼ 1 þ 6ð1−ε ÞD

−f

Foumeny, Benyahia, Castro, Moallemi and Roshani [43]

Lee and Ogawa [44] Liu, Afacan and Masliyah [45]

Hayes, Afacan and Boulanger [46]

B¼

Avontuur and Geldart [47]

29

2550 b Rem b 64,900

30

200 b Re b 13000

31

0.18 b Rem b 9.55 7 b D/dp

32

300 b Rem b 60,000

33

0.1 b Rem b 100,000

34

460 b Re b 14600

35

Rem b 10,000

36

1 b Rem b 100,000

37

1000 b Re b 100,000

38

N/A

39

Re1 b 1000

40

25 b Re b 900 3.5 b D/dp b 22

41

0.1 b Re b 4000

42


 D

dP where Y = A or B Y W ¼ Y W∞ −ae f(D/dP) = p(D/dP)3 + q(D/dP)2 + r(D/dP) Aw∞ = 5.34, a = 0.6545, p = 0, q = 0, r = 0.09034 and Bw = 0 Aw∞ = 172.9, a = 82.18, p = 0.0001125, q = −0.003931, r = 0.1314 Bw∞ = 1.871, a = 1.636, p = 0.0004908, q = −0.01665, r = 0.2925 Aw∞ = 213.7, a = 129.7, p = 0.00003852, q = −0.003376, r = 0.1510 Bw∞ = 1.569, a = 1.35, p = 0.0003688, q = −0.01465, r = 0.2646   Re  ð1−εÞ2 f P ¼ 130 þ B 1−ε ε3 Re

D=dP 0:335ðD=dP Þþ2:28

  ð1−εÞ2 1:56 f P ¼ 6:25 29:32 Re þ Ren þ 0:1 ε3 n = 0.352 + 0.1ε + 0.275ε2

 0:69 B Re3ε ð1−εÞ2 f p ¼ 85:2 A þ ε11=3 Re 162 þReε2   Re  ð1−εÞ2 f P ¼ 150 A þ 1:75 B 1−ε ε3 Re 1=2 1=2 1þð1−ε Þ Reε ¼ Re ð1−εÞε1=6
2
 dp π2 d 0:5d B ¼ 1− 24Dp 1− D p A ¼ 1 þ π 6ð1−ε ÞD i0:5  2  h   T 1−ε Bð3T−1Þ 1−ε f p ¼ T1 A þ T ð1−ε þ C 12 3T−1 Þð1−T Þ Re ε ε Re 2

O'Neill and Benyahia [48]

f2 ¼

1:87ε0:75 ð1−ε Þ0:26

3

ε = (3T − 1) /4T Theoretical: A = 456, B = 17.8, C = 2.6 Recommended: A = 850, B = 11.6, C = 1.3   Re  ð1−εÞ2 f P ¼ 141 þ 1:52 1−ε ε 3 Re   Re  ð1−εÞ2 f P ¼ A þ B 1−ε ε 3 Re

43

Re b 2.3 2.3 b Re b 80 2.3 b Re b 80 80 b Re b 408 80 b Re b 408 5 b Rem b 8500

44

1 b Re b 400,000

45 46

Rem b 1600 Rem b 1600

47 48

4 b Re b 450

50

Rem b 10,000

51

5 b D/dP b 25

52

0.01 b Re b 17,635

2

Eisfeld and Schnitzlein [6]

Yu, Zhang, Fan, Zhou and Zhao [50] Montillet [51]

A ¼ 521:26−22581:24=ðD=dP Þ B ¼ 1:12 þ 4:2=ðD=dP Þ
 Re  ð1−εÞ2 M ε3 Re f P ¼ 154M þ B1W 1−ε h i2 2 4dP M ¼ 1 þ 6ð1−ε BW ¼ 1:15ðdP =DÞ þ 0:87 ÞD   Re  ð1−εÞ2 f P ¼ 203 þ 1:95 1−ε ε 3 Re

53

750 b Re b 2500

54

30 b Re b 1500

491

E. Erdim et al. / Powder Technology 283 (2015) 488–504 Table 1 (continued) Author(s)

Relation

Montillet, Akkari and Comiti [5] Özahi, Gundoğdu and Çarpinlioğlu [16] Çarpinlioğlu and Özahi [52] Harrison, Brunner and Hecker [56]

f P ¼ 0:066


0:17
D dp


0:20


−ΔPdp

ε3

2

2ρV L ð1−ε Þð3−nÞ

:

ð22Þ

Here n is a variable between 1.0 and 2.0. The value of n and the friction factor fm are given in graphical form as functions of Re. The relation between fP and fm is shown in Table 1 (see Eq. (23)). Leva's correlation has been included in all editions of Perry's Chemical Engineers' Handbook published during the last fifty years, including the most recent one (Green and Perry [22]). Due to the fact that it has a graphical form, however, this correlation is not convenient to use in computer calculations. To facilitate its use, the following curve-fitting expressions were developed in this work using the curves given by Leva [21]: n¼

X6

B Rek k¼0 k

log f m ¼

X5

C Rek k¼0 k

ð24Þ ð25Þ

where Bk (k = 0, …, 6) = 7.60657, − 19.2986, 21.02695, − 10.96663, 3.02928, − 0.42867, and 0.02453 and Ck (k = 0, …, 5) = 1.982535, − 1.0218594, 0.0295464, 0.0269893, 0.0024996, and − 0.0008754, respectively. Eq. (24) is applicable if Re ≥ 11.5; otherwise n = 1. These admittedly bulky expressions were developed here only for computer calculations and represent Leva's curves very accurately. They are very useful when a large number of calculations (such as those in this work) need to be carried out.

1000 Re

þ 12 þ p60ffiffiffiffi Re

Equation

Range of applicability

55

10 b Re b 2500

56

708 b Rem b 7772

57

675 b Rem b 7772

60

0.32 b Re b 7700

ð1−εÞ ε3

 Þ f P ¼ 0:061 þ 12 ð1−ε þ p60ffiffiffiffi ε3 Re   Re  ð1−εÞ2 f P ¼ 160 þ 1:61 1−ε ε 3 Re h i0:4733 Þ L f P ¼ 139:52 ðε1−ε 7 Re d P
 Re 5=6  ð1−εÞ2 f P ¼ 119:8 A þ 4:63 B 1−ε ε 3 Re
2
 dp π 2 dp 0:5d A ¼ 1 þ π 6ð1−εÞD B ¼ 1− 24D 1− D p D dp

manual calculations. It was developed simply to facilitate the computer calculations required for the evaluation of the Rose equation. Morcom [19] was one of the earlier workers who suggested a twoterm equation to express the relationship between pressure drop and flow rate through granular materials (Eq. (20)). The effect of porosity is taken into account by a correction factor of the form (εn/ε)3, where ε is the actual porosity of the bed and εn is the porosity under “normal packing conditions.” Morcom [19] reported the value εn = 0.405 for spherical particles. Morcom [19] also concluded that container diameter did not have any noticeable effect “over and above its effect on void space” until the container to particle diameter ratio D/dP was as low as 5. Rose and Rizk [18] proposed a modified version (see Eq. (21)) of the original Rose equation...