A Modification on Ergun's Correlation for Use in ... - Science Direct

Advanced Powder Technology 19 (2008) 369–381 www.brill.nl/apt

Original paper A Modification on Ergun’s Correlation for Use in Cylindrical Packed Beds With Non-spherical Particles Emrah Ozahi, Mehmet Yasar Gundogdu ∗ and Melda Ö. Carpinlioglu Department of Mechanical Engineering, Faculty of Engineering, University of Gaziantep, 27310 Gaziantep, Turkey Received 10 July 2007; accepted 19 September 2007

Abstract An experimental study was conducted to determine the pressure drop characteristics of cylindrical packed beds through which turbulent pipe flow was passing. This study was planned to clarify the applicability of the well-known Ergun correlation proposed for beds composed with spherical particles on beds with nonspherical particles. The experimental packed beds were constructed by using two different irregular-shaped and one spherical-shaped packing materials for understanding the effects of particle shape or sphericity, particle size, bed porosity, and bed length to diameter ratio on the pressure drop. The beds were constructed by using zeolite, chickpea and glass bead materials to cover the particle sphericity range of 0.55 1.00. Systematic experiments and the data analysis procedure showed that the well-known Ergun correlation can be applied to all of the experimental beds composed with non-spherical and/or spherical particles with a maximum deviation of ±20%. This deviation can, however, be decreased to a negligible level of ±4% if some simple modifications, including the use of particle Reynolds number, Rep , a new form of particle friction factor, fp∗ , and some adopted empirical constants in the well-known Ergun correlation, are used. © Koninklijke Brill NV, Leiden and Society of Powder Technology, Japan, 2008 Keywords Packed bed, pressure drop, Ergun’s correlation, sphericity, non-spherical, bed porosity

Nomenclature a, b Ap Asp Dp D De *

correlation constants in the literature surface area of a single non-spherical particle (m2 ) surface area of the equivalent-volume sphere (m2 ) particle diameter (m) pipe inner diameter (m) equivalent particle diameter, 6Vp /Asp (m)

To whom correspondence should be addressed. E-mail: [email protected]

© Koninklijke Brill NV, Leiden and Society of Powder Technology, Japan, 2008

DOI:10.1163/156855208X314985

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fp fp∗ k1 , k2 L P1 P2 r Re Rep u(r) U Vp Xh

E. Ozahi et al. / Advanced Powder Technology 19 (2008) 369–381

particle friction factor modified particle friction factor, (P∗ /2)(Dp /L)(ε3 /(1 − ε)) empirical constants of the original Ergun’s correlation bed length (m) pressure at point 1 which is 100 mm below the lower bed limiter (Pa) pressure at point 2 which is 100 mm above the upper bed limiter (Pa) pipe radius, D/2 (m) flow reference Reynolds number, U D/υ particle Reynolds number, U D p /υ(1 − ε) local velocity (m/s) superficial mean fluid velocity (m/s) volume of a single non-spherical particle (m3 ) horizontal distance between the settling tank exit and the pitot tube location (m)

Greek α β |P |/H PErgun Pexp P ∗ ε μ ρ ρm υ

empirical coefficient of Montillet open area ratio pressure gradient (Pa/m) calculated pressure drop by Ergun’s correlation (Pa) measured pressure drop of packed bed (Pa) pressure coefficient bed porosity fluid dynamic viscosity (kg/ms) fluid density (kg/m3 ) packing material density (kg/m3 ) kinematic viscosity (m2 /s) particle sphericity

1. Introduction The determination of pressure drop through a packed bed as a function of fluid flow rate, geometrical constraints of the bed and physical properties of bed material is very critical for selection and use of an optimum fluid driving device such as a pump in a hydraulic system or a fan in a pneumatic system. The well-known equation used for that purpose was proposed by Ergun [1] as follows: PErgun (1 − ε)2 μU 1 − ε ρU 2 = 150 · + 1.75 · , L Dp 2 ε3 Dp2 ε3

(1)

where PErgun is the calculated pressure drop through the packed bed by Ergun’s correlation, L is the bed length or height, ε is the bed porosity, U is the superficial velocity at the exit of bed, ρ is the fluid density, μ is the fluid dynamic viscosity, is the particle sphericity and Dp is the particle diameter. If this equation is used


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for an actual bed with spherical particles, the value of can then be received as unity and the actual diameter of the particles used for Dp . Use of this equation for beds with non-spherical particles, however, requires the determination of the equivalent particle diameter which was defined in terms of the sphericity [2]: De =

6Vp 6Vp = , Ap Asp

(2)

where Vp is the volume of a single non-spherical particle, Ap is its surface area and Asp is the surface area of the equivalent-volume sphere. The sphericity indicates the ratio of the surface area of the equivalent-volume sphere, Asp , to that of the particle, Ap [2]. The equivalent particle diameter defined in (2) becomes also equal to the well-known mean volume-surface diameter frequently used in the literature. In the literature, many further studies to check the values of empirical constants 150 (= k1 ) and 1.75 (= k2 ) in (1) have been performed. A variety of empirical values have been found because of using different packing materials such as regularshaped spherical particles or irregular-shaped non-spherical particles. The first term on the right-hand side of (1) represents the viscous flow and so the change in pressure is proportional to (1 − ε)2 /ε3 together with an empirical proportionality constant of 150 at low flow rates [3]. The change in pressure at turbulent flow, resulting from kinematic energy loss, was proportional to (1 − ε)/ε 3 [4]. A constant of 1.75 was found to be relevant at the turbulent flow. In order to check the functional dependency upon the bed porosity, Ergun also varied the packing density for some materials to verify the term (1 − ε)2 /ε3 for the viscous loss part and the term (1 − ε)/ε 3 for the kinetic energy part, and found that a small change in ε had a large effect on the pressure drop [1]. Universal values of the Ergun constants k1 and k2 have been a subject of considerable speculation since 1952. The values of empirical constants in Ergun’s correlation have been, respectively, proposed as 200 and 1.75 [5], and as 180 and 1.8–4.0 [6] as cited by Niven [7]. The reason for variation in the constants was determined as the variations in particle shape for non-spherical particles. A generalized Blake–Kozeny equation for multi-sized spherical particles was also obtained [8]. It seems now that most researchers have satisfied themselves with the fact the values of the Ergun constants ought to be determined empirically for each bed. This philosophy arises from the belief that the values are not only dependent on the particle geometry, but in addition they can vary from macroscopic bed to bed (made of the same particles) due to different structures of the packing within the bed after repacking. It is hard to expect that the simple assumptions of the model can be describe the flow through packed beds made up of irregular-shaped particles with universal constants. However, if the problem is broken up to smaller parts that involve groups of particles of similar geometry some general principles regarding the Ergun constants may be found. The original Ergun correlation with values of viscous and inertial constants of 150 and 1.75, respectively, fits the pressure drop data for beds of spheres extremely well. This is in fact in agreement with a number of works

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Table 1. Alternative equations proposed for pressure drop calculation through packed beds Authors

Equation

Rose [13]

−0.5 + 12 fp = 1000Re−1 p + 60Rep

Rose and Rizk [14] Hicks [15]

Tallmadge

Rep validity range

−0.5 + 14 fp = 1000Re−1 p + 125Rep (1 − ε)1.2 −0.2 Rep fp = 6.8 ε3 150 (1 − ε)2 4.2(1 − ε)1.166 −1/6 fp = + Rep Rep ε3 ε3

rather large (not precisely given by authors) rather large (not precisely given by authors) 500–60 000

0.1–100 000

[16]

1 12.5 2 29.32Re−1 + 1.56Re−n + 0.1 1–100 000 (1 − ε) p p 2 ε3 2 Ogawa [17] with n = 0.352 + 0.1ε + 0.275ε 25 2 21Re−1 + 6Re−0.5 + 0.28 Kürten fp = (1 − ε) 0.1–4000 p p 4ε3 et al. [18] −0.5 + 12 Montillet fp = α 1000Re−1 10–2500 p + 60Rep D 0.2 1−ε et al. [19] with α = 0.061x Dp ε3

Sug Lee and

fp =

found in the open literature [6, 9–11], which claim that the original Ergun constants are able to predict the pressure drop in beds composed of spherical particles to within 10%. According to a recent study [12], Ergun’s correlation is only able to accurately predict the pressure drop for flow over spherical particles, whereas it systematically under-predicts the pressure drop for flow over non-spherical particles. In the literature [13–19] there are also some other approaches regarding pressure drop calculations through packed beds. They developed alternative equations for predicting pressure drop through packed beds. These equations were reported and listed in a recent study [19] as given in Table 1. These alternative equations were especially proposed to simplify the more complex form of Ergun’s correlation and to ensure practical use by introducing the particle friction factor, fp , as functions of Rep , ε, and D/Dp . For that purpose, fp is directly related to the pressure drop through the bed, P , as follows: fp =

|P |Dp . HρU 2

(3)

In view of the literature, the present experimental study was mainly directed to investigate the applicability of both the well-known form of Ergun’s correla-


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tion and the form of alternative correlations cited in the literature [13–19] (see Table 1) for the pressure drop determination of packed beds with non-spherical particles. The pressure drops through different types of packed beds, Pexp , and the cross-sectional velocity distribution at the bed exit, u(r), were measured systematically through 64 separate test cases. The Re and Rep numbers were changed in the ranges of 15 000 Re 33 000 and 708.2 Rep 7772.73, respectively, and the ranges of packed bed parameters ε, D/Dp , L/D, Dp /L and covers 0.36 ε 0.56, 5.72 D/Dp 17.16, 0.24 L/D 1.46, 0.04 Dp /L 0.72 and 0.55 1.00. 2. Experimental Set-up and Procedure An open-circuit pneumatic test set-up used in this study is shown in Fig. 1. It consists of horizontal and vertical rigid PVC pipe lines of inner diameter, D, of 103 mm. Air flow was generated by means of a blower-type fan coupled with an AC motor speed control unit (Siemens Micromaster MM 110). A pitot tube was located at the reference station 57.3 D downstream of the settling tank on the horizontal pipe line in order to set the Reynolds number, Re, of the fully developed turbulent pipe flow. The packed bed test section on the vertical pipe line was located at 9.7 D downstream of the 90◦ elbow in order to measure pressure drop through packed beds and the mean velocity at the exit. Four different packed bed lengths (L = 25, 40, 60 and 150 mm) were systematically changed for each of four different packing materials and four different Reynolds numbers. Bed limiters of the packed bed were made up of synthetic cloth as a screen with an open area ratio, β, of 0.73. The average porosities of the constructed packed beds were determined according to the mass and volume measurements of packing materials used in the test bed [20]. In order to determine the average porosity of any bed, it was filled

Figure 1. Sketch of the experimental test set-up.

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E. Ozahi et al. / Advanced Powder Technology 19 (2008) 369–381 Table 2. Physical properties of packing materials Packing materials

ρm (kg/m3 )

Dp (mm)

D/Dp

Zeolite 1 Zeolite 2 Chickpeas Glass beads

640 640 970 2675

6 18 9 16

0.82 0.76 0.55 1.00

17.16 5.72 11.44 6.43

with water up to the upper surface level of the packing material. The water was then drained off and its volume was measured. By taking the ratio of this volume to the total volume of bed, the average porosity was calculated for each of the packed beds. The physical properties of packing materials are given in Table 2 where the particle diameter and the mass density of packing materials were determined by means of the mean volume of a single non-spherical particle measured by using the well-known gas-flow method proposed by Abdullah and Geldart [21]. The static pressure drop, Pexp (Pexp = P1 − P2 ), through the packed bed was measured between the two static pressure tapings located 1 D upstream and 1 D downstream of the bed by using an inclined-leg alcohol manometer. The crosssectional velocity distribution was measured at 1 D downstream of the bed by using a pitot tube traverse constructed according to the British Standard BS 1042 [22]. A pitot tube with an outer diameter of 2.2 mm and an inner diameter of 1.1 mm was used. Dynamic pressures as low as 0.07 Pa could be measured by using this pitot tube and a coupled inclined-leg alcohol manometer, resulting in a ±0.17 m/s sensitivity in local velocity measurements. The experimental data based on alcohol column measurements were then corrected with respect to the standard atmospheric pressure and temperature of 100 kPa and 25◦ C. The present experimental study was systematically organised in 64 separate test runs by changing the bed material type, packing particle diameter (Dp ), bed length (L) and flow Reynolds number (Re) as can be seen from Table 3 for clarifying the effects of particle sphericity (), bed porosity (ε) and particle Reynolds number (Rep ) on pressure loss through the packed beds. For this purpose three different types of bed material (chickpeas, glass beads and two different sizes of zeolite) were used as the packing material. The cylindrical test beds were constructed with four different characteristic lengths (L) of 25, 40, 60 and 150 mm, and with a constant diameter D of 103 mm providing the L/D range of 0.24 L/D 1.46. The Reynolds number of turbulent air flow passing through the bed was also changed to four different values of 15 000, 25 000, 30 000 and 33 000. As given in Table 3, the code system MXXYY was used in order to classify packed beds. M denotes packing material type, XX denotes the equivalent diameter of packing particles (in mm) and YY denotes the bed length (in mm). For example, C0960 denotes a 60-mm length test bed constructed from the packing of 9-mm diameter chickpea particles.


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Table 3. Test run organization Packing material

Packed bed code

L (mm)

ε

Re

Zeolite 1

Z0625 Z0640 Z0660 Z06150

25 40 60 150

0.41 0.37 0.36 0.36

15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000

857, 1600, 1643, 1800 787, 1160, 1200, 1247 721, 997, 1128, 1180 708, 879, 957, 1059

Zeolite 2

Z1825 Z1840 Z1860 Z18150

25 40 60 150

0.56 0.48 0.45 0.40

15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000

3955, 6818, 6982, 7773 3000, 5631, 5815, 6046 2793, 4887, 5193, 5542 2400, 3960, 4360, 4600

Chickpeas

C0925 C0940 C0960 C09150

25 40 60 150

0.44 0.40 0.40 0.38

15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000

1307, 2464, 2593, 2721 1200, 1850, 2180, 2300 1200, 1850, 2150, 2280 1113, 1597, 1761, 1839

Glass beads

G1625 G1640 G1660 G16150

25 40 60 150

0.44 0.40 0.39 0.39

15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000 15 000, 25 000, 30 000, 33 000

2260, 4411, 4520, 4791 2066, 3979, 4199, 4250 2017, 3583, 3933, 4083 1967, 3000, 3300, 3500

Rep

Figure 2. Cross-sectional velocity distribution at Xh = 5.9 m.

The experimental procedure followed throughout the test runs may be summarized as follows. The flow Reynolds number firstly settled at the reference station by using a pitot tube fixed at the dimensionless radial position, r/R = 0.758 to obtain a fully developed turbulent flow. In the selected range of Reynolds number, 15 000 Re 33 000, the cross-sectional velocity distribution at the reference station on the horizontal pipe line were found to be symmetrical and in conformity with the Prandtl 1/7th power law as shown in Fig. 2.

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Figure 3. Cross-sectional velocity distribution at the exit of Z1825 for Re = 15 000.

After settling of Re, the pressure drop through the packed bed and the crosssectional velocity distribution at the exit were measured. The velocity distribution was determined 1 D downstream of the upper bed limiter by measuring the local velocities at 21 different radial positions through the pipe cross-section. Figure 3 shows the cross-sectional velocity distribution for the test bed Z1825 for Re = 15 000 as a sample. The cross-sectional mean velocity at the exit was then calculated by integrating the velocity profile throughout the pipe cross-section and dividing with the cross-sectional area of the pipe.

3. Results and Discussion 3.1. Proposed Modification of Ergun’s Correlation In view of the literature, it is clear that the form of the well-known Ergun correlation (1) is very complex and difficult to use for industrial applications. The equation includes a large number of parameters such as Pexp , L, μ, U, Dp and ρ. To replace these dimensional parameters with the non-dimensional ones and so simplify this equation, some non-dimensional parameters should be defined and used. The first term on the right-hand side of the (1) includes the parameters ε, U, Dp and . These parameters can be represented in the non-dimensional form of the particle Reynolds number, Rep (Rep = U Dp /(1 − ε)υ). The term on the left-hand side of (1) can be converted to a new modified particle friction factor, fp∗ . For that purpose, Pexp is first converted to the pressure coefficient P ∗ by dividing both sides of (1) with dynamic pressure (1/2ρU 2 ) and then fp∗ can be introduced as: fp∗

P ∗ = 2

Dp L

ε3 2 , 1−ε

(4)

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where P ∗ = Pexp /(1/2ρU 2 ). Then (1) can be simplified from its complex form to a modified case converting the dimensional parameters into a non-dimensional form step by step as follows. Both sides of (1) are first divided by dynamic pressure: Pexp (1 − ε)2 μU 1 1 − ε ρU 2 = 150 + 1.75 3 . (5.1) 2 3 2 L ε Dp ε Dp 1/2ρU 2 After that L is transferred to the right-hand side and the defined Rep is introduced: 1−ε 1 P ∗ 1−ε = 150L 2 3 + 1.75L 3 . 2 ε Dp Rep ε Dp

(5.2)

Then the reorganization is done for evaluating, the modified particle friction factor, fp∗ , on the left-hand side: 3 Dp ε P ∗ 150 2 = + 1.75. (5.3) 2 L 1−ε Rep Thus, the final modified form of Ergun’s correlation is evaluated as: fp∗ =

150 + 1.75, Rep

(6)

where the empirical constants, 150 and 1.75, are only valid for spherical particles. 3.2. Data Reduction and Fitting The measured pressure drop data, Pexp , for the all test beds in this study was initially compared with the calculated values, PErgun , through Ergun’s correlation (1) for the same beds and conditions. Figure 4 shows the variation of Pexp /PErgun against Rep (Dp /L) for the beds. It can be easily detected from Fig. 4 that the wellknown form of Ergun’s correlation proposed for beds with spherical particles fits the data with a considerable deviation range of ±20%. It is also understood from

Figure 4. Variation of Pexp /PErgun ratio with respect to Rep (Dp /L) for all packed beds.

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additional dada reduction studies that this deviation may be decreased as low as ±10% of the measured data by adopting k1 and k2 constants in the well-known form of 180 and 1.85, respectively, by considering the use of non-spherical particles in the sample beds. The minimum value (i.e. ±10%) of this deviation is still considerable for the beds with non-spherical particles. It can thus be considered that the original form of Ergun’s correlation should be modified for non-spherical particles. The modification procedure presented in Section 3.1 is followed for that purpose. First, the Pexp data are converted to fp data by using (3) and then compared with the alternative correlations proposed in the literature [13–19] (see Table 1). Second, the Pexp data are converted to fp∗ data by using (4) as a new approach in the literature and then compared with the modified form of Ergun’s correlation (6). It can be seen from Fig. 5 that fp data for the present test beds are well fitted with the following empirical correlation: 1−ε D −5 −1 Re − 66.487Re + 0.1539 . (7) 3 × 10 fp = p p Dp ε3 The deviation between the data and the correlation is ±3%. The empirical correlations of Rose [13], Rose and Rizk [14], Hicks [15], Tallmadge [16], Sug Lee and Ogawa [17], Kürten et al. [18] and Montillet et al. [19] listed in Table 1 are also compared with the present data in Fig. 5. It can be easily detected from Fig. 5 that the correlations of Rose [13] and Rose and Rizk [14] both overestimate the data

Figure 5. Comparison of fp data with the previously proposed correlations given in Table 1.


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for Rep > 4000 and underestimate it for Rep < 4000 due to the elimination of the effects of D/Dp and ε. The correlations of Hicks [15], Tallmadge [16], Sug Lee and Ogawa [17] and Kürten et al. [18] all underestimate the data through all the Rep range of the present study due to the elimination of the effect of D/Dp in these studies. The recent correlation of Montillet et al. [19], however, slightly underestimates the data for Rep < 2500 and may be acceptable for this range where it was proposed. The present correlation (7) fitted on fp data is similar to the correlation of Montillet et al. [19] by considering the effects of all Rep , D/Dp and ε, but the applicable Rep range of it is wider than that of Montillet et al. equation (7) well fits the fp data and may be used for the calculation of pressure loss data for the packed beds with non-circular particles in industrial applications; however, it is still complex and difficult to remember for practical applications. Figure 6 shows the variation of the fp∗ data for the packed beds of present study as a function of Rep and . It also shows the data fitted correlation and the results of the modified form of Ergun’s correlation (6) for the same beds and conditions as solid and dashed lines, respectively. It can be easily detected from Fig. 6 that the following correlation well fits the present data with a deviation of approximately ±4%: 276 fp∗ = + 1.762 . (8) Rep Equation (8) implies that the pressure drop through vertical-cylindrical packed beds with circular and/or non-circular packing particles can be calculated by using the originally proposed fp∗ definition in (4) and a simple first-order polynomial re-

Figure 6. Variation of fp∗ data for the present study as a function of Rep and .

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lationship between the terms fp∗ Rep and Rep 2 as proposed in (8) for practical purposes. However, (6) overestimates the data by 17%. This overestimate should be caused by the values of k1 = 150 and k2 = 1.75 in (6) proposed by Ergun for circular packing particles. If these coefficients are changed with the values 160 and 1.61 for considering the effects of non-circular packing particles in the test beds as in: 160 + 1.61. (9) fp∗ = Rep Its deviation from the experimental data for the G16 and Z18 beds is also decreased to ±4% as can be seen from Fig. 6. Thus, the modified form of Ergun’s correlation with adopted coefficients (9) should also be proposed for the pressure drop calculations of packed beds with circular and/or non-circular particles. 4. Conclusions In view of the systematic experiments and data analysis conducted in this study, it is shown that: (i) The well-known form of Ergun’s correlation (1) may be used for pressure drop calculation of packed beds composed of spherical and/or non-spherical particles with a maximum deviation of ±20%. (ii) The alternative approaches based on the use of the fp definition instead of P proposed in the literature may be used for the pressure drop calculation of the beds with a maximum deviation of ±3% as is proposed in (7). However, this equation is very complex and difficult to use for practical applications. (iii) A new approach based on the use of the fp∗ definition instead of P proposed in this study can be efficiently used for the pressure drop calculations of the beds with a maximum deviation of ±4% as proposed in (8). This correlation is a simple first-order polynomial equation between the terms fp∗ Rep and Rep 2 . It is very simple and useful for practical purposes. (iv) The modified form of Ergun’s correlation (6) may also be adopted for calculating the pressure drop through packed beds with circular and/or non-circular packing particles by means of changing the original values of the empirical constants k1 = 150 and k2 = 1.75 to 160 and 1.61, respectively, as proposed in (9). References 1. S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog. 48, 89–94 (1952). 2. M. Leva, M. Weintraub, M. Grummer, M. Pollchik and H. H. Storch, Fluid flow through packed and fluidized systems, Bureau Mines Bull. 504, 149 (1951). 3. P. C. Carman, Fluid flow through packed beds, Trans. IChemE 15, 150–166 (1937). 4. S. P. Burke and W. B. Plummer, Gas flow through packed columns, Ind. Eng. Chem. 20, 1196– 1200 (1928). 5. M. Leva, Pressured drop through packed tubes, Chem. Eng. Prog. 43, 549–554 (1947).


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