Predicting the drag coefficient and settling velocity of spherical particles

Powder Technology 239 (2013) 12–20

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Predicting the drag coefficient and settling velocity of spherical particles A. Terfous a,⁎, A. Hazzab b, A. Ghenaim a a b

IMFS, INSA Strasbourg Graduate School of Science and Technology, France Modelling and Calculation Methods Laboratory, Saïda University, Algeria

a r t i c l e

i n f o

Article history: Received 12 April 2012 Received in revised form 22 January 2013 Accepted 24 January 2013 Available online 30 January 2013 Keywords: Drag coefficient Settling velocity Spherical particle Fluid at rest Computational method

a b s t r a c t This paper presents a relationship for calculating the drag coefficient of a spherical particle. The results obtained were validated using experimental data, including comparison with results from other relationships such as certainty analysis. The validation indicated that the proposed relationship is easier to apply and produces the best results. In addition, a relationship based on the characteristics of both the fluid and the particles was established to calculate directly the settling velocity of a spherical particle in a fluid at rest. The settling velocity values obtained with the proposed relationship were validated using experimental data and the model proved to be reliable and precise. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The study of fluids containing particles or fibres involves a class of complex fluids and is very important and useful in both theory and application. The essential problem lies in correlating the local and global properties of the mixture. The presence of the solid phase mixed with the fluid phase leads to very complicated phenomena, especially considering the hydrodynamic aspect (Dallavalle [1], Graf [2], Streeter and Wylie [3]). Determining the local behaviour of flow and the interaction of phases (solid/liquid) is very important for understanding the natural phenomena involved (Baba and Komar [4], Van Rijn [5], Burr et al. [6], Gabriel et al. [7]) and for comprehending many industrial procedures (Clift et al. [8]). Analysis of studies published on the subject shows that the results related to global behaviour including velocity and average concentrations are relatively numerous compared to the results obtained on local behaviour to determine velocity and local concentrations. This indicates that in most cases, the studies are specific studies with limited fields of application (Fortier [9]). Flow modelling using mass diffusion equations remains approximate and usually ignores the mechanical and physico-chemical characteristics of particles as well as the rheological behaviour of the fluid phase and the mixture (White [10], Michaelides [11]). In this context, the terminal settling velocity of solid particles represents an important parameter in the equations governing complex fluid movement (Schlichting [12], Dietrich [13], Soulsby [14], Ahrens [15], Gan et al. [16]). Many relationships with a variety of choices have been proposed ⁎ Corresponding author. E-mail address: [email protected] (A. Terfous). 0032-5910/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.01.052

to determine this parameter. It is precisely this diversity that makes applications difficult (Heikanen [17], Brown and Lawler [18], Michaelides [11], Chhabra [19]). In experimental studies, special attention is paid to the influence of the characteristics of particles, like density and shape, on the sedimentation process (Gibbs et al. [20], Hawley [21], Dietrich [13], Chien [22], Hofman [23], Le Roux [24], Tang et al. [25], Agarwal and Chhabra [26]). Also, particular attention has been devoted to the study of the influence of certain fluid parameters like density, rheological characteristics, etc. (Lali et al. [27], Chhabra et al. [28], Kelessidis [29,30], Dhole et al. [31], Rajitha et al. [32]). In practice, the terminal settling velocity of a solid particle is usually associated with a spherical shape. The relationships proposed for spherical shapes are most often used as the basis in the case of more complex shapes (Renaud et al. [33], Hazzab et al. [34]). In these cases, the relationships of the settling velocity for spherical shapes are employed directly or indirectly to calculate the settling velocity for particles of different shapes (Ganser [35], Jimenez and Mandsen [36]). The settling velocity of spherical particles can be calculated by two methods. The direct method based on the velocity as a function of dimensionless numbers approximately equal to an Archimedes number (Valembois [37], Turton and Clark [38], Ganguly [39], Trim and She [40], Tsakalakis and Stamboltzis [41], Gabitto and Tsouris [42]). There is also the indirect method based on an iteration process using the drag coefficient Cd and Reynolds number Res (Zimmermann [43], Haider and Levenspiel [44], Chhabra et al. [45], and Camenen [46]). Bibliographic analysis reveals the empirical aspect of the relationships determining the drag coefficient Cd, and also shows that some of these relationships cover only limited intervals of Reynolds number values (Morsi and Alexander [47], Komar [48]). The theoretical and

A. Terfous et al. / Powder Technology 239 (2013) 12–20

semi-theoretical treatments are valid only for Reynolds number values Res ≤ 1 (Majumder and Barnwal [49]). In this context, the aim is to establish a relationship that covers a large interval of Reynolds number values, and find an arrangement that permits obtaining the most precise and easily usable relationship. This necessarily demands both knowledge and finer analysis of relationships by correlating the experimental data concerning the drag coefficient and Reynolds number Cd − Res. We achieve this goal by performing an experimental study followed by numerical analysis. The experimental results above all allow us to deduce an approach to determine the drag coefficient as a function of the Reynolds number Res, and then to establish a formulation to calculate directly the terminal settling velocity of a spherical particle in a liquid at rest.

13

2. Methods for calculating the settling velocity of spherical particles

However, it should be noted that the drag coefficient value of a free-falling sphere in a quiescent liquid is greater by 15–30% than the drag coefficient deduced when the particle is fixed in a moving fluid. This difference is caused by the adjustment of the particle's trajectory during its settling in the case of free fall (Boillat and Graf [66]). The analysis of relationships indicates that though certain relationships are simple and singular for the Reynolds number explored (Serafini [52], White [54], Brauer [56], Molerus [61], Chien [22], Di Filice [63]), other relationships have varied and sometimes complicated formulae (Morsi and Alexander [47], Weber [55], Clift et al. [8], Midoux [58]). We also noted a similarity between certain relationships (Weber [55], Midoux [58], Lali et al. [27], Haider and Levenspiel [44], Ganser [35]). The general aspect of the relationships mentioned implicitly expresses the hydrodynamic behaviour associated with particle movement. This hydrodynamic behaviour is characterised by three regimes (White [10]):

A particle of density ρs settles in a fluid at rest of density ρf and undergoes acceleration due to gravitation during a short period of time to attain a terminal settling velocity Ws. This velocity is attained when a balance between the drag force and the gravitational force is reached. Thus we obtain the following equation (Fredsoe and Deigaard [50]).

• Res b 1: The relative movement between the solid particle and the liquid is laminar. • 1 b Res b 10 3: Transitional regime • 10 3 b Res b 10 5: Entirely turbulent flow regime developed around the particle.


 ρs −ρf gV
2 Cd ¼ ρf W2s A

ð1Þ

A and V are the surface and the volume, respectively, of the particle under study.Cd is the drag coefficient that is dependent on the Reynolds number Res of the particle: Res ¼

dp W s g

ð2Þ

dp is the particle diameter and γ is the kinematic viscosity of the fluid. In practice, we often use the experimental results of the terminal settling velocity Ws in association with Eqs. (1) and (2) to deduce the typical curve Cd = f(Res). On the other hand, conventional methods employ experimental data only to determine the curve Cd = f(Res) and use an iterative calculation procedure (Hazzab and Miloudi [51]) to determine the value of Ws. Many empirical relationships have been used to describe the relationship Cd =f(Res) (Chift et al. [8], Chien [22], Lali et al. [27], Ganser [35], Turton and Clarck [38], Morsi and Alexander [47], Serafini[52], Govier and Azziz [53], White [54], Weber [55], Brauer [56], Cheremisinoff and Cupta [57], Midoux [58], Doron et al. [59], Swamee and Ojha [60], Molerus [61], Machac et al. [62], Di Felice [63], Tran-Cong et al. [64], Hölzer and Sommerfeld [65]). In addition, it is possible to determine the settling velocity value by direct calculation. In this case, we take into account the dimensionless parameter corresponding to the Archimedes number Grs. Therefore by modifying Eq. (1) we can obtain the following relationship:

Grs ¼


 ρs −ρf gd3p ρf γ 3

¼

3 2 C R : 4 d es

ð3Þ

We can deduce the value of the spherical particle velocity Ws by using the relationship Res = h(Grs) and considering relationship (2) at the same time (Valembois [37]). The majority of the relationships used to calculate the drag coefficient are obtained from measurements of the particle's settling velocity in a fluid at rest. Some relationships are obtained by measuring the drag force on a sphere located in a turbulent flow in a wind tunnel (Morsi and Alexander [47]).

The linearity in the Stokes domain is clearly noted in all the relationships. However, the slope of this linearity is not exactly the same for all the relationships. Even in turbulent flow, we can express that the drag variation depends only on the Reynolds number. In addition, in certain models the drag coefficient value somehow seems to be reduced considering the normal value of 0.44 (Govier and Azziz [53]). Some of these relationships were developed for a Newtonian fluid case while others were developed or adopted for non-Newtonian fluids, especially the case where the rheological behaviour obeys the power law (Machac et al. [62]). The adaptation to this type of fluid takes into account the generalised Reynolds number. In fact, the drag coefficient value varies in both cases. In the low Reynolds number regimes this value seems to decrease in the case of non-Newtonian fluid when compared with a Newtonian fluid (Kelessidis [29]). Bibliographic analysis shows that the number of models for the direct calculation of settling velocity is lower than that of models for calculating the drag coefficient. The latter models often include a unique correlationship (Ganguly [39], Tsakalakis and Stamboltzis [41]) or a diversified correlation (Govier and Azziz [53], Valembois [37]) and the majority of them cover exploration domains sufficiently adapted to practical applications. 3. Measurement of settling velocity Sedimentation tests were conducted on an appropriate experimental set-up (Fig. 1). They included a solid particle deposit mechanism and a measuring system based on space-time registration by video camera. The advantage of this method is that it does not disturb the flow. The deposition mechanism consists of a controlled system for particle injection and column of glass (thickness 10 mm), squared section (150 ∗ 150) mm and a height of 2 m. The height imposed was based on the need to establish the permanent movement of particles at the level of the measurement sections. Two digital-display stopwatches with synchronised start/stop permit measuring the travel time of particles in the measurement sections. A temperature stability system is built into the measuring device and includes heat interchange and a thermometer to maintain constant temperature during the experiment and ensure the reliability of the tests, since changes of temperature influence fluid viscosity and density. The errors in the values of these two parameters, induced by changing

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A. Terfous et al. / Powder Technology 239 (2013) 12–20

the effect of the wall decreases progressively as the Reynolds number increases. In this context, Andrieu [75] presented a literature review of studies dealing with this phenomenon and also developed an inventory of relationships to calculate or estimate the effect of walls on the settling velocity of particles. Di Filice [63] built a model applicable to all flow regimes. This model depends on the Reynolds number and on the ratio between particle dimension and the dimensions of the settling medium. By studying this model, Kelessidis [29] estimated that the error in the value of settling velocity due to the retardation caused by the wall effect is approximately 5.5%. Agarwal and Chhabra [26] estimated that in extreme cases this error can be about 7%. 4. Experimental results and discussion

the temperature by 1 °C are 0.2% for viscosity and 0.0002% for density (Majumder and Barnwal [49]). Particle deposit (movement) was registered by a digital video camera. Then the images registered were explored and analysed on a microcomputer by image-treatment software. The tests were conducted under specific conditions to eliminate all external influences on particle movement since impurities can affect the values. Cui et al. [67] noticed that even the presence of bubbles can cause a difference in the sedimentation value of about 15%. The particles used were spherical and composed of P.V.C (ρs = 1380 kg/m 3) and glass (ρs = 2640 kg/m 3). The diameters of these particles are given in Table 1. It is noteworthy that the dimensions of the measurement column were chosen as a function of the practical requirements of the laboratory, such as the stability of the experimental structure, optimisation of the field of vision for video measurements, and minimising the influence of the confining walls on the settling velocity of particles. Indeed, the boundary walls have a retarding effect on particle settlement (Chhabra [68]). This effect is even greater when the ratio of the size of the falling particle relative to that of the wall increases (Chhabra [69]). The settling velocity in infinite media is greater than that in a column (Eaton and Hoffer [70]). In an unlimited domain, the settling of particles induces velocities in the same direction at every point of a fluid and the entraining effect appears to be far enough from the particle. On the other hand, in a limited domain, the fluid flow is hampered. The entrainment effect induced near the particle generates the backward movement of the fluid associated by viscous friction at the boundary walls. This backward movement slows down particle settlement. The fluid pushed by the falling particle slides upwards along the sides of the latter and subjects it to additional friction (Lali et al. [27], Maus et al. [71], Blanc and Guyon [72], Valsak and Chara [73] and McKinley [74]). However, Chhabra [19] notes that

– the appearance of the curves Ws = f(dp) differs according to the fluid used. – the increase in the settling velocity according to the diameter is clearly observed both in water and petroleum solvent. – the curves in water have a rather exponential form while they show a parabolic form in the petroleum solvent. – the influence of the density of the solid material on the settling velocity is evident. The velocity of the glass particles is greater than that of the P.V.C. particles. Based on the results of the terminal settling velocity Ws obtained experimentally, we can determine the Reynolds number Res values and the associated drag coefficients Cd. The results obtained are illustrated in the form of the function Cd = f(Res) represented in Fig. 4 for water and in Fig. 5 for the petroleum solvent. It is noteworthy that the appearance of the two curves corresponds to those found in the literature. In fact, the general aspect of the two curves is similar to the standard curve Cd = f(Res) presented 0,8 0,7

Glass particles PVC particles

0,6 0,5

Ws (m/s)

Fig. 1. Experimental set-up.

Analysis of the records related to the tests revealed that different secondary movements (fluctuations and rotations) accompany the fall of the particle during its sedimentation in water. These movements are clearly observed for isometric particles (Ilic and Vincent [76]). The frequency of these secondary movements seems to lessen or fade if the sedimentation process occurs in a petroleum solvent. These fluctuations and rotations are less marked when the size of the particles increases (the corresponding Reynolds number also increases). Stout et al. [77] made similar observations during their studies. The test results are reported for the sedimentation of glass or P.V.C. particles in water (Fig. 2), and in a petroleum solvent (Fig. 3). The analysis of the figures reveals that:

0,4 0,3 0,2 0,1

Table 1 Characteristics of solid particles. Experiment no. Diameter (mm)

1 1

0,0 0,000 2 2

3 3

4 4

5 6

The physical characteristics of the fluids used are given in Table 2.

6 8

7 10

0,002

0,004

0,006

dp (m) Fig. 2. Settling velocity in water.

0,008

0,010

A. Terfous et al. / Powder Technology 239 (2013) 12–20

0,06

15

Glass particles PVC particles

0,05

Ws (m/s)

0,04 0,03 0,02 0,01 0,00 0,000

0,002

0,004

0,006

0,008

0,010

dp (m) Fig. 3. Settling velocity in petroleum solvent.

in the study by Hug [78] and Hölzer and Sommerfeld [65]. The linearity of the function Cd = f(Res) is verified experimentally for Reynolds number values obtained from the Stokes domain. This is clearly observed in the case of particles falling in the petroleum solvent, when the Reynolds number is relatively low (Fig. 5). 5. Modelling drag coefficient and settling velocity To find a simple aspect correlation covering a large interval of Reynolds number values, and more reliable precision, we began with a specific numerical treatment of the relationships Cd = f(Res) obtained from the experimental data. This treatment considers that every data set can be represented approximately by a function of the type: Y¼

n X

ai x

ki

i¼1

where n represents the number of terms in the approximation and ki (i = 0, 1, …n) are the real numbers of the distinct values. By applying the least-squares method, we can find an infinite set of approximations of the previous type recommended. This set depends on the choice of n and ki values. The precision of each of these approximations can be evaluated using the precision parameter ε that is equal to the maximal absolute value of the relative error. The

Fig. 5. Cd = f(Res) in the case of petroleum solvent.

error value was calculated by taking into account the experimental results as reference data. ki values were adjusted by the correction procedure until optimal values of the precision parameter ε were obtained. This adjustment can be applied successively, considering for each new operation, a new term in the approximation function Y until parameter ε appears to be no longer influenced by the correction. If the approximation remains insufficient, we can add a correction function to maintain the required precision. This function represents an approximation of curve Y − Y*, where Y is the experimental curve and Y* is the approximation obtained. To optimise this type of correlation, other numerical techniques like the optimisation method using the Genetic Algorithm (Dawar and Chase [79]) can be applied. However, the method applied allows easy accessibility and offers flexible manipulation. Its application allows calculating the drag coefficient as a function of the Reynolds number Cd = f(Res) for the experimental data. The correlation can be represented as: C d ¼ A1 þ

A2 A3 A A5 þ þ 4 þ 0:2 Res R2es R0:1 Res es

A1 ¼ 2:689; A2 ¼ 21:683; A3 ¼ 0:131; A4 ¼ −10:616; A5 ¼ 12:216:

ð4Þ In addition, considering the experimental results related to the Reynolds number and the associated Archimedes number, we can achieve an approach for Res = h(Grs). Considering Eq. (2), this approach enables determining the settling velocity of spherical particles directly:

10 Glass particles PVC particles

Grs Grs b 10 18 Grs ≥ 10 logRes ¼ α 1 þ α 2 ðGrs Þα3 α 1 ¼ −2:086 α 2 ¼ 1:772 α 3 ¼ 0:613 :

Cd

Res ¼

1

ð5Þ

6. Validation of the models developed 6.1. Experimental validation 0,1 10

100

1000

Res Fig. 4. Cd = f(Res) in the case of water.

10000

The reliability of the two models developed will be investigated. First, the results obtained from the two models are compared with the data available in the bibliography. We considered two approaches in this context. The first takes into account the experimental data as a support for comparison, while in the second, the results obtained

16

A. Terfous et al. / Powder Technology 239 (2013) 12–20

Table 2 Characteristics of the fluids used. Water

Petroleum solvent

Density (kg/m3) Kinematic viscosity (m2/s)

1000 1.0 × 10−6

793 2.19 × 10−3

from the models chosen in the bibliography are compared to those developed to calculate the drag coefficient and the settling velocity. For the experimental validation of the drag coefficient model, different experimental works are considered (Pettyjohn and Christiansen [80], Andrieu [75], Zimmermann [43] and Kelessidis [29]). The advantage of these studies is that they can be applied to different carrying fluids and to solid particles of different materials (Steel, Aluminium, Glass, Hostofarm, PVC). Also, these results cover a large interval of drag coefficient values sufficiently acceptable in the domains of application (50 to 7400). The principal characteristics of the materials used in the experiments related to these researches are illustrated in Tables 3 and 4. Fig. 6 represents the comparison between the measurement data of the drag coefficient for the different particles and the associated calculation results. The figure shows that the values of Cd obtained from the model are comparable with the measurement data for the different materials. To quantify the reliability of a model, we consider the absolute value of the relative error between the experimental data and the results obtained from the model developed. A comparison is made while taking into account the maximum and average values of the error for all the particles and for the various materials considered (Table 5). We noted that the average error does not exceed 2% and the maximal error is less than 3% for the variety of particles used. The same experimental results are used as the basis for validating the model for the direct calculation of settling velocity. Fig. 7 compares the values obtained from the proposed model with those of the measurement data for the different materials. The data sets are comparable with each other. The quantitative evaluation of the model's reliability is obtained from the error comparison values. These values are presented in Table 6. In this case the maximal error is less than 5% and the average error does not exceed 3%.

6.2. Numerical validation In addition to comparing the experimental data collected from the bibliography, the results obtained from the models developed to

1000

Value of Cd calculated

Carrying liquids

10

Reynolds number

Steel Aluminium Glass 1 Glass 2 Hostaform PVC

7760 2800 2680 2610 1360 1380

2–20 3–30 1–10 1–10 1–10 1–20

0.01–2070 0.02–1180 120–1270 160–7400 58–1960 60–3500

1000

Fig. 6. Comparison of Cd values.

Table 5 Error according to Cd values for each material. Material

Maximal error (%)

Average error (%)

Steel Aluminium Glass Hostofarm PVC

2.57 2.75 2.68 2.89 1.88

1.89 1.72 1.88 2.69 1.97

calculate the drag coefficient and the settling velocity were compared with those from models developed by other authors. Concerning the relationships calculating the drag coefficient, we chose models in which the results were judged acceptable by previous studies (Chhabra et al. [45], Hazzab and Miloudi [51], Tsakalakis and Stamboltzis [41], Kelessidis [29]). Considering the experimental data from 20 or so references, a study was performed to evaluate the reliability of models designed to calculate the drag coefficient of spherical and non-spherical particles (Chhabra et al. [45]). Considering the average and maximal error, the evaluation of studies indicates that the application of the models of Haider and Levenspiel [44], Ganser [35] and Chien [22] results in acceptable drag coefficient calculation values. The synthesis of this comparative work is illustrated in Table 7.

Settling velocity Ws calculated

Diameter (mm)

100

Value of Cd measured

10

Density (kg/m3)

100

10

Table 3 Particle characteristics. Material

Glass particles PVC particles Aluminum particles Hostofarm particles Steel particles

Glass particles PVC particles Aluminum particles Hostofarm particles Steel particles

1

0,1

Table 4 Fluid characteristics. Liquid Density (kg/m3) Viscosity (m2/s)

Water 1000 10−6

Glucose solution

Petroleum solvent

Petroleum oil

Petroleum mixture

1346 5.55 10−4

793 2.19 10−3

937 5.84 10−4

861 9.84 10−4

0,1

1

Settling velocity Ws measured Fig. 7. Comparison of settling velocity values.

10

A. Terfous et al. / Powder Technology 239 (2013) 12–20 Table 6 Error according to Ws values for each material.

17

Table 10 Error according to Ws values calculated for each model by Hazzab and Miloudi [51].

Material

Maximal error (%)

Average error (%)

Authors

Maximal error (%)

Average error (%)

Steel Aluminium Glass Hostofarm PVC

3.28 4.68 2.88 2.27 3.98

2.11 2.47 1.63 1.67 1.97

Govier and Azziz [53] Valembois [37] Hazzab and Miloudi [51]

35.07 17.27 07.79

15.42 04.69 04.02

Authors

Maximal error (%)

Average error (%)

Haider and Levenspiel [44] Ganser [35] Chien [22]

275.8 180.9 152.5

21.50 16.30 23.50

Using the work of Morsi and Alexander [47] as the basis for comparison, Hazzab and Miloudi [51] concluded that the Brauer model [56] produces reliable estimations for calculating the drag coefficient. The recent works of Hazzab et al. [34] suggest estimations of the error associated with certain models (Table 8), in particular the models of Haider and Levenspiel [44] and Chien [22]. On the basis of comparisons between seven models, Kelessidis [29] estimated that those of Haider and Levenspiel [44] and Lali et al. [27] were the best from the standpoint of precision in determining the drag coefficient value (Table 9). The works of Tsakalakis and Stamboltzis [41], confirmed the reliability of the models of Haider and Levenspiel [44], Ganser [35] and Chien [22]. Besides these models and in the light of the synthesis of all the comparative research mentioned previously, we decided to adopt the models of Brauer [56] and Lali et al. [27] for the numerical validation of the drag coefficient calculation model presented within the scope of this research. We also took into account the models of Tran-Cong et al. [64] and Hölzer and Sommerfeld [65] as they are up to date. We note that the exploration domain for Reynolds number values differs from one model to another. It is limited by the value of 1500 that corresponds to the model of Tran-Cong et al. [64] and reaches 10 7 for Brauer's model [56]. On the other hand, Hazzab and Miloudi [51] studied the reliability of certain models to perform validation by calculating settling velocity directly. The results of this study are given in Table 10.

Drag coeficent for experimental data and models

Table 7 Error according to Cd values for each model (Chhabra et al. [45]).

When studying the reliability of models that calculates settling velocity directly, and based on the results of Table 10, only those of Valembois [37] and Hazzab and Miloudi [51] were acceptable. The results of these models are comparable with the results of the models of Trim and She [40] and Tsakalakis and Stamboltzis [41] because they are up to date. The data of the experimental research of Morsi and Alexander [47] is the initial calculation basis for all the models. This work has the advantage of high precision and covers a relatively large Reynolds number interval (0.1 to 5 × 10 4). This calculation serves two purposes: the first is to compare the models used to calculate the drag coefficient and the second is to compare the models used to calculate settling velocity. Fig. 8 represents the comparison between the results of the model proposed and the experimental data. It also shows a comparison

Average error (%)

Govier et Azziz [53] Weber [55] Brauer [56] Midoux [58] Haider and Levenspiel [44] Chien [22]

36.27 11.13 09.71 14.28 20.22 5.08

11.17 05.49 04.02 04.68 12.19 07.73

Table 9 Error according to Cd values calculated for each model by Kelessidis [29]. Authors

Average error (%)

Di Filice [63] Cheremisinoff and Gupta [57] Doron et al. [59] Lali et al. [27] Haider and Levenspiel [44] Machac et al. [62]

08.45 03.38 12.80 02.48 01.30 03.06

1

10

100

Drag coeficient for proposed model Fig. 8. Comparison of drag coefficient between standard experimental measurements and the calculated values obtained from the model proposed.

1.5

Drag coefficient(Cd)

Maximum error (%)

10

1

Table 8 Error according to Cd values for each model calculated by Hazzab and Miloudi [51] and Hazzab et al. [34]. Authors

100

Exp. data, Morsi and Alexander [47] Model of Brauer [56] Model of Heider and Levenspiel [44] Model of Lali et al, [27] Model of Ganser [35] Model of Chien [22] Model of Tran-Cong et al, [64] Model of Hölzer and Sommerfeld [65]

Exp. data ; Morsi and Alexander [47] Model of Brauer [56] Model of Heider and Levenspiel [44] Model of Lali etal, [27] Model of Ganser [35] Model of Chien [22] Model of Tran-Cong etal, [64] Model of Hölzer and Sommerfeld [65] Proposed model

1

0.5

100

1000

10000

Reynolds Number(Res) Fig. 9. Drag coefficient as a function of the Reynolds number for experimental data and with the proposed model.

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A. Terfous et al. / Powder Technology 239 (2013) 12–20

Regarding the procedure of evaluating the reliability of a model used to calculate settling velocity directly, the approach considered consists in determining the value of Grs for each value of Res belonging to the exploration domain, using Eq. (3). We obtain a new value of Res for each model that we denote Res-f (f: final). Next, we evaluate the reproducibility of Res values for each calculation method considered initially Res-i (I: initial). The reliability of each calculation method increases when the calculated Res value is closest to its initially considered value. The relative error between the initial Res-i and the calculated value Res-f for each calculation method considered is obtained as:

Table 11 Error values according to Cd = f(Res) for each model. Authors

Maximal error (%)

Average error (%)

Brauer [56] Haider and Levenspiel[44] Lali et al. [27] Chien [22] Tran-Cong et al. [64] Hölzer and Sommerfeld [65] Proposed model

09.71 09.26 07.09 31.61 07.35 21.36 03.97

04.02 04.28 03.17 17.90 07.09 09.28 01.29

between the previous model and our model. From this figure we can infer that the results of our model are qualitatively comparable to both the results of other models and the experimental data. The results of the correlation Cd = f(Res) for all the models, besides those of the experiments, are shown in Fig. 9. An analysis of this figure indicates that the reproducibility of the experimental results varies from one model to another. However, the proposed model shows the best correlation with the experimental results. The error is evaluated for each model regarding the experimental values. The average and maximal error values within each model are given in Table 11. An analysis of these values indicates that some of the models appear subject to acceptable errors, while others are not. A higher degree of precision appears to be associated when using the model proposed. The average corresponding error does not exceed 1.5%.

  Resi −Resf   δ ¼ 100  Resi

; δ is expressed in %:

In Fig. 10 we compared the results of certain models with the experimental data. The figure shows that except for the results of Tsakalakis and Stamboltzis [41], the rest of the results are comparable with each other. The reproducibility of experimental data regarding the function Res = h(Grs) for all the models considered is presented in Fig. 11. Our model appears the best when correlated with the experimental results. In Table 12, the average and maximum values of relative error δ corresponding to the application of each calculation method are listed. The table, quantitatively, indicates the reliability of the models

Reynolds number for experimental data and models(Res)

100000 Exp. Data, Morsi and Alaxander [47] Model of Govier and Azziz [53] Model of Valembois [37] Model of Ganguly [39] Model of Hazzab and Miloudi [51] Model of Trim and She [40] Model of Tsakalakis and Stamboltzis [41]

10000 1000 100 10 1 0.1

0.1

1

10

100

1000

10000

100000

Reynolds number for proposed model (Res) Fig. 10. Comparison of Reynolds number between the standard experimental measurements and the calculated values obtained from the model proposed.

Reynolds number (Res)

10000 1000 100 Exp. data ; Morsi and Alaxander [44] Model of Govier and Azziz [53] Model of Valembois [37] Model of Ganguly'[39] Model of Hazzab and Miloudi [51] Model of Trim and She [40] Model of Tsakalakis and Stamboltzis [41] Proposed model

10 1 0.1 1

10

100

1000

10000

100000 1000000

1E7

Archimedes number (Grs) Fig. 11. Reynolds number as a function of Archimedes number.

1E8

1E9

A. Terfous et al. / Powder Technology 239 (2013) 12–20 Table 12 Relative error δ corresponding to each calculation method. Authors

Maximal error (%)

Average error (%)

Govier and Azziz [53] Valembois [37] Ganguly [39] Hazzab and Miloudi [51] Trim and She [40] Tsakalakis and Stamboltzis [41] Proposed model

35.07 17.28 22.16 04.03 38.33 35.93 04.69

14.30 04.19 08.05 01.71 14.58 21.95 01.38

explored. The average error obtained by applying the proposed model does not exceed 1.5%. 7. Conclusion This paper presented a numerical method for correlating standard experimental measurements. Its application on a Cd − Res experimental curve enabled us to propose a relationship for a spherical particle covering a large domain of Res (0.1 to 5 10 4). A comparative study was performed and demonstrated that the proposed relationship is efficient. Using the same approximation method, a procedure for evaluating the settling velocity of a spherical particle in a fluid at rest was presented. An approach for comparing the difference of settling velocity calculation methods was formulated. This approach was validated by experimental measurements taken from the literature and enabled evaluating the precision of each settling velocity calculation method. We observed from this evaluation that the settling velocity calculation method used thus far could at times prove highly inaccurate (up to 35% relative error). High precision was obtained using the settling velocity calculation method proposed in this paper (δmax = 5%, δaverage = 1.5%). Notation A Surface of particle (m 2) Coefficient ai, ki Drag coefficient Cd Diameter of the sphere dp Archimedes number Grs Reynolds number associated to the particle Res V Particle volume (m 3) Settling velocity (m/s) Ws Carrying fluid density (kg/m 3) ρf Particle density (kg/m 3) ρs ν Kinematical viscosity (m 2/s) δ Error (%)

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