Prediction of interphase drag coefficient and bed expansion using a variational model for fluidization of small spherical particles

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Prediction of interphase drag coefficient and bed expansion using a variational model for fluidization of small spherical particles ´ Mihal Ðuriˇs a,∗ , Zorana Arsenijevic´ a , Radmila Garic-Grulovi c´ a , b – Tatjana Kaluderovic´ Radoiˇcic´ a Institute of Chemistry, Technology and Metallurgy (National Institute), Department for Catalysis and Chemical Engineering, University of Belgrade, Njegoseva 12, Belgrade, Serbia b Faculty of Technology and Metallurgy, University of Belgrade, Karnegijeva 4, Belgrade, Serbia

a r t i c l e

i n f o

Article history: Received 25 September 2019 Received in revised form 19 November 2019 Accepted 20 November 2019 Available online xxx Keywords: Calculus of variations Isoperimetric problem Bed expansion Drag coefficient Fluidization Spherical particles

a b s t r a c t In this study, we applied the variational model to fluidization of small spherical particles. Fluidization experiments were carried out for spherical particles with 13 diameters between dp = 0.13 and 5.00 mm. We propose a generalized form of our variational model to predict the superficial velocity U and interphase drag coefficient ˇ by introducing an exponent n to describe the different dependences of the drag force Fd on fluid velocity for different particle sizes (different flow regimes). By comparing the predictions with the experimental results, we conclude that n=1 should be used for small particles (dp < 1 mm) and n = 2 for larger particles (dp > 1 mm). This conclusion is generalized by proposing n = 1 for particles with Ret < 160 and n = 2 for particles with Ret > 160. The average mean absolute error was 5.49% in calculating superficial velocity for different bed voidages using the modified variational model for all of the particles examined. The calculated values of ˇ were compared with values of literature models for particles with dp < 1.0 mm. The average mean absolute error of the modified variational model was 8.02% in calculating ˇ for different bed voidages for all of the particles examined. © 2019 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

Introduction The particulate expansion of liquid fluidized beds of uniform spherical particles has been extensively studied (Couderc, 1985; Di Felice, 1995; Epstein, 2003). Information on expansion properties of fluidized beds is necessary for designing equipment in such applications as particle mixing, particle separation and classification, solid coating, powder granulation, particle drying, chemical synthesis, and aerobic biological wastewater treatment. Knowledge of the superficial velocity–voidage relationship, U(ε), is also important for understanding fluid–particle interaction in fluidized and other fluid–particle systems. Investigations have been done on liquid–solid and gas–solid fluidization and expansion as well as different correlations for those systems (Akgiray & Soyer, 2006; Ðuriˇs, ´ Garic-Grulovi ´ ´ & Grbavˇcic, ´ 2016; Epstein, ´ Arsenijevic, c, Radoiˇcic, 2003; Garside & Al-Dibouni, 1977; Riba & Couderc, 1977; Grbavˇcic´ et al., 1991; Khan & Richardson, 1990).

∗ Corresponding author. E-mail address: [email protected] (M. Ðuriˇs).

Another approach to bed expansion is based on drag force measurements in fluidized beds. Some investigators have expressed drag data in terms of the ratio Fd /FdS of the drag force acting on a particle in a fluidized bed to that acting on a single particle at the same relative superficial velocity (Gibilaro, Di Felice, Waldram, & Foscolo, 1985; Richardson & Meikle, 1961; Rowe & Henwood, 1961). Interphase forces cause interaction between phases, and consequently the drag model is very important in two-phase flow. Different correlations for the drag coefficient are available in the literature, but the literature lacks correlations for determining the frictional force in fluid–particle fluidized systems. Most researchers have used empirical and semi-empirical approaches to derive correlations for the drag force (Du, Bao, Xu, & Wei, 2006; Esmaili and Mahinpey, 2011 ; Gidaspow, 1994). Morgan, Day, and Littman (1985) and Day, Morgan, and Littman (1987) first applied the classical isoperimetric problem of variational calculus to fluidization, spouting, and particle transport. Grbavˇcic´ et al. (1991) developed a model for the interphase drag coefficient for particulate fluidized beds using the classical isoperimetric problem of variational calculus as an optimization tool. Variational calculus is used in this paper to determine the drag

https://doi.org/10.1016/j.partic.2019.11.002 1674-2001/© 2019 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

Please cite this article in press as: Ðuriˇs, M., et al. Prediction of interphase drag coefficient and bed expansion using a variational model for fluidization of small spherical particles. Particuology (2019), https://doi.org/10.1016/j.partic.2019.11.002

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Nomenclature List of symbols C0 Integral constant in Eq. (5) C1 Variational constant in Eq. (22) C2 Variational constant in Eq. (10) dP/dz Pressure drop in the bed, Pa dp Particle diameter, mm Hydrodynamic drag force per unit volume of susFd pension, N/m3 FDs Hydrodynamic drag force for single particle, N H Bed height, m Gravitational acceleration, m/s2 g L Distance, m n Variational exponent Number of frames nf Re Particle Reynolds number Ret Terminal Reynolds number Time, s t Interstitial fluid phase velovity, m/s u Superficial fluid velocity, m/s U UmF Minimum fluidization velocity (superficial), m/s Ut Particle terminal velocity in a stagnant fluid, m/s v Particle phase velocity, m/s Greek letters ˇ Fluid–particle interphase drag coefficient, kg/m4 (n = 2), kg/(m3 s) (n = 1) ˇmF Fluid–particle interphase drag coefficient at minimum fluidization, kg/m4 (n = 2), kg/(m3 s) (n = 1) Bed voidage ε εMf Bed voidage at minimum fluidization  Lagrange multiplier Fluid viscosity, kg/(m s)  f Fluid density, kg/m3 p Settling particle density, kg/m3 , Coordinates

applying the classical isoperimetric problem of variational calculus to fluid–particle dynamics led Littman and Morgan (2010) to review this general approach. However, there is a lack of research on applying the variational model to fine particles. Grbavˇcic´ et al. (1991) applied the variational model with the assumed square dependence of the drag force Fd on velocity in the form Fd = ˇ(u − v)2 , which is generally used for large particles. Davidson (1961) assumed that drag force in fluid–particle systems with lower voidage is a linear function of the relative velocity and thus proportional to (u − v). Lefroy and Davidson (1969), Jackson (1971), and Gidaspow (1994) also observed that drag force is a linear function of the relative velocity for small particles (thus relatively low velocities). This work extends the variational model to predict the fluid–particle interphase drag coefficient ˇ and particulate expansion for fluidization, U = f(ε), of small spherical particles. The values of ˇ(ε) and U(ε) predicted by the proposed variational model are compared with those of other models. Basic mathematics The mathematics of the isoperimetric problem of variational calculus is given in the literature (Weinstock, 1952). The term “isoperimetric” is generally applied to all problems in which one integral expression is to be an extremum while another has a given value (Littman & Morgan, 2010). Many problems in analysis and in practical application can be regarded this way. This section covers the basic concept of the isoperimetric problem and its application to fluidized beds. The objective of variational calculus is to find extrema of functionals of a finite number of independent variables. A functional is a quantity or function that depends on an entire course of one or more functions. The domain of a functional is a set of admissible functions. A simple example is the length of a curve J whose coordinates are (,). The length of that curve (a functional) is represented by the integral

1

coefficient in a model we call the “variational model” because it was also used in our previous paper (Grbavˇcic´ et al., 1991). The model contains a single dimensionless parameter, UmF 2 /Ut 2 εmF 3 , which can be determined experimentally using particulate expansion data. Knowing the minimum fluidization velocity UmF and corresponding bed voidage εmF , we can determine the “effective” particle terminal velocity. The resulting model predicts the interphase drag coefficient over the range of superficial fluid velocities from minimum fluidization to terminal velocity and agrees very well with experimental data for higher bed voidages – ´ and Kaluderovi and coarse particles. Ðuriˇs, Arsenijevic, c´ Radoiˇcic´ (2019) examined the sensitivity of the variational model to UmF , Ut , and εmF values obtained from various correlations in the case when experimental values are not available. Additionally, Grbavˇcic´ et al. applied a variational model to the hydrodynamics of a vertical accelerating gas–solid – ´ Kaluderovi ´ Garic-Grulovi ´ ´ Ðuriˇs, & c´ Radoiˇcic, c, flow (Arsenijevic, ´ 2014; Garic, ´ Grbavˇcic, ´ & Jovanovic, ´ 1995; Garic-Grulovi ´ ´ Grbavˇcic, c, ´ Arsenijevic, ´ Ðuriˇs, & Grbavˇcic, ´ 2014; Grbavˇcic´ et al., 1992; Radoiˇcic, ´ Garic, ´ Jovanovic, ´ & Roˇzic, ´ 1997). A disadvantage of empiriGrbavˇcic, cal correlations is that the most empirical models were each derived for only one type of particle (such as glass spheres, sand, and catalyst particles). The main advantage of the variational model is that it is based on the intrinsic properties of a fluid–particle system, which allows simple scaling of fluidized beds using a variety of particle types and sizes. The success of Grbavˇcic´ et al. (1991) in

2 1/2

[1 + ( ) ]

J=

d,

(1)

0

where the prime represents the first derivative of . The value of J depends on the value of the argument function (), which may be taken to be an arbitrary continuous function with a piecewise continuous derivative. Isoperimetric problem The basic isoperimetric problem applied to fluidized and spouted beds involves a normalized functional J:

J=

1 

1 + (y )2 dx.

(2)

0

The aim of the calculations is to find the extreme of the integral J at the following end conditions: y(0) = 0,

(3)

y(1) = 1.

(4)

The integral constraint is

1 K=

ydx = C0 ,

(5)

0

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where C0 is constant (Grbavˇcic´ et al., 1991; Littman & Morgan, 2010). In Eqs. (2)–(5), x and y are normalized variables ranging from 0 to 1. The functional J is the length of a curve in the (x,y) plane connecting the end points (0,0) and (1,1). It is to be optimized subject to the value of the integral K, which represents the area under the aforementioned curve stretching from (0,0) to (1,1). When the isoperimetric problem is applied to fluidized beds, the integral J describes the length of the planar curve for all velocities from minimal fluidization UmF to terminal velocity Ut . In a single-constraint isoperimetric problem, y(x) must satisfy the Euler–Lagrange differential equation d dx



∂F ∂ y



∂F = 0, ∂y

−

(6)



3

Table 1 Basic particle characteristics and experimental values of fluidization parameters for 13 diameters of spherical glass particles. No.

dp (mm)

p (kg/m3 )

Ut (m/s)

εmF

UmF (m/s)

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13

5.00 4.50 4.00 3.00 2.00 1.200 1.000 0.910 0.645 0.500 0.300 0.200 0.130

2520 2550 2540 2510 2510 2640 2700 2650 2610 2600 2530 2500 2460

0.4513 0.4341 0.4183 0.3540 0.2519 0.1742 0.1612 0.1206 0.0933 0.0837 0.0569 0.0289 0.0128

0.4815 0.4378 0.4197 0.4102 0.4400 0.3957 0.4061 0.4071 0.3990 0.4094 0.4044 0.4050 0.3778

0.0553 0.0505 0.0423 0.0310 0.0205 0.0120 0.0124 0.0092 0.0053 0.0029 0.0017 0.0006 0.0002

where F = 1 + (y )2 + y, and  is the Lagrange multiplier. The mathematics of this problem is described in detail by Weinstock (1952). The Euler equation reduces the variational problem to a problem in differential equations. The solution of the Euler–Lagrange differential in Eq. (6) is





dy =

(x + C1 )

y(x) = C2 −

1 





1 1 − (x + C1 )

2

 dx,

2

1 − (x + C1 ) ,

(7)

(8)

where C1 and C2 are the constants of the first and second integrations, respectively. Satisfying Eq. (6) is a necessary but insufficient condition for J to be an extremum. The sufficient condition is for the second derivative of the variation of y(x) to be positive for all admissible variations. Solving Eqs. (3) and (4) gives the following result:



1 − C12 − C1 ,

=

(9)

 C2 =



1 − C12

1 − C12 − C1

(10)

.

Differentiating Eq. (8) yields the equation 

y (x) =

x + C1



1 − (x + C1 )



2 1/2

.

−

This paper only gives a short overview of applying the variational model to particulate fluidization. Our previous papers explain the model fully (Grbavˇcic´ et al., 1991; Ðuriˇs et al., 2019). To apply the isoperimetric problem to particulate fluidization, Grbavˇcic´ et al. (1991) introduced two non-dimensional parameters, the dimensionless drag coefficient and dimensionless voidage, which vary between 0 and 1. This way, the voidage and drag coefficient vary from the condition of minimum fluidization to that for the terminal velocity of a single particle owing to changes in the superficial velocity of the fluid. The normalized (dimensionless) voidage x is ε − εmF . 1 − εmF

(12)

The drag coefficient per unit volume, ˇ, is normalized as y(x): y(x) = 1 −

ˇ . ˇmF

Eqs. (12) and (13) satisfy the condition y(0) = 0 at minimum fluidization and y(1) = 1 at terminal velocity. Solving this problem requires deriving ˇ(ε). The overall momentum equation for a fluidized bed is

(11)

Applying variational calculus to particulate fluidization

x=

Fig. 1. Schematic of the experimental system (a: column; b: calming section; c: overflow; d: reservoir; e: pump; f: valve; g: flowmeter; h: piezometers; TI: temperature indicator).

(13)

  dP = p − f (1 − ε) g, dz

(14)

where P is the dynamic pressure, which represents the pressure arising solely from the motion of the fluid in the bed (purely hydrostatic variations have been removed); and z is the distance from the bed inlet. This fundamental fluidization equation has been experimentally verified (Couderc, 1985). The fluid momentum equation introduces the fluid–particle interphase drag coefficient ˇ:

 dP

ε −

dz

= ˇ(u − v)2 = Fd ,

(15)

where ˇ(u–v)2 is the interaction force per unit of bed volume; and Fd is the hydrodynamic drag force per unit volume of suspension, which includes the individual momentum balances for the fluid and particle phases. Because no particles leave the bed, the average particle velocity v is assumed to be zero and Eq. (15) becomes

 dP

ε −

dz

= ˇu2 = Fd ,

(16)

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Fig. 2. Dependence of U on ε (a and b) and ˇ/ˇmF on ε (c and d) for spherical particles with dp = 5.00 and 2.00 mm.

where u is the interstitial fluid velocity. It is important to note that this equation assumes a square dependence of the fluid–particle drag force on the interstitial fluid velocity. Grbavˇcic´ et al. (1991) applied variational calculus to Eq. (16) and obtained very good agreement between calculated and experimental values of the fluid–particle interphase drag coefficient and bed expansion. Their experiments used relatively large spherical particles with dp = 2.98, 1.94, and 1.20 mm. The drag force Fd for these particles is proportional to ˇ(u − v)2 . Although it is common to use (u − v)2 in the momentum equation (Grbavˇcic´ et al., 1991; Gibilaro et al., 1985; Syamlal and O’brien, 1988; Wen & Yu, 1966), other authors (Jackson, 1971; Lefroy & Davidson, 1969) have assumed that Fd can be approximated as a linear function of ˇ(u − v) for small particles, and thus for low velocities. Pigford and Baron (1965) and Murray (1965) also used a linear function of superficial velocity in the momentum equations for fluidized beds. Anderson, Sundaresan, and Jackson (1995) examined instabilities and bubble formation in fluidized beds, and used the equation Fd = ˇ(u − v) for drag force. It should be noted that none of these authors gave a criterion for using a linear [(u − v)] or square [(u − v)2 ] dependence. In this paper, Eq. (16) is generalized by introducing the exponent n so that it can be applied to different particle sizes:

 dP

ε −

dz

n

= ˇ(ε)u .

(17)

Combining Eqs. (14) and (17) to eliminate the dynamic pressure gradient and solving the obtained equation for ˇ(ε) yields



ˇ (ε) =



εn+1 (1 − ε) p − f g [U (ε)]n

.

(18)

In Eq. (18), the interstitial fluid velocity u is replaced with the U(ε) superficial fluid velocity U using u = ε . Eq. (18) can be written for the minimum fluidization state as

ˇmF =

− εmF ) εn+1 mF (1





p − f g

n UmF

.

(19)

By substituting Eqs. (18) and (19) into Eq. (13), we obtain

y(x) = 1 −

ˇ (ε) ˇmF



=1−



n εn+1 1 − ε UmF . n n+1 1 − ε mF U (ε) ε

(20)

mF

Differentiating Eq. (20) gives y (1) = −

=−

=

1 − ε

mF

ˇmF n UmF



dˇ   dε  ε→1  



p − f g ((n)ε − (n + 1)εn ) Utn

εn+1 p − f g mF

n UmF

εn+1 Utn mF



.

(21)

Finally, parameter C1 is derived as

 C1 =

 1+

n UmF

1

Utn εn+1 mF

2 − 12 (0 ≤ C1 ≤ 1).

(22)

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Fig. 3. Dependence of U on ε (a and b) and ˇ/ˇmF on ε (c and d) for spherical particles with dp = 1.00 and 0.5 mm.

The final solution of the modified variational model for predicting the bed expansion for fluidized particles of any size is

1−

n εn+1 1 − ε UmF εn+1 1 − ε U n (ε) mF





1 = C2 − 

 ε−ε mF

1− 

1 − εmF

+ C1

2 .

(23)

Parameters  and C2 are defined by Eqs. (9) and (10). Experimental methods To experimentally verify the proposed model, fluidization experiments were carried out using glass spherical particles of 13 sizes ranging from dp = 0.13 to 5 mm. The basic characteristics of the particles used are shown in Table 1.The fluidization parameters UmF , Ut , and εmF were measured for each particle size. The fluidization experiments were performed in two acrylic columns with an inside diameter of 64 mm for large particles and 40 mm for small particles and height of 2000 mm. The columns were equipped with a flow distributor 40 cm in height, piezometers, an Yamatake-Honeywell electromagnetic flow meter, and a Lutz centrifugal pumps (TMB 35-WR, TMB 65-WR and AM 06.10 P WR), as shown in Fig. 1. During the experiments, water temperature was measured (TI) and calculations of water density and viscosity were performed. The particle terminal velocity in water, Ut , was measured in an acrylic straight-walled cylindrical column 100 mm in diameter and 160 mm in height for each particle size. Terminal velocity was measured using a video camera positioned at 0.5 m from the column. The particles were discharged into the fluid from the top of the column and after a distance of 80 cm. When static conditions were achieved, the settling velocity along a distance of 20 cm was recorded using a video camera at 60 frames/s. Because the column

diameter (100 mm) was significantly higher than the particle diameter, the wall effects were considered negligible. Terminal velocity was determined from the video recording by taking the time t required for each particle to travel the distance L:

t = nf Ut =

1 , 60

L ,

t

(24) (25)

were nf is the number of frames required for the particle to travel the distance L. The recordings were made for between 50 and 100 randomly selected particles for each size, and the arithmetic mean values were calculated. Results and discussion Table 1 shows the experimental results for minimum fluidization velocity UmF , voidage at minimal fluidization velocity, εmF , and particle terminal velocity Ut . From these measurements in Table 1, the variational model was used to calculate U = f(ε) and ˇ/ˇmF = g(ε) for each particle size. Model analysis To determine the validity of the variational model for different particle sizes, both exponents n = 1 and n = 2 were used to calculate the bed expansion U = f(ε) and the fluid particle interphase drag coefficient ˇ/ˇmF = g(ε). The predictions were compared with the experimental measurements. Fig. 2(a) and (b) shows the dependence of the fluid superficial velocity U on bed voidage ε predicted by the variational model for

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Table 2 Mean absolute errors (MAE) between experimental values and predictions of U = f(ε) and ˇ/ˇmF = g(ε) for spherical particles using variational model for exponents n = 1 and n = 2. dp (mm)

5.00 4.50 4.00 3.00 2.00 1.20 1.00 0.91 0.65 0.50 0.30 0.20 0.13

MAE (U = f(ε)) (%)

MAE (ˇ/ˇmF = g(ε)) (%)

n=1

n=2

n=1

n=2

9.41 9.54 10.49 11.57 9.57 12.56 8.36 2.13 4.33 5.48 6.82 9.90 8.57

4.20 4.89 6.13 3.86 2.65 4.04 17.23 9.77 24.16 32.95 33.07 36.57 35.31

8.52 8.57 9.28 10.16 8.53 11.03 9.22 2.19 4.60 5.87 6.99 9.97 8.00

7.78 9.12 11.11 7.26 5.05 7.41 48.94 25.35 80.32 134.87 151.93 196.12 205.26

particles of diameters 5.00 and 2.00 mm together with the experimental data. We see that the variational model gives better results for these particle sizes when n = 2 is used. The mean absolute error (MAE) was 4.20% for n = 2 and 9.41% for n = 1 for spherical particles with dp = 5.00 mm. Similar results were obtained for dp = 2.00 mm: the model obtained a MAE of 2.65% using n = 2 and 9.65% using n = 1. We also examined the effect of n on the quality of the predicted momentum drag coefficient ratio ˇ/ˇmF in the variational model. Fig. 2(c) and (d) shows the dependence of ˇ/ˇmF on ε predicted by the variational model together with the experimental results. Because ˇmF (Eq. (19)) and ˇ (Eq. (18)) depend on n, different experimental values of ˇ/ˇmF are obtained depending on whether n = 1 or n = 2 is used in calculating them. The MAEs are somewhat lower for n = 2. For dp = 5.00 mm, the MAE is 7.78% for n = 2 and 8.52% for n = 1. Similarly, for dp = 2.00 mm, the MAE is 5.05% for n = 2 and 8.53% for n = 1. Fig. 3 shows the results obtained for smaller particles, dp = 1.00 and 0.5 mm. The results in these cases are opposite to the results obtained for larger particles. Fig. 3(a) and (b) shows the dependence of U on ε predicted by the variational model for dp = 1.00 and 0.5 mm along with the experimental results. The predictions for these smaller particles are better when n = 1 is used. For dp = 1.00 mm, the MAE is 17.23% for n = 2 and 8.36% for n = 1. For even smaller particles, dp = 0.5 mm, the MAE rises to 32.95% for n = 2 but is 5.48% for n = 1 (Fig. 3(a)). Fig. 3(c) and (d) shows the dependence of ˇ/ˇmF on ε for dp = 1.00 and 0.50 mm. The MEAs in this case are significantly lower for n = 1. For dp = 1.00 mm, the MAE is 48.94% for n = 2 and 9.22% for n = 1. Similarly, for dp = 0.5 mm, the MAE is 134.87% for n = 2 and 5.87% for n = 1. Table 2 summarizes the MAEs for the values of U = f(ε) and ˇ/ˇmF = g(ε) predicted by the variational model with exponents n = 1 and n = 2 for all of the particles used in this study. For particles with diameter larger than ∼1 mm, the variational model shows better predictions for both U and ˇ/ˇmF when n = 2 is used. The difference in prediction quality is especially visible in U, while slightly better predictions are also obtained for ˇ/ˇmF . Different behavior is observed for particles smaller than ∼1 mm; in this case, the variational model gives much better predictions when n = 1 is used for both U and ˇ/ˇmF . Fig. 4 shows the experimental and calculated values of Ut /UmF for spherical particles of different diameters. The terminal velocity Ut was calculated using the correlation of Turton and Levenspiel (1986), while the minimum fluidization velocity UmF was calculated using the equation proposed by Ergun (1952). The experimental values of Ut /UmF from this work agree well with the calculated ones.

Fig. 4. Dependence of Ut /UmF on particle diameter dp .

Fig. 5. Dependence of ˇmF on particle diameter dp .

Fig. 4 shows that Ut /UmF increases sharply at dp < 1 mm for glass (n+1)

n /U n ε spherical particles. Consequently, the value of y (1) = UmF t mF changes drastically at this point, which affects the variational model according to Eq. (21). A similar change in ˇmF can be observed in Fig. 5. This figure shows that the coefficient ˇmF starts to increase rapidly for spheres with diameters smaller than 1.0 mm. For the smallest particles investigated, dp = 0.13 mm, the experimental values of ˇmF rise to 7.86 × 109 . The changes in the slopes of Ut /UmF = f(dp ) and ˇmF = f(dp ) at (n+1)

n /U n ε dp ≈ 1 affect the value of the parameter y (1) = UmF used t mF in the variational model. We conclude from these slopes that the value dp ≈ 1 can be used to determine the value of the exponent n that should be used in the model. On the basis of the above observations, we propose the following criteria for the variational exponent for glass spherical particles:

fordp < 1 mm,n = 1 fordp > 1 mm,n = 2

.

(26)

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Fig. 6. Comparison of experimental values and predictions of variational model for (a) U = f(ε) and (b) ˇ/ˇmF = g(ε). Table 3 Average mean absolute error in predicting superficial velocity U for spheres in literature correlations. Correlation

Percent error in predicting U (%)

Wen and Yu (1966) Foscolo, Gibilaro, and Waldram (1983) Riba and Couderc (1977) Hartman, Trnka, and Havlin (1992) Akgiray and Soyer (2006) Di Felice (1994) Variational model proposed in this study

12.32 26.42 22.76 20.97 10.18 46.72 5.49

To generalize the conclusion, we propose using the following Ret ranges to determine the exponent n in the variational model for spherical particles: forRet < 160,n = 1 forRet > 160,n = 2

.

(27)

Thus, the variational model can be used to predict the fluidization behavior of small particles in addition to that of large particles, as has been previously proposed (Grbavˇcic´ et al., 1991; Littman & Morgan, 2010). Comparison with experimental results Fig. 6 shows the comparison of experimental values of U = f(ε) and ˇ/ˇmF = g(ε) with the values calculated using the chosen exponent n in the variational model for selected particles sizes. The calculated and experimental values agree well with an average MAE of 8.30% for ˇ/ˇmF and 5.49% for U for all particle sizes. Comparison with literature correlations for bed expansion There are several correlations in the literature for predicting bed expansion for spherical particles in fluidization. These equations are reviewed by Ðuriˇs et al. (2016) and Akgiray and Soyer (2006). Table 3 shows the MAEs of these correlations in predicting U, as well as the results obtained in this study. In addition to the average MAE of 5.49% achieved with the variational model for all of the investigated particle sizes, good results are obtained using the model of Wen and Yu (1966) with an average MAE of 12.32% and the model of Akgiray and Soyer (2006) with an average MAE of 10.18%. Although these models give good results, the main advantage of the proposed variational model for determining U is that Eq. n /U n ε(n+1) and (23) involves a single particle parameter y (1) = UmF t mF contains no adjustable constants.

Comparison with literature correlations for ˇ The results obtained in this study were also compared with literature correlations for the fluid–particle interphase drag coefficient ˇ. We chose the most commonly used models for ˇ: Ergun (1952), Wen and Yu (1966), Gidaspow (1994), Gibilaro et al. (1985), and Syamlal and O’brien (1988). The mean particle phase velocity v is always zero in particulate fluidization; thus, the slip velocity was neglected in the literature models. The comparison was performed only for particles with diameters below 1 mm, for which the exponent n = 1 is used in the variational model. This is because when n = 2 is used in the model (for larger particles), the units of ˇ are different from those used in the literature correlations, so they cannot be compared. Fig. 7 shows the comparison of ˇ values obtained using the variational model with those from literature correlations, together with the experimental values (the results for dp = 0.91 and 0.3 mm are shown in Fig. 7(a) and (b). The results of the variational model agree very well with the experimental data. Of the literature correlations tested, that of Wen and Yu (1966) agreed best with the experimental data. Furthermore, the mean error in predicting ˇ significantly increases with decreasing particle size for all of the models. Table 4 shows the MAEs of the different literature correlations in predicting ˇ, and the ˇ values predicted by the variational model for dp < 1 mm. The modified variational model proposed in this paper predicts ˇ in particulate fluidization with an average MAE of 8.02% for this particle size. The results of the literature correlations have MAEs from 26% to 59%. This can be explained by the fact that these models were predominantly used for gas–solid systems (Du et al., 2006; Esmaili & Mahinpey, 2011). Computational fluid dynamics simulations of liquid–solid systems have shown that the results of the different drag models differ significantly; therefore the model must be carefully selected according to the particle type (Cornelissen, Taghipour, Escudié, Ellis, & Grace, 2007; Koerich, Lopes, & Rosa, 2018; Visuri, Wierink, & Alopaeus, 2012).

Conclusions This paper has applied the variational model to fluidization of small spherical particles. Experiments were carried out for spherical particles of 13 diameters between dp = 0.13 and 5.00 mm. We generalized the variational model for the fluid–particle interphase drag coefficient and particulate expansion in solid–liquid fluidized beds by introducing a new exponent into the model. This exponent takes into account the different types of dependence of the drag force on the fluid velocity for different particle sizes (different flow regimes).

Please cite this article in press as: Ðuriˇs, M., et al. Prediction of interphase drag coefficient and bed expansion using a variational model for fluidization of small spherical particles. Particuology (2019), https://doi.org/10.1016/j.partic.2019.11.002

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Fig. 7. Dependence of interphase drag coefficient ˇ on bed voidage ε for (a) dp = 0.91 mm and (b) dp = 0.30 mm.

Table 4 Mean absolute error in predicting interphase drag coefficient ˇ for particles with dp < 1.0 mm. Correlation

Percent error in predicting ˇ (%)

Wen and Yu (1966) Ergun (1952) Gibilaro et al. (1985) Gidaspow (1994) Syamlal and O’brien (1988) Variational model proposed in this study

26.42 39.11 59.10 36.83 35.24 8.02

To determine the validity of the variational model for different particle sizes, the exponents 1 and 2 were used to calculate the bed expansion and fluid–particle interphase drag coefficient. Comparing the calculated and experimental results showed that the exponent should be 1 for small particles (dp < 1 mm) and 2 for large particles (dp > 1). To generalize the conclusion, we propose using n = 1 for particles with Ret < 160 and n = 2 for particles with Ret > 160. The difference in n in these two cases can be attributed to the different flow regimes for small and large particles. The superficial velocity required for fluidization of small particles is low, leading to laminar and low-transitional flow, while the conditions for large particles approach turbulent flow, leading to the square dependence of the drag force on velocity. This way, the variational model, which was originally proposed for large particles, is extended to the whole range of Ret values. Finally, the results obtained using the proposed modified variational model were compared with those of other models. The average mean absolute error in calculating the superficial velocity for different bed voidages was lower for the modified variational model than for the other models. Similarly, the variational model had a lower mean average absolute error than the other models in calculating the interphase drag coefficient for different bed voidages for small particles. Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work was supported by the Serbian Ministry of Education, Science and Technological Development (grant number ON172022).

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Please cite this article in press as: Ðuriˇs, M., et al. Prediction of interphase drag coefficient and bed expansion using a variational model for fluidization of small spherical particles. Particuology (2019), https://doi.org/10.1016/j.partic.2019.11.002