Kinetic theory for multi-particulate flow: Description of granular flow with rotary movement of particles

PTEC-14913; No of Pages 9 Powder Technology xxx (2019) xxx

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Kinetic theory for multi-particulate flow: Description of granular flow with rotary movement of particles M.F. Rahaman a, A.R. Sarhan b,c, J. Naser b,⁎ a b c

SLR Consulting Australia Pty Ltd, Spring Hill, QLD 4000, Australia Department of Mechanical and Product Design Engineering, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia Department of Mechanical Engineering University of Anbar, Ramadi, Anbar 31001, Iraq

a r t i c l e

i n f o

Article history: Received 21 March 2019 Received in revised form 4 November 2019 Accepted 11 November 2019 Available online xxxx Keywords: Kinetic theory Multi-particulate flow Granular mixtures Rotary movement

a b s t r a c t In this paper, the kinetic Theory (KT) for multi-particulate flow has been further extended by incorporating contributions from the rotary movement of particles on key flow parameters. The balance laws and constitutive relations for granular mixtures have been re-derived to incorporate contributions from both linear and angular movement of particles in determining the flow parameters. The computed results show similar trend with the existing KT based on equal granular temperature assumption. However, with different granular temperature, the change in flow parameters shows dissimilar trend compared with the previous KT. Flow properties like collisional stress, bulk viscosity of phase i decreases with the increase of the granular temperature of species j and shear viscosity and dilute viscosity increases with the increase of granular temperature of species j. A detail comparison of the computed results for a range of flow properties, based on both equal and unequal granular temperature has been presented in this paper. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Gas-solid fluidized beds technology has diverse industrial applications ranging from the gasification of coal in the power industry to chemical reactions for the plastic industry [1]. Due to their complex chaotic non-linear behaviour understanding the dynamics of fluidized beds is often limited to pressure drop measurements and a mass balance of the system. Experimental research into the hydrodynamics of a fluidized bed can be difficult and costly, especially for large industrial scale systems [2,3]. The process of designing an industrial scale system from an experimental rig requires knowledge of complex phenomena such as hydrodynamics, mass and heat transfer, particle mixing and reaction rates. With the advancement in computing power, at present Computational Fluid Dynamics (CFD) is a well-accepted and useful tool in understanding gas-solid flow systems. Three approaches, Eulerian-Eulerian, Eulerian-Lagrangian and Eulerian-Discrete Element Method (DEM) are used in modelling gas-solid systems. For large systems, the only currently viable approach is the Eulerian-Eulerian method [4–8]. Two-fluid model (TFM) for two-phase analyses has emerged as a very promising tool as a result of its compromise between computational cost and amount of detail provided [9]. The TFM is formulated by considering each phase as fully interpenetrating continua and are described by separate governing balance equations of mass, momentum ⁎ Corresponding author. E-mail address: [email protected] (J. Naser).

and energy. The interaction terms which couple the transport of mass, momentum and energy across the interface appear in the macroscopic field Equations [10]. The main challenge of the TFM is establish an accurate hydrodynamic and rheological descriptions of the solid phase. Gidaspow and Lu [11] showed that there exists a relation between particle pressure, granular temperature and bulk density. Savage and Jeffrey [12] used the term granular temperature to quantify the random motion of particles about the mean velocity. This theory also allows the computation of viscosity for the particle phase from values of granular temperature. To close conservation equations for the solid phase requires specification of the solid stress tensor. Earlier models [7,11,13–16] used a semi-empirical approach requiring a particular viscosity and solid pressure as an input and thus lack generality and make them difficult to apply to new systems. To overcome this difficulty, KT for two phase flows was developed based on the theory for nonuniform dense gases described by Chapman and Cowling [17]. The majority of models developed in the early 80's to late 90's use a Eulerian description of the phases, where the solid phase constitutive relations are based on the KT for granular flow [11,18–20]. In these models, particles are assumed to be identical, characterized by one diameter, one density and one coefficient of restitution. However, in real particle system, existence of particles with different size or density is predestined; furthermore, the operation of many industrial processes is strongly dependent on having different particle sizes in the reactor. Experimental results of Zhang and Arastoopour [21] showed that particles of different diameters possessed unequal turbulence energy in the

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Nomenclature ρ θ μ ξ μdil ε εs,max go A,B,D E,F,H c d e F m n r ω q

density granular temperature Particulate viscosity Bulk viscosity Dilute viscosity Volume fraction Maximum volume fraction Radial distributive function Auxiliary constants Auxiliary constants velocity vector Particle diameter Restitution coefficient Force Mass of the particle Number of particle Position vector Angular velocity Heat flux

Subscripts i ith phase j jth phase s Solid k Kinetic c collisional

riser of gas-solid dilute flow, which clearly indicate the existence of different granular temperature for different particle size/density groups. Based on this result, KT for granular flow was extended to capture the complex dynamics involved in multi-particulate flows using different granular temperature for different particle size groups [22–24]. Recently, various extensions have been proposed [3,5,6,25–28]. For example, the original theory developed by Goldshtein and Shapiro [29] for rough in elastic spheres combining the translational and rotational granular temperatures into a total granular temperature has improved by [3,27] for granular flow of rough spheres. Zhao, et al. [25] presented a KT for rough spheres. Although, they considered both particle friction and rotation for energy flux, the effect of rotation on the particle velocity distribution function was not included in their model. Chialvo and Sundaresan [26] proposed a modification to the KT model that originally was developed by [30] for shear stress and energy dissipation rate where they included a new expression for the radial distribution function at contact. The modified model relied on empiricism, particularly in the corrections for the shear stress. The simulation results showed that neglecting the rotational degrees of freedom leads to an overestimation of the granular temperature in the dilute region. Yang, et al. [6] presented a further development of a KT for rough spheres based on the work of [26]. In their modified model, particle friction and rotation were included. While the rheological properties of the solid phase are explicitly described in terms of the friction coefficient. Yang, et al. [5] further validated the modified model for a bubbling fluidized. In the majority of these above mentioned models, only the linear mean velocity and fluctuation of linear velocity components were used in evaluating the flow properties, assuming the contribution from the angular velocity components to be negligible. Although the particle rotation motions were taken into account by [5,6] in their model, particles are assumed to be identical, characterized by one diameter, one density and one coefficient of restitution. However, in real particle system, existence of particles with different size or density is predestined; furthermore, the operation of many industrial processes

is strongly dependent on having different particle sizes in the reactor. Other attempt was made by Songprawat and Gidaspow [31] to extend the kinetic Theory to unequal masses and unequal diameters. Their proposed model was only valid for single phase. For more than one phase the radial distribution function is not valid. This work focuses on investigating the effect of angular movement of multi-size solid particles on flow parameters like solids stress, bulk viscosity, shear viscosity etc. For this purpose, KT for multi-particulate flow of Rahaman, et al. [22] is further extended by incorporating angular movement of particles in addition to the linear movement. The computed results indicate similar trends with that of previous KT for particles with equal granular temperature. However, it clearly indicates significant changes when particles having different granular temperature have been considered. 1.1. Frequency of collision we considered two different types of particles i and j with diameters di and dj moving relative to each other (Fig. 1). The distance between their centroids at the time of collision can be defined as dij = (di + dj)/2 with a unit vector k defined to be the direction from the centroid of particle i to the centroid of particle j. Let ci and c/i be the linear velocities and ωi and ω/i be the angular velocities of particle i immediately before and after a collision, respectively. Also, let cj and c/j be the linear velocities and ωj and ω/j be the angular velocities of particle j immediately before and after a collision. It then follows that the relative linear and angular velocities between the particles are cij = ci − cj and ωij = ωi − ωj. Using conservation of linear and angular momentum, the combined linear velocity G and the angular velocity Ω can be given by; m0 G ¼ mi ci þm j cj

ð1Þ

I o Ω ¼ I i ωi þ I j ω j

ð2Þ

Considering the collision between particles of type i and j the probable number of collision per unit time per unit volume can be given by [17]; ZZZ Nij ¼

f

ð2Þ 

 2 ci ; r; ωi ; r þ dij k; c j ; ω j cji :kdij dkdci dc j dωi dω j

ð3Þ

where k is the unit vector along the line from the center of particle i to the center of particle j at collision. cji is the relative velocity of the particles. The collisional pair distribution function, f(2) has been defined by; f

ð2Þ 

 ci ;ωi r; r þ dijk; c j ; ω j   1 1 ¼ χ r þ dij k f ðr; ci ; ωi Þ:f r þ dij k; c j ; ω j 2 2

ð4Þ

Fig. 1. Geometry of a collision of two spheres of diameter di and dj.

Please cite this article as: M.F. Rahaman, A.R. Sarhan and J. Naser, Kinetic theory for multi-particulate flow: Description of granular flow with rotary movement of part..., Powder Technol., https://doi.org/10.1016/j.powtec.2019.11.031

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where χ is a function that is near one when the flow is dilute and becomes infinite when the flow is so dense that motion is impossible. Based on the single solid phase model, Mathiesen, et al. [32] considered the binary radial distribution function as;

E¼

    1 N g χ r þ dij k ¼ g ij ¼  o  ε i þ ε j 2 2 1−ε g

1.2. The conservation equations

ð5Þ

2

 1 εs g o ¼ 41− εs max

f ðr; c; ωÞ ¼ n

3

5

¼ g ij

ðmIÞ3=2 3

ð2πθÞ

ni n j

mi m j θi θ j

ð2πÞ3

!3

 2

Ii I j θi θ j

!3



2 2 mi c2i þ I i ω2i m j c j þ I j ω j exp − − 2θi 2θ j

Substituting the above equations into Eq. (3) the number of binary collisions per unit time and unit volume becomes; 2 πdij g ij ni n j 6

mi m j θi θ j

!3

 2

Ii I j θi θ j

!3



 ð2πÞ " # ZZ 2 2 mi c2i þ Ii ω2i m j c j þ I j ω j − cji exp − dc i dc j dωi dω j 2θi 2θ j

pffiffiffi  3 mi m j π 2 Nij ¼ dij ni n j g ij θi θ j 32

 2



Ii I j θi θ j

3

ð8Þ

f ij

1

"

A3=2 D2 1 ðEHÞ3=2

1þ

2

ð9Þ

S1 S2

# 3πB2 45πB4 þ  ⋯⋯ 2AD 8ðADÞ2

"

# 9π F 2 þ ⋯⋯ 1þ 4EH

and mi θ j þ m j θi 2θi θ j

B¼

ð10Þ



 1 1 r‐ dij k; ci ; ωi ; r þ dij k; c j ; ω j dkdci dc j dωi dω j 2 2 ð11Þ

and 1 2 N c;ij ¼ dij 2

Z

 = = ψ2 þ ψi −ψ2 −ψi cij :k

cij :kN0 ð2Þ

f ij

  1 r‐ dij k; ci ; ωi ; r; c j ; ω j dkdci dc j dωi dω j 2 ð12Þ

#

Equations describing the conservation of physical properties of the system are obtained by analogy to the single-phase conservation laws. Placing ψi = miin Eq. (10), the equation describing the conservation of mass for species i becomes; ∂ ðni mi Þ þ ∇:ðni mi vi Þ ¼ 0 ∂t

ð13Þ

  mi m j θi −θ j 2m0 θi θ j

D¼

  mi m j mi θi þ m j θ j 2m20 θi θ j

The volume fraction, ε, of species i can be defined as; εi ¼



where,

A¼

2I20 θi θ j

2

To evaluate this expression the variables of integration are changed from ci, cj to the variables G, cji and ωi and ωj to the variables Ω and ωij and expanded in a Taylor series. Performing the integral of Eq. (8), within the range of 0-2π, expression for number of collisions becomes;

S2 ¼

  I i I j I i θi þ I j θ j

 = ψi −ψi cij :kk ð2Þ

ð6Þ

"

2

Z

1 3 P c;ij ¼ − dij 2

cij :kN0

  mc2 þ Iω2 exp − 2θ

ð7Þ

S1 ¼

H¼

where based on the first order approximation used by Gidaspow and Lu [11], the collisional stress contribution can be given by;

where n is the number of particles, θ is the granular temperature of the particles, c is the fluctuating velocity and ω is the fluctuating angular velocity. It is assumed that each species in the mixture has a different granular temperature. Following the procedure outlined by Chapman and Cowling [17], the Maxwellian distribution for particles in three dimensions in terms of granular temperature can be obtained by using Eqs. (4), (5) and (6);

N ij ¼

  Ii I j θi −θ j 2I0 θi θ j

0 1 X ∂nhψi i ∂ψ @ P c;ij A ¼ bni F i i N þ Nc;ij þ ∇: ni bci ψi N þ ∂c ∂t j

 3−1

where, gij is the radial distribution function and εs is the total volume fraction of the solid phases, N is the total number of phases. Assuming a Maxwellian distribution [17], the velocity distribution of the particles having rotary movement can be given by;

f

F¼

Following Rahaman, et al. [22], a transport equation for the property ψ of species i, being any property of the system can be given by;

where

ð2Þ

I i θ j þ I j θi 2θi θ j

ni mi ρi

ð14Þ

and allows the mass conservation equation to be written as; ∂ ðεi ρi Þ þ ∇:ðεi ρi vi Þ ¼ 0 ∂t

ð15Þ

Setting ψi = mici, in Eq. (10), the conservation of momentum of species i becomes;   ∂ ðεi ρi vi Þ þ ∇:ðεi ρi vi vi Þ ¼ −∇: Pk;i þ Pc;i þ εi ρi Fi þ N M;i ∂t

ð16Þ

here Pki and Pci are the kinetic and collisional contributions to the stress tensor for species i and NM,i is the transfer on momentum between particle phases. Similarly, the force term, Fi, is external forces, which may include buoyancy and gas-particle drag. The detailed expressions of these variables are given in the next section. An equation for the conservation of translational energy for species i 1 can be obtained by setting ψi ¼ mi c2i , in Eq. (10). By multiplying the 2 conservation equation for momentum by the hydrodynamic velocity an equation for the mean translation energy is obtained. The mean translation energy equation is subtracted from the resulting total translation energy equation and with some additional manipulation and the

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introduction of granular temperature yields the conservation equation for the fluctuating translational energy for species I; 

9

     ∂ ðε i θi Þ þ ∇:ðεi θi vi Þ ¼ Pk;i þ Pc;i : ∇vi −∇ qk;i þ qc;i þ εi ρi h Fi Ci i þ NE;i ∂t

3

πdi

ð17Þ

The second part of Eq. (22) has been integrated over k and by using the integration identity given in [33], the expression becomes; ! pffiffiffi  2     " ( s   mi m j 3=2 Ii I j 3=2 2 π 4I m j ∇ vi 4 mi m j P2c;ij ¼ − g ij dij 1 þ eij ni n j mi þ 2i mo 2θi θ j 2θi θ j m θ 48 5 o i di ! ) ( ! ! )# 4I j mi ∇s v j 4I j mi ∇v j 1 4Ii m j ∇vi þ mj þ 2 þ mj þ 2 ð25Þ þ I S2 S4 mi þ 2 mo θ j mo θi mo θ j 3 d d d j

i

j

1.3. Constitutive relations For a Maxwellian distribution, the kinetic part of the stress tensor, for species i, can be obtained in a similar way as shown in Ferziger and Kaper [33]; Z

ð0Þ ni mi ci ci f i dci

Pk;i ¼

ð18Þ

where S4 ¼

Pk;i

ρθ ¼ i iI mi

ð19Þ

X

Z

1 ð0Þ n m c c2 f dci 2 i i i i i

ð20Þ

Since the integral is an odd function of C, the zero-order approximation to the heat flux gives; ð0Þ

qk;i ¼ 0

ð21Þ

9πB2 þ   ⋯⋯⋯ 2AD

P1c;ij þ P2c;ij ¼ P s;i þ εsi ζ i ∇  vi I þ μ i ∇s vi

ð26Þ

where the normal components of Eqs. (19) and (24) has been added to give the solid phase pressure; ρi θ i mi

where I is the unit tensor. From the transport equations, the kinetic heat flux can be given by; ð0Þ

D

½1 þ

j

Psi ¼

qk;i ¼

A

3

By analogy with single-phase flow the stress tensor can be separated into a normal pressure component and a shear stress component; Pk;i þ

when the pressure tensor is calculated from the zero-order approximation, it can be shown that;

1 3=2

  X π  3 mi m j ni n j mi m j 1 þ eij dij g ij þ 48 mo 2θi θ j j

3

 2



Ii I j 2θi θ j

3

 2

ð27Þ S3 S2

and the solid shear viscosity can be calculated from (25);    X  2μ i;dil 4 4π 3 m  3 πni di g ii  dij n j gij i 1 þ eij 1þ ð1 þ ð1 þ ei Þgii mo 15 15 j ! pffiffiffi  2  3=2  3=2 s   m m m m I I 4I i m j ∇ vi π i j i j i j 4 g ij dij þ 1 þ eij ni n j mi þ 2 S2 S4 mo 2θi θ j 2θi θ j mo θi 240 d μi ¼

i

ð28Þ The collisional part of the stress tensor for species i can be obtained considering ψi = mici in Eq. (11); Pc;i ¼

X j

"

Z

1 3 − g ij dij 2

  = mi ci −ci cij :k k f i f j dkdci dc j dωi dω j

where μi, dil is the dilute viscosity and can be expressed as;

cij :kN0

# Z

  fj 1 4 = − gij dij mi ci −ci cij :k kf i f j k∇ ln dkdci dc j dωi dω j 4 fi X 1 cij :kN0 2 Pc;ij þ Pc;ij ¼

ð22Þ

X j

To account for inelastic collisions between particles i and j, the restitution coefficient, eij has been introduced which is the ratio of relative velocity after a collision, c/ij to the relative velocity before a collision, cij. From the dynamics of a collision including two particles the change in velocity due to the collision is given by;   mj  c0i −ci ¼ − 1 þ eij cij  k k m0

ð23Þ

Integration of the first part of Eq. (22) can be performed using integral identities from [17] and the above change in velocity due to a collision. It can be shown that;   3 mi m j ni n j mi m j π 1 þ eij dij g ij 48 mo 2θi θ j

where S3 ¼

A

3=2

5=2

D

2

# 15π B2 þ ⋯⋯ . 1þ 4AD

"

1

 3 

Ii I j 2θi θ j

 3

2

ð29Þ

The bulk viscosity is obtain from the normal terms of Eq. (25); ζi ¼

j

P1c;ij ¼

24εsi θi ρi X 3 2 π di mi Nij j

μ i;dil ¼

! pffiffiffi  2       mi m j 3=2 Ii I j 3=2 π 4I m j 4 mi m j gij dij 1 þ eij ni n j mi þ 2i S2 S4 144θi mo 2θi θ j 2θi θ j d mo θi i

ð30Þ Collisional flux terms for species i, in Eq. (17) can be given by; qc;i ¼

X

q1c;ij þ q2c;ij

ð31Þ

j 3

q1c;ij ¼

dij m 2 i

Z

   =2 k cij :k f i f j ci −c2i dkdci dc j dωi dω j

ð32Þ

cij :k 4

dij q2c;ij ¼ − mi 4

Z

   =2 k cij :k f i f j ci −c2i ðk:ci Þdkdci dc j dωi dω j

ð33Þ

cij :k

S3 S2 I

ð24Þ

Following the same procedure as used by Ferziger and Kaper [33] the collisional flux terms becomes; q1c;ij ¼

   mi n j 1 3 2 3 πdij 1 þ eij mi 1 þ πni di g ij λð0Þ ∇θi 5 5 mo ni

ð34Þ

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M.F. Rahaman et al. / Powder Technology xxx (2019) xxx

where λ(0) is the coefficient of thermal conductivity. q2c;ij ¼

pffiffiffi  3=2     Ii I j 3=2 π 8 þ 3π  4 mi m j ∇ ln θi −∇ ln θ j S1 ni n j mi g ij dij 16 2θi θ j 2θi θ j 32

( mi þ

4Ii

!

2 di

mj ∇ ln θi þ 3mo θi

mj þ

4I j

!

2 dj

) #

mi ∇ lnθ j S3 S2 3mo θ j

ð35Þ

The momentum source term in Eq. (16) can be given by; NM;i ¼

X

φ1ij þ φ2ij

ð36Þ

5

Table 1 Operation conditions used in the present simulation. Riser dimensions (cylindrical) Diameter Height

0.032 m 1.0 m

Operating conditions Gas phase Solid phase I Solid phase II Initial bed height Pressure Inlet gas velocity Fraction of solids

ρg = 1.2 kg m−3; μg = 0.00001 Pa s ρsI = 2400 kg m−3; dI = 120 μm ρsII = 2400 kg m−3; dII = 185 μm 0.04 m Atmospheric pressure Jg = 1.2 m s−1 0.63

j

 mi m j 2  φ1ij ¼ − d 1 þ eij mo ij

φ2ij

 mi m j 3  ¼− d 1 þ eij mo ij

Z



2 k:cij k f i f j dkdci dc j dωi dω j

ð37Þ

Z



k:cij

k:cij N0

2

fj k f i f j k∇ ln dkdci dc j dωi dω j ð38Þ fi

Using the integral theorems from Ferziger and Kaper [33]:

φ1ij

pffiffiffi  3  mi m j π 2 mi m j  ¼ 1 þ eij ni n j g ij dij mo 2θi θ j 8

 2



Ii I j 2θi θ j

3

 2

  S1 S2 vi −v j ð39Þ

φ2ij ¼ −

       mi m j 3=2 I i I j 3=2 π mi m j 3  ni n j dij 1 þ eij g ij 48 m( 2θi θ j o ! !2θi θ j )  4I j mi ∇v j  4I m j ∇vi þ mj þ 2 mi þ 2i vi −v j S1 S2 I mo θi mo θ j d d i

j

ð40Þ The energy dissipation in Eq. (17) can be given by; NE;i ¼

 X

γ 1ij þ γ 2ij

ð41Þ

j

Z

 dij mi m j 2 eij −1 4 mo 4

γ 1ij ¼

2. Results and discussion

k:cij N0



3 cij :k f i f j dkdci dc j dωi dω j

Since experimental data are not available, the computed results from the proposed theory have been presented in similar way with previous works [22,23]. The computed results based on both equal and unequal granular temperature are presented and explanations of the results are given to establish the developed kinetic theory. Results obtained from the proposed kinetic theory are presented in the form of number of collisions, bulk viscosity, dilute viscosity, shear viscosity, and collisional stress component, as a function of granular temperature. The parameters used in the analysis are mi = 8.18 E-11 kg, mj = 1.13E-09, di = 5.0E-04 m, dj = 1.2E-03 m, θi has been kept constant at 2.18 E10 kg.m2.s−2. As part of the model validation, published experimental data of Mathiesen, et al. [32] were used to validate the accuracy of the proposed KT model for the selected set of conditions, then the model was used for further analysis. Operation conditions and summary of the model formulation used in the present work are introduced in Tables 1 and 2. Fig. 2 (a, b) shows that the radial distribution of the axial particle velocities of two different solid phases, have identical flow parameters, but different diameters and under different gas fluidization velocities. It can be seen from Fig. 2 (a, b) that the proposed KT model correctly captures the trend of the velocity in the core region. The magnitudes of the velocities are in reasonably good agreement with the experimental results. Results obtained from the present KT model are also compared with the results of the KT model proposed by Rahaman, et al. [22] (i.e. without rotational movement). The comparisons between the predicted

ð42Þ

cij :k

Z  3 fj mi m j 2 eij −1 cij :k kf i f j ∇ ln dkdci dc j dωi dω j γ 2ij ¼ 8 mo fi 2 dij

Table 2 Summary of the model formulation.

ð43Þ

cij :k

Commercial software

CFX

General

Linear solver type Pressure formulation Run mode Total time Gravitational body force Convergence criteria Mesh type

Models

Eulerian–Eulerian flow approach Drag model Viscous-standard k–ɛ, dispersed Number of phases = 2 Continuous phase = solid Secondary phase = gas Minimum volume fraction Inlet condition Outlet condition Wall condition

Using the integral theorems of [33] the energy source terms becomes; γ 1ij ¼

γ2ij

pffiffiffi

 m m m m 3=2  I I 3=2 π 2 i j i j i j dij ni n j e2ij −1 g ij S1 S2 mo 2θi θ j 2θi θ j 16

ð44Þ

      mi m j π 2 ∇vi ∇v j mi m j 3=2 Ii I j 3=2 dij ni n j g ij ¼ þ Þ  128 mo θi θj θi θ j θi θ j

( mi þ

! 4Ii m j ∇vi þ 2 mo θi d

mj þ

i

where S5 ¼

1 A3=2 D7=2

! )  4I j mi ∇v j 2 eij −1 S2 S5 I 2 mo θ j d

ð45Þ

Control

j

" 1þ

2

#

21πB þ ⋯⋯ 4AD

Boundary conditions

GSTB SIMPLE Unsteady, Δt = 0.001 s 32.5 Hrs Full body force–Y direction 0.0001 22,400 grids, symmetric grid Schiller–Neumann [34]

Gas, and solid

1E-006 Normal velocity Static pressure, 100,000 Pa No slip condition

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M.F. Rahaman et al. / Powder Technology xxx (2019) xxx

Fig. 4. Effect of species j granular temperature on the dilute viscosity of species i.

Fig. 2. Measured particle velocity profiles by Mathiesen, et al. [32] with the proposed KT model and the KT model of Rahaman, et al. [22], (a). solid1, (b). solid2.

results from the present KT model and from the previous KT model [22], shows that the current KT model provides a better agreement with experiment results. Fig. 3 shows the number of collisions as a function of granular temperature of species j while keeping the granular temperature of species i constant at 2.18E-10 kg.m2.s−2. Results obtained for particles with equal granular temperature (i.e. θi = θj) are also presented in

Fig. 3. Effect of species j granular temperature on number of collisions between species i and j.

this figure. Fig. 3 clearly indicates that at equal granular temperature, number of collisions increases with the increase of granular temperature (or fluctuating kinetic energy) of the mixture, and shows similar trend with that obtained by Gidaspow and Lu [11]. This is expected because, with the increase of granular temperature of both species, the relative velocity remains constant leading to a constant volume of collisional cylinder. Hence with the increase of granular temperature, the total energy of the system increases, leading to higher number of collisions. However, if granular temperature of species i kept constant, the number of collisions decreases with the increase of granular temperature of species j. This trend is dissimilar with that obtained by KT of multi-particulate flow, where effect of rotation was ignored. Considering the contribution of angular movement in particle collisions, the results observed in Fig. 3 is expected because at constant granular temperature of species i, with the increase of granular temperature of species j, loss of energy occurs during the collision, leading to reduction in the volume of collisional cylinder and hence lower number of collisions. The curve with unequal granular temperature intersects with the curve with equal granular temperature at the point θi = θj = 2.18E-10 kg.m2.s−2. Hence showing that for θi = θj, Eq. (9) for Nij, reduces to the one that is used for particles with equal granular temperature.

Fig. 5. Effect of species j granular temperature on the solids viscosity of species i.

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Fig. 6. Effect of species j granular temperature on the collisional stress component.

Fig. 4 shows dilute viscosity as a function of granular temperature of species j, keeping the granular temperature of species i constant at 2.18E-10 kg.m2.s−2. The figure clearly indicates that both dilute and solid's viscosity increases with the increase of granular temperature of species j. This is expected because of the dependence of dilute viscosity on Nij; furthermore, this can be explained in physical terms because, with the increase of granular temperature of species j, more loss in angular momentum occurs during collision, causing reduction in the relative movement of particles, leading to higher dilute viscosity. Similar trend was observed for solids viscosity and is shown in Fig. 5. In both of the cases, the curve with unequal granular temperature intersects with the curve with equal granular temperature at the point θi = θj = 2.18E-10 kg.m2.s−2, showing the feasibility of using this theory for any granular flows regardless of equal or unequal granular temperatures of the particles. Fig. 6 shows the collisional stress component of species I as a function of granular temperature of species j. In Fig. 6, particles having equal as well as unequal granular temperature have been considered. Results for collisional stress component behave similarly with number of collisions. For equal granular temperature the collisional stress contribution increases with the increase of granular temperature of the mixture. However, at unequal

Fig. 7. Effect of species j granular temperature on the bulk viscosity of species i.

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Fig. 8. Effect of rotation on number of collision with granular temperature of species j (θi = 4.0E − 11kgm2/s2.)

granular temperature collisional stress of species, i increases with the increase of granular temperature of species j. This trend is expected as collisional stress is directly related with the number of collisions. Hence with the increase or decrease of number of collisions, the collisional stress increases or decreases. The curve with unequal granular temperature intersects with the curve with equal granular temperature at the point θ i = θ j = 2.18E-10 kg. m2.s−2. At this intersection point, the collisional stress component in Eq. (22) reduces to the one, used for particles with equal granular temperature. Plot of bulk viscosity (Fig. 7) also shows similar trend because of the dependency of bulk viscosity on number of collisions. A comparison between the multi-particle kinetic theory of [22] (i.e. without rotation) and the newly derived kinetic theory with rotation has been made. The objective of this section is to study the effect of rotary movement of the particles on different flow parameters. Since any experimental data regarding this issue are unavailable, we have compared the results with the results from the existing kinetic theories to validate our result. The same concept of validation has been used by different researchers [1,22] to validate the updated kinetic theory. The

Fig. 9. Effect of rotation on collisional stress component with granular temperature of species j (θi = 4.0E − 11kgm2/s2.)

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with equal granular temperature. However, for particles with unequal granular temperature, the trends are quite different compare to those developed by previous researchers [22,23]. It was also found that there is a very minor effect of rotation on the number of collision when the granular temperature of the particles is unequal. Furthermore, a small change in the value of collisional stresses have been found when different granular temperatures have been considered. The results from the present work also showed that the effect of rotation has significant effect on the shear component of the collisional stress and hence solids viscosity while it has negligible effect on solids pressure. Declaration of Competing Interest There is no Conflict of Interest. Acknowledgements

Fig. 10. Effect of rotation on solids viscosity with granular temperature of species j (θi = 4.0E − 11kgm2/s2).

results are shown in the form of number of collisions and collisional stress component, as a function of granular temperature. The Parameters used in this analysis are mi = 1.60E-07 kg, mj = 2.21E-06 kg, di = 5.0E-04 m, dj = 1.2E-03 m and a has been kept constant at 4.0E11 kg-m2/s2. The effect of rotary movement of particles on the number of collision between i and j type particles has been shown in Fig. 8. From this figure it can be observed that there is no effect of rotation on the number of collisions between the particles when granular temperature of the particles are equal. There is a very minor effect of rotation on the number of collision when the granular temperature of the particles is unequal. The change in the values of the computed result is negligible. So effect of rotation on the number of Collisions can be ignored. The effect of rotary movement of particles on the collisional stress component between i and j type particles has been shown in Fig. 9. It has been observed that in the case of equal granular temperature, collisional stress component is independent of the particles rotary movement. A small change in the value of collisional stresses have been found when different granular temperatures have been considered. Again the change in the values of computed result is negligible. This is expected, as the rotary movement of the particles cannot have any significant effect on the normal component of the particulate properties. Therefore, the effect of rotation on the collisional stress component can be ignored. The effect of rotary movement of particles on solids viscosity has been shown in Fig. 10. From the figure, it can be observed that rotation has a significant effect on the solids viscosity. When particle rotation has been considered, there is a significant increase in the solids viscosity. The rotary movement of particles causes significant changes in the shear component of the collisional stresses. Since solids viscosity is directly related with shear stress, there is significant increase in solids viscosity when rotary movement of particles are considered. So in calculating the solids viscosity, the effect of rotation should be taken into consideration. 3. Conclusion Kinetic Theory for multi-particulate flow has been further extended by incorporating angular movement of particles. The concept of equipartition principle from dense gas kinetic theory was used in calculating the granular temperature. Particulate viscosities and solid stress caused by particle collisions are derived using the mathematical techniques of dense gas kinetic theory. The computed results shows similar trend with previously developed kinetic theory [11,22] for particles

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