Transverse flow at the flight surface in flighted rotary drum

Powder Technology 275 (2015) 161–171

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Transverse flow at the flight surface in flighted rotary drum Koteswara Rao Sunkara, Fabian Herz ⁎, Eckehard Specht, Jochen Mellmann Otto von Guericke University Magdeburg, Institute of Fluid Dynamics and Thermodynamics, Universitätsplatz 2, 39106 Magdeburg, Germany Leibniz Institute for Agricultural Engineering Potsdam-Bornim (ATB), Department of Postharvest Technology, Max-Eyth-Allee 100, 14469 Potsdam, Germany

a r t i c l e

i n f o

Article history: Received 22 May 2014 Received in revised form 21 January 2015 Accepted 24 January 2015 Available online 9 February 2015 Keywords: Granular material Rotary drum Drying Flights Kinetic angle Eulerian

a b s t r a c t Rotary drums, installed with longitudinal flights are often used to dry/cool granular materials in large quantities. The rate of solids falling down from the flights determines the amount of material responsible for the contact between the solids and hot air stream, which primarily depends on the angle of repose at the flight surface. In this study, the model of Schofield and Glikin has been extended by considering a flowing layer at the flight surface and inertial force acting in the cascading layer due to the rotation of the drum. The corresponding velocity profile for the layer has been predicted by following the Eulerian approach by treating the granular flow as a continuum similar to the transverse flow in rotating drums without flights. To validate the model, experiments were carried out at a laboratory drum of 0.5 m in diameter using quartz sand, glass beads and steel balls of different particle sizes at various drum rotations. The measured data is compared with the model predictions under different experimental conditions. The experimental data showed good agreement with the model predictions. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Rotary drums equipped with internal flights have a great importance in sugar, mineral processing, metallurgical, and in chemical industries. These drums are commonly used for drying or cooling the particulates in large quantities. The flights are installed to the interior of the drum in order to improve the contact between the hot gases flowing in an axial direction and the wet solids. The quality of the end product primarily depends on the time spent by the material against the hot conditions during transport from the upstream end to the downstream end of the dryer. It is mainly controlled by drum speed, flight design, drum inclination, properties of the material, and the flow properties of the gas. Drum speed and the flight dimensions regulate the cascading rate of the flights which in turn controls the amount of material distributed in the air-borne phase of the drum. All the parameters mentioned above determine the retention time of the material in the drum. Heat and mass transfer are mainly effected by the area of the material exposed to the hot or cold conditions in the dryer/cooler. The prediction of this contact area is, therefore, essential for modeling the dryer/cooler. As will be shown this can be achieved by a geometrical approach. Numerous contributions have been made during the last few decades in the context of flight unloading studies by determining the quantity of material in the flight [1–6]. Various sets of equations were developed to determine the flight holdup for angular and extended circular flights [7]. Kelly [8] further extended this approach by proposing a ⁎ Corresponding author at: Otto von Guericke University Magdeburg, Institute of Fluid Dynamics and Thermodynamics, Universitätsplatz 2, 39106 Magdeburg, Germany. E-mail address: [email protected] (F. Herz).

http://dx.doi.org/10.1016/j.powtec.2015.01.058 0032-5910/© 2015 Elsevier B.V. All rights reserved.

theory of equal angular distribution. In this approach of the flight design, an equal distribution of particles to cascade over the region of the flight discharge was considered by taking a constant surface angle over the flight discharge. An ideal and complex flight shape was proposed in this context. However, the flight geometry proposed is impractical. Revol et al. [9] further extended the approach from Kelly [8] by proposing model equations for three segmented flights. The power required to lift the solids was accurately predicted based on the flight hold-up. However, the measured unloading rate showed a large deviation from the predicted unloading rate of solids. This is attributed due to the assumption of a constant surface angle. Wang et al. [10] developed a differential approach to predict the behavior of the flights with arbitrary geometry, and an outline was also proposed to achieve the maximum drying efficiency. However, the unloading rate predicted in his study deviated from the measurements by 33% which is not acceptable. Blumberg and Schlünder [11] developed a normalized model by assuming linearity of the flight holdup with the angular position to depict the discharge characteristics of particles. Van Puyvelde [12] instrumented a model to describe the behavior of complex geometry flight profiles. He stated that an inclined flight occupied more material than flights whose radial length was perpendicular to the drum wall. In all these studies, the flight holdup depends on the surface angle of the flight to the horizontal termed as the kinetic angle of repose (γ). Schofield and Glikin [13] were the first who developed a relation for this angle from the equilibrium of gravitational, centrifugal, and frictional forces at the flight tip by neglecting the inertial force. For this purpose, a negligible thin active layer was assumed at the flight surface, although, the active layer at the flight tip already has a thickness of a few particles. The resultant forces balance equation was

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validated using an experimental drum filled with pumice granules [2], by taking the photographs of the drum. The angles of repose were calculated from the resultant images and compared with the theoretical values. Porter [1] validated this equation for Froude numbers greater than 0.18 by mounting closed rectangular shaped boxes to the inner shell of the drum filled with 3 mm glass spheres. However, under practical conditions the operated range of Froude number is much lower. The aim of the present study is to predict the kinetic angle of repose as this parameter is responsible for determining the rate of solids unloading from the flight. In the previous model of Schofield and Glikin, a single/thin layer of particles was considered in the cascading layer, and the inertial force acting on the rolling particles at the flight surface was neglected. A mathematical model was developed by considering a flowing cascading layer of few particle thicknesses at the flight surface similar to the case of rotating cylinder without flights by incorporating the inertial force based on the approaches followed by Mellmann et al. [14] and Khakhar et al. [15]. The velocity profile has been determined by treating the granular flow as a continuum. Experiments were conducted at a laboratory drum using various materials (quartz sand, glass beads, steel balls) of different sizes at different rotational speeds in order to validate the model.

FF A

FC sin(δ c − γ )

FC FC cos(δ c − γ )

FG cos γ

FI

FN

E

FG sin γ

FG

δc

γ

Fig. 2. Balance of forces acting on a particle in the active layer at the flight tip.

where δc = γ − θ ± 90, ‘−’ is for δ b γ and ‘+’ for δ N γ, and tanθ = xE/(− R cos ϵ*). The resultant force acting vertical to the bed surface is given by

2. Model development 2.1. Extended model of the kinetic angle of repose The model of Schofield and Glikin [13] assumed only a single particle layer to exist at the flight surface while determining the kinetic angle of repose by including the centrifugal force and neglecting inertial force, although a flowing region exists similar to cascading layer in drums without flights. Better prediction of this angle is truly important since this parameter is responsible for determining all phases of motion in the drum. Therefore, in the present study the model of Schofield and Glikin [13] has been extended by considering an active layer at the flight (see Fig. 1) and also incorporating the inertial term [16]. The modified forces balance diagram for this case is shown in Fig. 2 F F þ FI ¼ FG sin γ−FC cosðδc −γ Þ;

ð1Þ

FN ¼ FG cos γ−FC sinðδc −γ Þ:

The particles that are ready to fall from the layer over the flight surface will undergo centrifugal force 2

FC ¼ dm ω r dHS ;

ð3Þ

where rdHS is given by r dHS ¼

xE : sinθ

ð4Þ

The flow motion at the flight surface induces an inertial force which can be expressed as [17] FI ¼ dmvx

A

ð2Þ

dvx ; dx

ð5Þ

where vx is the average velocity of the particles in the active layer. After simplifications of the above equations and rearranging, the final form of the kinetic angle is given by  
r  v dvx μ þ Fr dHS ð cos δc −μ sin δc Þ þ Fr x R cosγ dx 
r  tanγ ¼ 1−Fr dHS ð sin δc þ μ cos δc Þ R

E

l1

l2



vx x and dv ¼ where vx ¼ Rω dx

xE

rE

ð6Þ

vx jA −vx jE xA −xE . The detailed description of the velocity

gradient term is presented in the following section.

rdHS

2.2. Analysis of transverse flow at the flight surface

rH

ε*

θ

δc

γ − R cos ε *

Fig. 1. View of transverse motion of particles in the cascading layer at the flight surface of flighted rotary drums.

The angle made by the surface of the material in the rotary drum with the horizontal is called the dynamic angle of repose (Θ). The bed remains stable until this angle approaches the upper angle of repose or the maximum angle of stability [18,19] and the avalanches begin when the dynamic angle exceeds this angle. The frequency of these avalanches increases as the rotational speed increases resulting in rolling motion. In a flighted drum, the material in the flight is tilted due to

K.R. Sunkara et al. / Powder Technology 275 (2015) 161–171

the rotation of the drum, and the avalanches can start when the tilting surface of the bed exceeds the stability angle [20–22], while the material at this point organizes itself to reach the angle of repose by transporting the additional material through avalanche and trying to achieve a stable position. However, due to the elevation of the flight by the drum rotation, the surface of the bed in the flight exceeds the maximum angle of stability and again discharges a discrete quantity of material [23]. Increasing the drum rotation leads to roll down the particles constantly over the surface of the static bed in the flight. A flowing layer develops as the material cascades over the flight surface. Continuous rolling of the particles can be observed on the flight surface, while the material is being supplied continuously by the static bed which is resting in the flight. This phenomenon can be expected similar to the transverse motion of material in rotating cylinders without flights, where a larger part of the bulk bed moving with the wall (plug flow) is transported into a flowing cascading layer on the surface of the bed [14,24]. However, an interfacial boundary exists between these two regions having few particle thicknesses (see Fig. 3). Above and below the vortex point (W) bed material interchanges between the two layers. The bed surface is considered to be flat and equal to the dynamic angle of repose in this case. Similarly, the material surface in the flight is assumed to be flat and inclined towards the horizontal at an angle equal to the kinetic angle of repose (γ). However, in this case the developing cascading layer breaks at the intermediate region at the flight tip to cascade the material into the airborne phase. Various authors studied the rolling motion behavior theoretically and modeled it successfully [14,24–29] in drums without flights. [25] proposed a model by considering the material to be non-Newtonian and the resulted mass and momentum equations were solved using FLUENT. The model of Khakhar et al. [26] described the flow motion in the active layer by considering the collisional interactions between the particles. However, this model needs a fitting parameter to determine the layer thickness in order to agree with the experimental data. Mellmann et al. [14] developed a model for the rolling motion without the necessity of fitting parameters. However, all these models regarded the granular flow motion as the continuum and followed the Eulerian approach to model the solids motion in the cascading layer. Usually, in the Eulerian approach the continuity and momentum based equations are coupled together to solve for the flow properties, in which, it is assumed that the particles are cohesionless, thin active layers when compared to the height of the bed, and the transverse velocity is several times greater than the axial velocity. A similar approach has been followed in the present study to track the developing layer at the flight surface by considering the rolling mode. The main objective of this

163

model is to determine the velocity profile along the surface layer of the flight over the flight discharge which can be used in Eq. (6). For this purpose, a mathematical model has been developed to study the transverse motion at the flight surface based on the earlier works of Mellmann et al. [14] and Khakhar et al. [26]. In the present study these models were applied to the drum equipped with flights to determine the flow properties at the flight surface. The schematic of the rolling bed at the flight surface is shown in Fig. 4 depicted for certain flight positions in the drum. The surface of the layer always intersects with the drum wall at point A in region I, whereas in region II it intersects with the flight wall as shown in Fig. 4. The transition between these two regions occurs when OA = l1 i.e., at δ = (γ + α + β n), where β n is given by tan βn ¼

l2 þ p : l1

ð7Þ

The expression for p will be presented later. The particles moving with the wall detach from the rigid bed at the boundary line to start flowing into the active layer. These particles are accelerated along the flow forming a thin layer at the surface until they reach the tip of the flight where they cascade into the air borne phase. The maximum layer thickness (tE) occurs at point E at the flight tip.

a

b

A O

●

●

W

active layer boundary line

C

passive layer

Fig. 3. Granular motion of rolling bed in drum without flights. [30].

B

Fig. 4. Flow motion at the flight surface in a flighted rotating drum, (a) region I: 0 ≤ δ ≤ (γ + α + βn) and b) region II: (γ + α + βn) b δ ≤ δL.

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The local thickness of the active layer at the flight surface t (x) is written as

2.3. Coordinate system and variable definition At a given position of the flight (δ), the transverse motion can be assumed to be uniform in the axial direction, hence, only the two dimensional case is considered. The origin of the coordinate system is always fixed at the center of the drum whose x-axis is parallel to the bed surface and y-axis is perpendicular to it. The coordinate system always changes with respect to the change in flight position. The position of the differential element in Cartesian coordinates (x, y) can be determined from the polar coordinates of position (r) and the angular coordinate (φ) as x ¼ −r cos φ; y ¼ −r sin φ:

ð8Þ

The coordinate system and the influencing variables of the problem are illustrated in Fig. 5. The dimensionless coordinates are given as follows 

x ¼

x  y r  t r rH : ;y ¼ ;ρ ¼ ;t ¼ ; E ¼ R R R R R Rcosα



t ¼ r sin φ−R cos ϵ ; and in dimensionless form it is calculated as 



t ¼ ρ sin φ−cosϵ :

ð10Þ

By applying this equation at point E, the auxiliary filling angle (ϵ*) can be computed as 



cosϵ ¼ ρE sin φ E −t E :

ð11Þ

2.4. Velocity and mass flow The particles in the static bed move along with the wall, hence, the no slip condition can be assumed within the plug flow region [24]. Therefore, the velocity of the particles in the passive layer of the flight is only a linear function of the radius at constant angular velocity (ω) vφ ðr Þ ¼ ωr:

a

ð9Þ

ð12Þ

The mass flow of the particles from the bulk static bed to the thin active layer in the differential element of length dr is given by 

dM s ¼ ρb;s vφ ðr Þdr L:

ð13Þ

a

b

Fig. 5. Schematic of the flighted rotary drum to model the cascading layer profile.

b

Fig. 6. Schematic of the flighted rotary drum to model the cascading layer profile in region-II.

K.R. Sunkara et al. / Powder Technology 275 (2015) 161–171

After substituting Eq. (12) in the above equation, the integration of Eq. (13) from r to R results in 

Ms ¼ ρb;s ω  L

ZR r dr; ¼ ρb;s ω 


 R2 −r 2 2

considering ρb,s = ρb,a = ρb [14]. Applying these conditions and transforming into a dimensionless form for the mass flow results in


ð14Þ

:



M ¼

1−ρ2

 ;

2

r  

¼ vx t :

The mass flow in the active layer is calculated as 

Ma ¼ ρb;a  L

165

ð18aÞ ð18bÞ

Finally from the above relation we obtain

Zt vx ðyÞdy;

ð15Þ

0




vx ¼

where vx is the velocity of the particles along the active layer. This velocity varies across the layers and can be assumed to be a simple shear [14, 26–28]. Therefore the velocity profile in the transverse motion distributed linearly with the layer thickness is y vx ð yÞ ¼ 2vx  ; t

ð16Þ

where, vx is the average velocity of the particles in the active layer. Substituting for vx( y) in Eq. (15) results in

2

1−ρ 2t 

ð17Þ

By defining the dimensionless mass flow as M  ¼ ρ



Ms 2 b;s ωR L

¼ρ



Ma

a;s ωR

2

L

:

and from the principle of continuity, Eqs. (14) and (17) become equal (Ṁa = Ṁs). Although the density of the cascading layer is different from the density of bulk static bed the change is neglected by

:

ð19Þ

2.5. Force balance in the active layer Fig. 5a shows a differential element of the layer of length dx, and radial length of dr, the thickness of which increased by dt [29]. This element is inclined to the horizontal at an angle η, and υ is the angle between the surface line and boundary line. The varying gradient of the layer is expressed as



Ma ¼ ρb;a  vx t: L



tanυ ¼ −

dy dt dt  : ¼ ¼ dx dx dx

ð20Þ

The relation between the increase in the radial length and transverse length is written as [29] dr cosðφ þ υÞ ¼− : dx cosυ

Fig. 7. Simulation flow sheet for the motion equations.

ð21Þ

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(5) (1)

(2)

(7)

(3)

(8) (4)

(6)

Fig. 8. Experimental setup of the flighted drum: 1) experimental drum, 2) rectangular flights, 3) gear motor, 4) table, 5) stand, 6) plumb bob, 7) light and 8) camera.

The particles in the flowing layer are subjected to frictional, gravitational, and inertial forces along the transverse direction. The flowing particles are less subjected to centrifugal force, hence it can be neglected. The length related force balance for the differential element is written as X

F ¼ F F þ FN þ FG þ FI ¼ 0

ð22Þ

Fr ¼ ωg R we obtain 2



vx

dvx 1 μ cos η−sin η ¼ 0: þ  m dx Fr μ m sinυ þ cosυ

Differentiating Eq. (18b), a relation can be derived for the mass flow gradient as

as shown in Fig. 5. The normal and frictional forces are given by FN ¼ FG cos η þ FI sin υ; F F ¼ FG sin η−FI cos υ;

ð23Þ

where, η = γ + υ. Gravity acting on the mass of the infinitesimal element dm of the active layer is given by FG ¼ ρb t dx  g;

ð24Þ

tan υ ¼

dvx : dx

  dvx 0 dM 0 dt 0 ¼t:  ¼ vx ;  ¼M ; dx dx dx

Then using Eqs. (18a) and (21) cosðφ þ υÞ ; cosυ ¼ ðρ cosφ−ρ sin φ  tan υÞ;    0 ¼ −x þ y t :

0

ð25Þ

ð26Þ

The coefficient of friction (μ m) varies at each location of the boundary, the calculation of which will be discussed later in Section 2.6. Substituting for FF and FN from Eq. (23) in the above equation, we obtain the following equation for the motion of particles after dividing with ρb t dx and rearranging as dvx g ½μ m cos η−sin η þ ¼ 0: dx μ m sin υ þ cos υ

M ¼ρ



vx ; Rω

ð30Þ

Substitute Eq. (29) in Eq. (30). 0

 y  0  0   −x þ   M −t vx : vx

ð31Þ

Table 1 Bed material properties used for the experiments. Material

ð27Þ

After transforming the above equation into dimensionless form and defining, vx ¼



Taking the differential terms as

M ¼

vx

ð29Þ 

From Coulomb's law of friction, the resultant force is (μm) times the normal force at the boundary F F ¼ μ m  FN :

  dM  1  dvx  −t    : dx dx vx

Further, the mass flow gradient can be expressed as dM ¼ dM  dρ . dx dρ dx

The rolling particles undergo the inertial force along the layer which is given as FI ¼ ρb t dx  vx

ð28Þ

Quartz sand Glass beads

Steel beads

dp

ρb (consolidated)

Θ

(mm)

(kg/m3)

(°)

0.2 0.7 1 2 3 0.8 2

1570 1560 1560 1549 1485 4630 4680

32.4 28.0 25.1 25.6 27.9 29.7 23

K.R. Sunkara et al. / Powder Technology 275 (2015) 161–171

To represent t′ in terms of v′, x substitute M′ from Eq. (32) in Eq. (29) after simplification

Table 2 Parameters operated for the experiments. S. no.

Parameter

Value

1. 2. 3. 4. 6. 7.

D l1/R l2/l1 n fD nF

0.5 m 0.2 1.0, 0.75, 0.375, 0 2, 4, 8 rpm 0.2 12

After simplification of the above equation we obtain an equation of the following form

ð1 þ μ m tan υÞ ¼

ð34Þ

To evaluate v′x, substitute t′ from Eq. (34) in Eq. (33b) and we obtained a quadratic equation  0 2 0 a vx þ b vx þ c ¼ 0:




ð32Þ 0 vx

Substituting for η in Eq. (28) and rearranging yields the following relation

dx

  0 x þ t vx :  y −vx

ð35Þ

The solution of this equation becomes

 x vx þ y t  v0x M ¼− : vx −y

  dv vx x



0

t ¼



0

167

1  ða2 þ a1 tan υÞ; Fr

1   0 0 0 vx vx 1 þ μ m t ¼  a2 þ a1 t ; Fr

ð33aÞ

¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −b− b2 −4ac 2a

;

where a, b, and c can be written as  

a¼

μ m vx t ; y −vx

ð37aÞ

b¼

 μ m vx x a1 t     þ vx −   ; y −vx Frðy −vx Þ

ð37bÞ

ð33bÞ

where

 a x 1 : c ¼ − a2 þ  1  y −vx Fr

a1 ¼ cos γ þ μ m sin γ; a2 ¼ sin γ−μ m cos γ:

a)

/

=

c)

/

=

ð36Þ

b)

d)

=

/

/

=

Fig. 9. Images of the experimental drum showing the calculation of kinetic angle at different flight length ratios: quartz sand (0.2 mm), Fr = 0.0011, nF = 12.

ð37cÞ

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K.R. Sunkara et al. / Powder Technology 275 (2015) 161–171

a

b

c

d

Fig. 10. Kinetic angle of repose versus discharge angle at different flight length ratio for quartz sand (Fr = 0.0011): a) l2/l1 = 1.0, b) l2/l1 = 0.75, c) l2/l1 = 0.375 and d) l2/l1 = 0. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

2.6. Coefficient of friction To determine the motion behavior at the flight surface, it is important to know the coefficient of friction at each position of the layer at a given position of the flight. According to Mellmann et al. [14], the frictional coefficient is considered to vary with the local position of the boundary line between the static bed and the cascading layer. Since the boundary profile does not vary linearly, the friction is not constant along the boundary line. Generally, in drums without flights the coefficient of friction is referred to the coulombic form as the tangent of the dynamic angle of repose at the vortex point (W, see Fig. 3), where there is no interchange of material between the two layers, see Mellmann et al. [14] and [30]

μ m x¼0 ¼ tanΘ: Similarly, in the case of rotary drum with flights assuming an auxiliary vortex point, the coefficient of friction can be written as

μ m x¼0 ¼ tan γ:

μ m x¼x ¼ tan ηA ¼ tanðγ þ υA Þ: A

By taking these points as reference, the friction coefficient is assumed to vary linearly along the boundary line which can be expressed as μm

sffiffiffiffiffi dp Fr; υA ¼ 0:32Θ ð1 þ f D Þ þ 1800 D

ð39Þ

where Θ is in [°] and υA is also in [°]. When δ N γ, it is assumed that υA decreases linearly with the flight position by a factor of sffiffiffiffiffi 3 2 dp δL −δ 4 0:32Θ ð1 þ f D Þ þ 1800 Fr5: υA ¼ D δL −γ

ð40Þ

3. Simulation methodology

Above this point, the friction at the boundary line increases and attains its maximum at the apex position (point A, see Fig. 2), hence, the angle at the boundary line should also increase. Therefore, the friction coefficient at point A is characterized as

x ¼ tan γ þ ½tanðγ þ υA Þ−tan γ ; xA

where υA is the angle between the surface line and boundary line at point A. To predict the coefficient of friction the characteristics of the material υA and γ should be known. Here, υA is a material property which can be measured with the aid of photos collected from the experiments. In the present study, an empirical equation developed by Liu [30] has been used, which is assumed to be independent of flight position if δ ≤ γ

ð38Þ

3.1. Initial conditions The initial positions (xA⁎, yA⁎) to solve the system of equations (Eqs. (32) and (36)) vary for the two regions of flight discharge (Fig. 4) which have been explained in Section 2.2. Therefore, they are described separately for each case in the following. However, the exit points (xE⁎, yE⁎) are unique in both the cases 

xE ¼ −ρE cos φE ;  yE ¼ −ρE sin φE where φE = (γ − δ).

ð41Þ

K.R. Sunkara et al. / Powder Technology 275 (2015) 161–171

3.1.1. Region-I: 0 ≤ δ ≤ (γ + α + βn) The position of point A in Fig. 4a is initially unknown, since the thickness of the layer is unknown. Depending on the layer thickness at the flight tip (δE), this initial point is given as 



xA ¼ −sin ε ;  yA ¼ −sin φ A :

ð42Þ

where   φA ¼ arcsin cos ϵ :

ð43Þ

3.1.2. Region-II: (γ + α + βn) b δ ≤ δL The schematic for the second region is shown in Fig. 6. It is clear from the figure that the boundary points are different from the first region. In the first case the surface line intersects the cylinder wall (see Fig. 4), whereas in the second region it intersects the flight. Therefore, ρA varies for each flight position as it continuously discharges the material from the flight, which can be calculated as follows in Fig. 6 ρA ¼ ρH þ ðh=RÞ; h¼ p¼

l2 þ p ; tan φA

t E t ¼ E : sinð90−φA Þ cosφA

where φA in this case is given by φA = δ − γ − α. The initial conditions for the two dependent variables of Eqs. (36) and (32) are defined as  vx x ¼ ρA ; A 0   M x ¼ vx x  t x : A

A

ð44Þ

A

In order to get proper numerical solution, t  jx is assumed to be a A

thickness equal to the half of one particle diameter [14,29]. Therefore t  jx ¼ dp =D , where dp is the particle diameter and D is the drum A

diameter. 3.2. Solution procedure The motion equations for vx ; M  (Eqs. (32) and (36)) were solved in MATLAB. After including the friction coefficient (Eq. (38)) the differential equations became more complex. Therefore, it was substantially difficult to solve explicitly for γ and t*. One would start by taking some initial guess for both γ and t*. The respective flow sheet to solve these expressions is shown in Fig. 7. For each γ a local convergence for the layer thickness at flight tip (tE) must be accomplished and then using this converged profiles for velocity a new γ can be obtained using Eq. (6). The motion profiles were calculated until a converged solution for both γ and tE was attained. 4. Experimental setup In order to measure the kinetic angle, a laboratory experimental drum of 500 mm in diameter and 150 mm in length has been designed as shown in Fig. 8. The drum was directly fixed at the shaft of a gear motor attached to a base frame standing on a table. The head end of the rotating drum was covered with a transparent glass plate in order to observe the motion of the falling particles and the rear end was a metal wall. The drum was horizontally arranged to maintain uniform axial profiles. A high definition camera focusing on the center of the cylinder was aligned perpendicular to the glass sheet. The data was collected from the images, which were extracted from the recorded videos of the camera. The drum was furnished with rectangular flights arranged

169

with an equal angular distance. The experiments were carried out in a dark room to avoid light reflections on the glass plate. A plumb bob was suspended vertically with the help of a stand at the center of the drum to locate the drum center position and to measure the dynamic and kinetic angles of repose. Three materials were used for the analysis and validation: quartz sand (0.2 mm), glass beads (0.7 mm), and steel balls (0.8 mm, 2 mm). The properties of these non-cohesive materials are given in Table 1. The experimental procedure and the methodology of data evaluation were followed according to the earlier works of Sunkara et al. [31]. The experimental settings are listed in Table 2. 5. Results and discussion The particle flow experiments were performed at different flight length ratios l2/l1 in order to validate the extended model. The data points of the kinetic angle of repose were calculated for a series of frames extracted from the videos of the experiments at different time intervals by focusing on a single flight until the emptying point was reached. The calculation of the kinetic angle from one such image is exemplified in Fig. 9-(d) at different flight length ratios. The center of the drum was located with the use of the plumb line as shown in the figure, which is used to measure the position of the flight. This procedure was repeated for several other images to measure the surface angle at the other positions of the same flight. In order to increase the accuracy, more data were collected from other flights which were randomly selected, and the procedure was repeated for the same experiment. The experimental findings were compared with the model predictions for quartz sand at Fr = 0.0011 as shown in Fig. 10. Fig. 10(a)–(d) represent the effect of the discharge angle on the kinetic angle of repose for different l2/l1 values 1.0, 0.75, 0.375, and, 0 respectively. The model predictions in the figure are represented as solid lines and the measured data as symbols (○). Each data point in the figure corresponds to one single image that has been collected for one flight at a specific position. As the measurements illustrate, the kinetic angle of repose increased by approximately 5° in the initial period of discharge and decreased at higher discharge angles, see Fig. 10(a). As the flight length ratio was reduced, this increase of the kinetic angle declined. At l2/l1 = 0, the kinetic angle of repose even decreased as the discharge angle was increased. As can be seen in Fig. 10, the model from Schofield and Glikin [13] calculated a constant kinetic angle of repose over the whole discharge range which is up to 15° lower than the measured values. This is because only a single particle neglecting the flowing layer at the flight surface. However, the extended model considering the inertial forces at the flight surface according to Eq. (6) well predicted the course of the kinetic angle of repose as the red colored solid lines depict. The slight deflection of the predicted results from the measurements can be attributed to measurement errors, the effect of the glass front wall, and the angle of the flight to the camera position. The percentage error was 5–10% which falls under the practical range. A similar behavior was observed for glass beads under comparable operating conditions as shown in Fig. 11. However, at a flight length ratio of l2/l1 = 0 the tendency of the kinetic angle predicted by the extended model differed from the measured ones. Fig. 12 shows the comparison between the experimental results of the kinetic angle and the predicted results for two different Froude numbers (0.0011 and 0.018). As can be seen in the figure, the kinetic angle of repose increased as the Froude number increased. This effect can be attributed to the centrifugal force pushing the particles in the static bed against the drum wall retarding their transition into the active layer. Thereby, the kinetic angle of repose was increased. The inflection point in the upper curve occurred at δ = γ. This is due to the pre-factor introduced in Eq. (40) for determining the inclination angle υA. The influence of the dynamic angle of repose has been illustrated in Fig. 13 for glass beads and steel balls. The solid lines and the dashed lines represent theoretical values for glass beads and steel balls of different

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a

b

c

d

Fig. 11. Kinetic angle of repose versus discharge angle at different flight length ratios for glass beads: (a) l2/l1 = 1.0, (b) l2/l1 = 0.75, (c) l2/l1 = 0.375 and (d) l2/l1 = 0. Symbols denote measurement values (○ for Θ = 28°, dp = 0.7 mm, ◊ for Θ = 25.1°, dp = 1.2 mm) and solid lines represent model predictions.

particle sizes determined for a Froude number of Fr = 0.0011 respectively, while the symbols denote the experimental values at different flight positions. The figure shows that, as the dynamic angle of repose was increased, the kinetic angle increased over the entire range of flight discharge. A good agreement can be observed between the theoretical and measured values during the initial discharge. Towards the end of the discharge, the model predictions provide higher values of the kinetic angle as compared to the measured values. This can be attributed to Eq. (40) which is over-predicting the angle υA at higher discharge angles

where the flight filling degree is rapidly deceasing. Further investigations are therefore necessary to improve this approach. 6. Conclusions A model was developed for the kinetic angle of repose to extend the model from Schofield and Glikin by incorporating the inertial effects acting on the particles rolling in the flight by considering a thin cascading layer at the flight surface. The Eulerian approach was followed to

50 Θ = 29.7o (SB – 0.8mm) Θ = 27.9o (GB – 3mm)

Kinetiic angle of re epose [o]

45

Θ = 25.6o (GB – 2mm) 40

Θ = 23o (SB – 2mm)

35 30 25 20 0

20

40

60

80

100

120

Discharge angle [o]

Fig. 12. Model validation: Kinetic angle of repose measured at different Froude numbers for l2/l1 = 1.0 as compared to predicted values.

Fig. 13. Influence of the dynamic angle of repose on kinetic angle for glass beads and steel beads for l2/l1 = 0.75. Symbols (○, Δ) represent measured values, lines denote model prediction (Fr = 0.0011).

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determine the velocity profiles and the layer thickness of the active layer at the flight surface. By this approach the mean residence time of the particles during the flight unloading alone can be determined. The influence of flight length ratio, the dynamic angles of repose and the Froude number on kinetic angles was presented. A higher dynamic angle of repose leads to a higher kinetic angle of repose over the entire unloading period of the flight. The experimental data of the kinetic angle are in good agreement with the model during the first region of the unloading period. In the second region of flight discharge, the agreement is good for quartz sand, however, for glass and steel beads the deviations increase due to over-prediction of the kinetic angle. This can be attributed to Eq. (40) which over-predicts the angle υA at higher discharge angles. Further research is necessary to improve the model e.g., by considering particle shape and other parameters characterizing the flowability of the bed materials. References [1] S.J. Porter, The design of rotary driers and coolers, Trans. Inst. Chem. Eng. 41 (1963) 272–280. [2] J.J. Kelly, P.Ó. Donnell, Dynamics of granular material rotary dryers and coolers, ICE Symp. Ser. 29 (1968) 33–41. [3] P.G. Glikin, Transport of solids through flighted rotating drums, Trans. Inst. Chem. Eng. 56 (1978) 120. [4] M.H. Lisboa, D.S. Vitorino, W.B. Delaiba, J.R.D. Finzer, M.A.S. Barrozo, A study of particle motion in rotary dryer, Braz. J. Chem. Eng. 24 (2007) 365–374. [5] O. Ajayi, M. Sheehan, Application of image analysis to determine design loading in flighted rotary dryers, Powder Technol. 223 (2011) 123–130. [6] M. Debacq, S. Vitu, D. Ablitzer, J.-L. Houzelot, F. Patisson, Transverse motion of cohesive powders in flighted rotary kilns: experimental study of unloading at ambient and high temperatures, Powder Technol. 245 (2013) 56–63. [7] C.G.J. Baker, The design of flights in cascading rotary dryers, Dry. Technol. 6 (4) (1988) 631–653. [8] J. Kelly, Flight design in rotary dryers, Dry. Technol. 10 (1992) 979–993. [9] D. Revol, C.L. Briens, J.M. Chabagno, The design of flights in rotary dryers, Powder Technol. 121 (2001) 230–238. [10] F.Y. Wang, I.T. Cameron, J.D. Litsler, Further theoretical studies on rotary drying processes represented by distributed systems, Dry. Technol. 13 (1995) 737–751.

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[11] W. Blumberg, E.-U. Schlünder, Transversale Schüttgutbewegung und konvektiver Stoffübergang in Drehrohren. Teil 2: Mit Hubschaufeln, Chem. Eng. Process. Process Intensif. 35 (1996) 405–411. [12] D. Van Puyvelde, Modelling the hold up of lifters in rotary dryers, Chem. Eng. Res. Des. 87 (2009) 226–232. [13] F.R. Schofield, P.G. Glikin, Rotary dryers and coolers for granular fertilizers, Trans. Inst. Chem. Eng. 40 (1962) 183–190. [14] J. Mellmann, E. Specht, X. Liu, Prediction of rolling bed motion in rotating cylinders, AIChE J. 50 (2004) 2783–2793. [15] D.V. Khakhar, A.V. Orpe, P. Andresen, J.M. Ottino, Surface flow of granular materials: model and experiments in heap formation, J. Fluid Mech. 441 (2001) 255–264. [16] K. Sunkara, Granular Flow and Design Studies in Flighted Rotating DrumsPh.D. thesis OVGU, Magdeburg, Germany, 2013. [17] J. Mellmann, E. Specht, Particle motion, filling degree and surface area of the flying curtain in cascading rotary dryers, EFCE Working Party on Drying, Magdeburg, 2002. [18] J. Mellmann, E. Specht, Mathematische Modellierung des übergangsverhaltens zwischen den Formen der transversalen Schüttgutbewegung in Drehrohren, Teil 1, ZKG Int. 54 (2001) 281–296. [19] X. Liu, E. Specht, J. Mellmann, Experimental study of the lower and upper angles of repose of granular materials in rotating drums, Powder Technol. 154 (2005) 125–131. [20] R.A. Bagnold, Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear, 225 (1954) 49–63. [21] H.M. Jaeger, C. Liu, S.R. Nagel, T.A. Witten, Friction in granular flows, Europhys. Lett. 11 (1990) 619–624. [22] J.M.N.T. Gray, Granular flow in partially filled slowly rotating drums, J. Fluid Mech. 441 (2001) 1–29. [23] A. Lee, M.E. Sheehan, Development of a geometric flight unloading model for flighted rotary dryers, Powder Technol. 198 (2010) 395–403. [24] A. Boateng, Boundary layer modeling of granular flow in the transverse plane of a partially filled rotating cylinder, Int. J. Multiphase Flow 24 (1998) 499–521. [25] J. Perron, R. Bui, Rotary cylinders: transverse bed motion prediction by rheological analysis, Can. J. Chem. Eng. 70 (1992) 223–231. [26] D.V. Khakhar, J.J. McCarthy, T. Shinbrot, J.M. Ottino, Transverse flow and mixing of granular materials in a rotating cylinder, Phys. Fluids 9 (1997) 31–43. [27] T. Elperin, A. Vikhansky, Granular flow in a rotating cylindrical drum, Europhys. Lett. 42 (1998) 619–624. [28] Y. Ding, R. Forster, J.P.K. Seville, D.J. Parker, Scaling relationships for rotating drums, Chem. Eng. Sci. 56 (2001) 3737–3750. [29] J. Mellmann, Zonales Feststoffmassestrommodell für flammenbeheizte Drehrohrreaktoren. Ph.D. thesis TU, Magdeburg, Germany, 1989. [30] X. Liu, Experimental and Theoretical Study of Transverse Solids Motion in Rotary KilnsPh.D. thesis Otto-von-Guericke-Universität Magdeburg, Germany, 2005. [31] K.R. Sunkara, F. Herz, E. Specht, J. Mellmann, Influence of flight design on the particle distribution of a flighted rotating drum, Chem. Eng. Sci. 90 (2013) 101–109.