Striking outwards reed OSCILLATION THRESHOLDS FOR "STRIKING OUTWARDS" REEDS COUPLED TO A RESONATOR

.

INTRODUCTION

In an important paper [START_REF] Wilson | Operating modes of the clarinet[END_REF] studied the oscillation thresholds for a clarinet-type reed exciting a cylindrical duct, using both theory and experiment. They especially showed that the control of oscillations with a playing frequency close to the a duct resonance requires a sufficient damping of the reed. Moreover they showed that the playing frequency is always smaller than the first reed resonance. More recently, studies have been published concerning a more general class of reeds, with a similar theory and new experimental results [START_REF] Tarnopolsky | Oscillating reed valves-an experimental study[END_REF][START_REF] Cullen | Brass Instruments : Linear Stability Analysis and Experiments with an Artificial Mouth[END_REF]].

In the present paper we will apply the method of Wilson and Beavers to the case of a "striking outward" reed, and exhibit major differences with a "striking inward" one. This kind of reed corresponds to the most common model of a lip-type valve, when the valve is assumed to be a one degree of freedom oscillator (for a discussion, see e.g. [START_REF] Cullen | Brass Instruments : Linear Stability Analysis and Experiments with an Artificial Mouth[END_REF]). Our goal is to derive approximate expressions, exhibiting some basic properties of such reed-tube system. We first present the model, then the characteristic equation to be solved, finally analytical and numerical results.

MODELLING A STRIKING OUTWARDS REED

Figure 1 : schematic representation of a striking outwards reed (after [START_REF] Fletcher | The Physics of Musical Instruments[END_REF] The model we use is very classical, and extremely simplified. A one degree of freedom oscillator is submitted to pressure forces : a pressure m p in the mouth, assumed to be constant, and an acoustic pressure p at the input of a resonator. This is described by the following equation:

µ ω ω p p y dt dy q dt y d m r r r - = + + 2 2 2 .
(1)

where y(t) is the reed displacement, q r the damping parameter, r ω the angular eigenfrequency, and µ the mass per unit length of the reed. For a lip-reed, the displacement has a vertical component, therefore the behavior is much more complicated than this of the spring shown in Fig. 1. Nevertheless we assume the model to be valid. We emphasize that in static regime, if the pressure p is zero, the pressure m p needs to be negative for the reed to be closed. Following [START_REF] Wilson | Operating modes of the clarinet[END_REF], see also [START_REF] Kergomard | Elementary considerations on reed-instruments oscillations[END_REF], we chose to use dimensionless variables. The origin for y(t) being the position when p m =p=0, we define -H as the position and -p M as the pressure for which the reed is closed, H and p M being positive quantities :

H p r M 2 µω = .
Writing γ= p m /p M , x=y/H -γ and replacing p/p M by p, Eq. (1) becomes :

p x dt dx q dt x d r r r - = + + 2 2 2 2 1 ω ω . ( 2
)
The following equation is nonlinear : it is the Bernoulli equation, written with certain hypotheses, relating the volume velocity to the opening area of the reed and the pressure difference. We write it directly using dimensionless variables and parameters :

p x u - + + = γ γ ζ ) 1 ( ( 3
)
where ζ is related to both the reed opening and stiffness :

M c p wH Z ρ ζ 2 =
. wH is the reed opening at rest, Z c = ρc/S the characteristic impedance of the tube. u is the volume velocity multiplied by Z c /p M . These equations are in accordance with the equations given by Wilson and Beavers, but two signs have to be changed, in order to consider an outwards instead of an inwards reed. Finally, the last equation gives the 2) to ( 4) give the complete model. In order to get the characteristic equation, we need to rewrite (2) in the frequency domain, and to linearize Eq. (3) :

) ( ) ( ) ( ω ω ω P Y U = (4) Equations (
) ( ) ( ) ( ω ω ω P D X - = where 2 2 1 1 ) ( r r r q j D ω ω ω ω ω - + = - . (5) and       - + + + = γ γ γ γ ζ 2 1 1 ) 1 ( p x u
. ( 6)

CHARACTERISTIC EQUATION

Substracting the d.c. component of ( 6), and using Eqs. ( 4) and ( 5) leads to the characteristic equation :

      + - - = γ γ ω γ ζ ω 2 1 ) ( ) ( D Y . ( 7
)
It can be compared to the equation for the case of an inwards reed :

.

2 1 ) ( ) (       - - = γ γ ω γ ζ ω D Y
The latter equation is in accordance with [START_REF] Wilson | Operating modes of the clarinet[END_REF]). In both cases the volume velocity due to the reed movement is ignored (see [START_REF] Kergomard | Analyse de quelques aspects du rôle de l'anche d'un instrument à vent cylindrique[END_REF]). The characteristic equation can be solved for a given set of parameters, searching for a complex frequency. This has be done numerically, and the instability threshold of the static regime can be deduced as follows : when the imaginary parts of all solutions but one are positive, all solutions but one are exponentially decreasing, and the instability threshold corresponds to one real frequency, obtained when one parameter varies ζ, γ, ω r ., etc… We do not discuss here the method for solving such equations, and give in the last section some examples of results.

We notice that the instability threshold of the static regime is not always the oscillation threshold : it is true only for direct bifurcation (see e.g. [START_REF] Grand | Oscillation threshold of woodwind instruments[END_REF]). This question is important, because for an inverse bifurcation, the pressure threshold can be slightly different, but hopefully the playing frequency does not vary too much with the parameters such as the blowing pressure γ. This question remains to be investigated.

ANALYTICAL APPROXIMATION

Solving Eq. ( 7) for real frequency gives the instability thresholds, as done by Wilson and Beavers. In order to get some basic results, we first ignore the losses in the resonator, so that Y(ω) is purely imaginary. Eq. ( 7) is split into two equations, written for the particular case of a cylindrical tube :

η η γ + - = 2 (8) ; ) 1 ( cot 2 ϑ η ϑ γ ζ ϑ - - = r r q L k (9) where 2 2 2 2 1 1 ϑ ϑ ϑ η - + - = r q .
L is the length of the resonator, c k . For a given resonator length, the two equations can be solved for the two unkowns, the mouth pressure γ and the playing frequency ϑ, at the threshold. The parameter γ can be eliminated from these equations, and the following equation is obtained for ϑ :

) ( ) Im( ϑ ζ G q Y r = where [ ] [ ] 2 / 1 2 2 2 2 2 / 1 2 2 2 2 ) 1 )( 3 ( ) 1 ( ) ( - - - - - - + - = ϑ ϑ ϑ ϑ ϑ ϑ ϑ r r q q G
(10). The last quantity in bracket, in order for the square root to be real, needs to be positive. As a consequence, the playing frequency lies within the following range :

2 1 ϑ ϑ ϑ < < where 2 1 4 1 1 r q + ≈ ϑ and ) 4 1 1 ( 3 2 2 r q - ≈ ϑ . ( 11
)
This important result shows, thanks to this simple model, that for a given reed eigenfrequency, the maximum range for the playing frequency is less than one musical sixth. Is it very different for the case of an inwards reed, for which the range lies between 0 and the reed frequency. There is no "duality" between the two kinds of reed. Moreover, in practice, the interval is shortened because the possibility to get a sound depends on the value of the pressure threshold for a given set of parameters. We notice that the existence of square roots in ( 10) is related to the Bernoulli equation.

Concerning the pressure threshold, taking now resonator losses into account, it can be shown to depend on both reed damping and tube losses, but the minimum of the threshold when the length of the resonator varies depends mainly on reed damping. This can be seen on the following equation :

      + ≈ 0 0 min 2 1 γ ζ γ γ n Y where       + ≈ r r q q 2 3 1 0 γ . ( 12
)
Y n is a minimum (real) value of the input (reduced) admittance of the resonator for a given mode (index n). As a consequence, losses in the tube can increase significantly the mimimum of pressure threshold, especially when the parameter ζ is small. We do not give here neither the derivation of Eq. ( 12) nor all the analytical results, which will be given in a future article.

NUMERICAL RESULTS

In what follows, we consider a resonator with a single mode and losses, i.e. with the following expression for the input admittance :

) ( 1 2 2 ω ω ωω - - = n n n n n j Y jQ Y Y .
Solving Eq. ( 7) by searching for complex frequencies, we are able to get results for various values of the parameters. Figure 2 shows an example of result for the frequency and pressure thresholds, with respect to the ratio

n r n ω ω ϑ / 1 = - .
When the length or the resonator increases (ω r decreases), the threshold frequency decreases to ϑ 1 , close to unity (see Eq. ( 11)), while when the length decreases, the threshold frequency increases to ϑ 2 , close to 3 , and cannot be larger because the threshold pressure tends to infinity. For the chosen values of parameters q r and ζ , the practical range of possible resonator eigenfrequencies is very narrow, and the minimum value of the threshold pressure lies just above q r . The influence of the losses in the resonator (Z n =1/Y n ) is very small because parameter ζ is not very small.

Figure 3 shows the same quantities for the case of an striking inwards reed, with the same values of the parameters. For large length of the resonator, the Figures 2 and 3: results for f r =440Hz ; q r =0.100 ; ζ=0.3, and different values of the impedance peak of the resonator. The abscissa is the ratio of the reed frequency to the resonator frequency. The upper and lower curves represent the ratio of the threshold frequency to the reed frequency, and the threshold pressure, respectively. Fig. 2(above) represents the outwards case, while Fig. 3(below) represents the inwards one.

threshold frequency reaches the resonator frequency, the curve becoming hyperbolic, and the pressure threshold tends to a value above 1/3, depending on the losses in the resonator. These properties are well known. For small resonator length, the threshold frequency tends to the reed frequency, but cannot become larger, because the reed becomes beating (γ>1). A similar behavior should be found for the outwards case if negative blowing pressures were considered. Finally the minimum pressure threshold lies just below q r : one can show that

      - = r r q q 2 3 1 0 γ
. It is smaller than the value for the outwards case. This comparison between the outwards and inwards cases are in qualitative accordance with the work by [START_REF] Cullen | Brass Instruments : Linear Stability Analysis and Experiments with an Artificial Mouth[END_REF].

Returning the the outwards case, Figure 4 shows the effect of reed damping, which can be compared to the inwards case (see [START_REF] Wilson | Operating modes of the clarinet[END_REF]) : in both cases weak damping makes easier the oscillation near the minimum value of the threshold pressure, and leads to a threshold frequency closer to the reed frequency. Figures 2 to 4 show results in a similar form than these of Wilson and Beavers. Figure 5 shows results in another form for the case of a cylindrical tube with a fixed length, L, and vaying reed frequency. The effect of the different modes of the resonator can be observed: using the ability to change his lips resonance frequency, the player can keep in tune by bending upward or downward the playing frequency. There are limitations to this action: first, as it can be seen on the upper curve, there exist frequency gaps, i.e. frequencies ranges that cannot be reached without varying the bore length. Then, especially at low frequencies for some values of ω r , the oscillation threshold is very high, so that a brass player is not able to produce such mouth pressure. Finally, in the higher register, the sensitiveness of playing frequency to lip resonance frequency decreases. Tuning by means of the lips only is more difficult in higher registers than in lower ones, this result corresponding to the feeling of a brass player. . CONCLUSIONS

The present work is based upon previous works, without any novelty concerning the model. What is new is probably the possibility to get analytical results, allowing to understand some essential features, such as the values of the minimum threshold pressures, and the maximum possible value of the threshold frequency compared to the reed eigenfrequency. Further work needs to examine the nature of the bifurcation, using the first harmonic approximation, or better, the small oscillation approach [START_REF] Grand | Oscillation threshold of woodwind instruments[END_REF].

  the tube, written in the frequency domain (we use capital letters for variables in the frequency domain) :

Figure 4 :

 4 Figure4: : results for f r =440Hz ; ζ=0.3 ; Z n =50, and different values of q r .. The abscissa is the ratio of the reed frequency to the resonator frequency. The upper and lower curves represent the ratio of the threshold frequency to the reed frequency, and the threshold pressure, respectively.

Figure 5 :

 5 Figure5: results for a fixed length L=1m ; q r =0.3 ; ζ=0.2. The abscissa is k r L, proportional to ω r . The upper curve gives the threshold frequency, the lower curve gives the threshold pressure.. CONCLUSIONS


Alex Stéphane

Bongo Njeng

Stéphane Vitu

Marc Clausse

Jean-Louis Dirion

Marie Debacq

A S Bongo Njeng

J.-L Dirion

Effect of Lifters Shape and Operating Parameters on the Flow of Materials in a Pilot Rotary Kiln: Part I. Experimental RTD and Axial Dispersion Study

Keywords: Rotary kiln, Rotary dryer, RTD, Axial Dispersion, Lifters, Lifting ights

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Through the years, rotary kilns, also often called rotary dryers, have become an inescapable benchmark in several industrial sectors. This is because they oer special possibilities regarding the wide range of materials which can be continuously processed with little or no labor to operate, especially when they are under automatic control. As wider are the scopes of applications as wider are the kilns design and size. Rotary kilns range from small cylinders with length-to-diameter ratio (L/iD) below 5 used mostly for mixing granular materials in the pharmaceutical manufacture; medium cylinders with 1 INTRODUCTION 2 L/iD up to 20, used for the drying or cooling of solid particles, for instance in the food processing industry when dealing with wet or dry granular materials; and long cylinders with L/iD above 20, often used at high temperature for calcination of limestone, petroleum coke, or production of cement. Other applications include: regeneration of spent catalyst, hazardous waste reclamation, deuorination and reduction of uranyl diuoride and so on. Kilns are usually equipped with internal xtures such as lifters, baes, constrictions or exit dam.

The complex hydrodynamic behavior of the granular materials within rotary kilns has led to extensive research over the years. Among the rst to investigate this subject, Sullivan et al. [START_REF] Sullivan | Passage of solid particles through rotary cylindrical kilns[END_REF] and others later [START_REF] Sherritt | The movement of solids through ighted rotating drums. Part II solids-gas interaction and model validation[END_REF], focused on the time of passage of solid particles. This has been the simplest approach to get some understanding of the solids transport.

However, this method can be inaccurate with regard to the distribution of the times that solid particles will undergo inside the kiln. Variations of times which may be observed are therefore represented by the residence time distribution (RTD) at steady state ow.

Because the knowledge and a good characterization of the RTD are necessary for the design, improvement and scale-up of manufacturing process, there have been some recent published works on the experimental and theoretical analysis of the solids transport using the RTD notion. Most of these publications report on the inuence of operating parameters (rotational speed, kiln slope, mass ow rate or exit dam height) [620] and others on the inuence of the bulk or tracer materials properties [START_REF] Sai | Residence time distribution and material ow studies in a rotary kiln[END_REF], or segregation phenomena [START_REF] Hatzilyberis | An Rtd Study for the Flow of Lignite Particles Through a Pilot Rotary Dryer Part Ii: Flighted Drum Case[END_REF][START_REF] Abouzeid | Flow of Non-Homogeneous Particulates in Rotating Drums[END_REF][START_REF] Bensmann | Residence Time Distribution of Segregating Sand Particles in a Rotary Drum[END_REF]. Fewer are those analyzing the eect of presence of lifters, their number or shape [START_REF] Venkataraman | Eect of lifter shape and conguration on material transport part I: In rotating drums[END_REF][START_REF] Hatzilyberis | An Rtd Study for the Flow of Lignite Particles Through a Pilot Rotary Dryer Part Ii: Flighted Drum Case[END_REF][START_REF] Li | Axial transport and residence time of MSW in rotary kilns: Part I. Experimental[END_REF]. However, some authors have focused on the study of lifters hold-up and discharging rate [START_REF] Schoeld | Rotary dryers and coolers for granular fertilizers[END_REF]2231].

RTD experiments usually involve tracer detection techniques. The stimulus response test is a common method used for measurements of the RTD [START_REF] Gao | A review of the Residence Time Distribution (RTD) applications in solid unit operations[END_REF][START_REF] Martin | Interpretation of residence time distribution data[END_REF]. In this test, the injection of either a step change or a pulse of tracer is performed at the inlet end. The tracer materials must share similar physical properties with the bulk materials. The tracer detection can be either inline by means of inline probes, therefore requiring fast sample acquisition, or oine, in which case the detection is performed either optically (uorescence analysis, spectroscopy, near infrared and ultraviolet), by measuring the conductivity, or by counting and weighing.

Several models have been used and developed to describe the experimental RTD of solid particles through the kiln. The most popular used is the axial dispersion model.

In addition to the mean residence time, this model provides a dispersion parameter, able to characterize the material transport behavior. A similar model is the tank in series model. These two models are usually called one-parameter models, as opposed to the multi-parameter models. Among the multi-parameter models are: the modied Cholette-Cloutier model used by Duchesne et al. [START_REF] Duchesne | Modeling of the Solids Transportation within an Industrial Rotary Dryer: A Simple Model[END_REF], and the models developed by Mu and Perlmutter [START_REF] Mu | The mixing of Granular solids in a rotary cylinder[END_REF], Sheehan et al. [START_REF] Sheehan | Transportation and axial dispersion of sugar in ighted rotary dryers[END_REF], Dinesh V and Sai [START_REF] Dinesh | A Model for Residence Time Distribution of Solids in a Rotary Kiln[END_REF], Sheehan et al. [START_REF] Sheehan | A model for solids transport in ighted rotary dryers based on physical considerations[END_REF], Britton et al. [START_REF] Britton | A physical description of solids transport in ighted rotary dryers[END_REF]. Generally these one-parameter or multi-parameter models can give quantitative predictions about the materials transport behavior, thus giving an insight on the deviation from the two ideal extremes. The ow in rotary kilns are likely closed to that of plug ow reactors. This paper is concerned with the eects of kiln rotational speed, kiln slope, mass ow rate, exit dam height, lifters shape and bulk materials properties on: the RTD of solid particles, determined from experimental stimulus response test; Predictions of other published correlations are also compared to the experimental results.

Materials and methods

Apparatus and materials

A pilot scale rotary kiln was used to carry out the investigation of solid particles motion. The layout of the experimental system is shown in Fig. 1. The main parts of this set up are the feeding system and the rotating kiln followed by a recovery zone.

Ṁ [kg.h -1 ]
0.68-2.5

The feeding system is mainly a combination of a hopper and a feed screw. The hopper acts as a storage where the bulk materials are manually poured. The solid particles ow by gravity from the bottom of the hopper through a feed screw. This later delivers the solid particles to a feed chute ending up at the kiln inlet end. There is no possible access to recover the solids at the kiln inlet end. Therefore the feed rate is controlled at the kiln outlet and regulated by adjusting the rotational frequency of a 180W DC motor, which drives the screw feeder by means of a belt drive. Thus the feed rate measurements were achieved when steady-state ow conditions were established.

The rotary kiln is a tube made of an Incoloy alloy 800 and measures 1.95 m in length and 0.101 m in (internal) diameter. For the purpose of the residence time distribution study, the smooth inner wall can be equipped with a network or a structure of lifters when necessary. Each structure is tted with 4 identical lifters equally distributed in the periphery. Two shapes of lifters were used: straight one-section lifters of 10 mm refer to as straight lifters (SL) and two-section lifters of 10 mm each, with a right angle cross section refer to as rectangular lifters (RL) as shown in Fig. 1. The kiln tube is supported on 4 rollers, and is driven through chain and sprocket and a variable speed 180W DC motor. This setting allows kiln operation at speeds ranging from 2 to 12 rpm. At the discharged end, constriction dams of dierent height can be installed in order to restrict the ow of solids, so that the lling degree within the kiln varies irrespectively of the solids ow rate or kiln rotational speed. The kiln slope is measured using an electronic level. The slope can be modied by means of an hydraulic jack with manual pump and an adjustment device. So, the kiln unit can be tilted to an angle of 5°. Table 1 summarizes the geometrical and operational characteristics of the kiln in this study.

In order to investigate the motion of solid particles of dierent properties within the kiln, two granular solids were used as bulk materials: (1) dry nodular sand characterized by a 1422 kg.m -3 bulk density, a mean particle size of 0.55 mm and an angle of repose of 39°; (2) broken rice characterized by a 889 kg.m -3 bulk density, a mean particle size of 3.8 mm in length and 1.9 mm in diameter, and an angle of repose of 36°. Table 2 shows some of the most important properties of the materials used. The bulk density was determined by measuring the weight of materials which occupied a volume of 100 mL in a graduated cylinder. In order to determine the tapped density this latter was compacted by hand tapping the measuring cylinder about a minute, approximately 200

Experimental technique

The experiments performed in this study were conducted at ambient temperature with a procedure mostly similar to Abouzeid et al. [START_REF] Abouzeid | The inuence of operating variables on the residence time distribution for material transport in a continuous rotary drum[END_REF]. The RTDs were determined by stimulus response tests using dyed sand and sodium chloride as tracers for sand bulk ow, and dyed rice as tracer for rice bulk ow. Depending on the bulk materials, the procedure may dier in few points as follows:

Step 1: Before starting a run, the variable parameters (N, S, Ṁ) are set to the desired values, the suitable lifters and exit dam are installed for the experiment. The rotary kiln, operating at atmospheric pressure, is then started and the feed hopper regularly fullled with the operating solids to keep it topped up till the end of the run.

Step 2: The system is run until it reaches steady-state conditions, usually after 4 hours. The steady state is assumed to be reached when at least three consecutive measurements of the ow rate at the kiln outlet are almost equal within a margin of ±0.05 kg.h -1 .

The ow rate was measured by weighing the materials collected at the kiln outlet during a period of 15 min.

Step 3: In the case of a bulk of sand, a preliminary test was performed with dyed sand tracer to determine approximately the starting and stopping time of the sampling. Then a known amount of NaCl tracer was injected in the feed chute at an arbitrary zero time.

Tracer injections lasted about 2 sec, which have been assumed short enough (with regard to measured residence times) to approximate an impulse input. Samples of 5

g were collected at the kiln outlet at constant time intervals of 30 sec. In the case of a bulk of broken rice, a known amount of dyed broken rice was injected in the feed chute at an arbitrary zero time. Samples were continuously collected at the kiln exit end with a sampling time of 30 sec until all tracer materials had discharged.

Step 4: The rotary kiln rotation was stopped and the screw feeder disabled at the same time.

Only the kiln rotation was started again and the solids could be discharged. The collected solids which constitute the kiln hold-up were weighted.

Step 5: Eventually, in the case of a bulk of sand, each collected sample was weighted, then mixed with 20 mL of distilled water in order to dissolve the NaCl particles if present.

The sample conductivity, which is proportional to the concentration of diluted NaCl tracer, was then measured. Whereas in the case of a bulk of broken rice, tracer The choice of tracer is important. Indeed, selected tracers were expected to share similar properties with the bulk materials, thus introducing as few disturbances as possible on the bulk ow. For the bigger particles size (broken rice) dyed particles were chosen, otherwise except for the color the tracer materials were identical to the bulk. And for the small particle size (sand), the tracers were sodium chloride particles of same size rather than dyed particles due to time consuming samples analysis. No less importantly, the amount of tracer to be injected should be well dened to provide the desired accuracy for analysis [START_REF] Abouzeid | The inuence of operating variables on the residence time distribution for material transport in a continuous rotary drum[END_REF]. Therefore experiments were conducted with dierent amount of dyed broken rice and resulting dimensionless residence time distribution were analyzed to assist decision-making. As shown in Fig. 2 for amount of tracer varying from 7.4 g to 15 g the dimensionless RTD was not signicantly aected, making easier the decision. Since the sampling analysis lasted longer with increasing amount of tracer, 7.4 g of tracer were used in experiments involving broken rice. To keep approximately a similar volume of tracer in all experiments, in the case of sand 10-12 g of tracer were needed. For a more accurate data 12 g of tracer were used in experiments involving sand.

Data processing

Data evaluation

When performing a stimulus test, the fraction of tracer that remains in the system at any time is described by the distribution function E(t) as follows :

E(t) = C(t) ´∞ 0 C(t)dt (1)
where C(t) represents the tracer concentration at the kiln exit end and the time integral of C(t) represents the total tracer concentration. In this study, for each sample the tracer concentration (or conductivity) was yielded, so that from the discrete sample, and associated sampling time ∆t i , i={1, 2, 3,..., N s }, the RTD curve or E-curve of each experiment can be determined as follows [START_REF] Levenspiel | Chemical reaction engineering[END_REF]:

E(t i ) ∼ = C(t i ) N S i C(t i )∆t i (2)
The Interpretation of the RTD curves is based on moment analysis which provides various aspect of the ow in the unit. The rst moment of the RTD function gives the mean residence time (MRT) t which is dened by [START_REF] Levenspiel | Chemical reaction engineering[END_REF]:

t = ´∞ 0 tC(t)dt ´∞ 0 C(t)dt ∼ = N S i t i C(t i )∆t i N S i C(t i )∆t i (3)
The second moment of the RTD function, σ 2 , gives the variance of residence times (VRT) about the MRT. It indicates the width or scatter of the distribution: the greater the value of this moment, the greater the distribution spread. This moment is dened as follows [START_REF] Levenspiel | Chemical reaction engineering[END_REF]:

σ 2 = ´∞ 0 (t -t) 2 C(t)dt ´∞ 0 C(t)dt ∼ = N S i t i 2 C(t i )∆t i N S i C(t i )∆t i -t2 (4)
Since the sampling time was constant and equal in all experiments, the sampling intervals in Eqs.3 and Eqs.4 can be simplied. The presented E(t) function and VRT can also be expressed in dimensionless form:

θ = t t (5) E(θ) = tE(t) (6)
σ 2 θ = σ 2 t2 (7)
respectively the dimensionless time, mean residence time and variance of residence time.

Axial dispersion model

The axial dispersion model is used to represent the RTDs. Originally used by Danckwerts [START_REF] Danckwerts | Continuous ow systems. Distribution of residence times[END_REF], this model considers the axial motion of uid element as two components [START_REF] Martin | Interpretation of residence time distribution data[END_REF]:

a convective component arising from the bulk motion of the uid and a diusive component arising from the random motion of solids in response to the decay of turbulent eddies. This is represented by the following one-dimensional Fokker-Planck equation, which describes the evolution of particle distribution in continuous systems [START_REF] Risken | The Fokker-Planck equation: methods of solution and applications[END_REF][START_REF] Sherritt | Axial dispersion in the three-dimensional mixing of particles in a rotating drum reactor[END_REF]:

∂C ∂θ = 1 P e ∂ 2 C ∂z 2 - ∂C ∂z (8)
D uL = 1 P e (9)
θ = t t = ut L ( 10
)
where D is the axial dispersion coecient in m 2 .s -1 , Pe is the dimensionless Peclet number, z is dimensionless and dened as the ratio of the axial position to the kiln length L, t and t are respectively the time and the MRT in s, u is the velocity in m.s -1 . This continuum model has been applied in rotary kilns [START_REF] Abouzeid | The inuence of operating variables on the residence time distribution for material transport in a continuous rotary drum[END_REF][START_REF] Venkataraman | Eect of lifter shape and conguration on material transport part I: In rotating drums[END_REF][START_REF] Lebas | Experimental study of residence time, particle movement and bed depth prole in rotary kilns[END_REF][START_REF] Sudah | Quantitative characterization of mixing processes in rotary calciners[END_REF][START_REF] Gao | Characterizing continuous powder mixing using residence time distribution[END_REF][START_REF] Sherritt | Axial dispersion in the three-dimensional mixing of particles in a rotating drum reactor[END_REF][START_REF] Gupta | Axial transport of granular solids in horizontal rotating cylinders[END_REF] to describe the transport of granular solids providing that some assumptions are satised. Among the most important [START_REF] Abouzeid | The inuence of operating variables on the residence time distribution for material transport in a continuous rotary drum[END_REF][START_REF] Sudah | Quantitative characterization of mixing processes in rotary calciners[END_REF][START_REF] Sheehan | Transportation and axial dispersion of sugar in ighted rotary dryers[END_REF]:

(i) Steady-state conditions must be achieved and maintained over the tracer stimulus test.

(ii) A delta-dirac tracer pulse insuring that tracer concentration is only function of time and axial position.

(iii) Constant axial convective velocity and axial dispersion coecient of tracer for stable operating conditions Analytical solutions for the RTD were published by several authors [START_REF] Abouzeid | The inuence of operating variables on the residence time distribution for material transport in a continuous rotary drum[END_REF][START_REF] Karra | Material transport in a continuous rotary drum. Eect of discharge plate geometry[END_REF][START_REF] Levenspiel | Chemical reaction engineering[END_REF][START_REF] Thomas | Longitudinal mixing in aeration tanks[END_REF] for the dierent type of boundary conditions. In this study the ow of materials was undisturbed at the inlet and outlet, so that the open-open boundary condition is considered for the system.

The analytical solution for Eq.8, acknowledging the open-open boundary condition, published by Levenspiel [START_REF] Levenspiel | Chemical reaction engineering[END_REF] is given in a dimensionless form as follows:

E(θ) = 1 2 P e πθ exp - P e(1 -θ) 2 4θ (11)
The mean and variance of this distribution are dened as:

too = 1 + 2 P e (12)
σ 2 θ,oo = 2 P e + 8 P e 2 (13)
There are two main methods to evaluate the Peclet number: the simplest one is to match the measured variance (Eq.4) to theory (Eq.13), the other one imply to t the experimental E-curve using the dispersion model (Eq.11). These two methods are considered in the coming section to evaluate the Peclet numbers and resulting axial dispersion coecients for varying experimental conditions.

Results and discussion

In this study, benchmark values of a range of operating conditions were dened, so that every operating parameter was kept constant except the one being studied. Whatever the materials being processed, in presence or not of lifters (SL or RL), the benchmark values were a rotational speed of 3 rpm, a kiln slope of 2°, a mass ow rate of 2.5 kg.h -1 (±0.05 kg.h -1 ) and a 23.5 mm exit dam height. The rotary kiln was either over-loaded or designloaded during the experiments. Hence a set of 69 experiments was performed. The main purposes of these latter being to investigate: (1) the residence time entire distribution,

(2) the mean residence time of the distribution which is discussed elsewhere separately [START_REF] Bongo Njeng | Eect of Lifters Shape and Operating Parameters on the Flow of Materials in a Pilot Rotary Kiln : Part II[END_REF], (3) the variance of residence time, (4) the validity of the axial dispersion model, [START_REF] Schoeld | Rotary dryers and coolers for granular fertilizers[END_REF] the Peclet number and axial dispersion coecient.

In order to test the reproducibility, an additional 20 experiments were performed. Some of these experiments are reported in Table 3. Fig. 4 displays some repeated experiments with a bulk of broken rice while using straight, rectangular, or no lifters at 10 rpm with other parameters kept at the benchmark values. From this, it is clear that the reproducibility is good, but greater variations were observed at high rotational speed, thereby supporting the chosen benchmark value. These variations derived from a variety of sources [START_REF] Abouzeid | The inuence of operating variables on the residence time distribution for material transport in a continuous rotary drum[END_REF] ranging from the feeding system to the sampling analysis and including the tracer injection and discharge sampling procedures. Results from the overall replicated experiments showed that the coecients of variation of the mean residence time and the variance of RTD are on average 1.2% and 16.2%, respectively. In light of this, results from the experimental campaign were considered as suciently reliable for the analysis of the inuence of operating parameters upon the materials transport in a rotary kiln equipped with lifters.

Axial dispersion model

The axial dispersion model presented above was used to represent the time dependent E-curves. Therefore, to determine the RTD equation, Eq.11 is rewritten in dimensional form using Eq.5 and Eq.6 as follows:

E(t) = 1 2 P e π tt exp - P e( t -t) 2 4 tt (14)
While the dimensionless form of the model needs only a unique parameter namely the Peclet number, in its dimensional form there is a second parameter to determine: the mean residence time of the distribution. As explained before there are two methodologies.

Both of them are considered:

Case 1: The MRT is obtained from experimental data using Eq.3. The Peclet number is obtained from the theoretical expression described in Eq.13 using the dimensionless VRT obtained from experimental data using Eq.7.

Case 2: The parameters are determined to minimize the mean sum of square error between the experimentally determined RTD values using Eq.2 and the predicted values from Eq.14. Parameters for the best tted curve were calculated through a MatLab script using the curve tting tool function.

Parameters deriving from the above methodologies are used in Fig. 7, which shows the inuence of operating parameters on the RTD of broken rice, to investigate the validity of the axial dispersion model. The simulated curves using the case 1 parameters, referred to as theoretical parameters, are dashed lines. The simulated curves using the case 2 parameters, referred to as tted parameters, are solid lines. It appears clearly that both sets of parameters give good agreement with experimental data; that was the case for experiments of the whole campaign. However, as one would have expected, tted parameters lead to better results than theoretical parameters. Although theoretical parameters are simple to obtain, they are not necessarily the best choice. A comparison of theoretical and tted parameters is given in Fig. 3. Experimental and tted MRT are perfectly correlated (see Fig. 3a), whereas theoretical and tted Peclet numbers exhibit some discrepancy. Consequently, it is recognized that the theoretical Peclet numbers reported in this study are likely underestimating the actual Peclet numbers as suggested in Fig. 3b. However, with regard to the validity of the chosen model, one may keep in mind that some irregularities can occur at inlet and outlet ends of the kiln, reducing the model accuracy.

t [min] E [min -1 ] NL-n°1 NL-n°2 NL-n°3 SL-n°1 SL-n°2 SL-n°3 RL-n°1 RL-n°2 RL-n°3
Fig. 4: Eect of presence and shape of lifters on the RTD for the ow of broken rice. Operating conditions: 10 rpm rotation speed, 2°slope, 2.5 kg.h -1 MFR , 23.5 mm exit dam height. In each case (NL, SL and RL), the experiment was repeated three times. Solid lines are the axial dispersion model of the n°1 replicated run with tted parameters.

The Peclet number expressed the ratio of the axial convection forces to the dispersion forces as given in Eq.9. Subsequently it is possible to obtain the axial dispersion coecient from theoretical and tted Peclet number. The calculated axial dispersion numbers were found in the range of 10 -7 to 10 -4 m 2 .s -1 highlighted by Sherritt et al. [START_REF] Sherritt | Axial dispersion in the three-dimensional mixing of particles in a rotating drum reactor[END_REF] for kiln diameters ranging from 0.076 to 0.144 m not always equipped with lifters and/or exit dam (from published data). Axial dispersion coecients calculated from theoretical Peclet numbers were found in general higher than those obtained from tted Peclet numbers (see Fig. 3c). From now on, only results obtained from the tted parameters are considered.

Inuence of operating variables and nature of materials on the RTD

There have been several research works investigating the same scope of this study.

However very few experiments on residence time using materials of dierent physical properties, were performed in rotary kiln equipped with dierent shape of lifters to study transport of particulate solids. Earlier studies [START_REF] Venkataraman | Eect of lifter shape and conguration on material transport part I: In rotating drums[END_REF][START_REF] Hatzilyberis | An Rtd Study for the Flow of Lignite Particles Through a Pilot Rotary Dryer Part Ii: Flighted Drum Case[END_REF][START_REF] Hatzilyberis | An Rtd Study for the Flow of Lignite Particles Through a Pilot Rotary Dryer Part I; Bare Drum Case[END_REF] dier from the present investigation with regard to the design of apparatus, the operating conditions, the experimental procedures, or the materials used.

Eect of lifters prole

Fig. 4 presents RTDs obtained while processing broken rice, with the kiln equipped with straight, rectangular and no lifters. The three curves are overlapping, but still a distinction can be found with regards to values of the MRT and the VRT, which are adequate for the analysis. as previously stated, the VRT gives an indication about the spread of the distribution.

The following orders apply in these experiments:

tNL < tSL < tRL and σ 2 N L < σ 2 SL < σ 2 RL .
The experiments performed in this study, for a given type of materials when all operating parameters but the shape of the lifters are kept constant, showed the exact same trend here observed for MRT and VRT of the distribution. Regarding the MRT: RL experiments always have higher MRT than those of SL experiments or NL experiments.

This may rst be explained by the maximum hold-up of loading lifters. As stated before, the kiln was either over-loaded or design-loaded during the experiments, so that lifters could be lled to their maximum capacity. Let's calculate the volume of solids in a loading lifter for both geometries at horizontal position following Debacq et al. [START_REF] Debacq | Transverse motion of cohesive powders in ighted rotary kilns: experimental study of unloading at ambient and high temperatures[END_REF] calculations. Assuming the slope drawn by solids in the lifters is equal to the static angle of repose, V SL = 75.8 cm 3 and V RL = 242 cm 3 , if the bulk materials are sand particles.

One may assume that while being regularly kept inside lifters, solid particles are somehow delayed until being cascaded from the lifters with respect to solids inside the kilning bed (which are not taking extra routes or trajectories). Hence the axial transport of particles occurs mainly in the kilning bed. This may be the simple reason why in the presence of lifters, the residence times of solids are increased. But this must not be a sucient analysis to understand the eect of the lifters' shape or prole. For that, it is necessary to study the discharge rate of lifters. Since there have been recent studies reporting on that eld [START_REF] Sunkara | Modeling the discharge characteristics of rectangular ights in a ighted rotary drum[END_REF][START_REF] Debacq | Transverse motion of cohesive powders in ighted rotary kilns: experimental study of unloading at ambient and high temperatures[END_REF], such a study was not necessary, but still it was possible to observe the angular position γ f of a lifter at the end of discharge (with respect to the horizontal). It was found that γ f SL < γ f RL . γ f SL was within the range of 20°to 40°whereas γ f RL was within the range of 120°and 140°. This implies that on average the SL were totally discharged before RL, every other operating parameter remaining constant. Thus, solid particles were less delayed when using straight lifters, explaining why residence times were higher when the kiln was equipped with rectangular lifters. It was observed that SL were already empty above 40°, while RL were still cascading solids.

It is therefore recognized that RL produce more eect of mixing than SL, because they discharge through almost the whole kiln cross section with a relatively sustained rate.

This explains why results showed σ 2 SL < σ 2 RL as displayed in Fig. 6.

Eect of materials of diering physical properties

The two materials used in the study, namely sand and rice, are very dierent as indicated by their properties given in Table 2. Fig. 5 presents examples of distribution curves obtained while processing these materials in similar operating conditions.

A rst remark to be made is that largest solid particles (broken rice) always have smaller residence times compared to the smaller solids particles (sand). In the presence of lifters, rolling motions were observed in the kilning bed. Assuming that lifters, in particular straight lifters, do not aect the ow regimes but only reduce the area of the central core in the bed at the bottom of the kiln as suggested by Grajales et al. [START_REF] Grajales | Mixing and motion of rice particles in a rotating drum[END_REF] analysis of Chaudhuri et al. [START_REF] Chaudhuri | Experimentally validated computations of heat transfer in granular materials in rotary calciners[END_REF] experiments and simulations, the main physical property responsible for the diering residence times is likely to be the angle of repose. That is because in this type of motion, as explained by Mellmann [START_REF] Mellmann | The transverse motion of solids in rotating cylinders-forms of motion and transition behavior[END_REF], the larger part of the bed is transported as a solid body upwards with the rotational speed of the wall to the surface layer where particulates roll on themselves from the apex, between or over the lifters. As mentioned by Friedman and Marshall [START_REF] Friedman | Studies in rotary drying part I-holdup and dusting[END_REF] the kilning eect decreases with increasing angle of repose of the materials. Thus, the higher the angle of repose, the Sand-SL Rice-SL Sand-RL Rice-RL Fig. 5: Eect of the bulk materials characteristics on the RTD. Operating conditions for both SL and RL: 3 rpm rotation speed, 2°slope, 2.5 kg.h -1 MFR, 23.5 mm exit dam height. Solid lines are the axial dispersion model with tted parameters.

higher the time for solid particles to roll down at the surface layer. This latter, combined with eects of gravity, can explain why the rice residence times are shorter compared to those of the sand particles.

Secondly the RTD curves displayed in general nearly the same width: over the whole experimental campaign the VRT obtained were generally of the same order of magnitude for both materials at equivalent operating conditions (see Fig. 6) except when varying the kiln slope and the feed rate. In the latter cases signicant deviations were observed.

Eect of operating parameters: rotational speed, kiln slope, ow rate and exit dam height Eects of prole of the lifters and nature of the bulk materials have been previously discussed, therefore Fig. 7 shows the inuence of the remaining operating parameters in this study in the case of a bulk of broken rice when the kiln is equipped with rectangular lifters.

Rotational speed. As shown in Fig. 7a, an increase of the kiln rotational speed from 2 to 12 rpm has caused a remarkable shift of the RTD towards lower and lower residence times, while the shape (width and height of the peak) of these RTD curves remains approximately unchanged. Within this range of rotational speed, had the kiln not been equipped with lifters, the motion of the particles in the rotary kiln could be described as rolling with regards to Mellmann [START_REF] Mellmann | The transverse motion of solids in rotating cylinders-forms of motion and transition behavior[END_REF] criteria. These results agreed with Abouzeid et al. [START_REF] Abouzeid | The inuence of operating variables on the residence time distribution for material transport in a continuous rotary drum[END_REF] experiments ranging from rolling to cataracting bed in a small cylinder, but industrial applications are usually within the rolling motion. For a given type of materials and shape of lifters, the VRT remained approximately steady (see Fig. 6a) as implied by the similar width displayed by the corresponding RTD curves. Kiln slope. Even though that some industrial applications have used horizontal rotary kiln [START_REF] Hehl | Longitudinal mixing in horizontal rotary drum reactors[END_REF][START_REF] Gupta | Axial transport of granular solids in horizontal rotating cylinders[END_REF][START_REF] Pan | Granule transport and mean residence time in horizontal drum with inclined ights[END_REF], most kilns are designed to operate in an inclined position. Therefore our investigations extended from 2 to 5 degrees slope. Fig. 7b shows that RTD curves shift left towards lower residence time regions, while demonstrating gradual changes in their shape. These latter changes may be explained by the gradual decrease observed on the VRT from 2.5°kiln slope. Results also show in Fig. 6b that sand VRT are slightly higher to those of broken rice.

Mass ow rate. It may be seen from Fig. 7c, that RTD curves are overlapping. When the solid ow rate decreases from 2.5 to 0.68 kg.h -1 , RTD curves gradually spread and atten, while slowly moving forward so that the residence time of the particles increases.

The VRT were found to diminish with increasing feed rate (see Fig. 6c). By qualitative comparison, these results agreed well with those published previously [START_REF] Abouzeid | The inuence of operating variables on the residence time distribution for material transport in a continuous rotary drum[END_REF][START_REF] Hehl | Longitudinal mixing in horizontal rotary drum reactors[END_REF]. Exit dam height. Eventually it was found that RTD curves shift on the right side (i.e toward region of higher residence time) and become atter and broader as the exit dam height increases from 0 to 33.5 mm. RTD curves of experiments with a 13.5 mm exit dam height or nothing installed at the kiln outlet, were completely overlapping, in fact nearly the same. This is shown in Fig. 7d. Thus, results suggest that dams covering less than 45% of the area at the kiln outlet will not have a signicant impact on the residence time of solid particles. However, that conclusion should likely apply to kilns having similar L/iD ratios to the one used in this study. The VRT exhibits a plateau before slowly increasing with the exit dam height as shown in Fig. 6d.

Inuence of operating variables and nature of materials on the Pe and D

The Peclet numbers obtained in this study are very high usually larger than 100. This is generally common when the ow behavior of the stream in a unit tends toward plug .68 kg.h -1 1.3 kg.h -1 1.9 kg.h -1 2.5 kg.h -1 0 mm 13.5 mm 23.5 mm 33.5 mm Fig. 7: Inuence of operating parameters (N, S, Ṁ and exit dam height) on the RTD for the ow of broken rice, when the kiln is equipped with rectangular lifters. Solid lines represent the axial dispersion model using tted parameters. Dashed lines represent the axial dispersion model using theoretical parameters. ow, as is usually the case for rotary kilns. Fig. 8 presents the Peclet numbers obtained by the tting method, as function of operating parameters. The rst observation is that the Peclet numbers obtained with rectangular lifters are always lower compared with those obtained while using straight lifters. This implies that rectangular lifters introduce more back-mixing into the system, conrming previous observations. The Peclet number was found to decrease with increasing kiln rotational speed and slope. However, a plateau seems to be reached in both cases for higher parameter values. As it has been previously observed while using straight lifters as well as forward and reverse spiraling lifters [START_REF] Venkataraman | Eect of lifter shape and conguration on material transport part I: In rotating drums[END_REF], the Peclet number increases with feed rate in the present study. It was found that the Peclet number remains quite constant, unlike Karra and Fuerstenau [START_REF] Karra | Material transport in a continuous rotary drum. Eect of discharge plate geometry[END_REF] who have detected a maximum value for the Peclet number, while varying the exit dam height. Regarding the materials used, no signicant tendency was found: at similar operating conditions the Peclet numbers are in general of the same order of magnitude.

The axial dispersion coecient was calculated using the Peclet number and the axial convective velocity in Eq.9. The eects of operating parameters on the dispersion coefcient are shown in Fig. 9. It can be seen that the eect of the prole of lifters on the dispersion coecient are highly pronounced with increasing kiln rotational speed and slightly signicant for higher kiln slope or smaller ow rate. Otherwise the axial dispersion coecient seemed to have relatively little direct dependence on the lifters shape. If speed, the kiln slope, and the ow rate, are the opposite of those mentioned for the Peclet number: it increases with an increase in the two rst (N, S) in agreement with Rutgers [START_REF] Rutgers | Longitudinal mixing of granular material owing through a rotating cylinder: Part II[END_REF] works, and decreases with the last parameter ( Ṁ) contrary to Venkataraman and Fuerstenau [START_REF] Venkataraman | Eect of lifter shape and conguration on material transport part I: In rotating drums[END_REF] tendency for that parameter. The dispersion coecient was quite stable when varying the exit dam height. Eventually, except for scarce cases, it appears that the axial dispersion coecient is higher in the case of larger particles: broken rice. This may be linked to the bouncing backward or forward at the bottom of the kiln of these particles after having been cascaded from a lifter, as observed during experiments. Some correlations intended to predict the axial dispersion coecient have been published previously [START_REF] Sai | Residence time distribution and material ow studies in a rotary kiln[END_REF][START_REF] Ang | Residence Time Distribution for a Cold Model Rotary Kiln[END_REF][START_REF] Sherritt | Axial dispersion in the three-dimensional mixing of particles in a rotating drum reactor[END_REF][START_REF] Fan | Axial dispersion of solids in rotary solid ow systems[END_REF]. Sai et al. [START_REF] Sai | Residence time distribution and material ow studies in a rotary kiln[END_REF] correlated the axial dispersion coecient with operating parameters as follows:

D/uL = 0.000562θ 0.79 t /S 0.67 N 1.06 ρ 0.25 t (15)

Tracer properties were considered of major importance along with the kiln rotational speed and slope. Sherritt et al. [START_REF] Sherritt | Axial dispersion in the three-dimensional mixing of particles in a rotating drum reactor[END_REF] proposed the following equation to correlate the axial dispersion coecient with operating parameters when the hold-up volume fraction is less than 0.5: D = 0.0011(N/N c ) 0.39 (iD) 1.15 (d p ) 0.46 X -0.43 with tting parameters determined for all modes of bed behavior from a set of published data. N c is the critical speed dened in Eq.17, and X is the hold-up, volume fraction.

The two correlations above allow one to determine the axial dispersion coecient. They are used to calculate the axial dispersion coecients for the operating conditions achieved in this study. Fig. 10a and Fig. 10b show comparison of predictions with experimental axial dispersion coecient values. Results do not fairly agree with the experimental data especially when using Eq.15. However both correlations results are mostly underpredicting. Therefore the axial dispersion coecient is correlated in terms of operating parameters. Acknowledging that the geometrical properties of the kiln [START_REF] Sherritt | Axial dispersion in the three-dimensional mixing of particles in a rotating drum reactor[END_REF], and as discussed previously N, S, and Ṁ, as well as the bulk materials properties, have stronger inuence on the axial dispersion coecient, the following correlation is proposed for kiln equipped with lifters:

D = k( Ṁ ρ bulk ) a (N • S) b (θ • d p ) c (Ls) d iD (18) Ls = n lif t -1 2 S horlif t (19)
The variables being expressed in the International System of Units, the axial dispersion coecient D is obtained in m 2 .s -1 , Ls accounts for the area occupied by solid particles contained in loading lifters m 2 , n lif t is the total number of lifters, and S horlif t represents the section of materials in a lifter at horizontal position m 2 . The values of the tted parameters k, a, b, c, and d were optimized using a MatLab program tting experimental and theoretical data. Results of the calculations are presented in Table 4. Predictions are in good agreements with experimental data as shown in Fig. 10c. As previously mentioned models of Sherritt et al. [START_REF] Sherritt | Axial dispersion in the three-dimensional mixing of particles in a rotating drum reactor[END_REF] and Sai et al. [START_REF] Sai | Residence time distribution and material ow studies in a rotary kiln[END_REF] appeared to be not truly realistic for the case of rotary kiln equipped with lifters while the model hereby presented should be valid and give acceptable results for that case as illustrated in Fig. 10c.

Conclusions

From the results obtained in this work, it is possible to draw the following conclusions:

The axial dispersion model used to represent the experimental data was found to be in very good agreement with the data set. There were no particular eects of segregation or dead volume, since RTD curves do not showed extended tails. The tted parameters were considered reasonably accurate to characterize the materials transport behavior within the system.

The RTD curves were found to shift towards lower residence time regions with an increase in either the rotational speed, the kiln slope, or the mass ow rate, or a decrease in the exit dam height. Changes in the shape of the curve were linked to the VRT value, which remained quite stable while varying the rotational speed and exit dam height. The smaller the VRT, the higher the peak of corresponding RTD curves. Broken rice used as the largest materials has generally lower residence time than sand particles. For a given type of bulk materials the VRT are always bigger when using rectangular lifters.

The Peclet numbers were often greater than 100, as usually observed in rotary kiln unit. They were found to be higher in the case of straight lifters. There was no signicant dierence between the two materials except when varying the kiln slope.

The exit dam height has no incidence on the Peclet number, whereas an increase in either the rotational speed or the kiln slope induces a decrease of the Peclet number.

The Peclet number increases with the feed rate. Regarding the axial dispersion coecient, results tendencies are the inverse of those reported for the Peclet number.

A correlation applicable to rotary kilns equipped with lifters is presented and showed a reasonable degree of quantitative accuracy with regard to the experimental data, compared to other published correlations.

List of symbols

Fig. 1 :

 1 Fig. 1: Layout of the experimental apparatus, including the structure and shapes of lifters.

Fig. 2 :

 2 Fig.2: Eect of the amount of tracer (dyed broken rice) upon the dimensionless RTD in a bulk of broken rice. Operating conditions: 3 rpm rotation speed, 2°slope, 2.5 kg.h -1 MFR , 23.5 mm exit dam height with rectangular lifters.

Fig. 3 :

 3 Fig. 3: Comparison of experimental/theoretical and tted values of the axial dispersion model parameters, i.e MRT, Pe, and resulting D. Solids lines are ±20% margins.

Fig. 6 :

 6 Fig.6: Inuence of operating parameters (N, S, Ṁ and exit dam height) on the VRT for the ow of sand and broken rice, when the kiln is equipped with either straight or rectangular lifters.

  of kiln exit dam height -RL and broken rice

Fig. 8 :

 8 Fig.8: Inuence of operating parameters (N, S, Ṁ and exit dam height) on the experimental Peclet number for the ow of sand and broken rice, when the kiln is equipped with either straight or rectangular lifters.

Fig. 9 :

 9 Fig.9: Inuence of operating parameters (N, S, Ṁ and exit dam height) on the experimental axial dispersion coecient for the ow of sand and broken rice, when the kiln is equipped with either straight or rectangular lifters. The axial dispersion coecients are obtained from the tted Pe.

Fig. 10 :

 10 Fig. 10: Comparison of axial dispersion coecient values obtained in the present study with predicted values from a) Sai et al. [11], b) Sherritt et al. [44], c) Present authors Eq.18. Solids lines are ±20% margins.

Table 1 :

 1 Geometrical characteristics of the rotary kiln

	Subsets	Parameters	Order of
			magnitude
	Rotary	iD [m]	0.1013
	Kiln	L [m]	1.95
		Exit dam	0-33.5
		height [mm]
		Lifters shape	None, SL, RL
	Operating	N [rpm]	2-12
	conditions	S [°]	2-5

Table 2 :

 2 Physical properties of materials

	Product	Shape	ρ bulk	ρ tapped	dp	θ
			[kg.m -3 ]	[kg.m -3 ]	[mm]	[°]
	Sand	Nodular	1422	1543	0.55	39
	NaCl	Nodular	1087	1184	0.6	35.4
	Broken rice Cylindrical	889	934	3.8×1.9 36
	times (on average) from 10 to 20 mm height and the new volume was measured. However
	the raising and lowering of the cylinder were done without reference to the height and
	arbitrary acceleration in both directions (upward and downward). The solids angle of
	repose were determined by means of the xed cone method similarly to Thammavong
	et al. [40].

Table 3 :

 3 Reproducibility of experiments

	Materials -Lifters	Operating conditions	t [min]	Cv [%]	σ 2 [min²]	Cv [%]
	Sand	3 rpm , 2.1°, 2.5	43.39		6.91
		kg.h -1 MFR , 23.5	43.64	0.76	6.72	11.2
	SL	mm exit dam height	42.99		8.22
	Broken	3 rpm , 2.5°, 2.5	31.77		8.57
	rice	kg.h -1 MFR , 23.5		0.67		5.9
	RL	mm exit dam height	31.47		7.88
	Broken	10 rpm , 2°, 2.5	13.38		2.98
	rice	kg.h -1 MFR , 23.5	13.14	2.1	3.37	19.2
	NL	mm exit dam height	12.83		2.28

Table 4 :

 4 Parameters for the axial dispersion coecient in Eq.18.

	Parameters	Value	Condence interval
	k×10 7	1.2023	0.1612	2.2434
	a	-0.8714 -0.9348	-0.8081
	b	1.6736	1.6329	1.7144
	c	0.4349	0.4120	0.4578
	d	0.2062	0.1834	0.2290