APPLICATION OF THE DISCRETE ELEMENT MODELLING IN AIR DRYING OF PARTICULATE SOLIDS Jintang Lia and David J. Masonb a Department of Chemical Engineering, UMIST, PO Box 88, Manchester M60 1QD, UK. Email: jintang.li@umist.ac.uk b School of Engineering, University of Brighton, Brighton BN2 4AT, UK. Email: d.j.mason@bton.ac.uk Key Words and Phrases: gas-solids flow; mathematical simulation; momentum, heat and mass transfer ABSTRACT The Discrete Element Method (DEM) has been widely used as a mathematical tool for the study of flow characteristics involving particulate solids. One distinct advantage of this fast developing technique is the ability to compute trajectories of discrete particles. This provides the opportunity to evaluate the interactions between particle, fluid and boundary at the microscopic level using local gas parameters and properties, which is difficult to achieve using a continuum model. To date, most of these applications focus on the flow behaviour. This paper provides an overview of the application of DEM in gas-solids flow systems and discusses further development of this technique in the application of drying particulate solids. A number of submodels, including momentum, energy and mass transfer, have been evaluated to describe the various transport phenomena. A numerical model has been developed to calculate the heat transfer in a gas-solids pneumatic transport line. This implementation has shown advantages of this method over conventional continuum approaches. Future application of this technique in drying technology is possible but experimental validation is crucial. INTRODUCTION Particulates are employed in a wide variety of industrial processes, such as in chemical, pharmaceutical and food engineering. In chemical plants, for instance, the production of two thirds of products involves particulate flows (Roco 1993). Whereas in the food production chain, almost all unit operations involve the handling of powdered and bulk foodstuffs. One of the major process during the production and utilisation of powdered and granular materials is the drying or cooling via a flowing gas, usually air. Due to its convenience in practical implementation, air drying has Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 1 been the most common drying technique for particulates in modern plants. Mujumdar (2000) states that more than 85 percent of industrial dryers are estimated to be of this type. The drying mechanism was originally defined by Sherwood (1929) as the manner in which moisture moves through a solid and thence out into the air during the drying process. Extensive theoretical and experimental works have been conducted to investigate the various mechanisms governing the coupled heat and moisture transfer phenomena since the1980’s (e.g. Mujumdar 1980). Although basic drying theories were well developed in the past two decades, Tsotsas (1998) stated that the design of connective dryers were still based on the scale-up models derived from single particle drying kinetics. Continuous static (steady state) operation is usually modelled in terms of an imaginary, average particle, which would be right for ideal plug flow or linear drying kinetics. However, most drying processes are highly non-linear and the distribution of solids moisture content at the dryer outlet, a potentially important aspect of product quality, are usually non-uniform. This may result in large errors when using continuous steady state models. Therefore, a dynamic modelling technique based on individual residence time and drying history would be of considerable importance in the future. Numerical techniques have been developed to solve the partial differential equations governing the coupled heat and mass transfer in a single particle (e.g. Abuaf and Staub 1986, Oliveira and Haghighi 1997 and Jumah et al. 1997). Although Oliveira and Haghighi discussed the multi-particle system and introduced a method to model the fluid flow around stationary particles (two soybean kernels), most contemporary models, such as thin-layer and deep-bed drying analyses (Mhimid and Nasrallah 1997 and Benet et al. 1997), are based on the assumptions of flow through a porous medium. In these models, a continuous and homogeneous behaviour of the medium has to be considered. This assumption may not accurately represent the distinctive distribution of solids in an aerated particle assembly. More importantly, the interactions between particles and surrounding fluid are difficult to take into account. These interactions play an important role when determining the particle behaviour and heat transfer in gas-solids pipe flows (Li and Mason 2000). Oliveira and Haghighi (1997) state that the analysis of drying of a multi-particle system may provide fundamental knowledge for understanding the effects of interaction on the hydrodynamics and heat and mass transfer characteristics of closely spaced particles. The mathematical modelling of gas-solids flow was originally built on the theories of fluid flow, either homogenous flow (ie. Julian and Dukler 1965, Michaelides 1984 and Chung et al. 1986) or two-phase flow (ie. Mason 1991 and Kuo and Chiou 1988), where the mixture or each of the two phases is considered as a continuum. Similar to the thin-layer and deep-bed drying analyses, in these models the assumption of a continuous change of variables for the solids phase is not realistic, particularly for large particles and flows with discontinuities in the concentration gradient. It also introduces problems when modelling particle-particle and particleDrying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 2 wall interactions, which have a great influence on the particle motion and the flow behaviour. A recently developing category of multiphase flow models uses the so called Lagrangian approach or the Discrete Element Method (DEM), which calculates the motion of each individual particle separately. This method is able to take into account the simultaneous occurrence of various kinds of movements and interactions of the particles with each other and with the surfaces of the boundary. Cundall et al. (1979) developed this method and used it in soil mechanics. It has been used to model a number of applications in gas-solids systems. Tsuji et al. (1993) and Pritchett et al. (1978) have examined gas-solids flow behaviours in fluidised beds and Langston et al. (1995, 1996 and 1997) have modelled granular flows in hoppers. Particle motions in the air stream and pneumatic pipelines have been simulated by Matsumoto and Saito (1970), Ottjes (1978), Yamamoto (1986), Summerfeld (1992), Tsuji et al. (1992), Frank et al. (1993), Peng et al. (1994 and 1996) and Salman et al. (1997). However, to date little report of this application in heat and mass transfer of gas-solids flows has been found in literature. The paper addresses the DEM method and its application to the modelling of gassolids flows. The various physical mechanisms that occur in air drying, including mass, momentum and heat transfer, are discussed and formulated as sub-models. Although there is potential to apply DEM to liquid-solid slurry droplets (such as Abuaf and Staub 1986 and Levi-Hevroni et al. 1995) and particle aggregation, this work will focus on the drying of particulate solids with an outer crust such as a kernel of cereal grain. A heat transfer model has been developed and simulations were conducted to examine the heat exchanges in gas-solids flows within a pneumatic pipeline. The modelled results are discussed in view of further implementation of this technique in drying process where the incorporation of a mass transfer model is also required. THE DISCRETE ELEMENT METHOD Pioneering work in the application of the Discrete Element Method was initially carried out by Cundall and Strack (1979) to model the behaviours of dense solids assemblies in soil mechanics. Their work concentrated on the use of ‘springs and dash-pots’ to represent particle interactions, as shown in Figure 1. This concept, which expresses the interaction with the use of a spring, dash-pot and friction slider analogy, has been adopted in most of the current DEM applications in particulate flows (e.g. Tsuji et al. 1992). Particles are usually assumed to be cohesionless elastic bodies and the microscopic particle-particle and particleboundary interactions are calculated with the evolution of particle trajectories. (Cohesive and inter-particle forces may also be considered as required by more complex applications. See discussions by Thornton and Yin 1991.) Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 3 The contact force between impacting particles is split into a normal force Fn and a tangential force Ft: (1) Fn K n vn Ft K t vt (2) where n and t are particle displacement in the normal and tangential directions, vn and vt are the relative velocities, K is the stiffness of the spring and is the coefficient of viscous dissipation (Determination of these properties may be found in the literature, e.g. Tsuji et al. 1992). If Ft is bigger than the limiting friction force, then the particles slide over each other and the tangential force is calculated by: (3) Ft f Fn It is noted that Equations (1) and (2) show a linear relationship between the contact forces and the displacements, which take the form of Hookes’s law. This assumption is in contradictory to the Hertzian contact theory. Cundall and Strack (1979) stated that a non-linear law, such as that of Hertzian contact, could equally well be employed. Seville et al. (1997) summarise that the linear spring model is the simplest mathematical form and is fairly widely used. The precise details of the contact mechanics may not be necessary, since in most real granular flow systems, particles are irregular and rough, it is unnecessary to model granular flow with ideal Hertzian contact. APPLICATION OF DEM IN GAS-SOLIDS FLOW The initial application of DEM in particulate systems has focused on the modelling of interactions between particles and between particles and boundary surfaces. This has been extended to take into account the interactions with the surrounding gases, such as Tsuji et al. (1992 and 1993) for plug flows and fluidised beds, where the interstitial gas flow plays an important role for the solids behaviours. In a gas-solids flow system, the particle motion and the flow structure are dominated by the interactions between gas, particle and boundary. These interactions are determined by the laws defining the mass, momentum and energy transfer mechanisms. To model the flow, the various interactions need to be calculated. (Details of these sub-models are discussed in the following sections.) Using DEM, the trajectory of each particle is evaluated together with the evolution of the flow over many time steps. Interactions between gas, particle and boundary wall are subsequently calculated using the local particle and gas parameters, and these interactions eventually determine the particle motion and the gas flow. Seville et al. (1997) state that the basic advantage of this method over continuum techniques is that it simulates effects at particle level. Individual particle properties, such as size and shape variation, can be specified directly and the assembly response is a direct output from the simulation. There is less need for global assumptions, such as uniform stress Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 4 at a certain depth in the assembly. Certain phenomena such as particle size distribution can be simulated directly, whereas in the continuum method this is far more difficult. In a DEM model, particle trajectories are computed by considering the various forces acting on each particle. The gas field is solved simultaneously with the particle movements considering the particle presence as a source of mass, momentum and energy and as a change to its volume fraction (See Crowe et al. 1977. Very fine space discretization for the gas field may be used to model fluid flow around particles but, as discussed by Oliveira and Haghighi 1997, this requires much more computational resources for multi-particle systems and may not complete a case in a reasonable time scale with the power of current computers.). As the location of a particle in the fluid is known all the time, this gives an opportunity to determine which cell in the fluid domain contains the particle and, therefore, the interactions between gas and particles can be computed using the local fluid conditions. This results in a more comprehensive coupling between the gas phase and the discrete particles and hence a more realistic gas and solids flow field. The importance of these coupling effects (temperature of gas, gas properties, solids properties, solids behaviour and mass and heat transfer between gas and solids) in the pneumatic drying of grains has been addressed by Matsumoto and Pei (1984). A variety of applications of DEM in gas-solids flow systems have been reported in literature. These works may be categorised in accordance with their applications in specific industrial processes, such as fluidised beds, hopper flows and pipe flows. The application of DEM in fluidised beds by Tsuji et al. (1993) has focused on the flow behaviours such as gas velocity distribution, particle flow patterns and bubbling and pressure fluctuation across the bed. In Tsuji’s model, a finite difference approach has been used to model the gas flow and a soft-contact model with reduced particle stiffness has been adopted for the particle-particle impact. A similar model was developed by Hoomans et al. (1996) but using a hard-sphere collision approach (Wang and Mason 1992), which is commonly encountered in the field of molecular dynamics. More recently, Li et al. (1999) developed a coupled CFD (Computational Fluid Dynamics) and DEM model to simulate the gas-liquid-solids fluidisation systems. A close-distance interaction model is introduced for the particle-particle collision, which considers the liquid interstitial effects among particles. In the above models, it has been shown that the fluid-particle interactions are usually modelled by the fluid drag approach but the particle-particle interactions may be considered in accordance with actual particulate properties. Langston et al. (1995, 1996 and 1997) have carried out extensive works in granular flows in hoppers. Their works have concentrated on the dynamic hopper discharge rates and hopper wall stresses with different filling methods. The effects of interstitial air flow have also been examined (Langston et al. 1996). It demonstrates that the DEM method has shown certain transient and oscillatory features of the flow field, which has not been produced by continuum theories. The simulated results have Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 5 been compared with experimental data and good agreements have been reported between the simulations and the imaging data from photography and gamma-ray tomography. There are wide spread applications of DEM in pneumatic pipelines. Matsumoto and Saito (1970) examined the mechanisms of particle suspension using a Monte Carlo simulation. Salman et al. (1997) discussed the simulation of particle movement in dilute pneumatic conveying and Tsuji et al. (1992) presented a DEM model for the study of plug flow in a horizontal pipe. Summerfeld (1992) modelled particle-wall collisions and Peng et al. (1994) analysed the effect of particle collision and particle rotation for the transport of coarse particles. The applications of DEM in the various gas-solids systems have shown potentials of this fast developing technique in analysing the flow behaviours and the interactions within the mixture and with the boundary. However, the incorporation of a heat transfer model in this technique has not been tackled until recent years. Assuming a random motion of particles and neglecting the interstitial fluid flow and interactions with boundary, Hunt (1997) developed the first DEM model to predict temperatures and effective thermal conductivities for flows of granular materials (in an imaginary two dimensional domain). Apart from this over simplified model, to date little report of this application in heat transfer of gas-solids flows has been published. EVALUATION OF MOMENTUM TRANSFER Particles in the gas flow field may be subjected to forces due to gravity, fluid drag, and collisions with other particles and the pipe wall. The buoyancy force can usually be ignored since the densities of the particles are normally two or three orders of magnitude larger than air. Inter-particle forces, such as van der Waals forces, capillary forces and electrostatic forces may also be neglected if the particle size is bigger than 100 m when those inter-particle forces become insignificant (Seville et al. 1997). The fluid drag force on a particle is given by: FD CD Ap 1 2 g ( u g u s ) u g us (4) where Ap is the projected area of the particle. Extensive experiments have been conducted to correlate the drag coefficient CDs, for a single particle in a finite fluid, with the particle Reynolds number Res. The relation is divided into three regimes (not considering the critical range at Res approaching 105): i) ii) for Res > 1000, CD 0.44 and is roughly constant (Newton’s law); in the intermediate range, 0.25 < Res < 1000, CDs is significantly dependent on Res. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 6 for Res < 0.25, CD /Res. This is the ‘creeping flow’ regime and known as Stokes’ law. A general correlation was proposed by Schiller and Naumann (1933) for the three regions: iii) 24 0.687 (5) CDs max 1 0.15 Re s ,0.44 Re s For very dilute phase flow, this formula has given good prediction of the drag force. However, when applying to particles in flows with high solids concentration it was found that it underestimated the drag force. The presence of neighbouring particles alters the local gas flow field and eventually influences the fluid drag on the particle. The drag coefficient CD for a particle in a group of particulate assembly is computed using the value for a single particle CDs modified according to the presence of neighbouring particles (Richardson and Zaki 1954). The Richardson-Zaki exponent n was derived by equating the Richardson-Zaki relationship with Gibilaro’s equation (Gibilaro et al. 1985), an improved Ergun equation (Ergun 1952). Thus, CD CDs gn (6) 1.8 7 0.4 600 (1 0.4) 1 n 2.8 log 1 log g g Re s 3 C Ds 7 where g is the gas volume fraction. (7) MODELLING OF HEAT TRANSFER The main heat transfer mechanism in an air dryer is the heat exchange between gas and particles. The contribution of conductive heat transfer between particles in contact may affect the drying process to some extent depending the concentration of particulate solids in the dryer (Sharma 1997). (Heat exchanges with boundaries, such as between gas and boundary walls, may be modelled by boundary-layer models according to practical boundary conditions, e.g. adiabatic or isothermal, or using empirical equations such as formulae for pipe flows proposed by Wrangham 1961.) A Particle in an Infinite Fluid For a stationary particulate sphere in an infinite fluid, it is easily shown from the heat conduction equation that: Nus hs d / k 2 (8) Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 7 When there is a relative motion between the particle and fluid, the most commonly used correlation is due to Ranz and Marshall (1952), which was developed by experiments on evaporation from water droplets. Nus 2 0.6 Re s1/2 Pr 1/3 (9) This formula was found sufficiently accurate when applied to particles with small Reynolds numbers (less than 200). However, when a higher Reynolds number is encountered, such as at the solids feeding area of a gas dryer, where the relative velocity between gas and particles is high (Mason and Li 2000a), the above equation was found only adequate to cover a limited range (Li and Mason 2000). After comparing with experiments using 3mm polyethylene pellets, a modified equation was adopted for 200<Res<1500: Nus 2 0.5 Re s1/ 2 Pr1/3 0.02 Re 0s .8 Pr1/3 (10) When the particle Reynolds number is bigger than 1500, a significant enhancement of heat exchange has been observed and another correlation, which is similar to the equation of Frantz (1962) but with a higher exponent, has been introduced: Nus 2 0.000045 Re s1.8 (11) It is noted that although this equation has been tested with a variety of flow conditions for 3mm polyethylene pellets at Res > 1500, further experimental validation may be necessary for different particle sizes. A Particle in a Particle Assembly In a similar manner to the fluid drag force, the heat transfer between gas and particle is also affected by the presence of neighbouring particles. This effect has been found more significant with the increase of solids concentrations. There are few works in literature that examine this subject (Bandrowski and Kaczmarzyk 1978). Using the similar concept as for the fluid drag, the following correlations have been proposed with the modification by gas volume fraction g: Nu s 2 0.6 gn Re s 1/ 2 Nu s 2 0.5 gn Re s 1/ 2 Pr 1 / 3 Pr 1 / 3 0.02 gn Re 0s .8 Pr 1 / 3 Nu s 2 0.000045 Re s n g Res <= 200 (12) 200 < Res <= 1500 (13) 1.8 Res > 1500 (14) The exponent n = 3.5 was found suitable for the 3mm polyethylene pellets when modelling dilute flows. However, a temperature discrepancy was found at the solids inlet in flows with higher solids concentrations. Again, this exponent needs to be reevaluated in the light of more sophisticated experiments. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 8 Gas-particle Heat Transfer Model Boothroyd (1971) stated that, for calculation purposes, it is often sufficiently accurate to use the steady-state heat transfer coefficient at all times if the suspension is not too dense. A uniform particle temperature distribution has been assumed in the present work. More sophisticated considerations, such as the assumptions of a crust temperature and wet core temperature by Abuaf and Staub (1986), may be adopted for the drying of slurry droplets or large particles. Then the rate of heat transfer qs from the gas to the particle can be calculated by: qs hs A(Tg Ts ) (15) where A is the surface area of the particle. The heat transfer coefficient hs can be determined by hd Nus s s kg (16) The amount of heat transferred in a time step t is Qs qs t (17) This heat transfer between gas and particle is considered as an energy source to the gas phase, to be added to the source term of the energy conservation equation, and a balance of the internal energy of the particle. The thermal balance in a time step t for an evaporating droplet moving in the gas stream can be written as: (ms c ps ml c pl )Ts Qs Lml (18) The terms on the left-hand-side of the equation express the heat capacity of both liquid and solid portions, whereas the right-hand-side denotes the heat exchange with the gas and the latent heat of liquid evaporation. The mass of evaporated liquid water ml is calculated by the mass transfer model. After heat exchange, the temperature change of the particle can be calculated by Qs Lml Ts ms c ps ml c pl (19) Particle-particle Heat Transfer Model For solid particles, when two bodies impact each other or a moving particle impacts on a stationary surface, heat conduction occurs through the contact area of the particles. Based on both numerical and analytical solutions of the heat conduction equation with time-dependent boundary conditions, Sun and Chen (1988) produced a quantitative calculation of this conductive heat transfer due to particle impact. For cases where the contact time is short, it is sufficient to use a simple one-dimensional calculation for this problem. The interface heat flux is given by: Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 9 q imp (T2 T1 )t c1/ 2 ( 1c ps1k1 ) 1/ 2 ( 2 c ps 2 k 2 ) 1/ 2 (20) This enables the particle-particle and particle-boundary heat transfers be evaluated at the resolution of a particle time step. As the time step used for the calculation of particle motion in DEM is very small, which is only a fraction of the total contact time (Li 2000), it is assumed that if particles or particles and surface contact at a certain time step the contact will occur over the whole period of the time step. Therefore in the above equation the contact time tc can be replaced by the length of a time step t. Then the energy exchange in a time step can be computed by Qimp (T2 T1 )rc2 t 1/ 2 ( 1c ps1 k1 ) 1/ 2 ( 2 c ps 2 k 2 ) 1/ 2 (21) where rc is the instantaneous contact radius considering the particle deformation as shown in Figure 2, which can be computed by: R12 rc2 R22 rc2 R1 R2 R12 rc (22) 4 R12 R22 (( R1 R2 R12 )2 R12 R22 )2 4( R1 R2 R12 )2 (23) For particle and surface contact, the surface deformation may be neglected. rc can be calculated by replacing R2 with infinity in the above equations. R12 rc2 R1 R12 rc (24) R12 ( R1 R1 ) 2 (25) The total energy exchange during the entire contact time is the sum of energy exchange for all time steps during the course of contact. Similar to the heat exchange between gas and particle, the energy transfer on impact also contributes to the particle internal energy, ie. a change of temperature. MASS TRANSFER MODEL One class of model describing the drying of a particulate solids is known as the ‘single-kernel drying model’ (Puiggali and Quintard, 1992). Most of these models were developed based on the application of non-equilibrium thermodynamics theories (Luikov 1966) in porous media. Drying models for generic hygroscopic porous media were reviewed by Bories (1989), Puiggali and Quintard (1992) and Waananen et al. (1993). The evaporation rate of liquid in a moist particle is usually formulated as (Abuaf and Staub 1986 and Levy and Mason et al. 1998): dml (26) hd d 02 (Cvg Cvs ) dt Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 10 where Cvg and Cvs are the water vapour concentrations at the outer diameter of the particle and the ambient air respectively. They are given by the following expressions: M p (27) Cvs w vo RT s C vg M w p vg (28) RT g The partial pressure of the water vapour in the ambient air, pvg, is determined by the proportion of water vapour content in the air, which is calculated by the continuum modelling of the gas and the water vapour phases. For a liquid-droplet or at the constant-rate drying period of a slurry droplet (Mujumdar 1983), the water vapour pressure pvo at the outside periphery of the droplet is straightforwardly assumed to be the saturation pressure of the water vapour at the droplet temperature Ts. However, when the drying approaches the falling-rate period, which is the case for the drying of moist particulate solids, the process becomes much more complicated and various assumptions have to be made. The moisture transport process in a porous medium is usually much slower than the heat transfer. It is common to assume a uniform temperature distribution inside a small solid particle but this assumption is not applicable to moisture distribution. Abuaf and Staub (1986) developed a model assuming that, during the falling-rate stage of the drying process, a dry crust surrounding a wet core was formed. The amount of liquid evaporated that diffuses through the dry porous crust is calculated by a Stephen-type diffusion equation: dml 1 1 p pvi 1 D p v ln (29) dt 2 d 0 d i RTave p pvo where di and d0 are the diameters of the wet core and the dry crust respectively. Assuming a constant dry crust diameter, i.z. the moist particle diameter, d0, and a uniform solids fraction through the particle, Abuaf and Staub derived the following equation to calculate the change in the diameter of the receding wet core: d (d i ) d0 dml (30) 1/ 3 2 / 3 dt dt 3ml 0 ml In equation (29), the porosity of the dry crust, , can be determined by actual measurement for the dry material. As a uniform temperature is assumed, here the average temperature, Tave, shall be equivalent to the particle temperature Ts. The partial pressure of the water vapour, pvi, in the wet core is assumed to be the saturation pressure of water vapour at the particle temperature Ts. Combining equation (26) with (29) to eliminate the partial pressure of water vapour in the dry crust, pvo, and obtain an implicit equation for the water evaporation rate: dml d i d 0 2Dv p p pvi ln 2 dt di d0 RT s p ( RT s / hd d 0 M w )( dml / dt ) pvg Ts / Tg Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. (31) 11 The mass transfer coefficient hd is calculated from the well known Sherwood number correlation: h d 1/ 2 (32) Sh d 2 0.6 Re s Sc1 / 3 Dv The inclusion of the Spalding number in the above equation was used by Abuaf and Staub (1986). This effect has been found minor (error less than 5%) if the temperature of the drying media is not significantly higher (within 100oC) than the particle temperature. SIMULATIONS FOR HEAT TRANSFER IN PIPE FLOWS As part of a program investigating gas-solids flow and heat transfer in dilute and dense phase pneumatic conveying both experimentally and numerically (Li 2000), a two-dimensional transient model has been developed to describe heat transfer in gassolids pipe flows (Li and Mason 2000). In the model, the gas phase has been modelled as a continuum, but the solid phase has been modelled as a set of discrete particles by DEM. The influence of solid particles on the gas flow is considered as a source of mass, momentum and energy and as a change to its volume fraction. Here a brief introduction of the model is presented and details are referred to Li (2000). Gas Phase Model The gas flow is governed by the following mass, momentum and energy conservation equations. Mass conservation ( g g ) ( g g ug ) Sm t (33) where Sm is the mass source due to the mass transfer with solids and boundaries. Mass transfer has been neglected in the current model. Momentum conservation ( g g ux ) p ( g g u g u x ) ( g g (u x )) g Sx t x (34) where Sx is the momentum source in the x direction, which is due to momentum transfer with the particles, and pipe wall friction. A similar equation may be written for the y direction. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 12 Energy conservation ( g g H ) ( g g u g H ) (g g ( H )) S e t where the diffusion coefficient is: g k g / cg (35) (36) and the stagnation enthalpy: H Tg cpg 12 ug 2 (37) The energy source term Se includes the heat transferred between gas and particles and gas and pipe wall and the mechanical work done due to friction at the boundary wall. Numerical calculation The gas flow is calculated using the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) method developed by Patankar (1980), taking into account the interactions with particles and boundary wall. The procedure of calculation follows the physical process of experiment, i.e. it starts from the singlephase gas flow and, when a well converged gas field is obtained, the particles are incorporated. Gas turbulence Because of the random nature of gas turbulence, now it is still difficult to model the physical modes of turbulence (such as eddies) in a gas-solids flow. In this model, the gas turbulence was modelled by employing an average effective turbulent viscosity, which was obtained from running the same problems in a commercial CFD package – PHOENICS, CHAM, UK. The numerical models and their solutions are described by Spalding (1980). In PHOENICS the gas turbulence is modelled by k- model. This effective viscosity was evaluated by comparing the pressure drop with experimental data. Solids Phase Model Particles in the gas flow field may be subjected to forces due to gravity, fluid drag, and collisions with other particles and the pipe wall. The buoyancy force is ignored since the densities of the particles are significantly larger than air at the pressures commonly used in pneumatic conveying systems. The Saffman lift and Magnus forces are also neglected because they are of the order between 10-2 and 10-3 Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 13 of the fluid drag force for the particles modelled in the present work. In addition, inter-particle forces, such as van der Waals forces, capillary forces and electrostatic forces have been neglected since the particle sizes (3mm sphere) used in this study are large and these inter-particle forces are insignificant (Seville et al. 1997). Fluid drag forces and collision forces are calculated in accordance with the sub-models described previously. Particle trajectories were computed by applying Newton’s second law of motion to each particle. Heat transfer between the gas and the particle is computed simultaneously with the evolution of particle trajectory. Simulations The numerical model was set up to simulate the gas-solids flow and heat transfer in an 8m section of a 3in (80mm bore) pneumatic transport line (Figure 3). The simulations used the same gas and particle properties and boundary conditions as adopted in an experimental study (Mason and Li 2000b). Validation of the model and comparisons with experimental data are presented in Li (2000). A common industrial material, 3mm polyethylene pellets, has been used in the experiments. Details of the pipe geometry and gas and solids properties are given in Table 1. The boundary conditions used in the model were specified to match those in the experimental system as illustrated in Figure 3 (See details in Li 2000). The gas was supplied from a straight pipe of sufficient length to ensure that the flow was fully developed. Particles were then introduced through the top of the pipeline, falling from a rotary valve through a dropout box (with a length of approximately 235mm). At the pipe outlet the pressure was maintained at a constant value. A steel pipeline with no thermal insulation was used for the flow of the gas-solids mixture. Therefore, the following boundary conditions were constructed: Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 14 Inlet Outlet Wall Gas was introduced with developed turbulent velocity and temperature profiles in the axial direction with a constant mass flow rate (a fairly constant pipe Reynolds number at the inlet since the change of gas density within the range of absolute pressure variation was negligible), supplied by a set of choked nozzles and heated by an electrical heater. Particles were introduced with a small velocity in the direction of gravity but zero axial velocity for a horizontal pipe. This velocity was computed from the actual height of the dropout box. All particles were assumed at ambient temperature (21oC). Pressure at the pipe exit was specified. A no slip condition was taken for the gas flow. A no slip condition was taken for the point on the particle in contact with the wall. This did not preclude particles from rolling along the wall. The surface temperatures of the pipe walls were assigned equal to the fluid temperature adjacent to the wall. A free convection thermal boundary at the wall was adopted. This study focused on the microscopic motion and heat transfer of individual particles and how these affect the gas-solids flow patterns and heat transfer. Figure 4 shows the solids distribution in the pipe at Reynolds number ReD = 79,400 and solids loading ratio SLR = 2, a typical dilute phase pneumatic conveying. (It is noted that the co-ordinates are not in scale in the radial and the axial direction.) Figure 5 traces a particle motion and its temperature variation in relationship to the surrounding fluid temperature. Due to the large thermal capacity of the 3mm particle, the temperature change of such a large particle during the drying in such a low fluid temperature (<50oC) is relatively slow. As the fluid temperature decreases along the pipeline, the heat transfer rate with the particle falls. Although a higher gradient of particle temperature change is observed at the particle inlet, the particle temperature shows a steady increase after the accelerating region. A similar simulation was conducted for a dense flow at ReD = 49,400 and SLR = 50. Figure 6 shows a much unsteady particle motion due to frequent particle collisions. A sharp gas temperature decrease is observed at the solids inlet, which indicates a rapid cooling of the gas due to the high solids concentration. The particle temperature change is very small after passing through the 8m long pipeline. This shows the difference of heat transfer mechanisms between the air drying in a dilute phase solids flow and that in a dense solids assembly or a moving particle bed. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 15 The above simulations show a slow temperature change for the 3mm pellets, which means that to dry such a large particle needs a much higher gas temperature and/or longer resident time. Using similar flow conditions (ReD = 79,400 and SLR = 2), computational experiments were also conducted for smaller particles: 1mm polyethylene pellets. Although a more pronounced temperature gradient was observed as plotted in Figure 7, there still remains a temperature difference between the particle and the gas. This instant temperature difference determines the heat transfer rate between the gas and the particle as shown in Figure 7, where more pronounced particle temperature changes happen at higher gas temperatures. This phenomenon can also be seen in Figure 5 for the 3mm pellet though not so obvious as in Figure 7 for the 1mm pellets. This reveals a quite different thermal behaviour of the large particles in a gas compared with a previous work of the Authors (Li and Mason 1998) on fine particles as shown in Figure 8. (A very small particle, 0.05mm aluminium sphere, in a very dilute phase flow at ReD = 47,200 and SLR = 1.1 in a 2in, 50mm bore, electrically heated pipe.) It demonstrates that, although there is always a temperature difference between the particle and the gas except some critical points, fine particles tend to follow the temperature of the gas. Therefore, a rapid temperature change is usually expected when drying fine particles. The thermal behaviours of particles with different sizes in a gas shall be of great interest to the study of drying of particulates. Figure 8 also clearly shows that by traversing across the pipe section and exchanging heat with the gases, particles actually indirectly enhance the heat transfer between the gas near the boundary wall and the gas in the middle of the pipe and, consequently, alter the flow and thermal structure of the gas. The integration of temperature difference between the particle and the fluid over time indicates the energy transferred during a certain period of time in correspondence to the exact particle trajectory. This is important in any gas-solids flow system where heat transfer is of primary concern. Although the present work only concerns the heat transfer in gas-solids flow through pipes, Figures 4 to 8 show that the application of DEM provides an opportunity to study the various flow and heat transfer phenomena in discrete particle level, including the solids distribution, the particle trajectory and particle temperature, the gas temperature, and the influences of particle diameter and particle concentration. These are also key parameters for the drying of particulate solids since the moisture transfer processes are largely dependent upon the conditions, temperature and moisture distribution, in the flow field surrounding the particles. Further development of the model to incorporate the mass transfer is envisaged. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 16 CONCLUDING REMARKS The application of the Discrete Element Method in gas-solids flow systems has been discussed in detail. A numerical model has been developed to simulate heat transfer in gas-solids flows through pipes. The implementation of this method for heat transfer in gas-solids systems has shown advantages over traditional continuum models. Further application of this technique in the drying of particulate solids has been evaluated. This evaluation shows the potential of using this method to model the coupled heat and mass transfer process in drying but validation by experiment is crucial. Future work is expected to focus on the implementations of the mass transfer model and its validation procedure. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 17 NOMENCLATURE Area (m2) Vapour concentration (kg/ m3) Fluid drag coefficient (-) Specific heat capacity (J/kg K) Diameter of pipe(m); Diffusion coefficient of water vapour in air within pore (m2/s) ds Particle diameter (m) F Force (N) f Coefficient of sliding friction (-) H Stagnation enthalpy (J/kg) h Heat transfer coefficient (W/m2 K) hd Mass stagnation enthalpy (J/kg) K Stiffness (N/m) k Thermal conductivity (W/m K) L Latent heat of water evaporation (J/kg) M Molecular weight (-) m Mass (kg) p Pressure (N/m2) Q Energy (J) q Heat transfer rate (W) R Gas constant (J/kg K) R Particle radius (m) r Radius (m) S Source term (Mass: kg/m3; Momentum: N/m3; Energy: W/m3.) SLR Solids Loading Ratio (kg/kg) T Temperature (K) t Time (s) u Velocity (m/s) v Relative velocity (m/s) A C CD cp D Greek letters Density (kg/m3) Dynamic viscosity (kg/m s) diffusion coefficient (kg/m s) Volume fraction (m3/m3); Porosity (m3/m3) Coefficient of viscous dissipation (kg/s) Displacement (m) Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 18 Subscripts: 0 Initial condition c Contact D Fluid drag; pipe l Liquid g Gas n Normal direction s Solids t Tangential direction w Water Dimensionless groups: Pipe Reynolds number Re D g Du g / Particle Reynolds number Re s g d s | u g u s | / Particle Nusselt number Nus hs d s / k g Prandtl number Pr c g / k g Schmidt number Sc / g Dv Sherwood number Sh hd d / Dv Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 19 REFERENCES Abuaf, N. and Staub, F.W., 1986, Drying of liquid-solid slurry droplets, pp.277-284, in A.S. Mujumdar (ed.), Drying ’86, Hemisphere Publishing Corportation, London. Bandrowski, J. and Kaczmarzyk, G., 1978, Gas-to-particle heat transfer in vertical pneumatic conveying of granular materials, Chemical Engineering Science, 33(10-A) pp. 1303-1310. Benet, J.C., Berthomieu, G., Naon, B. and Saix, C.,1997, Modelling convective drying of granular materials application to natural rubber, pp.415-437, in I. Turner and A.S. Mujumdar (ed.), Mathematical modeling and numerical techniques in drying technology, Marcel Dekker Inc., New York,. Boothroyd, R. G., 1971, Flowing gas-solids suspensions, Chapman and Hall, London, 289 pp. Bories, S., 1989, Recent advances in modelisation of coupled heat and mass transfer in capillary porous-bodies, pp.46-62, in A.S. Mujumdar (ed.), Proceedings of Sixth International Drying Symposium (IDS’89), Hemisphere, Washington. Chung, M.K., Sung, J.H. and Lee, K.B., 1986, Computational study of turbulent gassolids flow in a venturi, ASME J. of Fluids Engineering, 108 pp. 248. Crowe, C.T., Sharma, M.P. and Stock, D.E., 1977, The particle-source-in cell (PSICELL) model for gas-droplet flows, Journal of Fluids Engineering, Transactions of the ASME, June 1977, pp. 325-332. Cundall, P.A. and Strack, O.D., 1979, A discrete numerical model for granular assemblies, Geotechnique, 29(1) pp. 47-65. Ergun, S., 1952, Fluid flow through packed columns, Chem. Eng. Prog., 48(2) pp. 89-94. Frank, T.H., Schade, K.-P. and Patrak, D., 1993, Numerical simulation and experimental investigation of a gas-solid two-phase flow in a horizontal channel, International Journal of Multiphase Flow, 19 pp. 187-198. Frantz, J.F., 1962, Chemical Engineering, 69 pp.89. Hoomans, B.P.B., Kuipers, J.A.M., Briels, W.J. and Van Swaail, W.P.M., 1996, Discrete particle simulation of bubble and slug formation in a two-dimensional gasfluidised bed: A hard-sphere approach, Chemical Engineering Science, 51(1) pp. 99118. Julian, F.M. and Dukler, A.E., 1965, An eddy viscosity model for friction in gassolids flow, AIChE J., 1(5) pp. 853. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 20 Jumah, R.Y., Mujumdar, A.S. and Raghavan, G.S.V., 1997, A mathematical model for constant and intermittent batch drying of grains in a novel rotating jet spouted bed, pp.339-380, in I. Turner and A.S. Mujumdar (ed.), Mathematical modeling and numerical techniques in drying technology, Marcel Dekker Inc., New York. Kuo, J.T. and Chiou, C.H., 1988, Momentum and heat transfer of gas-solids suspensions in vertical pipes, Heat Transfer-Houston 1988 AIChE Symposium Series, 84 pp. 31-37. Langston, P.A., Nikitidis, M.S., Tuzun, U., Heyes, D.M. and Spyrou, N.M., 1997, Microstructural simulation and imaging of granular flows in two- and threedimensional hoppers, Powder Technology, 94 pp. 59-72. Langston, P.A., Tuzun, U. and Heyes, D.M., 1995, Distinct element simulation of granular flow in 2D and 3D hoppers: dependance of discharge rate and wall stress on particle interactions, Chemical Engineering Science, 50(6) pp. 967-987. Langston, P.A., Tuzun, U. and Heyes, D.M., 1996, Distinct element simulation of interstitial air effects in axially symmetric granular flows in hoppers, Chemical Engineering Science, 51(6) pp. 873-891. Levi-Hevroni, D, Levy A. and Borde, I., 1995, Drying Technology, 13(5-7) pp. 11871201. Levy, A., Mason, D.J., Levi-Hevroni, D. and Borde, I., 1998, Drying of wet solid particles in a steady-state one-dimensional flow, Powder Technology, 1 pp. 15-23. Li, J., 2000, Heat transfer in gas-solids flows through pipes, PhD thesis, Glasgow Caledonian University, UK. Li, J. and Mason, D.J., 1998, Simulation of gas-solids flow in pipes with heat transfer using the distinct element method, 1998 Research Event, Proceedings of the Institute of Chemical Engineers, at University of Newcastle Upon Tyne, UK, Paper 71. Li, J. and Mason, D.J., 2000, A computational investigation of transient heat transfer in pneumatic transport of granular particles, Powder Technology, 112(3) pp. 273-282. Li, Y., Zhang, J. and Fan, L-S, 1999, Numerical simulation of gas-liquid-solid fluidisation systems using s combined CFD-VOF-DPM method: bubble wake behaviour, Chemical Engineering Science, 54 pp. 5101-5107. Luikov, A.V., 1966, Heat and mass transfer in capillary-porous bodies, Pergamon Press, London, 522 pp. Mason, D.J., 1991, A study of the modes of gas-solids flow in pipelines, PhD Thesis, Thames Polytechnic (now University of Greenwich), UK. Mason, D.J., and Li, J., 2000a, A numerical study of transient heat transfer in the developing zone of gas-solids flow through pipes using the Distinct Element Method, Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 21 Research 2000, Proceedings of the Institute of Chemical Engineers, University of Bath, January 2000, Paper 109. Mason, D.J. and Li, J., 2000b, A novel experimental technique for the investigation of gas-solids flow in pipes, Powder Technology, 112(3) pp.203-212. Matsumoto, S., and Pei, D.C.T., 1984, A mathematical analysis of pneumatic drying of grains – I. Constant drying rate, Int. J. Heat Mass Transfer, 27(6) pp. 843-849. Matsumoto, S. and Saito, S., 1970, On the mechanism of suspension of particles in horizontal pneumatic conveying: Monte Carlo simulation based on the irregular bouncing model, Journal of Chemical Engineering of Japan, 3 pp. 83-92. Mhimid, A. and Nasrallah, S.B., 1997, Theoretical study of heat and mass transfers during drying of granular products, pp.381-413, in I. Turner and A.S. Mujumdar (ed.), Mathematical modeling and numerical techniques in drying technology, Marcel Dekker Inc., New York. Michaelides, E.E., 1984, A model for the flow of solid particles in gases, International Journal of Multiphase Flow, 10(1) pp. 61-67. Mujumdar, A.S., 1980, Advances in drying, Volume 1, Hemisphere Publishing Corporation, London, 301 pp. Mujumdar, A.S., 1983, Advances in drying, Volume 2, Hemisphere Publishing Corporation, London, 301 pp. Mujumdar, A.S., 2000, Classification and selection of industrial dryers, pp. 23-36, in Mujumdar’s practical guide to industrial drying, S. Devahastin (Ed.), Exergex, Montreal. Oliveira, L.S. and Haghighi, K., 1997, Finite element modeling of grain drying, pp.309-338, in I. Turner and A.S. Mujumdar (ed.), Mathematical modeling and numerical techniques in drying technology, Marcel Dekker Inc., New York. Ottjes, J.A., 1978, Digital simulation of pneumatic particle transport, Chemical Engineering Science, 33 pp. 783-786. Patankar, S.V., 1980, Numerical heat transfer and fluid flow, Hemisphere Publishing Corporation, New York. Peng, X., Tomita, Y., and Tashiro, H., 1994, Effect of particle-particle collision and particle rotation upon floating mechanism of coarse particles in horizontal pneumatic pipe, JSME. Int. J., Series B, 37 pp. 485-490. Peng, X., Tomita, Y., Tashiro, H. and Funatsu, K., 1996, Motion of single particle in a horizontal pipe, Advanced Powder Technology, 7 pp. 101-109. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 22 Pritchett, J.W., Blake, T.R. and Garg, S.K., 1978, A numerical model of gas fluidized beds, Fluidization: Application to coal conversion process - AIChE symposium, 74(176) pp. 134-148. Puiggali, J.R. and Quintard, M., 1992, Properties and simplifying assumptions for classical drying models, Vol. 5, pp.109-143, in A.S. Mujumdar (ed), Advance in Drying, Hemisphere, Washington. Ranz, W.E. and Marshall, W.R., 1952, Evaporation from drops (Parts I and II), Chem. Eng. Pro., 48(3) pp. 141-146 and 48(4) pp. 173-180. Richardson, J.F. and Zaki, W.N., 1954, Sedimentation and Fluidization, Part I, Trans. Inst. Chem. Eng., 32 pp. 35. Roco, M.C., 1993, Particulate Two-Phase Flow, Butterworth-Heinemann, 1002 pp. Salman, A.D., Gorham, D.A. and Verba, A., 1998, Particle movement in dilute pneumatic conveying, Proceedings of 1st International Symposium on On-Line Flow Masurement of Particulate Solids, Organized by School of Engineering, University of Greenwich, UK. Schiller, V.L. and Naumann, A., 1933, Uber die grundlegenden Berechnungen bei der Schwerkraftaufbereitung, Zeitschrift des Vereines deutscher Ingenieure, 77 pp. 318320. Seville, J.P.K., Tuzun, U. and Clift, R., 1997, Processing of Particulate solids, Blackie Academic & Professional, London, 372 pp. Sharma, K.R., 1997, Relative contributions from particle conduction and gas convection to the heat transfer coefficient between dense gas-solid fluidized beds and surfaces, Powder Technology, 91 pp. 75-80. Sherwood, T.K., 1929, Ind. Eng. Chem., 21 pp.12. Spalding, D.B., 1980, Numerical computation of multi-phase fluid flow and heat transfer, in C.Taylor and K. Morgan (Eds.), Recent Advances in Numerical Methods, Pineridge Press Ltd, Swansea, UK, 1(5) pp. 139-167. Summerfeld, M., 1992, Modelling of particle-wall collisions in confined gas-particle flows, International Journal of Multiphase Flow, 18 pp. 905-926. Sun, J. and Chen, M.M., 1988, A theoretical analysis of heat transfer due to particle impact, International Journal of Heat and Mass Transfer, 31(5) pp. 969-975. Thornton, C. and Yin, K.K., 1991, Impact of elastic spheres with and without adhesion, Powder Technology, 65 pp.153-166. Tsotsas, E., 1998, Achievement, challenges and perspectives in modelling and design of convective dryers, Volume A, pp11-22, in C.B. Akritidis et al. (Ed.), Drying ‘98, Ziti Editions, Thessaloniki, Greece. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 23 Tsuji, Y., Kawaguchi, T. and Tanaka, T., 1993, Discrete particle simulation of twodimensional fluidized bed, Powder Technology, 77 pp. 79-87. Tsuji, Y., Tanaka, T. and Ishida, T., 1992, Lagrangian numerical simulation of plug flow of cohesionless particle in a horizontal pipe, Powder Technology, 71 pp. 239250. Waananen, K.M., Litchfield, J.B. and Okos, M.R., 1993, Classification of drying models for porous solids, Drying Technology, 11(1) pp.1-40. Wang, Y. and Mason, M.T., 1992, Two-dimensional rigid-body collision with friction, Journal of Fluid Mechanics, 59 pp. 635. Wrangham, D.A., 1961, The Elements of Heat Flow, Chatto and Windus Ltd, London, 486 pp. Yamamoto, F., 1986, A study of motion of a sphere in air flow through horizontal pipe, Bulletin of JSME, 29 pp. 2055-2061. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 24 Pipe: Pipe internal diameter (m) Pipe length (m) Gas: Gas used Density (kg/m3) Thermal conductivity (W/m K) Specific heat capacity (J/kg K) Particles: Material Shape Diameter (mm) Particle density (kg/m3) Stiffness (N/m) Coefficient of friction (-) Coefficient of restitution (-) Thermal conductivity (W/m K) Specific heat capacity (J/kg K) 0.080 8.0 Air By ideal gas law 0.02564 1006.1 Polymer Spherical 3.0 880 2E+5 0.3 0.8 0.17 2300 Table 1: Data used by model Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 25 dash-pot spring spring slider slider dash-pot (a) (b) FIGURE 1. Models of contact forces: (a) normal force; (b) tangential force. Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 26 R1 R2 rc R12 FIGURE 2. Contact between two particles Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 27 Cold solids in 235mm Un-insulated pipe Hot gas in 80mm 8m FIGURE 3. Gas/Solids loading and flow boundaries Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 28 Particle 3mm 0.08 0.07 Radial (m) 0.06 0.05 0.04 0.03 0.02 0.01 0 0 1 2 3 4 5 6 7 8 Axial (m) FIGURE 4. Solids distribution in pipe Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 29 0.08 320 Particle trajectory Particle temperature 315 Fluid temperature 310 0.04 305 300 0.02 Temperature (K) Radial (Pipe dia. m) 0.06 295 0 290 0 2 4 Axial Distance (m) 6 8 FIGURE 5. Particle (3mm) trajectory and temperature in a dilute phase flow Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 30 Particle trajectory Particle temperature Fluid temperature 330 0.06 320 0.04 310 0.02 300 0 Temperature (K) Radial (Pipe dia. m) 0.08 290 0 2 4 Axial Distance (m) 6 8 FIGURE 6. Particle (3mm) trajectory and temperature in a dense phase flow Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 31 0.08 325 Pronounced increases 0.06 320 315 310 0.04 305 300 0.02 Slow increases Temperature (K) Radial distance (m) Particle trajectory Particle temperature Fluid temperature 295 0 290 0 1 2 3 4 5 Axial distance (m) 6 7 8 FIGURE 7. Particle trajectory and temperature for a 1mm pellet Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 32 297 Particle Trajectory Particle Temperature Fluid Temperature 0.04 0.03 294 291 0.02 288 0.01 0 285 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Axial distance (m) FIGURE 8. Particle trajectory and temperature of a 0.05mm aluminium sphere Drying Technology - An International Journal, ISSN 0737-3937 2002 vol.20, no.3, pp.255-282. 33 Temperature (K) Radial distance (m) 0.05