POWDER TECHNOLOGY ELSEVIER Powder Technology93 (1997) 253-260 A comparison of analytical and numerical models with experimental data for gas-solid flow through a straight pipe at different inclinations Avi Levy *, Thomas Mooney, Predrag Marjanovic, David J. Mason Centrefor Industrial Bulk Solids Handliag, Department of Physical Sciences, GlasgowCaledonian University, CowcaddensRoad, GlasgowG40BA, UK Received I 1 July 1996; revised 4 March 1997; accepted 27 May 1997 Abstract An analytical model for gas-solid suspension flow through an inclined section of pipe was developed. This model predicts the ratio of the total pressure drop in an inclined pipe to that of a horizontal pipe. The model has been used to predict the critical pipe angle, which is defined as the angle at which the maximum pressure drop for a given solids flow rate is achieved. This angle differs from 90° (found in a single-phase flow) and is directly proportional to the ratio between the gas superficial velocity and the particle terminal velocity. The three-dimensional conservation equations for steady-state two-phase flow in an inclined pipe were solved numerically for constant solids and gas flow rates at different pipe inclinations. This model was based on the continuum theory for describing the mass and momentum balance equations for the fluid and solid phases. A packing model, describing the shear stress of the solid phase as a function of its volume fraction, is suggested in order to limit the maximum value of the solid volume fraction. A new model for particle-wall interaction was developed taking into account the angle of inclination of the pipe. The prediction of the numerical model was compared with experimental data obtained in a specially designed test rig. In general, the agreement between the experimental data and the models was satisfactory. The results of the numerical simulation also confirmed that the critical pipe angle for gas-solid flow is lower than 90°. The assumptions made during the development of the models were assessed in order to explain the differences between the predicted and measured values of the flow parameters for different flow regimes. © 1997 Elsevier Science S.A. Keywords: Inclinedpipelines; Two-phase flow;Pneumatic conveying c 1. I n t r o d u c t i o n The designer of a pneumatic conveying system always has to make a few decisions when designing the various components of a system. One o f these decisions relates to the design of a pipeline layout which includes a change o f level. T w o basic alternatives exist to reach point B from point A (Fig. I): (a) the level change can be designed as two sections ( A C vertical, CB horizontal), or (b) the level change can be designed as one inclined section AB. Considering that the whole pipeline lies in the vertical plane, defined b,' points A, B and C, the first solution requires two 90 ° bends (points A and C) and the second solution requires two ( 1 8 0 - a ) ° bends (points A and B ). At the same time it is obvious that the overall length o f the inclined pipeline is smaller than that of the vertical-horizontal solution. * Corresponding author. Tel: +44 141 331 3711; fax: +44 141 331 3448; e-mail: a.levy@gcal.ac.uk 0032-5910/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved PIi S0032-591 O( 97 ) 03 280-4 A I :IV B I A Fig. I. Pipelnne layout. For a single-phase flow the solution using an inclined section would be preferable as it results in a smaller energy loss (shorter straight section, higher bend angle). However, the nature o f two-phase flow, either gas-solid or liquid-solid, influences energy loss in a way that, with inclined sections, could turn out to be the worst possible scenario. Although a number o f plant operating problems have been reported (high energy losses, high fluctuations of all flow parameters, even blockages o f pipelines), caused allegedly 254 A. Levy et aL / Powder Technology 93 (1997) 253-260 by inclined sections, only a few studies have been dedicated to investigating the cause of this kind of performance. This paper gives an indication of the possible reason, based on the analytical analysis of the Laost influential forces which act on the particulate phase in gas-solid flow, along with a numerical and experimental investigation. Most of the published work detailing methods for the calculation of energy loss in gas-solid flow either relates to horizontal flow or to vertical flow. Examples of various approaches are given in Ref. [ 1]. They normally start from the assumption that the total energy loss is the sum of the energy loss for gas flow and additional energy loss which takes into account the influence of the presence of particles in the gas stream. The determination of this additional energy loss is the main problem and a number of proposed formulae exist in the literature, e.g. Refs. [2,3]. Most, if not all, rely heavily on one or a few empirical parameters, hence they are not straightforwardly applicable to the design of a system transporting a 'new' material. Very few experimental studies of gas-solid flow in inclined pipes have been carded out. For example, the study of flow regimes in vertical and inclined (up to 18° from the vertical) pipelines is reported by Ginest et al. [4]. This study gives a classification of various flow regimes which can develop due to changes in pipeline inclination. Experimental results obtained for dolomite and cement in a 120 mm bore inclined pipeline are reported by Salt et al. [5]. In their study they measured the solids flow rate at a constant pressure drop for different angles of inclination. One of the first detailed theoretical approaches, which covers all pipe inclinations, is given by Muschelknautz [ 2,3 ]. This model considers only gas-solid flow at a low concentration of solid particles in a gas stream, i.e. low solids loading ratio (SLR=n~Jtfig). This uses an analogy to Darcy's formula and the full model takes into account all forces which act on a cluster of particles (gravity, buoyancy, aerodynamic drag, aerodynamic lift, friction). Marjanovic [ 6] presents a method for the comparative analysis of the influence of pipeline inclination on the energy loss in a pneumatic conveying system. This method is based on the assessment of the angle at which a solid particle moves in a pipeline and impacts on the wall. This model includes, implicitly, only some of the most influential terms. The results obtained, however, differ by less than 5% from the full model of Muschelknautz and Wojahn [3] for all conveying conditions. This simplified model describes the overall pressure drop as the sum of two terms. The first is the pressure drop for gas flow and the second is the pressure drop caused by the presented solid phase. This was written as A p t = A p s + A p ~ Pi,-P~.....x m~ L I , 2p~ - Ag+ A -~p~v; tively; pg and vg are the density and superficial velocity of the gas, calculated at mean pressure p~; As is the well-known parameter for calculating the pressure drop caused by gas flow and the effective solids friction factor, A~,is an additional parameter which takes into account all effects arising from the presence of solid particles in the gas stream. This factor is obtained through analysis of all the mentioned forces which act on a cluster of particles: A~= A~ ÷ (vs/v~)Fr2 (2) where W is the solid phase velocity, A* is a factor which takes into account both impact and friction (to be determined experimentally ), Fr ( - v~/ (gD) ' / 2) is the Froude number, D is the pipe bore and/3 is a parameter which takes into account the effects of the incline pipeline, defined by fl = sin(a) + 14 cos(a) v~ (3) where t~ is the terminal velocity of a particle falling vertically, and t~ is the angle of inclination of the pipe to the horizontal. Determination of the slip velocity ratio represents a major problem, particularly if the influences of particle-particle impact and the gas velocity profile are to be taken into account. The relationship given in this model is rather complex and a detailed analysis can be found in Refs. [ 2,3 ]. It should be pointed out here that the slip velocity ratio depends upon several parameters, e.g. vJv~=f(A*, 14, D, a, v~). The terminal velocity 14 is often used in such correlations since it is a function of the drag coefficient of the particle, as is the slip velocity, and is rel:aively simple to measure. Muschelknautz and Wojahn [ 3 ] tested a number of products in order to determine the factor A* and the ratio between the particle terminal velocity and the gas velocity. Their values varied within the range 0.001-0.040 and 0.3-0.7, respectively, depending on both the material and the pipeline characteristics, as well as the gas velocity. An empirical correlation for the velocity ratio was presented by Hinkle as a function of the particle density and diameter [ 7 ]. Note that the range of the panicle terminal velocity to gas velocity ratio can be estimated for differem particles. For example, polyethylene pellets (mean size 3 ram, terminal velocity 8.1 m/s) would normally be transferred in dilute phase flow with an air superficial velocity between 14 and 35 m/s. This gives a velocity ratio between 0.23 and 0.58. Based on the Hinkle [7] equation, the value of the velocity ratio for 3 mm polyethylene pellets was estimated as 0.66. Note that the lower limit of the air superficial velocity range is the minimum usually chosen in order to convey in a dilute phase flow and the upper limit is the maximum air superficial velocity necessary to minimise particle damage. (l) where Pi,, P~ and P8 are, respectively, the pressures at the inlet and exit of the pipeline and their mean value; d~ and n~ are the mass flow rates of the gas and solid phases, respec- 2. Present study In the present study, analytical, numerical and experimental investigations were conducted. The analytical model for A. Levv et al. / Powder Technology 93 (1997) 253-260 gas-solid suspension flow through an inclined section of pipe was developed. Unlike Marjanovic [6], whose analysis is based on the forces acting on a single particle and the angle at which the particle impacts the pipe wall, this model is based on the dimensional analysis of the major forces acting on a dispersed solids phase. The three-dimensional conservation equations for steadystate two-phase flow in an inclined pipe were solved numerically for constant solids and gas flow rates at different pipe inclinations. This model was based on the continuum theory for describing the mass and momentum balance equations for the fluid and solid phases. A packing model, describing the shear stress of the solid phase as a function of its volume fraction, is suggested in order to limit the maximum value of the solid volume fraction. A new model for wall-particle interaction was used during the numerical simulations. In order to validate the model and the prediction of the numerical simulations an experimental investigation was carried out. The prediction of the numerical model was compared with experimental data obtained in a specially designed test rig. 3. Analytical study In order to determine the overall effect of the pipe's angle of inclination, a, on the total pressure drop it was decided to investigate Eq. ( 1). Note that the total pressure drop in Eq. ( 1) differs by less than 5% from the full model of Muschelknautz and Wojahn [ 3 ] for all conveying conditions, as noted by Marjanovic [61. Substituting Eqs. (2) and (3) into Eq. ( I ) yields LI ~ LI , + r,p~ sin(a) L + r,p.,g cos(a) L t~ vs (4) where r, is the solid volume fraction. In order to find the extreme value of the pressure drop, for given gas and particle flow rates, as a function of the angle of inclination, tr, Eq. (4) was differentiated with respect to a and compared with zero. c3pt 0p~ -----'~=r,p.,gcos(oO L-r,p.,gsin(a) L t&---0 0a 0or v~ (5) This resulted in tan(a¢~t) =v~ (6) Differentiating Eq. (5) with respect to the angle of inclination, a, gives i~p, O2p.~ 0tx,2=~---~= - r , o• , •g s i n ( a ) L - r ~ cos(or) L tIJg t<0 (7) 255 For angles of inclination between 0 ° and 90° this second derivative is always negative, hence a maximum value of the pressure drop will be achieved in this range. The angle which results in the maximum pressure drop, amax,can be estimated for representative values of the particle density, volume fraction, and velocity ratio. For large particles in a dilute phase flow the particle terminal velocity to gas velocity ratio varies between 0.3 and 0.7. Hence, the maximum pressure drop will occur at ~maxbetween 55 ° and 73 °. For similar particles conveyed in a dense phase mode of flow where the solid volume fraction is much higher and the superficial gas velocity is much lower the critical angle will be close to 0 °. Conversely, a critical angle close to 90 ° will be achieved by small particles conveyed in a dilute phase mode at very high velocities. From this analysis it can be seen that the critical angle can vary between 0 ° and 90° depending upon the particle characteristics (e.g. particle diameter, particle density, etc.) and the flow regime. 4. Numerical study A three-dimensional model based upon the concept of interdispersed continua was used to model this flow. This model solves the conservation equations for mass and momentum for the gas and solid phases by using a finitevolume numerical method. The three-dimensional model employed in this work is incorporated in the PHOENICS software by CHAM, UK. The models and their solution are described by Spalding [8]. This model uses a finite-volume formulation of the conservation equations for mass and momentum for two phases. PHOENICS provides solutions to the discretised version of a set of differential equations having the general form O( riPA°i) t- V. ( ripiv~o~ + r~F,,V~o~) = r,S,, Ot (8) where t is the evolution time, r~, p, and v~ are the volume fraction, the density and the velocity of phase i, ~p~ is the conserved property of phase i, such as enthalpy, momentum per unit mass, etc., and F,, and S,, are the diffusion coefficient and source term of ~o~.Since we are looking for a steady-state solution the transient term in Eq. (8) was set to zero. The concept of interpenetrating continua is used to obtain the two-phase conservation equations by time and space averaging. These equations are solved using the Inter Phase Slip Algorithm. This is an iterative method: • starting with a guessed pressure field, • the conservation equations are solved in the following order: energy, mass, momentum; • finally, a pressure correction equation is solved and the velocities are corrected in order to ensure satisfaction of the overall mass conservation equation. This cycle is performed many times until the errors remaining in all the equations are acceptably small. 256 A. Levy et aLI Powder Technology 93 (1997) 253-260 4.1. Inter-phase momentum transfer Source terms in each of the conservation equations provide the means of specifying the nature of the transfer of mass and momentum between the phases. In this case the inter-phase momentum is based upon that used for fluidisation processes, since the range of solids concentrations experienced in pneumatic transport systems is similar. This has been employed by both Patel and Cross [9] and Kuipers [ 10] for modelling gas-solid fluidised beds. The inter-phase friction force F is evaluated using F = lpvVceH( Ug-- Us) (9) where Vceu is the volume of a control volume. For solids concentrations greater than 0.2 the model computes the interphase friction coefficient. IpF. using the Ergun [ 1 1] equation: Sk = - 2p~rgr~ 7E+k ¢p (14) S~ = - 2c~pjgr, ca + rp (15) where c~ is an empirical constant ( 1.0 in this case). ~-pand eE are, respectively, time factors characterising the particle response and large-scale turbulent motion: Tp=pr,v,-- vJ I I (16) k TE=CT-E (17) where the empirical constant CT= 0.35. 4.3. WaU friction model lpF = 150 ~ ~2 + 1.75r~ ~ pgl ug-- vsl rgd~ ( l 0) where/L is the dynamic viscosity of the gas phase and d~ is the average diameter of the particles. For solids concentrations less than 0.2, the model computes the coefficient as follows: 3r~ I2 rspgl vR- v,I IpF = Core- 2.65 2d. (11) where the single-particle drag coefficient, Co, is given by CD = max ( 1 + 0.15Re°6sT), 0.44 (12) and is modified to take account of multiparticle effects using the method of Richardson and Zaki [ 12 ]. The particle Reynolds number is given by Re=P~d~(rsl v s - v,I ) (13) The friction force between the conveying gas and the pipe wall was modelled by adding a source term to the gas phase momentum equation for those control volumes adjacent to the pipe wall. This term assumed no slip at the wall and a logarithmic velocity profile, and calculated the wall friction factor based upon a smooth pipe correlation. The friction force between the solid phase and the pipe wall was modelled by adding a source term to the solid phase momentum equation for those control volumes adjacent to the pipe wall. In order to take into account the inclined pipe effect, a solid phase-wall friction source term was developed. This was developed by usi:tg Eq. (4) and subtracting the pressure drop caused by gravity from the pressure drop of the solid phase, i.e. Ap.~ - r~p~g sin(a) L Ll ffi A*r.~ -~ ~ p~v.~2 + r~p.~gcos(t~) L 14= ew v~ (18) As a consequence, the three-dimensional source term for the solid phase-wall friction was written as 4.2. Turbulence model S w = r ~ p , A w , , , ( k , f f g •.n p A*Iv,. v.) + ~ The k-e turbulence model was used in this work to model the gas phase turbulence, with additional source terms to take account of the influence of the presence of particles. Several extensions of the k-e turbulence model have been proposed, and PHOENICS provides two options, the models of Chen and Wood [ 13,14] and of Mostafa and Mongia [ 15]. The effect of these source terms is to reduce the turbulence kinetic energy k and the turbulence dissipation rate e. However, depending on the relative extent of the reductions of k and e, the turbulent viscosity may be either reduced or increased by the presence of particles. In this work the Mostafa-Mongia model of Ref. [ 15 ] was used. The additional volumetric source terms in the k and e equations are ( 19) where Aw,o is the wall area of the control volume, kv ( - vt/ vg) is the particle terminal velocity to gas velocity ratio and t~p is a unit vector perpendicular to Aw,u. 4.4. Other relationships The gas was assumed to behave as a perfect gas, and the flow was assumed to be isothermal. Thus the conservation equations for energy were not solved. A fixed temperature was assigned so that values such as the gas density could be calculated. Normally the mass conservation equations are solved for both phases, which can lead to problems when the concen- 257 A. Levy et al. / Powder Technology 93 (1997) 253-260 tration of one of the phases is significandy less than the other. If this is the case then only the equation for the dispersed phase is solved, with the volume fraction of the continuous phase determined from In this study this second option was used with only the solids mass conservation equation being solved. When solving the momentum conservation equation for one of the phases, the code solves the conservation equations for the velocity component in each coordinate direction. The turbulence model requires the solution of two conservation equations, one for the turbulence kinetic energy and one for the turbulence dissipation rate. Thus a set of ten equations must be solved for this three-dimensional two-phase problem ( nine conservation equations and the pressure correction equation). 5. Experimental study The influence of pipeline inclination on pressure gradient for two-phase gas-solid flow was investigatedexperimentally in order to validate the predictions of the physical model. The experiments were conducted in an industrial-scale pneumatic conveying system at the Centre for Industrial Bulk Solids Handling in the Department of Physical Sciences at Glasgow Caledonian University. A schematic illustration of the experimental system is shown in Fig. 2. This system was operated as a batch process. Air was supplied from a Roots type blower capable of delivering up to 0.33 m3/s of free air when running in a positive pressure mode. The bulk material was fed into the pipeline via a variable-speed rotary valve from a 0.7 m 3 feed hopper. The gas-solid mixture was then transported through a pipeline approximately 100 m long, with an internal diameter of 81 ram, to the receiving hopper which was mounted directly above the feed hopper. This arrangement simplifies the task of conveying a batch of material at a number of operating conditions. Receiving F~t Hopper Hopper Rotary Valve InclinedSection Material Polymer pellets Mean diameter 3 mm Particle density 880 kg/m 3 Bulk density 550 k g / m 3 (20) rcontmuou s "~- 1 - rdlsper,~ d Filter Table 1 Basic material properties J ~ Fig. 2. Isometric sketch of pipeline loop. The blower supplying air to the system operated at a fixed speed, and so a bleed valve was used to control the air flow rate to the system. Because of the way it was operated, air also leaked through the solids feeder. Thus the air flow rate was measured at the end of the pipeline after the filter using a thermal mass flow meter capable of measuring up to 0.2 kg/s with an accuracy of + 2% of the reading. The solids mass flow rate was also determined at the end of the system by measuring the mass of solids collected in the receiving hopper. This hopper was mounted independently on three 500 kg load cells with an accuracy of + 0 A kg. The solids mass flow rate was determined by differentiatingthe load cell transient. Pressure and temperature measurements were made at a number of locations around the system. Pressure was measured using single-ended transducers with a range of 02.5 bar and an accuracy of ±0.006 bar. Temperature was measured using probes with a range of 0-100°C with an accuracy of +0.5°C. Data from all the ~:~;msducers were recorded by a computer-controlled data logger, usually at a rate of I Hz. Polyethylene pellets were used as the test material in this programme. Table 1 lists the properties ofthesepellets. Pressure gradients were measured at 0 °, 45 °, 60°, 75 ° and 90 ° orientations, superficial gas velocities ranged from 10 to 20 m/s and solids throughputs from 0.25 to 0.5 kg/~. The experimental data for the pressure gradients data, as measured for 10 different angles of inclination and solids and air mass flow rates, are presented in Table 2. 6. Comparison between the numerical prediction and the experimental result and parametric investigation In order to validate the physical model (described in Section 4), it was solved numerically for several different initial conditions and angles of inclination and compared with experimental results. A comparison between the predictions of the numerical simulation and the analytical model with experimentally determined pressure gradients at various initial conditions is presented in Table 3. From this it can be seen t~mt the numerically predicted pressure gradients are up m 20% lower than the experimental values. The predicted pressure gradient is mainly influenced by models for the drag force on a particle, the solids-wall friction, and the turbulence reduction due to particles. A combination of one or more of the following would result in an increase in the predicted pressure gradient: an increase in the particle drag coefficient, a higher coefficient of particle-wall friction, or a decrease in the turbulence reduction. From this it is clear that some refinement of these models is required in order to predict the flow more accurately. Since the difference between the predicted 258 A. Levy et al, / Powder Technology 93 (1997) 253-260 Table 2 Experimental data for the pressure gradients as measured tbr different angles of inclination and mass flow rates Test no. Air mass flow rate (kg/s) Solids loading ratio Angle of inclination ( deg. ) Experimental data for pressure gradient (mbar/m) I 2 3 4 5 6 7 8 9 10 0.131 0.114 0.123 0.109 0.126 O.126 0.127 0.103 O.113 0.106 3.24 5.02 3.01 3.85 2.26 4.45 4.59 6.1 4.47 4.06 0 0 45 45 60 60 75 75 90 90 1.89 1.93 2.79 2.85 1.89 3.33 2.72 3.3 2.56 2.32 Table 3 Comparison between the predictions of the numerical simulation and the analytical model with the experimental results for the pressure gradient Test no. ! 2 3 4 5 6 7 8 9 I0 Pressure gradient (mbar/m) Error ( % ) Experimental data Numerical prediction Analytical prediction Numerical prediction Analytical prediction 1.89 1.93 2.79 2,85 1.89 3.33 2.72 3.3 2.56 2.32 1.97 1.86 2.24 2.75 1.72 2.70 2.94 2.69 2,65 2.22 I. I0 1.41 1.65 2.01 1.35 2.45 2.41 3.45 2.10 1.88 4.23 3.63 19.7% 3.51 8.99 18.99 8.09 18.48 3.52 4.31 41.9 27. I 40.9 29.6 28.6 26.4 I 1,3 4.5 18.1 19.2 and measured pressure gradients for horizontal and vertical pipes was much lower than for the inclined pipes, the wall friction model is probably the most likely cause of error. Unlike the numerically predicted pressure gradient, the error in the analytically predicted gradient is much greater (up to 42%) for angles of inclination between 0° and 60 °. It should be noted that, in calculating the pressure gradient, Eq. (4) was used with the same values for the parameters as used in the numerical study. One of the key parameters in the analytical model is the particle velocity. A number of correlations for this parameter were examined and the correlation of Muschelknautz and Wojahn [ 3] was selected, since it was the only model which took into account the angle of inclination. One of the main reasons for the analytical model underestimating the pressure gradient was that the particle velocity was overestimated. in general, the comparison showed satisfactory agreement between the predictions of the numerical simulations and the experimental data for the pressure gradients. Since it is not possible to change the angle of inclination, or, continuously during the experiments and to find the critical angle ~,,..,xfor a constant set of initial conditions, it was decided to find the critical angle numerically for different solids loading ratios. The physical model was solved numerically for three different soiids loading ratios, 5, 10 and 20. This was done by keeping a constant air flow rate, having an inlet superficial velocity of 9.2 m/s and changing the mass flow rate of the solids. Note that due to the compressibility of the air the superficial velocity changes along the pipe. For different solids loading ratios the inter-phase exchange rates of momentum and energy change significantly. This results in a different average superficial gas velocity and hence a change in the critical angle am~x. The difference between the total pressure gradient for an inclined pipe and that for a horizontal pipe, under the same flow conditions, as a function of the angle of inclination is shown in Fig. 3. Data are presented for solids loading ratios of 5, 10 and 20. It should be noted that the pressure gradient for a horizontal flow is also dependent on the solids loading ratio and grows continuously as the solids loading ratio increases. From this figure it can be seen that by doubling the solids loading ratio the maximum difference in the pressure gradients increases by almost the same amount. Hence, one could conclude that this maximum is proportional to the solids mass flow rate. Fig. 4 represents the normalised pressure gradient (pressure gradient for a particular angle of inclination divided by that for the horizontal) as a function of the angle of inclination for different solids loading ratios. This figure shows that the normalised pressure gradient ranges from 1 for horizontal flow to 1.5-1.6 at the critical angle C~m,,x. The normalised pressure gradient increases when the solids loading ratio is A. Levy et al. / Powder Technology 93 (1997) 253-260 3.5 8 ] •.-e-SLR = 5 3 :-4-SLR = 10 ~ ] I SLR = 20 2.5 i g 15 ~ o. 0.5 0 0 10 20 30 40 50 60 70 80 90 Inclination Angle [Deg.] Fig. 3. The total pressure gradient difference between that of an inclined pipe and that of a horizontal pipe for different solids loading ratios (SLR) as a function of the inclination. _o "0 1.6 -e-SLR = 5 ~ " ~ 1 ~ ] 8. List of symbols ~ 1.3 R, I 1.2 I shear stress of the solid phase as a function of its volume fract[on, is suggested in order to limit the maximum value of the solid volume fraction. A new model for particle-wall interactions was developed, taking into account the angle of inclination of the pipe. The predictions of the nu~.rical model were compared with experimental data obtained in a specially designed test rig. In general, the agreement between experimental data and the models was satisfactory. The results of numerical simulation also confirmed that the critical pipe angle for gas-solid flow is lower than 90 °. The analytical and numerical models developed in this study are not restricted to a particular particle size, although they have only been validated with experimental data for relatively large particles. The authors plan further experiments with other bulk materials to extend the validation of the models over a range of particle sizes. 1.5 0 -= ,,,,= 1.4 IA o 10 20 30 40 50 60 70 80 90 Indinstion Angle [Dog.] Fig. 4. The normalised pressure gradient (pressure gradient at an angle divided by that at the horizontal) as a function of the angle of inclination for different solids loading ratios (SLR). A,,~. CD d~ D F Fr g IPF k wall area of control volume drag coefficient particle diameter diameter of pipe section inter-phase friction force Froude number gravity acceleration inter-phase friction factor turbulence kinetic energy changed from 5 to 10. A further increase to 20 results in a reduction of the normalised pressure gradient. This was due to a significant increase in the predicted horizontal pressure gradient at this solids loading ratio. k~ q/v~ L th ~p p r 7. Conclusions S, S~, Sk S~ v V~. x length of pipe section mass flow rate unit vector perpendicular to Aw~, pressure volume fraction Reynolds number solid phase-wall friction source term source term of q~ source term for particle effects on k source term for particle effects on E velocity volume of control volume axial distance Re An analytical model for gas-solid suspension flow through an inclined section of pipe was developed. This model predicts the ratio of the total pressure drop in an inclined pipe to that of a horizontal pipe. The model has been used to predict the critical pipe angle which gives the maximum pressure drop for given solids flow rate, or the minimum solids flow rate for a given pressure drop. This angle differs from 90 ° ( found in a single-phase flow) and is directly proportional to the ratio between the gas superficial velocity and the particle terminal velocity. Further experimental studies would be required to confirm this result for other bulk materials. The three-dimensional conservation equations for steadystate two-phase flow in an inclined pipe were solved numerically for constant solids and gas flow rates at different pipe inclinations. This model was based on the continuum theory for describing the mass and momentum balance equations for the fluid and solid phases. A packing model, describing the 259 Greek letters o[ O[ma X /3 E A~ A, A.* /x inclination angle critical pipe angle defined in Eq. (3) diffusion coefficient of ~Pi turbulence dissipation rate gas friction factor Muschelknautz effective solids friction factor Muschelknautz solids-wall friction factor dynamic viscosity of gas phase 260 P TE l"w A Levy et al. / Powder Techmdogy 93 (1997) 253-260 density time factor for large-scale turbulent motion time factor for particle response to turbulence wall shear stress conserved property of phase i Subscripts g s i in ex t gas phase property solid phase property phase i property inlet property exit property total or terminal property References [ 1] R.D. Marcus, L.S. Leung, G.E. Klinzing and E Rizk, Pneumatic Conveying of Solids, Chapman and Hall, London, UK, 1990. [2] E. Mu~helknautz, VDI Forschungsh., 25 (1959) 476. [3] E. Muscbelknautz and H. Wojahn, Chem.-Ing.-Tech., 46 (6) (1974) 223-236. [4] A. Ginestet, J.F. Large, P. Guigon and J.M. Beeckmans, A new classification of flow regimes in vertical and inclined pneumatic transport, 2nd Int. Syrup. Reliable Flow of Particulate Solids, Oslo, Norway, 1993, EFChE Publication Series No. 96, Powder Science and Technology Research, Porsgrunn, Norway, pp. 685-71 I. [5l P.E. Salt, S.M. Cohen and J.C. Whalen, Affect of inclined lines on pneumatic conveying systems, Proc. 14th Int. Conf Powder and Bulk Solids Handling and Processing, Rosemont, IL, USA, 1989, Cahners Exposition Group, Des Plaines, IL, pp. 670-680. [6] P. Marjanovic, Pneumatic conveying in an incline pipe: inappropriate, unfortunate or wrong concept? Proc. Powder and Balk Solids Conf., Rosemont Convention Center, Rosemont, IL, USA, 1994, Reed Exhibition Companies, Des Plaines, IL, pp. 235-251. [ 7l B.L. Hinkle, Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, 1953. [8] D.B. Spalding, Numerical computation of multi-phase fluid flow and heat transfer, in C. Taylor and K. r~organ (eds.), Recent Advances in Numerical Methods, Vol. 1, Pinefidge Press, Swansea, UK, 1980, Ch. 5, pp. 139-167. [9] M.K. Palel and M. Cross, The modelling of fluidised beds for ore reduction, in C. Taylor, P.N. Gresho, R.L Sani and J. Hanser (eds.), Numerical Methods in Laminar and Turbulent Flow, Pineridge Press, Swansea, UK, 1989, pp. 2051-2090. [10] J.A.M. Kuipers, A two-fluid micro-balance model of fluidized beds, Ph.D. Thesis, Twente University of Technology, Enschede, Netherlands, 1990. [ l i I S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog., 48 (2) (1952) 89-94. [12] J.F. Richardson and W.N. Zaki, Sedimentation and Fluidization, Part I, Trans. Inst. Chem. Eng., 32 (1954) 35. { 13] C.P. Chen and P.E. Wood, A turbulence closure model for dilute gas-particle flows, Can. J. Chem. Eng., 63 (1985) 349. [ 14l C.P. Chen and P.E. Wood, Turbulence closure modelling of the dilute gas-particle axisymmetric jet, AIChE J., 32 ( 1) (1986) 163. [ 15] A.A. Mostafa and H.C. Mongia, On the interaction of particles and turbulent fluid flow, Int. J. Heat Mass Transfer, 31 (10) ( 1988 ) 2063.