Effect of particle size and sphericity on the pickup velocity in horizontal pneumatic conveying L.M. Gomes a,n, A.L Amarante Mesquita b a School of Natural Science, Federal University of Pará,-Quadra 04, Lote Especial, CEP 68505-080, Marabá, PA, Brazil b School of Mechanical Engineering, Federal University of Pará, R. Augusto Correia, 01, CEP 66075-110, Belém, PA, Brazil H I G H L I G H T S � Analysis of the influence of particle sphericity on pickup velocity. � Analytical model taking into account the influence of particle sphericity. � Experimental setup for pickup velocity measurements. � Was demonstrated that the pickup velocity increases when sphericity decreases. � The pickup velocity increases with the pipe and particle diameter. a r t i c l e i n f o Article history: Received 17 April 2013 Received in revised form 11 August 2013 Accepted 27 August 2013 Available online 7 September 2013 Keywords: Particle sphericity Pickup velocity Dilute phase Pneumatic conveying a b s t r a c t This paper presents a theoretical and experimental analysis of the influence of particle size and sphericity on the pickup velocity in horizontal pneumatic conveying. An analytical model was developed to predict the pickup velocity based on force and moment balances taking particle size and sphericity into account. In this work it was demonstrated analytically that the pickup velocity of a particle increases when particle sphericity decreases. An experimental apparatus was developed in order to verify the influence of the diameter particle and pipeline on pickup velocity, and to validate the proposed analytical model. The obtained equations to the pickup velocity prediction cover a particle diameter range of 20–4000 μm, densities range 1000–5000 kg/m3 and pipeline diameter ranging from 25 to 150 mm in the prediction of the minimum gas velocity required to pickup particles. These results agree with the experimental data available from the literature. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction The minimum velocity required for particle entrainment is a key parameter for a successful operation of a pneumatic conveying system. The minimum transport velocity is defined as the lowest velocity at which particles can be transported inside a pipeline in a stable manner without deposition on the bottom of the pipe. The pickup velocity is defined as the gas velocity necessary to suspend the particles initially at rest at the bottom of the pipeline (see Fig. 1a and b), or it may be defined as the fluid velocity required to initiate a sliding motion, rolling and suspension of particles. In addition to these definitions, many designers of pneumatic con- veying systems use the term “pickup” to refer to the velocity required to maintain particles suspended at the feed point. This velocity is critical for an accurate design of a pneumatic conveying system. If the fluid velocity is much larger than is necessary, the system is conducted towards unnecessary energy losses, particle attrition and excessive pipe erosion. If the fluid velocity is below the values of these critical velocities, the result can be clogged pipelines. The pickup velocity is relevant in a wide range of applications. Examples include dry powder inhalers for drug delivery in some pharmaceutical industries, the movement of sand dunes and soil deposition in river and ocean flows (Kalman et al., 2005) and understanding erosion of silt on riverbeds (Kalman and Rabinovich, 2007). Likewise, pickup velocity is an important parameter in dust control applications. A good estimate of the minimum conveying velocity must be known in order to optimize any solids conveying system. Despite the existence of advanced modeling for turbulent gas– solid flows (Huilin et al., 2003) and powerful modern numerical methods for solving this complex flow model (Chu and Yu, 2008; Pirker et al., 2010; Stevenson et al., 2002), the empirical correlation for predicting the pickup velocity still remains as a practical, though realizable issue for industrial pneumatic conveying systems design. Several attempts were made in the past to predict the minimum conveying velocity, but unfortunately, the majority of existing approaches have several limitations and contradictions (Yi et al., 1997; Rabinovich and Kalman, 2008). In the literature there are Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ces Chemical Engineering Science 0009-2509/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2013.08.055 n Corresponding author. Tel.: þ55 94 2101 5900; fax: þ55 94 2101 5901. E-mail addresses:
[email protected] (L.M. Gomes),
[email protected] (A.LA. Mesquita). Chemical Engineering Science 104 (2013) 780–789 www.sciencedirect.com/science/journal/00092509 www.elsevier.com/locate/ces http://dx.doi.org/10.1016/j.ces.2013.08.055 http://dx.doi.org/10.1016/j.ces.2013.08.055 http://dx.doi.org/10.1016/j.ces.2013.08.055 http://crossmark.crossref.org/dialog/?doi=10.1016/j.ces.2013.08.055&domain=pdf http://crossmark.crossref.org/dialog/?doi=10.1016/j.ces.2013.08.055&domain=pdf http://crossmark.crossref.org/dialog/?doi=10.1016/j.ces.2013.08.055&domain=pdf mailto:
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[email protected] http://dx.doi.org/10.1016/j.ces.2013.08.055 various correlations for determining pickup velocity. Among these correlations can be highlighted the correlations proposed by Cabrejos and Klinzing (1992, 1994), Hayden et al. (2003), Kalman et al. (2005) and Kalman and Rabinovich (2009). In a theoretical and experimental study, Gomes (2011) performed a critical analysis on the main correlations to determine the pickup and saltation velocities. From this work, the recommended correlations to predict the pickup velocities are those according to Cabrejos and Klinzing (1994) and Kalman et al. (2005), presented in Table 1. In according with Hilton and Cleary (2011) the particle shape is a crucial factor in systems with gas–grain interactions but has so far been overlooked in models of pneumatic conveying. Stevenson et al. (2002) developed a model for incipient movement of sand grain. They choose a hemisphere in your model to represent the shape of a sand grain. In according with them the use of a hemisphere formulates a relevant well-posed problem, capable of mathematical solution. However, the flow field around real particles, e.g. sand, is complex, because of their irregular shapes, but the hemisphere work gives a qualitative indication of the behavior of irregular particles. This work presents a theoretical and experimental analysis of the influence of the particle size and the pipe diameter on the pickup velocity in horizontal pneumatic conveying, taking the particle sphericity into account. Nomenclature a deformation radius of spherical particles (m) A′ area A′ of a sphere with volume VT (m2) aþ dimensionless particle radius AH Hamaker constant (N m) Ap surface area of the particle (m2) Ar Archimedes number CD drag coefficient D50 inside diameter of 2″ pipe (m) dp particle diameter (m) dp′ relative particle diameter (m) DT pipe diameter (m) Fa adhesion force (N) Fd drag force (N) Fe buoyancy force (N) Ff friction force (N) Fg gravity force (N) Fn normal force (N) f friction coefficient Fs lift force (N) Fsþ dimensionless lift force (N) g acceleration due to gravity (m/s2) M Moment (N m) mp particle mass (kg) n coefficient of wall effect on the drag force R particle radius (m) R′ radius equivalent of a sphere (m) Re Reynolds number related to pipe diameter Rep Reynolds number related to particle diameter Repn Reynolds number modified S separation length between particle and the wall (m) U mean gas velocity (m/s) un threshold shear velocity (m/s) Up minimum pickup velocity (m) Up0 minimum pickup velocity for single particle (m/s) VC volume of the spherical cap (m3) VE sphere volume (m3) Vp volume of the particle (m3) VT volume of the sphere whose (m3) y longitudinal coordinate (m) Greek symbols δ laminar sub-layer (m) Δm weight loss (kg) θ angle for modification of the particle sphericity (rad) μ gas dynamic viscosity (kg/ms) ν fluid kinematic viscosity (m2/s) ρ fluid density(kg/m3) ρp solid particle density (kg/m3) Ψ particle sphericity Fig. 1. (a) Particles at rest at the bottom of the pipeline. Air velocity less than the pickup velocity. (b) Particles are captured by the air stream. Air velocity equal or greater than the pickup velocity. Table 1 Correlations of Cabrejos and Klinzing (1994) and Kalman et al. (2005) to predict the pickup velocity. Cabrejos and Klinzing (1994) UPffiffiffiffiffiffiffiffiffi g dp p ¼ 0:0428 Re0:175p DTdp � �0:25 ρp ρ � �0:75 (1) where Rep is the Reynolds particle number. Valid for 25oRepo5000, 8o(DT/dp)o1340e 700o(ρp/ρg)o4240 Kalman et al. (2005) First zone: Repn ¼ 5 Ar 3=7 ; for Ar416:5 Second zone: Repn ¼ 16:7; for 0:45 oAro 16:5 Third zone: Repn ¼ 21:8 Ar1=3 ; for Aro0:45 (2) where, Repn ¼ ρ UP dp μð1:4�0:8: e�ðDT =DT50=1:5ÞÞ Valid for: 0.5oRepno5400, 2�10�5oAro8.7�107, 0.53odpo3675 μm, 1119oρpo8785 kg/m3 and 1.18oρo 2.04 kg/m3 L.M. Gomes, A.LA. Mesquita / Chemical Engineering Science 104 (2013) 780–789 781 2. Material and methods 2.1. The experimental setup An experimental apparatus was developed to determine the pickup velocity. The pipeline (Fig. 2) has total length of 7.5 m. It consists basically of a 1.5-meter-long, 50 mm in diameter, a horizontal steel pipeline, three horizontal PVC pipelines (each 6-meter-long and 50, 75 and 100 mm in diameter) with a butterfly valve at the end of each pipeline, three transparent sections (placed in the middle of the PVC pipelines) where the visual observations were carried out, a roots blower (controlled by a frequency inverter) that provides the gas velocity and pressure necessary for picking up the particles, and a solids collector with a paper filter bag placed on top of it. A complete description of this experimental setup can be found in Gomes (2011). Experiments were carried out using air as the conveying gas under ambient conditions. The experimental methodology starts with a stationary layer of particles placed in the center section of a transparent pipe. Air flow is initiated at a constant volumetric flow rate through the pipe. As the free cross-sectional area of the pipe increases, due to removal of particles (Δm), the air velocity decreases. When the velocity is no longer sufficient to entrain any additional particles, a final state of equilibrium is automatically reached. This procedure is repeated until nearly 95 percent of the whole material has been captured. Plotting the amount of entrained particles (weight reduction of the layer) as a function of operating gas velocity made it possible to determine the pickup velocity by the intersec- tion of the extrapolated curve passing through the measured points and abscissa (see Fig. 3). Obviously, a higher number of measurements improves the accuracy, especially if measurements with very small weight losses can be achieved. In order to assess the effect of pipe diameter in the pickup velocity, three PVC pipelines were used in an alternate manner. Air flow was initiated at a constant volumetric flow rate through the pipeline with one of the three butterfly valves opened. The layer started to erode slowly as the gas stream picked up the top particles. The experimental was carried out using sand and alumina particles. The particle diameters used in the tests were obtained by the method of separation in sieves. The mean diameter was calculated by the arithmetic mean of adjacent sieve diameters. Table 2 presents the properties (density, size distribution, mean diameter and shape) of the particles used in the measurements. The density was measured by a pycnometer. All tests were conducted with a narrow range of particle sizes prepared by sieving. Fig. 3 shows the pickup velocity of the sand particles obtained with the methodology used in this work. The best straight line that intersects the points is obtained through the technique of least squares. To improve accuracy of the result, the experiment was repeated seven times. Thus, by extrapolating this curve, the point that intercepts the abscissa axis was obtained. This point corre- sponds to the pickup velocity. For this experiments the uncertainty analysis gives a result of 4.870.5 m/s. For all experiments the uncertainty was at this order of magnitude. Fig. 4 present the effect of the particle diameter. Thus, other parameters are kept constant. Measurements were carried out with sand in a pipe with a 50 mm in diameter. The average diameters of sand particles used in the measurements were 22, 48.5, 63.5, 89.5, 179.5 and 253.5 μm. For each particle diameter, three velocity measurements were performed. An important result obtained was the minimum point on the pickup velocity curve as a function of particle diameter, a fact which was also verified by Cabrejos and Klinzing (1992), Hayden et al. (2003) and Kalman et al. (2005). This minimum point appears in the particle diameter of 63.5 mm (see dashed vertical line). As expected, particles of larger sizes also require higher pickup velocities due to the inertia effect that is dominant in this region. For smaller particles, the pickup velocity is also high. However, in this case, this is due to the fact that the particle–particle interactions are quite significant in this region. Fig. 2. The experimental setup: Schematic representation. Fig. 3. Weight loss as a function of the average air velocity. Experimental data dp¼179.5 μm. Table 2 Properties of particles tested in this research. Particle Density (kg/m3) Size Mean Sphericity Range (μm) Diameter (μm) Sand 2636 50–90 70 ψ¼0.7 90–150 120 150–250 200 250–430 340 600–430 510 600–850 730 850–1000 930 1000–2360 1680 2360–3350 2860 3350–4360 3860 Alumina 3750 53–90 71.5 Ψ¼0.9 90–125 107.5 125–150 137.5 150–205 177.5 L.M. Gomes, A.LA. Mesquita / Chemical Engineering Science 104 (2013) 780–789782 To illustrate the effect of particle density, the pickup velocity is plotted as a function of particle diameters (Fig. 5). As shown, the pickup velocity of alumina particles is greater, a fact which confirms Cabrejos and Klinzing (1994). This is because the domi- nant effect of inertia requires a higher gas flow rate in order to drag the particles with higher density. 3. Theory and calculation 3.1. Sphericity model The proposed model for the incipient movement of a particle considers the influence of particle sphericity on the pickup velocity of a particle at rest at the bottom of a horizontal pipe. In order to introduce the sphericity in the model was employed the concept similar to the hemisphere theory, which is discussed in detail in Stevenson et al. (2002). According to Stevenson et al. (2002), because sand grains have irregular shape, they may be simply modeled using the hemisphere shape in order to reduce the mathematical analysis for a well-posed problem. Thus, in the development of a useful analysis, the particle shape must be chosen so that the physical behavior of the sand grains is approached whilst generating a mathematically well-posed problem. According to them, a hemisphere with its flat side facing the pipe wall is an appropriate model since the flow around this hemisphere is capable of analysis and a first approximation to the actual industrial problem of predicting the incipient motion of a sand grain in a pipe wall. The implementation of this model considers the scheme showed in Fig. 6. In this way it is possible to obtain a relationship for the sphericity, by modeling a non-spherical particle as a sphere whose cap was removed. Sphericity of a particle, Ψ, is a measure of how spherical an object an d by definition is the ratio of the surface area of a sphere (with the same volume as the given particle) to the surface area of the particle: ψ ¼ the surface area of a sphere ðwith the same volume as the given particleÞ the surface area of the particle ð3Þ We have that h is given by the relation, h¼ R�g ¼ R 1� cos θ 2 � �� � ð4Þ Making some algebraic manipulations and simplifying the results, we get the volume of the spherical cap Vc VC ¼ 1 3 πR2 1� cos θ 2 � �� �2 3R�R 1� cos θ 2 � �� �� � ð5Þ We obtain the radius R′ would have a sphere with volume V ′ð ¼ ð4=3ÞπR′3Þ, R′¼ Rffiffiffi 43 p 2þ3cos θ 2 � � � cos 3 θ 2 � �� �1=3 ð6Þ Thus, after some algebraic manipulation we have obtained the numerator A′ (the surface area of a sphere with the same volume as the given particle) and denominator Ap (the surface area of a particle) of Eq. (3) as, A′¼ 4π R 2ffiffiffiffiffiffi 163 p 2þ3 cos θ 2 � � � cos 3 θ 2 � �� �2=3 ð7:aÞ Ap ¼ 4πR2�2πR2 1� cos θ 2 � �� � þπR2 sin 2 θ 2 � � ð7:bÞ Substituting Eqs. (7.a) and (7.b) in Eq. (3), simplifying the result, the sphericity as function of the cap angle θ is obtained. The resultant Fig. 4. Effect of the particles diameter. Dominant forces. Fig. 5. Pickup velocity as a function of particle diameter. Alumina and sand particles. Fig. 6. Geometric elements in a sphere used for determining sphericity as function of the angle θ. L.M. Gomes, A.LA. Mesquita / Chemical Engineering Science 104 (2013) 780–789 783 relationship is given by: ψ ¼�ð22=3Þð2� cos ðθ=2ÞÞ 2=3ð cos ðθ=2Þþ1Þ1=3 cos ðθ=2Þ�3 ð8Þ As expected, the sphericity depends only on angle θ. Fig. 7 shows a plot of sphericity as function of angle θ. Thus, for example, we can verify that the shape of a hemi- sphere, treated in modeling Stevenson et al. (2002), is a particular case of our model of a sphere whose cap was removed, consider- ing θ¼π. Solving Eq. (8) for θ, we obtained, θ¼ 2arccos 3ða 1=3þ1þ i ffiffiffi 3 p Þ a1=3 þ�a 2=3�16�16i ffiffiffi 3 p �8a1=3þ i ffiffiffi 3 p a2=3 a1=3b ! ð9Þ where i is the imaginary unit and, a¼�8þ10ψ3�ψ6þ4ψ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψðψ3�1Þ q þψ4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ ðψ3�1Þ q ð10:aÞ and, b¼ 4þψ3 ð10:bÞ In following section the force balance equations for large and small particles are employed in order to determine the incipient velocity of a particle at rest at the bottom of the horizontal pipeline, considering the influence of the particle sphericity. 3.2. The force balance analysis Fig. 8 shows a spherical particle and a slightly deformed spherical particle under a flow in the boundary layer development. This deformation enables the normal force (FN) to change the particle center so as to develop a point opposite the drag force. From the moment balance relative to the contact point O (Fig. 8b), and according to Kalman and Rabinovich (2009), the mechanism of the initial movement occurs when the moment caused by the lift and drag forces is superior to those from the gravitational and adhesion forces. The sliding mechanism begins when the drag force becomes equal to the force of friction. The mechanism of the initial movement in the vertical direction occurs when the lift is equal to the sum of gravitational forces of adhesion. Table 3 shows the main forces acting on a single particle at rest. The adhesive force is due mainly to van de Waals attractive force type and was employed the model proposed by Hamaker (1937), which was also used by Cabrejos and Klinzing (1992) and Kalman and Rabinovich (2009). In this model AH is the Hamaker constant and s is the separation length between the particle and the wall surface. In this work, following the analysis from Kalman and Rabinovich (2009) and from several numerical tests performed in this present work, was considered a value of AH¼10�10�20 J and s¼4�10�10 m for fine particles, and AH¼6.5�10�20 J and s¼8�10�8 m for coarse particles. For the lift force is employed a model based in the Saffman (1965, 1968) theory. However, as analyzed by Stevenson et al. (2002) this model underestimates the lift force when used directly, as performed by Cabrejos and Klinzing (1992). Then, some corrections are necessary, as carried by Mollinger et al. (1995) and Kalman and Rabinovich (2009), where the parameters p and q are dependent on the Reynolds number. In this work was considered the values presented in Kalman and Rabinovich (2009). For the drag force is employed the classical approach (see Table 3), where n is a correction coefficient to take into account the influence of the wall and the velocity profile (Kalman and Rabinovich, 2009). For the friction force the standard form is employed (see Table 3), where f is the friction coefficient, defined as the tangent of the angle at which the pipe must be raised above the horizontal to cause the particle, initially at rest, to begin to slide (Cabrejos and Klinzing, 1994). In the work was considered a value of f¼0.6 and 0.71 for large and small particles, respectively. An important consideration in this model is how to use a single particle model to represent a flow over a layer of particles. Cabrejos and Klinzing, 1994 perform this using a semi-empirical correlation with a multiplicative factor to correct the pickup velocity. Otherwise, Kalman and Rabinovich (2009) use an implicit correction through experiments performed from a layer of parti- cles. In this present work the influence of the layer particles is introduced by an empirical correlation for the parameter n in theFig. 7. Sphericity as function of the angle θ (in radians). Fig. 8. Main forces acting on a particle completely immersed in the laminar sublayer (Adapted from Kalman and Rabinovich, 2009). L.M. Gomes, A.LA. Mesquita / Chemical Engineering Science 104 (2013) 780–789784 drag force expression. This correlation was obtained from experi- ments carried out in the specific test bank developed for the work and also from experimental data available in the literature (Cabrejos and Klinzing,1994; Kalman et al., 2005). 3.2.1. The velocity profile The velocity profiles have a major influence on the incipient motion velocity. The velocity profile for turbulent flow is com- monly divided into viscous sub-layer (laminar sub-layer), inter- mediate and logarithmic zones. Schlichting and Gersten (1989) defined the viscous sub-layer as: uðyÞ un ¼ yun v for Reno5 ð17Þ where the dimensionless Ren is the number of Reynolds shear, given by the relationship: Ren ¼ dpun ν ð18Þ where dp is the particle average diameter, un is the shear velocity and ν is the kinematic viscosity. The logarithmic zone is defined as: uðyÞ un ¼ 2:5In yun v � � þ5:5 for Ren470 ð19Þ According to Schlichting and Gersten (1989) the laminar sub- layer δ is equal to: δ¼ 5 ν un ð20Þ The intermediate zone is very narrow, and is therefore usually neglected and the velocity profile is described only by the laminar and logarithmic zones. The relationship between the shear velocity and the superficial velocity can be found by using the well-known Blasius equation for a smooth surface: un ¼ ð0:03955U7=4v1= 4D�1=4T Þ0;5 ð21Þ From Eqs. (17)–(19) it is possible to conclude that the velocity profile over a particle smaller than the laminar sub-layer is linear and over a particle larger than the laminar sub-layer is combined by two profiles: the linear and the logarithmic. For particles much larger than the laminar sub-layer a logarithmical profile can be assumed. 3.2.2. Particle motion Cabrejos and Klinzing (1992) and Kalman and Rabinovich (2009) showed that the horizontal motion occurs before the vertical motion. On the other hand, Hayden et al. (2003) observed that the pickup velocity is equal to the fluid velocity when the particle, at the bottom of a horizontal pipe, begins its movement in the vertical direction. The verification of the type of initial move- ment of the particle (whether horizontal or vertical) can be made by computing the resultant forces (both directions) and moment. The highest intensity indicates the initial direction of motion of the particle. In this paper, it is considered that the following assumptions may occur � The initial movement of the particles occurs by dragging � The initial movement of the particles occurs in the vertical direction � The initial movement of the particles occurs by rolling. Fig. 9 shows the plot of the ratio between the sum of the forces that will put the particle in movement (RXþ and RYþ) and the sum of the resistive forces (RX� , and RY�), in the x and y directions, and the ratio between the moment to produce rolling (RMþ) and the moment due to the resistive forces (RM�), as a function of gas superficial velocity, for coarse sand particles. In this analysis we used particles with an average diameter of 1200 μm, (AH¼6.5�10�20 J, s¼8�10�8 m and n¼1.8). It was verified that the forces promoting particle movement in the horizontal direc- tion (drag force) dominated the process. We thus conclude that for this case analyzed the initial movement occurs in the horizontal direction. For easy viewing of the results presented in graphs, symbols are used to represent experimental data and symbols with lines (dashed or not) to represent only simulations. Fig. 10 shows the forces acting on a particle at rest. In this simulation we adopted the average velocity of the gas equal to 6 m/s. The range of particle size used was between 600 and 4000 μm. (AH¼6.5�10�20 J, s¼8�10�8 m and n¼1.8). The grav- itational force is the more significant component (which is obvious, since the particles are coarse and therefore the inertial effects are important). Then, in order of relevance appear the drag and lift forces. Fig. 11 shows the same procedure showed in Fig. 9, but for fine particles of sand. In this analysis we used particles with an average diameter of 60 μm (AH¼10�10�20 J, s¼4�10�10 m and n¼1.8). In this case, the initial movement of the particles occurs by rolling (Uo¼1.25 m/s). Table 3 Main forces acting on a particle at rest. (Source: Kalman and Rabinovich, 2009). Forces Equation Gravitational Fg ¼mg¼ VpρPg (11) Vp is the particle volume Fb ¼ Vρg (12) Buoyant V is the displaced volume Van der Waals Fa ¼ AHdp 12s2 (13) Lift FS ρv2 ¼ p d 2 un v � �q (14) where un is the shear velocity, p and q are parameters adjustment Drag Fd ¼ nCDAp ρU2 2 (15) where U is the relative velocity, n is the wall effect coefficient, CD is the drag coefficient, Ap is the particle projected area Friction Ff ¼ f Fn (16) f is the friction coefficient Fig. 9. Ratio between forces and moments as a function of superficial velocity for coarse sand particles. L.M. Gomes, A.LA. Mesquita / Chemical Engineering Science 104 (2013) 780–789 785 In Fig. 12 the same procedure of Fig. 10 was adopted. The data are those used in Fig. 11. However, the average velocity of the gas was 6 m/s. The range of average particle diameters was between 10 and 2000 μm. The relevance of adhesion force is verified, since the choice of values for the Hamaker constant and for the length of separation was made to produce the greatest possible adhesion force. Thus, one can analyze the minimum (Fig. 10) and maximum (Fig. 12) values of the adhesion force on the chosen configuration. It is more likely that the real value of the adhesion force lies between the maximum and minimum values obtained. Therefore, based on this analysis, the initial movement of both particles (fine and coarse) will be considered in the horizontal direction. However, for fine particles, analysis will be done through the moment balance, as verified in the simulation shown in Fig. 11. 3.2.3. Large spherical and non-spherical particles Consider an air flow surrounding a large non spherical particle at rest at the bottom of a horizontal pipe. Applying a force balance on the particle (equilibrium condition) in the horizontal direction, we have: Fd�Ff ¼ 0 ð22Þ According to the vertical force balance: FnþFsþFe�Fg�Fa ¼ 0 ð23Þ Since the projected area of the particle Ap, given by AP ¼ 1 2 R2ð2π�θþ sin ðθÞÞ ð24Þ Isolating R in Eq. (6), R¼ R′2 2=3 ð2� cos ðθ=2ÞÞ1=3ð cos ðθ=2Þþ1Þ2=3 ð25Þ We have substituted Eqs. (24) and (25) in Eq. (15) By simplify- ing the terms, results, Fd ¼ 0:0249nCDR′2ð2π�θþ sin ðθÞÞρ U7=4ν1=4 2:5lnðð0:1989yU7=8=ν7=8DT 1=8ÞÞþ5:5 � �2 ð2� cos ðθ=2ÞÞ2=3ð cos ðθ=2Þþ1Þ4=3D1=4T ð26Þ Isolating Fn in Eq. (23) and substituting the results in Eq. (22) we have: Fd�f ð�Fs�FeþFgþFaÞ ¼ 0 ð27Þ Substituting Eqs. (11)–(14), (26) and Eqs. (19) and (21) (loga- rithmic velocity profile) into Eq. (27) and making the necessary manipulations, we have 6:2287�10�3nCDd2p ð2π�θþ sin ðθÞÞρ Up7=4ν1=4ð2:5 lnðð0:09943dpUp7=8=ν7=8D1=8T ÞÞþ5:5Þ 2 ð2� cos ðθ=2ÞÞ2=3ð cos ðθ=2Þþ1Þ4=3D1=4T þ þ f 0:101d 2:31 p U 2:02 p ρf ν0:021D0:289T �16ρpπd 3 pg � � ¼ 0 ð28Þ Finally, making more simplifications, we obtain: 6:2287�10�3nCDd2pρ Up7=4ν1=4A B2=3C4=3DT 1=4 ð2:5 lnðð0:09943dpU7=8p =ν7=8DT 1=8ÞÞþ5:5Þ2þ þ f 0:101d 2:31 p U 2:02 p ρ ν0:021D0:289T �16ρpπd 3 pg � � ¼ 0 ð29Þ where, A¼ 2π�θþ sin ðθÞ B¼ 2� cos ðθ=2Þ C ¼ 1þ cos ðθ=2Þ ð30Þ This equation shows the dependence of incipient velocity on particle properties such as density, average particle diameter and sphericity, pipe diameter and density and viscosity of the gas stream. Fig. 10. Main forces acting on coarse particles. Particle diameter range: 600–4000 μm. Fig. 11. Ratio between forces and moments as a function of superficial velocity of air to fine particles. Fig. 12. Main forces acting on fine and coarse particles. Particle diameter range: 10–2000 μm. L.M. Gomes, A.LA. Mesquita / Chemical Engineering Science 104 (2013) 780–789786 3.2.4. Small spherical and non-spherical particle The model developed for the small non-spherical particle in our work considers that the incipient motion of a small non- spherical particle occurs by rolling. Based on the fact that the mechanism for incipient motion of fine particles is rolling (Kalman and Rabinovich, 2009) the balance of angular momentum is developed considering the rotation of forces around the point A (Fig. 8b). Thus, we can write: ∑MA ¼ 0 ð31Þ Consider the drag force acting on the equivalent point Ld¼1.4/ 2dp (measured from the bottom of the pipe) for fine particles. Thus, we can write: 1:4 dp 2 FdþaFS�aðFg�FeÞ�aFa ¼ 0 ð32Þ In Fig. 8b, j is obtained by the relationship, j¼ 2R�h ð33Þ Then, in Eq. (32), dp and a can be replaced by j and k (see Fig. 6), respectively. Thus, we have: 0;7jFdþkFs�kðFg�FeÞ�kFa ¼ 0 ð34Þ Making the appropriate mathematical calculations and substi- tuting Eqs. (9)–(17) and (21) (linear velocity profile) into Eq. (34) and making the necessary algebraic simplifications, we have 8:6221� 10�5d5pnCDρU7=2p A B5=3C1=3ν3=2 ffiffiffiffiffiffi DT p þ 1 B1=3C2=3 0:6028dp 287 100ν 291 800Up 1309 800 ρD DT 187 800 �0:13229dpπgd3pðρp�ρÞD� 0:06614dp2DAH s2 � ¼ 0 ð35Þ where D¼ sin θ 2 � � ð36Þ The coefficient n which considers the influence of the wall was adjusted with the use of the relationship obtained in this research and with experimental data on glass (ρp¼2480 and 2834 kg/m3) and sand particles (ρp¼2636 kg/m3) with average particle dia- meters ranging between 0.022 and 3.86 mm and pipe diameters ranging from 26 to 152 mm. The adjustment was made with respect to the dimensionless number Up� dp0.5/(g0.5�DT) (see Fig. 13) and obtained the following relationship: n¼ 0:5759 Up DT ffiffiffiffiffi dp g s !0:3248 ð37Þ In the next item results obtained with the use of Eqs. (29) and (35) and its agreement with experimental and literature data will be analyzed. 4. Results and discussion In order to assure the accuracy of the relationship developed in this work the results obtained with Eqs. (29) and (35) were compared with the experimental data available from the literature and with the pickup velocity measurements made in this work. Fig. 14 shows the pickup velocity of solid particles as a function of average diameter of spherical and non-spherical particles (spherical and irregular glass, irregular iron oxide, irregular alumina and sand), over the range of particles sizes from 30 to 450 μm. It is observed that the results obtained through the use of relationships developed in this work (Eqs. (29) and (35)) are in agreement with experimental data in the literature. What is noteworthy is the fact that there are reasonable differences between the experimental data (sand and glass particles). This discrepancy could be explained by the different measurement methodologies and experimental setups employed in these works. Fig. 15 shows the pickup velocity of solid particles as a function of the average diameter of spherical and non-spherical particles (spherical and irregular glass, irregular iron oxide, irregular alumina, sand and spherical Zirconium) over the range of particle sizes (from 200 to 4000 μm). The pipeline diameter used in Fig. 13. n As a function of dimensionless number Up� dp0.5/(g0.5�DT). Fig. 14. Pickup velocity as function of the average particle diameter: dp¼0–450 μm. Fig. 15. Pickup velocity as function of the average particle diameter: dp¼450–4000 μm. L.M. Gomes, A.LA. Mesquita / Chemical Engineering Science 104 (2013) 780–789 787 calculations was 52.4 mm. The results show agreement of the equation obtained in this work with experimental data obtained by Cabrejos and Klinzing (1992, 1994) and Kalman et al. (2005). The experimental data obtained are on average 27 percent below the calculated data with the relationships developed in the present work. This is due to the fact that we used a large amount of literature data to adjust the parameter n. These data, as we already stated, were obtained using various experimental benches and different measurement methodologies. It should also be noted that the experimental data used have a high degree of dispersion, a fact that complicates obtaining a better accuracy in the determination of pickup velocity through the relationships developed. Fig. 16 shows a simulation that represents the plot of the pickup velocity as a function of the sphericity of the fine (dp¼20 μm) and coarse (dp¼750 μm) particles. It can be seen that the capture velocity of the particle increases as an inverse function of the sphericity. It thus decreases as the particle sphericity, its contact area with the surface (bottom of pipe) increases and it is therefore also necessary to increase the pickup velocity to move the particle, due to the fact that the cohesive forces become more intense. In the case of a sphere, the contact with the surface is at only one point, therefore the cohesion is smaller, and consequently the pickup velocity. This result is confirmed by Cabrejos and Klinzing (1994), Hayden et al. (2003), Kalman et al. (2005) and Kalman and Rabinovich (2009). Fig. 17 shows the plot of the pickup velocity of irregular salt and spherical glass as a function of average particle diameter obtained in this work and also from the Cabrejos and Klinzing (1994) and Kalman et al. (2005) correlations. It can be seen that the pickup velocity increases with the average particle diameter, a fact that corroborates all analyses performed on pickup velocity as a function of average particle diameter in recent years (Cabrejos and Klinzing, 1992, 1994; Hayden et al., 2003; Kalman et al., 2005; Kalman and Rabinovich, 2009). As already noted, the correlation of Cabrejos and Klinzing (1994) overestimates the results, particularly for pickup velocity values corresponding to average particle diameter above 1500 μm. The concordance of results obtained with the relationships obtained in this work with experimental data is very good. For coarse particles, Fig. 18 shows the plot of the pickup velocity as a function of particle diameter. The graph shows pickup velocity calculated with the use of the relationships (Eqs. (29) and (35)) obtained in this work for spherical and non-spherical glass particles (ρp¼2480 kg/m3) with sphericity equal to 1 and 0.85, respectively. The results are compared with experimental data from Cabrejos and Klinzing (1994). We found that the non- spherical particles have higher pickup velocities. The model developed shows a good agreement with experimental results. Fig. 16. Pickup velocity as a function of particle sphericity (fine and coarse). Fig. 17. Pickup velocity as a function of average particle diameter. Comparison with correlations by Cabrejos and Klinzing (1994) and Kalman et al. (2005). Fig. 18. Pickup velocity as a function of average particle diameter. Glass particles. Fig. 19. Pickup velocity as a function of the pipe diameter. L.M. Gomes, A.LA. Mesquita / Chemical Engineering Science 104 (2013) 780–789788 For fine particles, Fig. 19 shows a plot of pickup velocity as a func- tion of the pipe diameter for glass spheres (dp¼0.45 mm), irregular alumina (dp¼0.45 mm) and irregular polyester (dp¼3mm). It is observed that the relationship developed for determination of pickup velocity gives good results when compared to experimental data. 5. Conclusion A new relationship was proposed for predicting pickup velocity in horizontal pneumatic conveying and was successfully tested. This new relationship was developed based on a model for incipient motion of a single particle at rest at the bottom of a horizontal pipeline. The pickup velocity for different solids cannot be easily predicted because this parameter is influenced by many diverse variables. Among these, the characteristics of the material itself, such as particle size, density and shape, the coefficient of sliding friction, and the particle interaction with other particles are the most important variables affecting pickup velocity. Therefore, the relationship developed in this work presents good results for particles with diameters ranging from 20 to 4000 μm, densities ranging from 1000 to 5000 kg/m3 and pipeline diameter ranging from 25 to 150 mm. These results agree with the experimental data available from the literature (Cabrejos and Klinzing, 1992; Hayden et al., 2003; Kalman et al., 2005; Kalman and Rabinovich, 2009). This expression was developed using an analytical approach (although using parameters which have been obtained with the use of correlated data) to consider the influence of particle sphericity for spherical and non spherical particles, with fine and large sizes, showing generally a good concordance with the experimental data. This is a contribution from this research, given that other works simply correlated particle sphericity based on empirical data with considerable limitations. Considering the large range size and density and the precision of numerical data for the evaluation of this expression, it is expected that it can be an efficient tool for predicting pickup velocity in horizontal pneumatic conveying. Acknowledgements The authors would like to express their gratitude for the financial support from CNPQ (doctoral grant) without which this paper would be impossible. References Cabrejos, Francisco J., Klinzing, G.E., 1992. Incipient motion of solid particles in horizontal pneumatic conveying. Powder Technology 72, 51–61. Cabrejos, Francisco J., Klinzing, G.E., 1994. Pickup and saltation mechanisms of solids particles in horizontal pneumatic transport. Powder Technology 79, 173–186. Chu, K.W., Yu, A.B., 2008. Numerical simulation of complex particle-fluid flows. Powder Technology 179, 104–114. Gomes, L.M., 2011. Contribution to the Dilute-Phase Pneumatic Conveying Analysis. Ph.D. Thesis. Federal University of Pará, Brazil. Hamaker, H.C., 1937. The London—van der Waals attraction between spherical particles. Physica 4 (10), 1058–1072. Hayden, K.S., Park, K., Curtis, J.S., 2003. Effect of particle characteristics on particle pickup velocity. Powder Technology 131, 7–14. Hilton, J.E., Cleary, P.W., 2011. The influence of particle shape on flow modes in pneumatic conveying. Chemical Engineering Science 66, 231–240. Huilin, L., Gidaspow, D., Bouillard, J., Wentie, L., 2003. Hydrodynamic modelling of gas–solid flow in a riser using the kinetic theory of granular flow. Chemical Engineering Journal 95, 1–13. Kalman, H., Rabinovich, E., 2007. Pickup, critical and wind threshold velocities of particles. Powder Technology 176, 9–17. Kalman, H., Rabinovich, E., 2009. Incipient motion of individual particles in horizontal particle–fluid systems: B. Theoretical analysis. Powder Technology 192, 326–338. Kalman, H., Satran, A., Meir, D., Rabinovich, E., 2005. Pickup (critical) velocity of particles. Powder Technology 160, 103–113. 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