GasâSolid Flow and Energy Dissipation in Inclined Pneumatic

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Gas-solid flow and energy dissipation in inclined pneumatic conveying Shibo Kuang, Ruiping Zou, Renhu Pan, and Aibing Yu Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 15 Oct 2012 Downloaded from http://pubs.acs.org on October 16, 2012

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Industrial & Engineering Chemistry Research

Gas-solid flow and energy dissipation in inclined pneumatic conveying

S. B. Kuang1, R. P. Zou1, R. H. Pan2 and A. B. Yu1*

1

Laboratory for Simulation and Modelling of Particulate Systems, School of Materials

Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia

2

Xiamen Longking Bulk Materials Science and Engineering Co., Ltd, Xiamen, 361000, China

* Corresponding author. Tel: +61 2 93854429; Fax: +61 2 93855856; Email: [email protected].

1

ACS Paragon Plus Environment


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Abstract This paper presents a numerical study of inclined pneumatic conveying by the combined approach of discrete element model (DEM) for particles and computational fluid dynamics (CFD) for gas. In the numerical model, periodic boundary condition (PBC) is applied to both gas and particles in the conveying direction for computational efficiency. The validity of the model is first examined by comparing the calculated and measured results in terms of solid flowrate and gas pressure drop in the pneumatic conveying with an inclination angle of pipeline varying from 0º to 90º. On this base, the effects of inclination angle, solid flowrate and gas velocity on gas pressure are quantified. The contributions of different forces including the particle-wall friction force, particle gravitational force, and fluid-wall friction force to the pressure drop are examined. Finally, the energy dissipation as a result of interactions between particles, between particles and wall, between particles and fluid, between fluids, and between fluid and wall is studied in details. The results show that the energy loss in a steady-state inclined pneumatic conveying is mainly attributed to the particle-fluid energy dissipation, gravitational potential energy, particle-wall friction energy dissipation and fluid-wall viscous energy dissipation. These energy dissipations vary significantly with inclination angle and flow regime.

Keywords: Discrete particle simulation, Inclined pneumatic conveying, Pressure behavior, Energy dissipation

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1. Introduction Pneumatic conveying is an important operation used in various industries for the transportation of granular materials from one place to another. Some advantages associated with this method of solid transportation include relatively high levels of safety, low operational costs, flexibility of layout, ease of automation and installation, and low maintenance requirements. On the other hand, pneumatic transport has some disadvantages with respect to equipment wear, product degradation and high power consumption.1 Usually, conveying pipelines can be arranged horizontally or vertically, or in an inclined matter. An inclined pipeline has two potential benefits over horizontal and vertical pipelines including a shorter transport distance and a smaller number of elbows, resulting in overall decreases of wear and pressure drop in the entire pneumatic conveying system.2,3 However, in an inclined pipe, high energy loss, high pressure drop, high fluctuation of all flow parameters, even pipe blockage may occur.4 It is important to understand the fundamentals governing this flow system and find effective methods to overcome these problems. In comparison with the numerous studies conducted for vertical or horizontal pneumatic conveying, studies of inclined pneumatic conveying are few. They have been mainly focused on pressure behaviors, but the findings are somehow not comparable. For example, Zenz and Othmer5 experimentally found that for given solid flowrate and gas velocity in inclined pneumatic conveying, the pressure drop largely increases with the increase of inclination angle from 0º (horizontal) to 90º (vertical). This was partially confirmed by the experimental work of Klingzing et al.6 where the inclination angles considered are relatively low, ranging from 0º to 45º. However, Ginestet et al.2,3 reported that the pressure drop at high inclination angles above 72º is significantly greater than that at an inclination angle of 90º. The above studies were mainly focused on the dispersed or suspension flow. For such a flow mode, Levy et al.4 found that the pressure drop initially increases to a maximum and then decreases with increasing inclination angle from 0º to 90º. A similar behavior was also observed in other flow regimes, e.g. by Hong and Zhu7,8 according to the so called two-layer analytical model for the stratified flow, and by Pan9 according to the force balance model for the slug flow. Moreover, for slug-flow transport of cohesive powder, Azia and Klinzing10 showed that 3



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the pressure drop along a horizontal or 45º inclined pipe is governed by the wall stress. Conversely, Hirota et al.11 reported that both the wall stress and gravity contribute to the pressure drop along inclined pipelines. Clearly, the pressure behaviors in inclined pneumatic conveying are very complicated and not well understood. It is important to find how they are linked to the internal flow. In recent years, research has been made in this direction, focused on the flow transitional instabilities in inclined pneumatic conveying using tomographic techniques12 and acoustic probes.13 However, those studies are limited to a narrow range of inclination angles. In recent years, the combined approach of computational fluid dynamics and discrete element method (CFD-DEM) has been increasingly used to study particle-fluid systems including pneumatic conveying (see, for example, the reviews by Zhu et al.,14,15 van der Hoef et al.,16 and Tsuji17). As summarized by Kuang et al.,18,19 earlier CFD-DEM studies of pneumatic conveying are mainly made based on simplified models where three-dimensional (3D) particle flow is sometimes treated as 2D and gas flow is 2D or 1D. These models are useful to generate fundamental, often qualitative, understanding of pneumatic conveying but may have problems in reproducing some key flow behaviors due to their inherent deficiencies. For example, in 2D DEM simulations, particles lose their motion in the third dimension and the fluid drag force has to be based on unrealistic porosity. On the other hand, 1D CFD misses the radial pressure and flow characteristics. Therefore, recent efforts have made to develop CFD-DEM models to quantitatively study various phenomena in pneumatic conveying. In particular, Chu and Yu20 demonstrated that this approach can reproduce key flow phenomena related to particles roping and pulsing clusters in a bend. Kuang et al.19,21 on the other hand developed a 3D CFD-DEM model to study the gas-solid flow regimes in horizontal and vertical pipes. Brosh and Levy22 studied particle attrition by incorporating an attrition model into a CFD-DEM model. Hilton and Cleary23 simulated pneumatic conveying of non-spherical particles in a square-sectioned pipe. Moreover, Kuang et al.24 proposed improved periodic boundary condition (PBC) for CFD-DEM modelling of pneumatic conveying. These studies clearly demonstrate that CFD-DEM is an effective method to study the gas-solid flow in pneumatic conveying. However, almost all of the previous CFD-DEM

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studies are focused on horizontal and vertical pneumatic conveying, with few reported on inclined pneumatic conveying. For example, Lim et al.25 and Zhu et al.26 studied the influence of electrostatic field on reversed flows for dense-phase flows through a 45º inclined pipe by 2D CFD-DEM model. Kuang and Yu27 conducted a 3D CFD-DEM study of inclined pneumatic conveying, focused on the pressure behavior in dilute-phase gas-solid flows. In that work, however, the flows are not fully developed for those with relatively high solid loadings due to the use of a relatively short pipe; so the analysis is largely preliminary. Energy efficiency is a major industrial concern in inclined and other conveying pipes. To minimize power consumption, the energy efficiency of a pneumatic conveying process is usually assessed according to pressure drop, gas flowrate, and other process parameters.28-32 To date, it is not clear how a gas-solid flow dissipates its energy along a pipeline, although such information is directly related to power consumption. In a CFD-DEM model, the trajectories of particles and the transient forces between particles, between particles and wall, and between particles and fluid can be readily retrieved from simulations to calculate detailed energy dissipations in a system considered, as demonstrated by different investigators.33-36 However, such studies are largely made on pure particle systems, not many for particle-fluid flows. In fact, to date, little information is available about the energy dissipation in a pneumatic conveying system. In this work, a 3D CFD-DEM model,19 facilitated with PBC for gas and solid phases in the conveying direction, will be extended to study inclined pneumatic conveying. The paper is organized as follows. First, the numerical model is briefly introduced. Then, the validity of the model is examined by comparing the numerical and experimental results in terms of solid flowrate and pressure behavior. On this basis, the effects of inclination angle and gas velocity on gas pressure are studied in association with the forces acting on particles. Finally, the energy dissipation as a result of different interactions among particles, fluid, and wall is quantified. The results should be useful not only to establishing a comprehensive picture about this gas-solid flow system but also to designing and controlling pneumatic conveying. 2. Simulation and Method The detail of the present CFD-DEM model has been given elsewhere.19 This model has been 5



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used to study the flow behaviors under horizontal and vertical pneumatic conveying conditions.19,21,24,37 For brevity, therefore, we only describe the key features of the model below. 2.1 Governing equations for particle flow The solid phase is treated as a discrete phase described by DEM, where the translational and rotational motions of a particle can be described by the following equations:

mi

ki dvi = f pgf ,i + fdrag ,i + ∑ (fc ,ij + fd ,ij ) + mi g dt j =1

(1)

and

Ii

dωi ki = ∑ (Tt ,ij + Tr ,ij ) dt j =1

(2)

where mi, Ii, vi, and ωi are the mass, moment of inertia, translational and angular velocities of particle i, respectively. According to Zhou et al.,37 the forces involved in the particle-fluid flow modeling are: (1) the pressure gradient force, given by f pgf ,i = −∇ PV p ,i , where P and Vp,i are the fluid pressure and the volume of particle i, respectively; (2) the fluid drag force, calculated by f drag , i = f drag 0 , iε −f ,χi , where fdrag0,i and εf are the fluid drag force on particle i in the absence of other particles and the local porosity for the particle respectively;38 (3) the gravitational force, mig; and (4) the inter-particle forces between particles i and j, which include the elastic contact force, fc,ij and viscous contact damping force, fd,ij. The torque acting on particle i due to particle j includes two components. One arises from the tangential forces given by Tt,ij=Ri,j×(fct,ij+fdt,ij), where Ri,j is a vector from the centre of mass to the )

contact point, and another is the rolling friction torque given by Tr ,ij = µr ,ij di f n,ij ωt ,ij , where µr,ij is the (dimensionless) rolling friction coefficient and di is particle diameter.39 The second torque is attributed to the elastic hysteresis loss and viscous dissipation in relation to particle-particle or particle-wall contacts, and it causes the decay in the relative rotational motion of particles. For viscoelastic material, it is recently reported that µr,ij, and other parameters can be evaluated as a function of materials properties.40,41 For a particle 6


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undergoing multiple interactions, the individual interaction forces and torques are summed over the ki particles in contact with particle i. The inter-particle and particle-wall forces are calculated based on the non-linear models commonly used in DEM.15 2.2 Governing equations for gas flow The gas flow is treated as a continuous phase and modelled in a similar way to the one in the conventional two-fluid modelling. Thus, its governing equations are the conservation of mass and momentum in terms of local mean variables over a computational cell, given by

∂( ρ f ε f ) + ∇ ⋅ (ρ f ε f u ) = 0 ∂t

(3)

and ∂ (ρ f ε f u ) ∂t

+ ∇ ⋅ (ρ f ε f uu ) = −∇P − Fp - f + ∇ ⋅ (ε f τ ) + ρ f ε f g

(4)

where ρf, u, P, τ and Fp-f are the fluid density, fluid velocity and pressure, fluid viscous stress tensor, and the volumetric forces between particles and fluid, respectively. Note that kc

Fp − f = ∑ (f drag,i + f pgf ,i ) / ∆V , where kc and ∆V are the number of particles in a considered i =1

computational cell and the volume of the computational cell, respectively. τ is given by an expression analogous to that for a Newtonian fluid. That is

[

τ = η (∇u) + (∇u)−1

]

(5)

2.3 Boundary conditions The interaction between a particle and a wall is calculated by the force models between particles, assuming that the wall is a rigid sphere with an infinite diameter, having no displacement and movement attributed by particle-wall interaction. No-slip condition is applied to gas phase as used in a CFD study. In the conveying direction, PBC is applied to both solid and gas phases. Thus, a particle re-enters the pipe from the inlet with its velocity and radial position the same as those at the 7



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outlet of the pipe. A similar consideration applies to the gas phase, as detailed elsewhere.18 The implementation of PBC is straightforward for particles but complicated for gas phase. The latter can be done by two methods. The first method was suggested by Patankar et al.42 In that algorithm, the pressure terms in gas momentum equations are required to be decomposed into an average pressure gradient and a reduced pressure on condition that the average pressure varies linearly along the fluid flow direction. The average pressure gradient can be specified as constant or iteratively adjusted to adapt to the gas flowrate given.43 The second method directly implements PBC for gas phase by mutual replacement of gas velocities at the inlet and outlet.44 The method was recently modified by Kuang et al.18 to consider the presence of particles in gas flow. In the algorithm, the inlet velocity is adjusted according to the given gas flowrate to satisfy the global mass conservation after each replacement, and the decomposition of the pressure terms is not needed. Considering the non-linearity of pressure drop in the flow direction encountered in pneumatic conveying, the second method is used to implement PBC for gas phase in the present CFD-DEM model. Accordingly, both gas and solid flowrates are set at the inlet, corresponding to the normal practice in pneumatic conveying. PBC allows us to use a short pipe section to simulate the developed flow zone which is usually much longer than that used in simulation, hence reducing significantly computational efforts. It also eliminates the problem that the fully developed flow may not be achieved for a short pipe simulated without PBC.27 In fact, PBC has been widely adopted in the previous studies of pneumatic conveying.18,23,25,26,45-49 However, in a CFD-DEM simulation with PBC, the number of particles is pre-set, which gives different solid flowrates when the gas velocity and other pneumatic conveying conditions vary. This leads to difficulties in conducting simulations at a given solid flowrate as used in the physical experiments. To overcome this 8


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problem, the recently developed iterative method based on Eq. (6)24 is adopted in this work:

Np =

wp ρ f kw f ρ p + wp ρ f

⋅

Vpipe Vp

(6)

where Np, wf, wp, Vp, and Vpipe are particle number, gas and solid flowrates, and volumes of particle and pipe section, respectively; k is a factor associated with the flow structures of gas and solids in the pneumatic conveying pipe considered and cannot thus far be determined theoretically. The iterative method mainly involves the following steps: (1) k is initialized with a value between 0 and 1 before simulation; (2) k and the target solid flowrate wp are used to estimate particle number Np according to Eq. (6); (3) CFD-DEM simulation is performed with Np particles determined in Step 2, and its macroscopically steady-state outputs are used to obtain a new time-averaged solid flowrate wnew, and (4) the entire simulation procedure stops if wp is close to wnew within a pre-set error. Otherwise, the wnew and Np are used to calculate a new k using Eq. (6), and then go back to step 2. 2.4 Solution and coupling schemes Facilitated by the above boundary conditions, the CFD-DEM problem here can be numerically solved by the well-established methods reported in the literature. In this work, for the DEM model, an explicit time integration method is used to solve the translational and rotational motions of discrete particles.50 For the CFD model, the finite volume method with the SIMPLE velocity-pressure coupling method51 is used to solve the governing equations for gas phase by means of non-staggered grid arrangements in a body-fitted coordinate system.52 The two-way coupling of particle and fluid flows is numerically achieved as follows. At each time step, DEM provides information, such as the positions and velocities of individual particles, for the evaluation of porosity and volumetric particle-fluid forces in a computational cell. CFD then uses these data to determine the gas flow field, yielding the particle-fluid forces acting on individual particles. Incorporation of the resulting forces into DEM enables the determination of the motion of individual particles for the next time step. This solution scheme has been well established since the work of Xu and Yu,53 and is used in this work.

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To calculate the fluid drag forces on particles using local fluid velocity on particle position, the point-locating scheme proposed by Kuang et al.54 for 3D hybrid grids is used to locate the host cell where a considered particle is residing, and then the fluid velocity stored on the centre of the host cell is mapped to the particle position using a least-square interpolation method. 2.5 Simulation conditions Table 1 shows the parameters used in the present study. Two typical cases are simulated, involving three variables: inclination angle, gas velocity, and solid flowrate. To reproduce the key features of inclined pneumatic conveying such as pressure behaviors, inclination angle is varied in Case I from 0º to 90º while all other parameters remain constant. Different solid flowrates are also considered for each inclination angle, as listed in Table 1. In Case II, gas velocity is varied from 6.2 to 46 m/s for a constant solid flowrate of 0.9 kg/s in a 45º inclined pipe. This allows us to obtain typical flow regimes in inclined pneumatic conveying to study the corresponding gas pressure behavior and energy dissipation. The gas involved is incompressible, with a density of 1.205 kg/m3, and viscosity of 1.85×10-5 kg/m/s. For Case I, a 3D 1-m long pipe section with an internal diameter of 0.081 m is chosen as the computational domain, as schematically illustrated in Fig. 1. The particles used are spherical, with a diameter of 3 mm, density of 880 kg/m3, and restitution coefficient of 0.8. This setup is done following the experimental work of Levy et al.4, so as to examine the applicability of the present CFD-DEM model. For Case II, bigger particles (5 mm) and a smaller pipe (0.05 m) are considered to reduce computational effort, while all other material properties and geometry parameters are the same as those in Case I. 3. Results and Discussion 3.1. Model validation In order to examine if the present CFD-DEM model can obtain a given solid flowrate and reproduce the flow characteristics in inclined pneumatic conveying, ten experimental tests carried out by Levy et al.4 for a pipe with inclination angle varying from 0º to 90º are first simulated. In these tests, the solid and gas flowrates to some degree vary with inclination 10


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angle, as listed in Table 2. Other simulation parameters are the same as those of Case I (see Table 1). The simulated and measured pressure drop and solid flowrate are given in Table 2. It can be seen from the table that the present CFD-DEM model with PBC can achieve the given solid flowrates for all the considered tests, with errors less than 1%. Here, the solid flowrates in the simulations are obtained by counting particles passing through the cross-sectional area of the pipe. Table 2 also shows that the CFD-DEM model can well capture the key behaviors of pressure drop observed in the experiments. For most of the cases, a lower gas velocity or larger solid flowrate leads to a larger pressure drop for a given inclination angle. As expected, in both the experiment and simulation, inclined pipes give larger pressure drops compared to horizontal and vertical pipes. However, the present model is found to under-estimate the pressure drop in most of the cases simulated. The prediction errors are generally less than 9% except for the third and sixth tests where the errors are respectively 19% and 15%. Similar prediction errors have also been reported by Levy et al.4 who studied inclined pneumatic conveying using a two-fluid model (TFM) (see Table 2). Levy et al.4 attributed the under-estimation to some uncertainties related to the equations to calculate particle-fluid interaction force, and the selection of parameters for simulation. This should apply to the present CFD-DEM model. However, another fact may also significantly contribute to the under-estimation. In the experiment, a relatively short (8-m) pipe connected with two bends was considered. It is well established that particles may take considerable distance of downstream of a bend to accelerate to a steady status.55 This acceleration causes extra pressure loss that may be to some degree included in the measured results, but not in the calculated results which involve only the developed flow. Nonetheless, the above results manifest that the CFD-DEM model is valid for the study of inclined pneumatic conveying, at least qualitatively. It should be pointed out here that the applicability of the present CFD-DEM approach has been verified in the studies of horizontal and vertical pneumatic conveying, where the same CFD-DEM model was successfully used to predict complicated behaviors in relation to flow regime and flow transition.19,21 With the same model framework, the complicated phenomena in a bend can also be reproduced.20

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3.2. Basic flow characteristics 3.2.1. Effect of inclination angle Fig. 2 shows the influence of inclination angle on the average pressure drop under different solid flowrates, corresponding to Case I. It can be seen from the figure that when inclination angle increases from 0º to 90º at a given gas velocity, the pressure drop of the empty pipe nearly remains constant, whereas with the particles loaded at a constant flowrate, the pressure drop initially increases to a maximum and then decreases. Therefore, inclined pneumatic conveying has a higher pressure drop compared to horizontal and vertical pneumatic conveying, as reported by different investigators.4,8,9 Fig. 2 also shows that with increasing solid flowrate, the pressure drop changes more significantly with the variation of inclination angle. This has also been observed by Levy et al.4 using a TFM model integrated with a specific particle-wall model for considering the effect of inclination angle. Note that such treatment is not needed in the present model. These results confirm that the CFD-DEM approach can capture the key behavior of pressure drop due to the change of inclination angle. In the process of pneumatic conveying, the transport of particles is driven by the flow of gas phase. Thus, its energy dissipation Ef in a given pipe over a period of time can be determined using the energy loss of gas phase:56 Ef =

1 T

T

kc1

kc 2

0

i =1

i =1

∑ [(∑ Pi,1si ,1ε i ,1ui,1 − ∑ Pi,2 si, 2ε i ,2ui, 2 )∆t ]

(7a)

where subscripts 1, 2, and f represent the inlet and outlet of the pipe (see Fig. 1) and the gas phase, respectively; kc1 and kc1 are the numbers of cell faces at the inlet and outlet, respectively; T is the simulation time; and si is the area of CFD cell face i. For practical use, the gas pressure on a cross section is approximately uniform. Thus, Eq. (7a) can be simplified as

Ef =

1 T ∑ (∆PQ∆t ) T 0

(7b)

where Q=UπD2/2. Eq. (7b) indicates that energy dissipation in pneumatic conveying is a 12


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function of pressure drop and gas flowrate which can both be measured readily. Thus, this equation is usually used to assess energy efficiency in pneumatic conveying.28-32 In this work, to be accurate, Eq. 7(a) is used. Fig. 2 plots the energy dissipation as a function of inclination angle at wp=1.31 kg/s in Case I. It can be seen from the figure that the energy dissipation has a similar trend to that observed for the pressure drop due to the constant velocity considered. This is also the case for other solid flowrates considered, although their results are not included in Fig. 2 for brevity. These results show that inclined pneumatic conveying has a higher energy loss compared to horizontal or vertical pneumatic conveying. Fig. 3 shows the typical particle flow patterns at different inclination angles. Here, for convenience, all the pipes are shown horizontally, and the “top” and “bottom” sides are defined according to the gravitational direction, as schematically illustrated Fig. 1. This treatment also applies to other spatial results in the following discussion. It can be seen from Fig. 3 that two flow modes, i.e. the stratified flow regime (at low inclination angles) and dispersed flow regime (at high inclination angles) are obtained as expected. When inclination angle is increased, the flow is developed smoothly from the stratified flow regime to the dispersed flow regime. Accordingly, the radial distributions of time-averaged gas and solid velocities change gradually from asymmetrical to symmetrical (see Fig. 4). It is of interest to note that under the present condition, the vertical pneumatic conveying more possibly forms clusters compared to others, as highlighted by a black circle in Fig. 3. Such clusters present only at a relatively high solid loading ratio (SLR), i.e. SLR=10 and 15, leading to a large pressure fluctuation, as shown in Fig. 5. However, it should be pointed out that if clusters are not formed, the pressure fluctuation in an inclined pipe should be larger than that in a vertical or horizontal pipe. This can be demonstrated by the results for the cases β=0°, β=30° and

β=60° in Fig. 5, where the horizontal pipe has the smallest pressure fluctuation. This is in line with the general observation of inclined pneumatic conveying.4,13

3.2.2. Effect of gas velocity Fig. 6 plots the average pressure drop as a function of gas velocity for the transport of particles in a 45º inclined pipe at a constant solid flowrate, corresponding to Case II. Their typical particle patterns are given in Fig. 7. In general, three typical flow regimes are 13



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observed, including the slug flow regime (U=6-6.5 m/s), stratified flow regime (U=14-34 m/s), and dispersed flow regime (U=34-46 m/s). Note the boundaries between flow regimes are largely determined according to Fig. 6, as done elsewhere.19 With increasing gas velocity, the flow in the 45º inclined pipe experiences transitions from the slug flow regime, to the transition flow regime, to the stratified flow regime, finally to the dispersed flow regime, as observed in the horizontal pneumatic conveying.57,58 Note that the transition flow regime in the so called unstable zone is very sensitive to the change of particle number or solid flowrate57 and cannot be reproduced by the present CFD-DEM model with PBC. To overcome this problem, a long pipe has to be used without PBC.19 In this work, for simplicity, the transition flow regime is not considered. Another relevant phenomenon not considered in this work is the so-called reversed flow which may occur when the particle-fluid force is not strong enough to transport particles in the main flow direction. It is widely observed in fluidization59,60 and may be encountered in inclined and vertical pneumatic conveying25,26 and vertical bend.20 Such reversed flow is observed within the settled layer ahead of a slug at the minimum gas velocity (U=6 m/s) in Case II. However, it is not observed under other conditions because large gas velocities (U≥6 m/s) are used in this work. The reversed flow and associated phenomena will probably be studied in the future. Fig. 6 shows that when gas velocity is increased, the pressure drop initially decreases sharply in the slug flow regime, and then slows down and decreases to a minimum in the stratified flow regime, followed by an increase in the dispersed flow regime. The right region of the minimum is usually referred to as the dilute-phase flow and the left region is called the dense-phase flow. Overall, the pressure drop in the slug flow regime is much larger than those in the stratified and dispersed flow regimes. Here, the decrease of pressure drop in the slug flow regime is attributed to the decrease of slug length as shown in Fig. 7, similar to those experimentally observed in horizontal pneumatic conveying.61,62 Furthermore, because the gravity still contributes significantly to particle segregation in the radial direction in the 45º inclined pipe, the pressure drop-gas velocity phase diagram obtained here is similar to that for horizontal pneumatic conveying where the gravity is the largest in the radial direction. Note that in this work, the critical velocity corresponding to the minimum pressure drop is

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relatively large (Ucritical,p=34 m/s), as compared to that obtained in horizontal pneumatic conveying where similar gas and solid flowrates were used.19 This may be due to the coarse particles (5 mm) and inclined pipe simulated. Fig. 6 also shows that when gas velocity is increased, the energy dissipation decreases initially to a minimum and then increases. Because of the varying gas velocity in Case II, unlike Case I, the energy dissipation here experiences different behaviors from that observed for the pressure drop, including: (1) it changes sharply in all the flow regimes obtained and is not limited to the slug flow regime as for the pressure drop, (2) the critical velocity corresponding to the minimum energy dissipation (Ucritical,E=27 m/s) is smaller than that corresponding to the minimum pressure drop (Ucritical,p=34 m/s), and (3) the energy dissipation in the slug flow regime can be as small as those in the dispersed flow regime, which is however conditionally valid, although the slug flow regime is recognized as an energy efficient transport mode.57,63 The above results suggest that both energy and pressure drop should be taken into account in the design and operation of pneumatic conveying, although they are related to each other.

3.3. Relation between pressure drop and forces acting on particles Pressure drop is one of the most important process parameters needed to be considered in the design and operation of pneumatic conveying. It can be used to qualitatively assess energy efficiency under certain conditions (e.g. in Case I). According to Eq. (4), gas pressure drop can be expressed as follows: ∇P =

Eq.

(8)

manifests

that

d (ρ f ε f u ) dt

pressure

+ ρ f ε f g − Fp - f + ∇ ⋅ (ε f τ )

drop

is

contributed

by

(8) four

factors,

i.e.

acceleration/deceleration of fluid, fluid gravity, presence of particles (and hence the resulting particle-fluid interaction), and fluid shear. Modelling of pressure drop in steady-state pneumatic conveying usually neglects the unsteady acceleration and convective acceleration and only considers the contributions by particles and fluid-wall stress.64

∇P ≈ −Fp -f + ∇ ⋅ (ε f τ w )

(9)

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To better understand the pressure drop behaviors in Figs. 2 and 6, we first examine Eq. (9) using the simulation results obtained. For such a purpose, the average volumetric particle-fluid force over a period of time is calculated by Fp - f =

 1 1  ∑ T ∆t V pipe

 N p   ∑ f i   i =1   

(10)

where fi represents the fluid drag force or pressure gradient force acting on particle i, Np and Vpipe are the number of particles in a considered pipe and the volume of the pipe, respectively. The second term in the right side of Eq. (9) can be rewritten according to the Gauss theory:52 Nw ∑ (ε f s ⋅ τ w ) i =1

V pipe

(11)

where Nw is the number of wall surfaces in CFD cells, and s is the area vector of a wall surface. Since the normal fluid stress at the wall is equal to zero (ττwn=0),52 τw is equal to the wall shear stress τwt, i.e. gas-wall friction force. Fig. 8 shows the relationship between pressure drop and total volumetric force consisting of the drag force, pressure gradient force, and gas-wall friction force in Cases I and II. It can be seen that the pressure drops well match the total forces in both cases. This result provides direct evidences to support the validity of Eq. (9), i.e. the pressure drop can be determined using the gas-wall friction force and particle-fluid force for steady-state pneumatic conveying. The latter force represents the contribution of particles to gas pressure. Fig. 8a also shows that the particle-fluid force is responsible for the pressure behaviors due to the change of inclination angle. This is in line with the result that the presence of particles in gas stream leads to a higher pressure drop in the inclined pneumatic conveying compared to the vertical and horizontal pneumatic conveying. The fluid drag force can be further analysed in relation to the gas and solid flow structures. By definition, it is a function of solid concentration and relative velocity between gas and particle. Fig. 9 shows that when inclination angle is increased, both the solid concentration and relative velocity increase to a maximum and then decrease. Consequently, the fluid drag force and hence the particle-fluid force follow the same trend. That is, the results in Fig. 8 can be explained from the gas and solid flow structures. 16


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To further understand the pressure behaviors, we analyse the volumetric forces acting on particles in the pipe section (see Fig. 10), and the results are given in Fig. 11. In the figure, only the axial forces governing the conveying of particles are considered. Fig. 11 shows that in the flow direction, the positive particle-fluid forces as the driving force balance with the negative particle-wall friction and gravitational forces as the resistance forces under all the conditions considered in this work. Figs. 8 and 11 together manifest that for a steady-state transport, the pressure drop is essentially contributed by the particle-wall friction force, gravitational force, and gas-wall friction force. The last force becomes important only when gas velocity is high. Nonetheless, the contributions of different forces to pressure drop, which have not been discussed in the previous studies, can now be quantified using the present analysis. It can be seen from Fig. 11a that when inclination angle is smaller than the critical angle (βcritical=50°) corresponding to the minimum pressure drop, the particle-wall friction force is the dominant resistance force which maintains almost constant, the magnitude of gravitational force increases with the increase of inclination angle. Conversely, when β>βcritical, the gravitational force becomes the dominant resistant force and is nearly constant, and the magnitude of the particle-wall friction force decreases with increasing inclination angle. Summarily, when inclination angle is increased, the gravitational force is mainly responsible for the increase of pressure drop at relatively low inclination angles, whereas the particle-wall friction force mainly accounts for the decrease of pressure drop at relatively high inclination angles. This is valid to other considered solid flowrates not included in Fig. 11a. Fig. 9 shows that as inclination angle increases, the solid velocity decreases to a minimum and then increases. This interesting behaviour can be well explained in terms of the governing forces. As shown in Fig. 10, at the pipe scale, particles in conveying are governed by the aforementioned resistance and driving forces in the flow direction, and the two forces balance with each other for macroscopically steady-state transport. When inclination angle is increased, the magnitude of the resistance force first increases because of the increased gravitational force and then decreases because of the decreased particle-wall interaction force. A similar trend must be observed for the driving force, which acts in opposite direction, to 17



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establish force balance and hence steady state flow, as shown in Fig. 11a. In Case I, the pressure gradient force is negligible and the driving force is the fluid drag force. It is known that the fluid drag force is proportional to solid concentration and the relative velocity between gas and particles. Thus, with inclination angle increasing, the solid concentration and the relative velocity may have the following options in order to produce the corresponding variation in the fluid drag force: (a), the solid concentration increases to a maximum and then decreases; (b), the relative velocity increases to a maximum and then decreases, and (c), both the solid concentration and relative velocity increase to a maximum and then decrease. For option a, because of a constant solid flowrate (=solid velocity × cross-sectional area of the pipe × solid concentration), the trend of solid velocity must be opposite to that of the solid concentration. For option b, because of a constant gas flowrate and hence velocity, the solid velocity must follow the trend in Fig. 9 so that the relative velocity increases to a maximum and then decreases. Option c simply demonstrates that options a and b work simultaneously. Therefore, all the three options will lead to the same outcome that the solid velocity decreases to a minimum and then increases as inclination angle increases. Which option is more effective may be condition-dependent. Our numerical results show that option c is effective for the case considered. Fig. 11b shows that with increasing gas velocity, the magnitudes of the particle-wall friction force and gravitational force initially decrease sharply in the slug flow regime, and then slow down in both the stratified and dispersed flow regimes. In the slug flow regime, the particle-wall friction force is dominant at relatively low gas velocities but becomes much closer to the gravity at relatively high gas velocities. This indicates that the relative importance of the gravity and particle-wall friction force in the slug flow regime depends on the conditions involved. This may be the reason why the pressure is found to be governed by the wall stress by Azia and Klingzing10 but up to both the wall friction force and inclination angle by Hirota et al.11 It is of interest to note that the gravity and particle-wall friction force are nearly close to each other in the stratified flow regime and dispersed flow regime in Case II. This can also be observed in Case I when β is around 45°.

18


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3.4 Energy dissipation in inclined pneumatic conveying Energy efficiency is of great importance to pneumatic conveying. Taking advantage of the CFD-DEM approach, the information about energy can be obtained. Here, we extend the energy analysis proposed by Wu et al.35 for particle systems to the present particle-fluid system under steady-state conditions. In the process of pneumatic conveying, the energy loss of gas flow calculated by Eq. (7) is mainly used to replenish the energy dissipations as a result of the interactions between particles, particles and wall, particles and fluid, fluids, and fluid and wall, as well as the energy for accelerating gas and solid phases and lifting particles in the gravitational direction. Accordingly, these energy dissipations are: (a) the so called impact energy dissipation ec as a result of the relative velocity at a contact point between particles, and between particles and wall; (b) the friction energy dissipation es as a result of the relative sliding between particles and between particles and wall; (c) the rolling energy dissipation due to the rolling friction arising from asymmetrical normal traction distribution on the contact area between particles and between particles and wall er; (d) the gravitational potential energy eg due to the transport of particles along the gravitational direction; (e) the elastic potential energy ec due to the contact between particles and between particles and wall; (f) the change of the transitional kinetic energy of particle ek,v; (g) the change of the rotational kinetic energy of particle ek,ω; (h) the change of the kinetic energy of gas phase ek,gas, which is not considered in this work as gas density is very small compared to particle density and stead-state flows are considered; (i) the particle-fluid energy dissipation ep-f as a result of the fluid resistance to the relative motion between particle and fluid;33 and (j) the viscous energy dissipation ef due to the viscous shear between fluids and between fluid and wall.56 These energy dissipations in a given pipe over a period of time are calculated according to the equations listed in Table 3. Their time-averaged values are considered using E pi =

1T ∫ ei (t )dt , where T is the simulation T0

time. To facilitate our discussion, the total energy change due to particle-particle and particle-wall collisions is defined as the collision energy dissipation, Ecollision (see Table 3). Fig. 12 compares the total energy dissipation calculated microscopically using the equations

19



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listed in Table 3 with that calculated macroscopically by Eq. (7a). It can be seen from the figure that as expected, in both cases considered, the energy dissipations obtained from two different ways well match each other. On this basis, the mechanisms underlying the energy dissipation in inclined pneumatic conveying are quantified and the results are given in Figs. 13-15. Figs. 13a and 15a show that the kinetic energy due to the transitional and rotational motions of particles does not change and the fluid-fluid viscous energy dissipation is negligible because only steady-state channel flows are considered here. Thus, the relevant energy dissipations are not discussed further in the following. The two figures also show that for steady-state

pneumatic

conveying,

its energy loss

is mainly attributed to the

particle-particle/wall, particle-fluid, fluid-wall interactions and the transport of particles along the gravitational direction. The resulting energy dissipations are respectively Ecollision, Ep-f, Ef-w, and Eg, which vary significantly with inclination angle (in Case I) and gas velocity (in Case II). Fig. 13a shows that when inclination angle is increased, both the fluid-wall viscous energy dissipation Ef-w and particle-fluid energy dissipation Ep-f increase to a maximum around the critical angle (βcritical=50°) and then decrease. Conversely, the gravitational potential energy Eg rapidly increases from zero and slows down at relatively high inclination angles. The collision energy dissipation Ecollision increases to a maximum when βEf-w>Ecollision≈Eg. Fig. 13b shows the details of the collision energy dissipation Ecollision in Case I. It can be seen from the figure that the collision energy dissipation is mainly attributed to the particle-wall friction energy dissipation Es,p-w because here the stratified flow regime and dispersed flow regime are considered. The particle-particle impact energy dissipation Ed,p-p and friction energy dissipation Es,p-p have a maximum in inclined pneumatic conveying (β=30°). However, these two energy dissipations are much smaller than the particle-wall friction energy dissipation. Fig. 14a shows how solid flowrate affects the major energy dissipations in Case I. It can be 20


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seen from the figure that the effects of solid flowrate on different energy dissipations are very different, depending on inclination angle. In general, the increase of solid flowrate leads to an increased fluid-wall viscous dissipation, collision energy dissipation, fluid-wall viscous dissipation, and gravitational potential energy. The most significant effect of solid flowrate on the gravitational potential energy appears at β=90°, however, on the fluid-wall viscous dissipation at β=30°, on the particle-fluid energy dissipation at β=50°, and on the collision energy dissipation at β=20°. The effect of solid flowrate is small if not negligible at β=0° for the gravitational potential energy, and for both the fluid-wall viscous dissipation particle-fluid energy dissipation at β=0° and 90°, and for the collision energy dissipation at β=90°. Fig. 15a shows how gas velocity U affects the major energy dissipations in the 45° inclined pipe in Case II. When U=6-6.5 m/s (corresponding to the slug flow regime), the collision energy dissipation and particle-fluid energy dissipation are dominant and the former is much bigger, whereas the gravitational potential energy is small and the fluid-wall viscous dissipation is negligible. When 14 m/s≤U≤Ucritical,E=27 m/s (corresponding to the stratified flow regime), the particle-fluid energy dissipation is the most significant and the roles of the collision energy dissipation, gravitational potential energy and the fluid-wall viscous energy dissipation become more and more important with increasing gas velocity. When U>Ucritical,E (corresponding to the stratified flow regime and dispersed flow regime), the fluid-wall viscous dissipation is the most significant and other three energy dissipations to some degree contribute to the total energy loss. It is of interest to note that the gravitational potential energy is nearly uniform in all the flow regimes. Furthermore, the collision energy dissipation becomes dominant only in the slug flow regime and has a minimum at the critical velocity Ucritical,E. At U=Ucritical,E, the four major mechanisms for energy dissipations, i.e. the particle-fluid energy dissipation, fluid-wall viscous energy dissipation, gravitational potential energy, and collision energy dissipation are close to each other. Fig. 15b shows that in all the flow regimes considered, the particle-wall friction dissipation is the major part of the collision energy dissipation. In the slug flow regime, the particle-particle rolling energy dissipation, impact dissipation, and friction dissipation become relatively important, although they are relatively small compared to the particle-wall friction energy 21



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Page 22 of 45

dissipation. However, with increasing gas velocity, these three energy dissipations decrease sharply and become negligible at high gas velocities. 4. Conclusions Pressure drop and energy dissipation are two major concerns in the design and operation of inclined pneumatic conveying. They are related to each other. A CFD-DEM model, facilitated with PBC for gas and particle phases, has been used to study this gas-solid flow system, focused on pressure behaviour and energy dissipation. The results from the present work can be summarized as follows: (1) The key flow behaviors in inclined pipelines can be reproduced by the present CFD-DEM

model. These include (a) the pressure drop initially increases to a maximum and then decreases with increasing inclination angle from 0° to 90°, (b) the variation of the pressure with inclination angle becomes more significant with increasing solid flowrate, and (c) typical flow regimes and their flow transition with the increase of gas velocity from the slug flow regime, to the stratified flow regime, finally to the dispersed flow regime at a given inclination angle. (2) The mechanisms underlying the pressure drop are studied with respect to the particle-fluid force, gravitational force, particle-wall friction force, and gas-wall friction force. The results show that the pressure drop can be well represented by the volumetric particle-fluid force and fluid-wall friction force. The former balances with the gravity and particle-wall friction force in the flow direction. The increase of pressure drop at relatively low inclination angles is attributed to the increase of gravitational force, while the decrease of pressure drop at relatively high inclination angles is due to the decrease of friction force. In the 45° inclined pneumatic conveying, the pressure drop is dominated by the particle-wall friction force and the gravitational force in the slug flow regime, but by the gravitational force, particle-wall friction force and gas-wall friction force in the stratified and dispersed flow regimes. (3) Energy dissipation in inclined pneumatic conveying has been analyzed. The results show that for steady-state pneumatic conveying, the energy dissipation macroscopically calculated from pressure drop and gas flowrate mainly consists of the particle-fluid energy dissipation, gravitational potential energy, collision energy dissipation, and fluid-wall energy dissipation 22


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at a microscopic level. The most significant energy dissipation is the collision energy dissipation in the slug flow regime, the particle-fluid energy dissipation in the stratified flow regime, and the fluid-wall viscous energy dissipation in the dispersed flow regime. The collision energy dissipation is mainly contributed by the particle-wall friction energy dissipation in all the flow regimes considered. However, it is to some degree by the particle-particle friction energy dissipation, impact energy dissipation and rolling friction dissipation in the slug flow regime. The gravitational potential energy in the 45° inclined pneumatic conveying is nearly uniform with varying gas velocity. Finally, it should be pointed out that one of aims of the present study is to establish some basic equations to determine energy dissipation particle-fluid flows. Gas-solid flow in inclined pneumatic conveying is specifically considered because of its importance in application. As the first step in this direction, the analysis in this work is focused on simple but typical cases. In fact, energy dissipation in pneumatic conveying is much more complicated, affected by many variables related to the geometry, material properties and operational conditions. A systematic study is needed in future studies in order to understand the effects of these variables and their interplays, and finally generate more accurate and reliable techniques for designing and controlling pneumatic conveying in a more energy-efficient way. Acknowledgements The authors are grateful to the Australia Research Council (ARC) and Fujian Longking Co., Ltd for the financial support of this work, and to National Computational Infrastructure (NCI) for the use of its computational facilities.

23



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Notations dp

Particle diameter, m

D

Diameter of pipe, m

f

Particle scale forces, N

e

Energy, J

E

Time-averaged energy, J

F

Volumetric force, N⋅m-3

g

Gravitational acceleration, m⋅s-2

I

Moment of inertia of particle, kg⋅m2

k

A factor in Eq. (6)

kc

Number of particles in a considered computational cell

ki

Number of particles in contact with particle i

L

Length of pie, m

m

Mass of particle, kg

Np

Particle number, kg

P

Pressure, Pa

q

Number of contact points between particles or particle and wall

Q

Gas volumetric flowrate, m3⋅s-1

si

Area of CFD cell face i, m2

R

Vector from the mass centre of the particle to the contact point, m

t

Time, s

T

Simulation time, s

∆t

Time step, s

T

Torque, N⋅m

v

Translational velocities of particle, m⋅s-1

Vp

Volume of particle, m3

Vpipe Volume of the pipe section considered, m3 ∆V

the volume of the computational cell, m3

u

Gas velocity, m⋅s-1

Ug

Superficial velocity, m⋅s-1 24


Page 25 of 45

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wp

Solid mass flowrate, kg⋅s-1

wf

Gas mass flowrate, kg⋅s-1

Y

Young’s modulus, Pa

Greek β

inclination angle

εf

Local porosity

η

Gas viscosity, kg⋅m-1⋅s-1

µ

Friction coefficient

ρf

Gas density, kg⋅m-3

ρp

Particle density, kg⋅m-3

ω

Particle angular velocity, s-1

) ω

Unit vector defined by ω = ω / | ω |

τ

Fluid viscous stress tensor, kg⋅m-1⋅s-2

)

Subscripts c

contact

d

damping

drag fluid drag force f

fluid

g

gravity

i

particle i

ij

between particles i and j

j

particle j

n

normal component

p

particle

p-f

particle-fluid

pgf

pressure gradient force

r

rolling friction

s

sliding

t

tangential component

25



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(60) Van der Meer, E. H.; Thorpe, R. B.; Davidson, J. F. Flow patterns in the square cross-section riser of a circulating fluidised bed and the effect of riser exit design. Chemical Engineering Science 2000, 55: 4079. (61) Borzone, L. A.; Klinzing, G. E. Dense-phase transport-vertical plug flow. Powder Technology 1987, 53: 273. (62) Li, J.; Pandiella, S. S.; Webb, C.; McGlinchey, D.; Cowell, A.; Xiang, J.; Knight, L.; Pugh, J. An experimental technique for the analysis of slug flows in pneumatic pipelines using pressure measurements. Particulate Science and Technology 2002, 20: 283. (63) Pan, R.; Wypych, P. W. Pressure drop and slug velocity in low-velocity pneumatic conveying of bulk solids. Powder Technology 1997, 94: 123. (64) Ratnayake, C.; Datta, B. K.; Melaaen, M. C. A unified scaling-up technique for pneumatic conveying systems. Particulate Science and Technology 2007, 25: 289.

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Tables Table 1 Parameters used in the present simulations. Table 2 Comparison of the measured and calculated results at different inclination angles. Table 3 Energy dissipations of solid and gas phases in the considered pipe section. Figure Captions Fig. 1 Schematic illustration of computational domain. Fig. 2 Pressure drop vs. inclination angle in Case I Fig. 3 Snapshots showing typical solid flow patterns at different inclination angles at wp=1.31 kg/m3 in Case I. Fig. 4 Radial distributions of (a) gas and (b) solid velocities, corresponding to Fig. 3. Fig. 5 Temporal pressure drops in the pipe with different inclination angles at wp=1.31 kg/m3 in Case I. Fig. 6 Pressure drop-gas velocity phase diagram in Case II. Fig. 7 Snapshots showing typical particle patterns when wp=0.9 kg/s and β=45º, corresponding to Fig. 6. Fig. 8 Average volumetric fluid drag force, pressure gradient force and gas-wall friction force as a function of: (a), inclination angle at wp=1.31 kg/m3 in Case I; and (b), gas velocity in Case II. Fig. 9 Average solid concentration and relative velocity between gas and particles as a function of inclination angle at wp=1.31 kg/m3 in Case I. Fig. 10 Schematic illustration of forces acting on a particle assembly in pneumatic conveying. Fig. 11 Average volumetric particle-fluid force, gravity and particle-wall friction force as a function of: (a), inclination angle at wp=1.31 kg/m3 in Case I; and (b), gas velocity in Case II. Fig. 12 Comparison of energy loss obtained macroscopically and microscopically in: (a), Case I; and (b), Case II. Fig. 13 Energy dissipations as a function of inclination angle at wp=1.31 kg/s in Case I: (a) major energy dissipations; and (b) energy dissipations due to particle-particle and particle-wall collisions. Fig. 14 Effects of solid flowrate on different energy dissipations: (a), gravitational potential energy; (b), fluid-wall viscous dissipation; (c), particle-fluid energy dissipation; and (d), collision energy dissipation. Fig. 15 Energy dissipations as a function of gas velocity in Case II: (a) major energy dissipations, and (b) energy dissipations due to particle-particle and particle-wall collisions.

30


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1 Parameters used in the present simulations. Parameter a

Case I

Value b Case II

Pipe and flow Pipe diameter, D (m) 0.081 0.05 Pipe length, L (m) 1 1 Inclination angle, β (º) 45 (0–90) 45 Solid flowrate, wp (kg/s) 1.31 (0.655–1.965) 0.9 Inlet gas velocity, Ug (m/s) 21.0 6 (6–46) Solid Loading Ratio (SLR) 10 (5–15) 64 (8–64) Gas Gas used Air Air 1.205 1.205 Density, ρf (kg/m3) Viscosity, η (kg/m/s) 1.85×10-5 1.85×10-5 Particles Shape Spherical Spherical Diameter, dp (mm) 3 5 Density, ρp (kg/m3) 880 880 Young’s modulus, Y (Pa) 1.0×108 1.0×108 Poisson ratio, ν 3.3 3.3 0.5 0.5 Sliding friction coefficient, µs Rolling friction coefficient, µr 0.02 0.02 Restitution coefficient 0.8 0.8 Time step, ∆t (s) 5.0×10-6 5.0×10-6 a The wall is assumed to have the same properties as particles. b

Base value and varying range in the bracket.

31



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Page 32 of 45

Table 2 Comparison of the measured and calculated results at different inclination angles. U (m/s)

β

21.1

0

Solid flowrate (kg/s) Calculated Error Measured by (%) CFD-DEM 0.424 0.424 0.04

Measured 189

Pressure drop (Pa) Calculated Error Calculated Error by (%) by TFM4 (%) CFD-DEM 179 4.23 182 3.55

18.4

0

0.572

0.576

0.67

193

196

1.66

186

3.63

19.8

45

0.370

0.371

0.13

279

226

19.07

224

19.70

17.6

45

0.420

0.421

0.28

285

267

6.24

275

3.51

20.3

60

0.285

0.283

0.47

189

199

5.36

172

8.99

20.3

60

0.561

0.562

0.18

333

282

15.30

270

18.99

20.3

75

0.583

0.582

0.10

272

270

0.80

294

8.09

16.6

75

0.628

0.632

0.55

330

322

2.42

269

18.48

18.2

90

0.505

0.506

0.16

256

234

8.65

265

3.52

17.1

90

0.430

0.429

0.32

232

216

6.76

222

4.31

32


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Table 3 Energy dissipations of solid and gas phases in the considered pipe section. Energy

Correlations n

∑ (m g ⋅ v ∆t )

Gravitational potential energy (eg)

i

i

i =1

n

Change of transitional kinetic energy (ek,v)

1

∑ [( 2 m

1 2 2 v i ) t + ∆t − ( m i v i ) t ] 2

i

i =1 n

Change of rotational kinetic energy (ek,ω)

1

∑ [( 2 I

2

i

ω i ) t + ∆t − (

i =1

n

∑ [f

Particle-fluid energy dissipation (ep-f)

drag , i

1 2 I i ω i )t ] 2

⋅ ( u i − v i )∆ t ]

i =1

Elastic potential energy between particles or between particles and wall (ec,p-p / ec,p-w) Impact energy dissipation between particles or between particles and wall (ed,p-p / ed,p-w) Friction energy dissipation between particles or between particles and wall (es,p-p / es,p-w)

q

∑ [f

cn , ij

⋅ v cn ,ij ∆ t + f ct ,ij ⋅ v ct ,ij ∆ t ] l

dn , ij

⋅ v dn ,ij ∆ t + f dt ,ij ⋅ v dt ,ij ∆ t ]

l =1 q

∑ [f l =1

l q

∑ [f l =1

s , ij

⋅ v t , ij ∆ t ] l

q Rolling energy dissipation between particles [ T r , ij ⋅ ω ij ∆ t ] ∑ l =1 l or between particles and wall (er,p-p / er,p-w) N Viscous energy dissipation between fluids or ( τ : ∇ u )∆ Vi∆ t ∑ between fluid and wall (ef-f / ef-w) i =1 Collision energy (ecollision) ec, + ed + es, + er, where n and Ncel are, respectively, the number of particles and the number of CFD cells in the considered pipe section, and q is the number of contact points between particles or particle and wall. cel

33



Fig. 1 Schematic illustration of computational domain.

6

Energy

wp = 1.31 kg/s

wp = 1.31 kg/s, SLR=10 wp = 0.655 kg/s, SLR=5

0.8

5 4

wp = 0 kg/s, SLR=0 U = 21 m/s

-4

1.0

Pressure drop wp = 1.965 kg/s, SLR=15

0.6

3

0.4

2

0.2

1

0.0

-20

0

20

Energy (10 J)

1.2

Pressure drop (kPa/m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 45

40

60

80

0 100

Inclination angle (Degree) Fig. 2 Pressure drop vs. inclination angle in Case I

34


Page 35 of 45

Fig. 3 Snapshots showing typical solid flow patterns at different inclination angles at wp=1.31 kg/m3 in Case I.

(a) Gas velocity (ug/Ug)

1.2 1.0 0 30 60 90

0.8 0.6 0.7 (b)

Solid velocity (vs/Ug)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6 0.5 0.4 0.3 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

0.2

0.4

0.6

0.8

1.0

Radial position (y/R)

Fig. 4 Radial distributions of (a) gas and (b) solid velocities, corresponding to Fig. 3.

35



600

β = 0° β = 60°

550

β = 30° β = 90°

Pressure drop (Pa)

500 450 400 350 300 250 200 0

1

2

3

4

5

Time (s)

Dense-phase flow Unstable zone

4

3

-4

1

5

Energy (10 J)

10

Dilute-phase flow

Fig. 5 Temporal pressure drops in the pipe with different inclination angles at wp=1.31 kg/m3 in Case I.

Pressure drop (kPa/m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 45

0.1

2 wp = 0.9 kg/s, β = 45° Pressure drop

Energy

0.01 5

10

15 20 25 30 35 40 4550 Gas velocity (m/s)

Fig. 6 Pressure drop-gas velocity phase diagram in Case II.

36


Page 37 of 45

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Fig. 7 Snapshots showing typical particle patterns when wp=0.9 kg/s and β=45º, corresponding to Fig. 6.

37



0.7

Gas-wall friction force Pressure gradient force Drag force Overall force Pressure drop

3

Volumetric forces (N/m )

0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

20

40

60

80

100

Inclination angle (Degree)

(a)

10

Drag force Pressure gradient force Gas-wall friction force Overall force Pressure drop

3

Volumetric force (KN/m )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 45

1

0.1

0.01 5

10

15 20 25 30 35 40 4550 Gas velocity (m/s) (b)

Fig. 8 Average volumetric fluid drag force, pressure gradient force and gas-wall friction force as a function of: (a), inclination angle at wp=1.31 kg/m3 in Case I; and (b), gas velocity in Case II.

38


Page 39 of 45

20

15 0.02 10

0.01 Gas velocity Solid velocity Relatively velocity between gas and particle Solid concentration

5

0 0

20

40

60

80

Solid concentration

0.03

Average velocity (m/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.00 100

Inclination angle (degree)

Fig. 9 Average solid concentration and relative velocity between gas and particles as a function of inclination angle at wp=1.31 kg/m3 in Case I.

Fig. 10 Schematic illustration of forces acting on a particle assembly in pneumatic conveying.

39



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Page 40 of 45

(a)

(b) Fig. 11 Average volumetric particle-fluid force, gravity and particle-wall friction force as a function of: (a), inclination angle at wp=1.31 kg/m3 in Case I; and (b), gas velocity in Case II.

40


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(a)

(b) Fig. 12 Comparison of energy loss obtained macroscopically and microscopically in: (a), Case I; and (b), Case II.

41



10 Ep-f Ecollision Eg

8

Ef-w

-5

Energy (10 J)

Ef-f Ek,v

6

Ek,ω

4

2

0 0

20

40 60 80 Inclination angle (Degree)

100

(a) 40

Ed,p-p Es,p-p Er,p-p

-6

Energy (10 J)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 45

Ec,p-p

30

Ed,p-w Es,p-w Er,p-w Ec,p-w

20

10

0

0

20


100

(b) Fig. 13 Energy dissipations as a function of inclination angle at wp=1.31 kg/s in Case I: (a) major energy dissipations; and (b) energy dissipations due to particle-particle and particle-wall collisions.

42


Page 43 of 45

wp=0.655 kg/s wp=1.31 kg/s wp=1.965 kg/s

-5

Eg (10 J)

8

4

0 0

20


100

(a) 12 10 8 -5

Ef-w (10 J)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6 wp=0.655 kg/s wp=1.31 kg/s

4

wp=1.965 kg/s

2 0

0

20


100

(b)

(continued)

43



-5

Ep-f (10 J)

12

8

4 wp=0.655 kg/s wp=1.31 kg/s

0

wp=1.965 kg/s

0

20


100

(c) wp=0.655 kg/s wp=1.31 kg/s

8

wp=1.965 kg/s

-5

Ecollision (10 J)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 44 of 45

4

0 0

20


100

(d) Fig. 14 Effects of solid flowrate on different energy dissipations: (a), gravitational potential energy; (b), fluid-wall viscous dissipation; (c), particle-fluid energy dissipation; and (d), collision energy dissipation.

44


Page 45 of 45

36 Ep-f Ecollision Eg Ef-w Ef-f

-5

Energy (10 J)

27

Ek,v

18

Ek,ω

9

0 5

10

15 20 25 30 35 40 4550 Gas velocity (m/s)

(a) 90 Ed,p-p Es,p-p

80

Er,p-p

-6

Energy (10 J)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Ec,p-p Ed,p-w

70

Es,p-w Er,p-w

30

Ec,p-w

20 10 0 5

10

15 20 25 30 35 40 45 50 Gas velocity (m/s)

(b) Fig. 15 Energy dissipations as a function of gas velocity in Case II: (a) major energy dissipations, and (b) energy dissipations due to particle-particle and particle-wall collisions.

45


GasâSolid Flow and Energy Dissipation in Inclined Pneumatic

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