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Algorithm design and the development of a discrete element method for simulating particulate flow and heat transfer

Algorithm design and the development of a discrete element method for simulating particulate flow... Keywords: DEM; algorithm; particulate flow; solid mechanics; heat-transfer model algorithm was originated by Cundall in 1971 [26] and ap- Introduction plied to simulate the behaviour of soil particles under dy- Granular materials in nature and particulate technology in namic loading conditions by Cundall and Strack in 1979 process industries are globally researched and developed [27]. The DEM algorithm is more efficient in quasi-static because the interactions among individual units in par - systems because it is capable of handling multiple particle ticulate systems are so complicated that the behaviours contacts. The particle deformation can be explicitly deter - of particulate-laden gases are quite different from those of mined by the interaction force of collisions. The transient conventional fluids. Thus, it is important for real interdis- contacting surface of particle-to-particle collisions can be ciplinary research on particulate systems to understand confirmed by the particle positions determined by the well- the mechanism at the microscopic level. established Newton’s laws of motion. That is to say, the With the rapid development of computer science and DEM algorithm is a powerful numerical method, in which technology, the discrete element method (DEM) [1 2, ] has the motion of each individual particle is determined by the been applied globally in recent years to gain knowledge of net force acting upon it. Another advantage of the DEM al- the microscopic mechanisms in particulate systems [3–6]. gorithm is that it can present more detailed information Thus, the soft-sphere model of the DEM has been conveni- on each particle, such as its position, trajectory, velocity ently applied to simulate grain processing in many fields and contact force, which are usually not easy to gain from of the processing industry, such as particle mixing/seg- experiments. In addition, the transient contacting surface regation [7–15] and screw transportation [16–18], and has can be modified to consider the heat-transfer area of two even been used to simulate the gas-solid two-phase flow colliding particles with different temperatures. Thus, the system in fluidization engineering coupled with computa- DEM algorithm coupled with thermodynamics and heat- tional fluid dynamics (CFD) [19–25]. transfer models can simulate the transient heat-transfer As a well-documented numerical tool and a prime or phenomena in a particulate system. Furthermore, the outstanding example of a dynamic algorithm, the DEM Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 143 Net force → Translation motion Total moment → Rotating motion The status analysis The kinematics analysis (Mechanics model) (Kinematical model) (Thermodynamic model) (Heat transfer model) Time driven: t = t + d t Particles attached in grids Searching technology Fig. 1: The foundational principle of the particulate three-dimensional soft-sphere model of the DEM for computing particle flow and heat transfer particulate macro-chemical-reaction kinetics also can be and metal particles heating in a calciner. Finally, to prepare introduced to the kinetics-analysis program to describe the the simulation of the coal-pyrolysis process in a downer variance of density or size at the particle scale. Therefore, reactor using the developed DEM algorithm in the next the soft-sphere model of the DEM algorithm can provide step, some necessary physical-property parameters of the an effective numerical method to simulate the particle be- fuel particles of coal and the heat-carrier particles of sand haviour of flow and heat transfer in a particulate system. have been validated and confirmed through comparing the To utilize the DEM algorithm to serve the simulation of experiments of particle filling in the tube, particle heaping particulate processes extensively and openly, for example, on the platform and particle flowing through an hourglass the mixing and heat transfer of coal and sand during the for timing. coal-pyrolysis process in a downer reactor or screw reactor, an integrated DEM program package has been developed 1 Development of the DEM algorithm and presented in detail. For one thing, the detailed struc- 1.1 Hypothesis and principle ture of the DEM algorithm, such as the searching algorithm for determining the particle collisions, has not been intro- In the DEM algorithm, each element can be considered as duced in previous publications. For another, the parameter one rigid body with a fixed shape, such as a sphere, a billet values are meaningless if the contact-mechanics model or a pyramid. A  little overlap is allowed to happen when or heat-transfer model is unknown or undocumented in two elements contact each other, which is calculated by some simulation tools. The contact-mechanics model and a force-interaction model. As a dynamic algorithm, the heat-transfer model should be stated clearly and the phys- status of all elements is unchanged in each time step, such ical meaning and values of the parameters should be well as force analysis and thermodynamics analysis. The foun- understood. dational principle or big picture of the DEM algorithm is During the algorithm design of the DEM, a new method described in Fig. 1. for cubic-grid formation and connection was developed, As shown in Fig. 1, the DEM algorithm, as a Lagrangian especially for application to irregularly shaped simula- method for modelling the individual trajectory of each ele- tion domains, which can save the overhead and computa- ment, can be described as a dynamic algorithm, whose tion time of systems. The particle-dynamics model in the structure consists of three parts: the status analysis, the kinematics analysis of the DEM program has been supple- kinematics analysis and an algorithm for the prediction mented by the particle-rotation model controlled by the of particle collisions. In the particle-status analysis, the rigid-body rotation’s law to simulate the angle of repose particle-mechanics model calculates the particle-force of a particle pile accurately. Most importantly, to simu- analysis and the particle-torque analysis, and the particle- late coal pyrolysis in a downer reactor, the particle-heat- thermodynamics model confirms the particle temperature transfer model and particle-thermodynamics model have according to the heat balance of one single particle. In the been coupled to the developed DEM algorithm. Second, the particle-kinematics analysis, the particle-dynamics model correctness and reasonability of the developed DEM al- takes charge of simulating the particle position and the gorithm have been validated and checked through simu- velocity of the particle in the next time step, and the heat- lating the conditions compared with experiments. In this transfer model can calculate the quantity of heat transfer research, the mechanics model for particle-status analysis between particles or from a particle to its surroundings. For and the heat-transfer model for particle-kinematics ana- a large number of particles in the system, it is a major task lysis of the developed DEM algorithm have been validated to detect the particle collisions and to calculate the contact by an experiment with glass beads discharging from a silo forces acting on the particle. A  search algorithm based on Searching Association Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 144 | Clean Energy, 2021, Vol. 5, No. 2 cubic grids for particle collisions is adopted in the whole In the Hertz theory, the normal stiffness coefficient k is program to realize the particulate three-dimensional soft- associated with the physical parameters of particles, such sphere model of the DEM on the computer more efficiently. as elastic modulus E, Poisson’s ratio ν and the particle ra- The whole calculation process of the dynamic algorithm of dius of two colliding spherical particles. e is the normal- the three-dimensional soft-sphere model of the DEM will not unit vector of the contact surface, which can be obtained be finished until the calculation step approaches the end. by the position vectors of two colliding particles as x − x j i 1.2 Particle-status analysis e = (3) x − x¯ j i Each particle has its own status parameters, such as The calculation method of the normal-unit vector can be the space-position vector x =(x , x , x ), phase-position 0 1 2 obtained from Fig. 2. The normal damping coefficient η vector ϕ =(ϕ , ϕ , ϕ ), translational-movement-velocity 0 1 2 can be calculated by [29]: vector v =(v , v , v ), rotating-movement-velocity vector 0 1 2 ω =(ω , ω , ω ), net-force vector F =(F , F , F ), total- 0 1 2 0 1 2 F = −η · v dn,ij n,ij n,ij  √  ∗ 1 moment vector M =(M ,M ,M ), particle temperature T  m k ·ln 0 1 2 n.ij ( ) p e η = 2 · n,ij and its enthalpy value H . All the status parameters of each 2 1  π + ln p [ ( )] m m (4) particle simulated in the DEM algorithm should be deter - ∗ i j m = m +m i j mined at the end of each time step. v  −ev n n,0 v = −v = − v · e e n,ij n,ji ji n n 1.2.1 Mechanics model where the reduced mass m can be calculated by two col- Normal-contact-force model. liding particles; e represents the recovery coefficient ac- It is very important to calculate the net force and total moment cording to the initial velocity and the rebound velocity of of each particle when particles contact each other, as illustrated one particle; and v is the normal relative velocity of par - n,ij in Fig. 2, because accurate particle motion is determined by the ticle i relative to particle j. exact particle-mechanics model introduced in this section. To calculate the normal contact force acting on each Tangential-contact-force model. particle, the Hertz theory [28] is applied to obtain the rela- Vemuri theory [30] is employed to obtain the tangential tionship between the normal contact force and the particle interactions between the colliding particles, which can be deformation, which is summarized as expressed as follows: F = F + F (1) n,ij cn,ij dn,ij (5) F = F + F t,ij ct,ij dt,ij F = −k · δ · e  cn,ij n,ij n However, the calculation method of the tangential force be- n,ij  √ 4 ∗  ∗ k = E r tween two colliding particles is complicated because the n,ij 2 (2) physical parameters cannot be tested easily in experiments. 1−υ 1−υ 1 j  i  = +  E E E i j Therefore, the following methods [3132 , ] are used to calcu-  1 1 1 = + r r r late the tangential force between the colliding particles: i j  ï ò ¶ © 3/2    δ t,ij F = −μ F  1 − 1 − min ,1 e ct,ij s cn,ij t  t,max (6) δ = v · Δt t,ij t,ij 2−ν δ = μ δ t,max s n,ij 2(1−ν ) →→ i τ where μ is the coefficient of side friction; F is the same cn,ij as mentioned above as the normal contact force of two col- ji → lision particles; δ is the relative displacement of two colli- j t,ij sion particles at contact time t; and Δ e is the unit tangential ji vector of the contact surface, which can be obtained by j τ ν ji τ → v − v · e e i n n ji ji ν e = (7) →→ v − v · e e ji ji n n → X – X j i e = – – The calculating method for the normal-unit vector can be X X j i →→ → → obtained from Fig. 2. η is the tangential damping coeffi- t, ij ν – ν •e ji ji n () cient, which can be calculated by e = →→ →→ – ν •e e 1 ν Å ã ji n n ji ()  δ t,ij  ∗ F = −η 6m μ F  1 − /δ v s t,max  dt,ij t,ij cn,ij t,ij t,max (8) Fig. 2: The solutions of unit normal vector and unit tangential vector 2 η = η t,ij n,ij during the contact collision of two spherical particles with different v = −v = − v · e e = − v × e × e t,ij t,ji ji t t ji n n diameters Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 145 According to the assumption mentioned above, the accel- where v is the tangential relative velocity of particle j t,ji eration of the particle motion in a small-enough time step relative to particle i. is a constant and the angle of acceleration of the particle Conservative forces, such as gravity or electromagnetic rotation in a small-enough time step is also a constant. forces, acting on the centre of a homogeneous spherical The velocity vector of the particle translational movement particle cannot cause the particles to rotate. However, v and the angular velocity vector of the particle rotation the line of total contact force acting on the particle - sur ω at the next time step can be expressed by the first-order face does not go through the centre of a spherical particle, Taylor series expansion at t = t . The spatial-position which makes the spherical particle rotate. The total rota- vector of particle x and the phase-position vector of par - tion moment is composed of the tangential contact torque ticle ϕ at the next time step can be derived by the second- and the rotational damping moment: order Taylor series expansion at = t t : (9) dv (t ) M = M + M . .. (9) i 0 i t,i r,i v (t)= v (t )+ · Δt i i  dt (15) dx (t ) d x (t ) M = r × F i 0 1 i 0 2 t,i i t,ij x (t)= x (t )+ · Δt + · · Δt 0 2 i i dt 2! dt (10) μ F ω r| cn,ij| ij M = − r,i | | ij dω (t ) i 0 ω (t)= ω (t )+ · Δt i i dt 2(16) where r is the vector of the radius of particle i, whose dir - dϕ (t ) d ϕ (t ) i i 0 1 i 0 2 ϕ (t)= ϕ (t )+ · Δt + · · Δt 0 2 i i dt 2! dt ection is from the centre of the spherical particle to the The particle-kinematics equations of translational motion contact point of collision; F is the tangential component t,ij and rotational motion can be discretized into a Verletleap- of the contact force from particle j to particle i; and is the μ frog scheme as in Equations (16) and (17): damping coefficient of the rolling friction. −→ a (t )= F (t )  i 0 i 0 −→  Δt Δt 1.2.2 Thermodynamics model v t + = v (t )+ a (t ) ·  i 0 i 0 i 0 2 2 In this research, the enthalpy of a solid particle is assumed Δt x (t + t)= x (t )+ v t + · Δt(17) i 0 i 0 i 0 to be a function of the temperature, i.e. the enthalpy H of  −→ a (t +Δt)= F (t +Δt) 0 0 i i  m a particle at temperature T can be calculated from that at  −→ Δt Δt v (t +Δt)= v t + + a (t +Δt) · i 0 i 0 i 0 2 2 the specified temperature T , i.e.: −→ j (t )= M (t )  i 0 i 0  i H (T)= H (T )+ m · c dT(11) 0 p p   −→  Δt Δt ω t + = ω (t )+ j (t ) ·  i 0 i 0 i 0 2 2 Δt (18) ϕ (t + t)= ϕ (t )+ ω t + · Δt i 0 i 0 i 0 where m is the mass of the particle; and is the heat cap c -  −→ p p   1  j (t +Δt)= M (t +Δt)  i 0 i 0 acity of the particle at constant pressure, which can be cal-  i −→  Δt Δt ω (t +Δt)= ω t + + j (t +Δt) · culated as c = a + bT + cT . Therefore, the enthalpy change i 0 i 0 i 0 2 2 ΔH of the particle can be calculated as Å ã Ä ä Ä ä 1 1 3 3 2 2 (12) ΔH = m · c T − T + b T − T + a (T − T ) p 0 1.3.2 Heat-transfer model 0 0 3 2 The calculation method for the heat transfer between two –1 –1 –1 –2 –1 –3 where parameters a/kJ· kg  ·K , b/kJ· kg  ·K and c/kJ ·kg  ·K colliding particles or between a particle and a wall with are constants collected in the thermodynamics hand- different temperatures can be obtained by the theoretical books. Therefore, the change in particle temperature can T analysis of Fourier’s law, which can be described as follows: be calculated from the enthalpy difference Δ thr H ough the heat balance of particle i: ΔT ij Q = (19) ΔH i α A c,ij c,ij T = T + (13) i i,0 m · c i p,i where |ΔT | is the absolute temperature difference be- ij tween two colliding particles or between a particle and a wall standing for the impetus of heat transfer through 1.3 Particle-kinematics analysis thermal conduction; A is the area of the thermal conduc- ij 1.3.1 Dynamics model tion between two colliding particles or between a particle The translational movement of the particle is driven by the and a wall; α is the contact coefficient of thermal con- c, ij net force acting on it, and accords with Newton’s Second duction; and the factor of stands for the resistance α A c,ij c,ij Law of Motion. The rotating movement of the particle is of heat transfer. It is worth noting that the contact coeffi- driven by the total moment acting upon it and is described cient of thermal conduction is affected by the roughness of by Newton’s Law of Rotation: the contacting surface, the acting force on the contacting 2 surface, the gas pressure in the gap between two colliding −→ dv d x i i F = F + F + F + F + m g = m a = m = m i cn,ji dn,ji ct,ji dt,ji i i i i i dt dt contact surfaces, etc. It cannot be tested by experimenta- −→ dω d ϕ i i M = M + M = I j = I = I i t,i r,i i i i i dt dt tion for most engineering materials except for some highly polished metals. Therefore, the calculation method for (14) Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 146 | Clean Energy, 2021, Vol. 5, No. 2 in Fig. 3a–c. At this time, the heat transfer through thermal conduction is only controlled by the overlap of the gas r r r 1 2 2 layers of the two particles, whose thermal resistance R can be calculated as follows: = dS l =+ r r + l =+ r r 12 1 2 12 1 2 R Δ l g Π (a) (d) 2πr sin θ = k d (r sin θ) g i l − 2r cos θ a a 12 i a 1 2 β β 1 2 sin θ cos θ = 2k πr dθ l − 2r cos θ r ++ r l r r + δ l r + r << < 1 2 12 1 2 12 1 2 − cos α l i 2r (b) (e) i = k πr (cos α − 1)+ ln (i = 1, 2) i i R 2r i − 1 g1 2r R R R R R (20) s1 g s2 s1 s2 where k is the thermal-conduction coefficient of the gas g2 layer; and Δl is the distance between two relevant particle (c) (f) surfaces along the direction of the centre line in metres. Fig. 3: Illustration of the calculation of the thermal resistance of α is illustrated in Fig. 3b and can be calculated as follows: thermal conduction between two colliding particles Ç å 2 2 l + r − (r + δ) −1 12 i α = cos (i = 1, 2)(21) 2r l the contact coefficient of thermal conduction is the core A similar calculation method can be applied to the thermal of the modelling of particulate thermal conduction. The resistance between a particle and a wall with different equivalent thermal-resistance model of the heat transfer temperatures through overlapped gas layers, as follows: between contact particles with different temperatures is  Ñ é illustrated in Fig. 3. l pw − cos α 1 l i pw   As shown in Fig. 3, the particle can be assumed to = 2k πr (cos α − 1)+ ln (22) i i pw R r g i − 1 be surrounded by a thin gas membrane with thick- i ness δ, which is much less than the particle diameter where l presents the distance between the particle centre pw d . According to the conclusion of Delvosalle from the and the wall surface along the direction of the normal measurement in experiment [33], the thickness of the vector of the wall surface. assumed gas layer can be considered as 0.1 times the (ii) The thermal conduction will be enhanced when two particle diameter, i.e.δ = 0.1d . Thus, the thermal con- particles with different temperatures collide with each duction between two colliding particles with different other, i.e. l ≤ r + r . The heat can be conducted through 12 1 2 temperatures is affected by the thin gas layer around the not only the overlapped gas layers, but also the tiny gas particle. Further assumption should be supplemented to interval d between the contacting surfaces of two par - gap build up the model of particulate thermal conduction. ticles with different temperatures. Thus, the thermal re- First, the contacting surface is smooth, whose normal sistance is composed of two parts: the thermal resistance vector is parallel with the centre line of the two colliding of the tiny gas interval R and the thermal resistance of g1 particles, the direction of the heat transfer through the annular gas layer R. Therefore, the thermal resistance g2 thermal conduction. Second, the area of the contacting under this condition can be calculated as follows: surface is equal to that of the overlap circle based on 1 1 d k gap g + = + dS the assumption of DEM modelling. Third, there is a min- R R k A Δ l g g g gap Π 1 2 i –10 ˆ i imum tiny gap interval with a thickness of 4 × 10 m −10 4.0 × 10 2πr sin θ = + k d (r sin θ) g i between the contacting surfaces. Last, the thermal con- l − 2r cos θ k π(r sin β ) 12 i g i i duction can be transferred through the internal min- ˆ i −10 4.0 × 10 sin θ cos θ = + 2k πr dθ imum tiny gap and the annular overlap of the gas layer g l − 2r cos θ k π(r sin β ) 12 i g i i around it. Therefore, the heat transfer through thermal −10 12 − cos α 4.0 × 10 l 2r 12 i conduction between two particles with different temper - = + k πr (cos α − cos β )+ ln (i = 1, 2) g i i i l12 2r − cos β k π(r sin β ) i g i i i 2r atures can be discussed from two conditions, as follows. (23) (i) The heat transfer based on thermal conduction will where α and β (illustrated in Fig. 3e) can be calculated as follows: i i not occur when two particles are far away from each other, 2 2 2 l +r −(r +δ) −1 12 i i.e. l ≥ r + r + δ. l is the distance between two particle α = cos 12 1 2 12 i 2r l i 12 Ä ä (i = 1, 2)(24) centres; r and, r are the radii of two particles separately. −1 l  12 1 2 β = cos 2r However, thermal conduction will not begin until the A similar calculation method can be applied to the overlap of the gas layers of two particles with different tem- thermal resistance between a particle and a wall with peratures appears, i.e. r + r < l < r + r + δ, as illustrated 1 2 12 1 2 Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 147 different temperatures through the gap interval and In a cuboids space with a constant volume, the scale the annular overlapped gas layers during the contact of one cubic unit determines the distribution density of of a particle with a wall with different temperatures as the cubic grids. The scale of one cubic unit is determined follows: by its side length. The distribution density of cubic grids  Ñ é in a constant volume can be determined by the particle- lpw −10 − cos α 1 1 4.0 × 10 pw i   + = + 2k πr (cos α − cos β )+ ln size distribution and the density of the particulate phase. i i i 2 l pw R R r g g i 1 2 k π(r sin β ) − cos β g i i i Generally, a small scale for the cubic unit is suitable for (25) the dense phase of particle flow and a larger side length of the cubic unit is acceptable with the dilute particle flow. where l presents the distance between the particle centre pw According to the search algorithm based on cubic grids, and the wall surface along the direction of the normal the side length of the cubic unit is usually larger, but not vector of the wall surface. too much larger, than the maximum particle diameter, i.e. As known to all, the thermal flux can be also affected by l > d . box p, max the solid resistance of the particle’s interior, which can be Each cubic unit has its own serial number (ID). As pre- calculated as follows: sented in Fig. 4, each current cubic unit coloured in pink has 1 1 1 its own 26 neighbours including the 8 units in the same layer R = − s (26) 2πk r r p,i i,m i (NW, N, NE, E, SE, S, SW, W), the 9 units in the upper layer (U, UNW, UN, UNE, UE, USE, US, USW, UW) and the 9 units in where k is the thermal-conduction coefficient of the par - the lower layer (D, DNW, DN, DNE, DE, DSE, DS, DSW, DW). ticulate material; r presents the radius of the particle; and –1/3 However, it is obvious that half of the neighbours of one cubic r = r . The spherical surface a radius of r divides the i, m i m unit should be recorded in order to save computer-memory internal energy of the particle equally. space, because the neighbour relationship is mutual. For ex- Above all, the total thermal resistance of heat transfer ample, cubic unit U is the neighbour of the current box unit through thermal conduction between two particles with coloured in pink and the current box unit is also the neigh- different temperatures can be calculated as follows: bour of cubic unit U at the same time. When the neighbours +∞l > r + r + δ||l > r + δ 12 1 2 pw i of cubic unit U are searched, the current box unit coloured in R + R + R r + r ≤ l < r + r + δ||r ≤ l < r + δ s g s 1 2 12 1 2 pw pink can be omitted, because cubic unit U has been searched R = 1 2 i i total  R R g g 1 2  when the cubic unit coloured in pink is treated as the cur - R + + R l < s s 12 r + r ||l < r 1 2 1 2 pw i R +R g g 1 2 rent one. Therefore, those in the 26 neighbour units whose (27) ID numbers are larger than (or less than) that of the current cubic unit can be considered as the neighbours recorded in the data structure of the current one. In this research, the 1.4 Search algorithm based on cubic grids uncoloured neighbours whose ID numbers are less than that of the current cubic unit coloured in pink (D, DNW, DN, DNE, 1.4.1 Formation of cubic grids DE, DSE, DS, DSW, DW, W, SW, S, SE) are omitted to save calcu- Detecting colliding particles is an important component lation time. In other words, the spatial structure consisting of the algorithm of the three-dimensional soft-sphere of cubic units can be built up and described by the 13 neigh- model of the DEM, which makes up a large percentage of bours of the current unit coloured in pink (NW, N, NE, E, U, the computation time. In the particle-status analysis, cal- UNW, UN, UNE, UE, USE, US, USW, UW) in the memory of the culation of the total contact force acting on one particle computer and the criteria for the formation rule mentioned requires calculation of the contact forces from all the par - above have been listed in Table 1. ticles colliding with it. Therefore, it is essential to deter - In Table 1, ID_BOX indicates the series number of the mine all particles that collide with the current one at each cubic unit; COLUMN shows the column number of the time step. To save calculation time, a series of search algo- cubic unit in one layer of the whole cuboids space; LAYER rithms have been developed and applied in the soft-sphere represents the number of the cubic number in one layer of model of the DEM, such as the Neighbour List Method [34], the whole cuboids space. If one of the recorded neighbours Bounding Box Method [35, 36] and Boxing Method [36]. In does not exist, the linker field of the corresponding neigh- this research, a search algorithm based on cubic grids has bour of the current cubic will be set as NULL. been developed to execute the task of determining particle collisions in the three-dimensional soft-sphere model of Splicing cubic grids between two cuboids spaces. the DEM. The number of cubic units in the cubic grids seriously af- fects the memory overhead and the calculation speed of Cubic-grid formation in a cuboids space. the computer. The spatial domain of the particle motion Before the calculation program of the three-dimensional is usually an irregular one, i.e. the space of particle mo- soft-sphere model of the DEM is executed, the entire cu- tion is not usually a cuboids space. If the cubic grids are boids space of the particle motion will be meshed into a meshed all at once in a larger cuboids space involving an series of cubic units, as shown in Fig. 4. Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 148 | Clean Energy, 2021, Vol. 5, No. 2 (k + 1) layer (k+1) layer k layer (k–1) layer k layer UNW UN UNE UW UE (k–1) layer USW US USE NW N NE W E SW SSE DNW DN DNE DW DDE DSW DS DSE Fig. 4: The structure of cubic grids and the illustration of 26 neighbours of one cubic unit irregular shape space of particle motion, a large number subspaces of particle motion. Each cubic unit in the frac- of redundant cubic units will be generated in the memory ture surface of segment part W is connected with UNE, of the computer and a large quantity of computational UE, E, NE and USE as neighbours from segment part E, time will be wasted. Therefore, an irregularly shaped space and the one in the fracture of segment part E is needed can be divided into several regularly or quasi-regularly to link to USW, UW, UNW and NW as neighbours from shaped subspaces, in which the cubic grids can be meshed. part W. The third cutting direction can be perpendicular Therefore, the whole irregularly shaped space of particle to S–N and the subspaces of segment part S and seg- motion can be recombined through splicing the cubic grids ment part N are separated from the irregularly shaped in the regularly or quasi-regularly shaped space of particle space of particle motion. All the cubic units in the frac- motion together, as described below. ture surface of segment part S are attached to UNW, UN, The design of the splicing algorithm for cubic grids UNE, NW, N and NE as neighbours from segment part is completely determined by the partitioning of the ir - N, and those of segment part N are needed to adjoin regularly shaped space of particle motion that can be USW, US and USE as neighbours from segment part S. It segmented along the direction vertical to D–U into two can be concluded from these three segmentation cases parts: the segment part U and the segment part D. Each that nine neighbour relationships should be connected. cubic unit in the fracture surface of segment part D is However, the first segmentation case is beneficial for the spliced with UNW, UN, UNE, UW, U, UE, USW, US and design of the splice algorithm, because the cubic units USE as neighbours from the segment part U, and the on the fracture surface of only one segment part D are one in the fracture surface of segment part U does not matched with the neighbours from segment part U. The need to be spliced with any cubic unit in segment part algorithm can be described as below. D according to the cubic-grid-formation rule mentioned In the fracture surface, the series number of the first in the ‘Cubic grids formation in a cuboids space’ section. If cubic unit in segment part D is begin ; step and step are 0 E0 N0 the segmentation direction vertical to W–E is adopted, count steps along the direction of E and N in segment the segment part W and segment part E are the two part D. Similarly, the series number of the first cubic unit Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 149 Table 1: The criteria for the existing 13 neighbours of the cur - Table 2: The criteria condition of the splice cubic unit be- rent cubic grid in the cuboids space of particle motion tween two segment parts of an irregularly shaped space of particle motion Neighbours Criteria Neighbours Criteria E ①[(ID_box + 1)/column]==[ID_box/column] N ①[(ID_box + column)/layer]==[ID_box/layer] UE ①[(obj1 + stepE1)/stepN1]==[obj1/stepN1] NE ①[(ID_box + 1)/column]==[ID_box/column] UW ①[(obj1-stepE1)/stepN1]==[obj1/stepN1] ②[(ID_box + column)/layer]==[ID_box/layer] ②(obj1-stepE1)>=begin1 + i × stepN1 NW ①[(ID_box-1)/column]==[ID_box/column] UN ①[(obj1 + stepN1)/ ②[(ID_box + column)/layer]==[ID_box/layer] (lengthE× lengthN)]==[obj1/ ③(ID_box-1)>=0 (lengthE× lengthN)] U ①(ID_box + layer)<=end US ①[(obj1-stepN1)/(lengthE× lengthN)]==[obj1/ UE ①(ID_box + layer)<=end (lengthE× lengthN)] ②[(ID_box + layer + 1)/ ②(obj1-stepN1)>=begin1 + i × stepN1 column]==[(ID_box + layer)/column] UNE ①[(obj1 + stepE1)/stepN1]==[obj1/stepN1] UW ①(ID_box + layer)<=end ②[(obj1 + stepN1)/ ②[(ID_box + layer-1)/ (lengthE× lengthN)]==[obj1/ column]==[(ID_box + layer)/column] (lengthE× lengthN)] UN ①(ID_box + layer)<=end UNW ①[(obj1-stepE1)/stepN1]==[obj1/stepN1] ②[(ID_box + layer + column)/ ②(obj1-stepE1)>=begin1 + i × stepN1 layer]==[(ID_box + layer)/layer] ③[(obj1 + stepN1)/ US ①(ID_box + layer)<=end (lengthE× lengthN)]==[obj1/ ②[(ID_box + layer-column)/ (lengthE× lengthN)] layer]==[(ID_box + layer)/layer] USE ①[(obj1-stepN1)/(lengthE× lengthN)]==[obj1/ UNE ①(ID_box + layer)<=end (lengthE× lengthN)] ②[(ID_box + layer + 1)/ ②(obj1-stepN1)>=begin1 + i × stepN1 column]==[(ID_box + layer)/column] ③[(obj1 + stepE1)/stepN1]==[obj1/stepN1] ③[(ID_box + layer + column)/ USW ①[(obj1-stepE1)/stepN1]==[obj1/stepN1] layer]==[(ID_box + layer)/layer] ②(obj1-stepN1)>=begin1 + i × stepN1 UNW ①(ID_box + layer)<=end ③[(obj1-stepN1)/(lengthE× lengthN)]==[obj1/ ②[(ID_box + layer-1)/ (lengthE× lengthN)] column]==[(ID_box + layer)/column] ③[(ID_box + layer + column)/ [] is floor operation. layer]==[(ID_box + layer)/layer] USE ①(ID_box + layer)<=end segmentation direction. Additionally, eight neighbours in ②[(ID_box + layer + 1)/ segment part N of the current cubic unit in segment part column]==[(ID_box + layer)/column] U can be gained according to the criteria list in Table 2. The ③[(ID_box + layer-column)/ neighbours will be NULL if the conditions are not satisfied layer]==[(ID_box + layer)/layer] with the criteria listed in Table 2. USW ①(ID_box + layer)<=end ②[(ID_box + layer-1)/ column]==[(ID_box + layer)/column] 1.4.2 Association between particles and cubic grids ③[(ID_box + layer-column)/ The program of the three-dimensional soft-sphere model layer]==[(ID_box + layer)/layer] of the DEM will launch the dynamic mechanism after the formation of cubic grids. In each time step, all particles are [] is floor operation. associated with the corresponding cubic units according to their spatial-position vectors, in which particle collisions in segment part U is begin1; stepE1 and stepN1 are count will be determined. In this section, two problems are ad- steps along the direction of E and N in segment part U. If dressed: (i) how to find the series number of the cubic unit the fracture is a rectangle with row lengthN and column with which the current particle is associated; (ii) how to lengthE, the splice units on the two fracture segment parts record or remark the associations between particles and are obj and obj , which can be calculated as: cubic units. 0 1 obj = begin + j × step + i × step (i = 0, 1, . .. , length ) 0 0 E0 N0 N−1 Mapping from the position of a particle to the series number of obj = begin + j × step + i × step ( j = 0, 1, . .. , length ) 1 1 E1 N1 E−1 the cubic unit. (28) It is not difficult to imagine that a particle can be assigned For the same value of i and j for the cubic grids of two frac- to the corresponding cubic unit according to the pos- ture surfaces, the corresponding series number of cubic ition of the particle. The mapping from the position of the units obj and obj in two segment parts are just the two particle to the series number of the corresponding cubic 0 1 splice units, also a neighbour relationship, along the D–U unit can be built up based on the cubic grids created, as Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 150 | Clean Energy, 2021, Vol. 5, No. 2 described in Section 1.4.1. In a regularly shaped space of The particles are associated with cubic grids all over particle motion, the series number of the cubic unit is in- again in each new time step because of the change in the creased from the W direction to the E direction stepped by spatial position of all particles. To avoid influence from 1, from the S direction to the N direction stepped by the the last association between particles and cubic grids, all column number, and from the D direction to the U direc- pointers of a particle type in the data structure of particles tion stepped by the layer number. Therefore, the mapping and cubic units should be set as NULL before the associ- from the position of the particle to the series number of ation mechanism of the next time step is launched. When the cubic unit can be built up as: the series number of the cubic unit is calculated from Equation (29) according to the spatial position of the par - (x − x ) (x − x ) (x − x ) 0 a 1 b 2 c ID = + n + n(29) box column layer ticle, the particle unit will be linked to the pointer of the l l l a c particle type in the cubic-unit data structure. If there are where (x , x , x ) is the position of the vertex of units D, W, a b c some other particles associated with the same cubic unit, and S of the first cubic unit in the cubic grids whose series a chain table of particle units whose heads are linked to number is 0; l , l and l are the three side lengths of a cubic a b c the associated cubic unit’s pointer of particle type should unit; n stands for the column number in each layer of column be formed in ascending order of the series number of cubic units; and n represents the number of cubic units layer the particle unit. The flow sheet of the pseudo code is in each layer of the cubic grids. presented as Fig. 6. An example has been prepared for illustrating the as- Algorithm of association between the particles and the cubic sociation between the particles and the cubic units, as grids. shown in Fig. 7. The data structures of the particle and the cubic unit are When the subroutine that records the association be- illustrated in Fig. 5. The data structure of the particle not tween particle units and cubic grids is launched, a tree struc- only records the basis information of the particle, such ture like that in Fig. 7 will be generated and updated in the as the radius, density, mass, rotational inertia, position computer memory, which is beneficial for the realization vector, translational-velocity vector, rotating-velocity and design of the algorithm for detecting particle collisions. vector, force vector and moment vector, but also in- As represented in Fig. 7a, the cubic unit of 628 contains par - cludes a pointer member of the particle type. When one # # # # # # # ticle units of 428, 432 , 620 , 656 and 697 ; those of 617 , 751 cubic unit contains more than one particle, the particle- # # # and 758 are associated with the 629 cubic unit; the 630 type pointer field will link other particle units to form cubic unit involves only the 302 particle unit. The records of a chain table in a series of increasing particle numbers. the association between the particle units and cubic grids in The structure of the cubic unit has the pointer members the computer memory can be shown as in Fig. 7b . It is clear of 13 neighbours of the cubic-unit type and a pointer that the series number of particle units in the chain table member of the particle type. The pointer member of the is increased from head to tail. It is worth noting that the particle type is linked to the particle unit of the min- # # 629 cubic box unit is the east (E) neighbouring unit of 628 , imum series number of all particles matched with this and the 628 cubic unit is the west (W) neighbouring unit of current cubic unit. 629 . The west (E) neighbouring unit is invalid according to the principle mentioned in Section 1.4.1.Therefore, the 628 cubic unit is not linked to one of the 13 neighbour pointer # # # fields of 629 . The same principle acts on 629 and 630 . ID r m 1.4.3 Algorithm for detecting particle collisions den I *next ID *p[0] *p[1] A particle is associated with the corresponding cubic unit according to the spatial coordinate values of its centre. The cubic grids will be beneficial for detecting particle collisions x[0] x[1] x[2] *p[2] *p[3] *p[4] efficiently. It is not difficult to imagine that two particles close to each other may be associated with the same cubic v[0] v[1] v[2] *p[5] *p[6] *p[7] units or two neighbouring cubic units separately, as shown in Fig. 7a. Therefore, the algorithm for detecting particle w[0] w[1] w[2] *p[8] *p[9] *p[10] collisions has two steps. The particles associated with the cubic unit containing the current particle (in the case of a cubic unit containing more than one particle) should be as- F[0] F[1] F[2] *p[11] *p[12] *next sessed to determine whether they will collide with the cur - rent particle. In addition, the particles associated with the (a) particle (b) cubic unit M[0] M[1] M[2] neighbours of the cubic unit containing the current particle should be also assessed as to whether they will collide with the current particle or not. The flow sheet of a pseudo code Fig. 5: The data structure of a particle and cubic unit in the three- dimensional soft-sphere model of the DEM for detecting particle collisions is presented as Fig. 8. Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 151 Sub begin boxno=boxnof(ball[balli].x0); //calculate the series number of the cubic box unit matched with the current particle head=&box[boxno]; //get the head pointer of the chain table of particle unit node no head->next==NULL yes head->next =&ball[balli]; _pball=head->next; //link the current particle to the pointer field of //get the pointer of the first particle node of the the corresponding cubic box unit directly chain table yes _pball->next==NULL no _pball->next =&ball[balli]; _pball=_pball->next; Sub end //link the current particle to the pointer field of //move the pointer to the next particle node the end of the chain table of particle unit Fig. 6: The algorithm for recording the association between particles and cubic grids It will be helpful to take the case as shown in Fig. 7b as an ball example. When the 428 particle is the current particle, the 428 432 620 656 697 box # particle units associated with the same cubic unit of 628 628 ĂĂ ĂĂ ĂĂ ĂĂ ĂĂ will be searched along the chain table after the current par - *next *next *next *next *next *next # # # ticle unit of 428 first, which are the particle units 432 , 620 , *p[0] ĂĂ # # ball 656 and 697 . Afterwards, the particle units associated with *p[1] ĂĂ 617 751 758 # # box the 629 cubic unit, the east neighbour of 628 containing *p[2] ĂĂ 629 ĂĂ ĂĂ ĂĂ # the current particle unit of 428 , will be searched along the *p[3] ĂĂ *next *next *next *next chain table to assess whether they collide with the 428 par- # # # *p[4] *p[0] ĂĂ ĂĂ ticle or not, which are particle units 617 , 751 and 758 . ball *p[5] *p[1] ĂĂ 302 However, it is not necessary to search the particles asso- box *p[6] ĂĂ *p[2] ĂĂ 630 ĂĂ ciated with the same cubic unit whose series numbers are *p[7] ĂĂ *p[3] ĂĂ *next *next less than that of the current particle, or whose positions in the chain table are in front of the current particle. As shown *p[8] ĂĂ *p[4] ĂĂ *p[0] ĂĂ # # # in Fig. 7b, particle units 620 , 656 and 697 will be searched *p[9] *p[5] *p[1] ĂĂ ĂĂ when particle 432 is the current one. Thus, the particle unit *p[10] ĂĂ *p[6] ĂĂ *p[2] ĂĂ # # of 428 will not be searched because particle 432 has al- *p[11] *p[7] *p[3] ĂĂ ĂĂ ĂĂ ready been searched when 428 was the current particle. *p[12] ĂĂ *p[8] ĂĂ *p[4] ĂĂ The conclusion is that the current particle does not have *p[9] ĂĂ *p[5] ĂĂ to be judged as to whether or not it will collide with all *p[10] ĂĂ *p[6] ĂĂ other particles in the space of particle motion. Therefore, *p[11] *p[7] ĂĂ ĂĂ the computation time spent on detecting particle colli- *p[12] *p[8] ĂĂ ĂĂ sions can be saved through introducing the search algo- (a) *p[9] ĂĂ (b) rithm based on cubic grids. *p[10] ĂĂ *p[11] ĂĂ 628 629 630 1.5 Algorithm of the three-dimensional soft- 656 617 302 *p[12] ĂĂ sphere model of the DEM As mentioned in the discussion of the hypothesis Fig. 7: An example for the explanation of the association between par - ticles and cubic grids recorded in the computer memory and principle, the algorithm of the three-dimensional Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 152 | Clean Energy, 2021, Vol. 5, No. 2 Sub begin boxno=boxnof(ball[balli].x0); _pball=&ball[balli]; //calculate the series number of the cubic box //get the pointer of the current particle node of unit matched with the current particle the chain table yes ip=0; _pball->next==NULL //the first neighboring unit yes no ip<13 _pball=_pball->next; no //the 13 neighboring units //move the pointer to the next particle node yes no box[boxno].p[ip]==NULL Collision yes no yes head=box[boxno].p[ip]; Status analysis //get the pointer of the head particle node of (Mechanical model) the chain table (Thermal dynamics model) ip++; //the next neighboring unit Sub end Fig. 8: The algorithm of determining particle collisions in cubic grids soft-sphere model of the DEM contains three key parts: from the particles to the cubic grids should be found. The the particle-status analysis, the particle-kinematics tree structure describing the association between particle analysis and the search algorithm for particle collisions units and cubic units will be constructed and it will be up- based on cubic grids. The algorithm flow chart is illus- dated at each time step according to the new mapping be- trated in Fig. 9, the calculation process of which can be tween the particles and the cubic grids. presented as follows. (iv) The particle collisions will be searched based on the (i) Before the main program of the three-dimensional constructed tree structure describing the association be- soft-sphere model of the DEM is launched, the initializa- tween the particles and the cubic grids. The particle unit tion information of all particles will be scanned from an linked to the current cubic unit will be first searched and original data file, which is formatted by an initialization judged whether or not collisions with current particles will program or a backup of the results of the main program for occur along the chain table of particle units. The particle the continuing calculation. The initialization information units linked to the neighbours of the one containing the involves all the detailed records of each particle, such as the current particle unit will be searched along the chain table radius r , density ρ , temperature T, the spatial-position of particle units. p p p coordinates x =(x , x , x ), the translational-velocity (v) If the current particle does not collide with any other 0 1 2 vector v =(v , v , v ) and the rotating angular velocity particle in the space of particle motion, the status ana- 0 1 2 ω =(ω , ω , ω ). lysis of the current particle will not be changed, i.e. the 0 1 2 (ii) The space of particle motion will be meshed to cubic current particle-kinematics analysis will remain the same grids. As preparative work, the formation of cubic grids in the as in the last time step. Otherwise, the new status analysis space of particle motion is indispensable to the whole algo- will be launched and the results of dynamic analysis of the rithm of the three-dimensional soft-sphere model of the DEM. current particle will be changed according to the current (iii) All particles at any time step should be associated new status analysis. with the corresponding cubic unit of the cubic grids ac- (vi) The iteration will be continued until the number of cording to the particle spatial position, i.e. the mapping time steps exceeds the maximum value. Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 153 Begin Particle initialization …… Grids formation t = t …… no t ≤ t max yes Associating particles with grids Neighbor searching no Particle collision yes Status analysis (Mechanical model) (Thermodynamics model) Kinematics analysis t = t (Kinematical model) t = t 0 0 (Heat transfer model) Particle info Particle info Particles initialization End t = t programme Particle info Results calculation Results analysis Analysis 1 Analysis 2 Analysis 3 …… Analysis n …… Result 1 Result 2 Result 3 Result n Fig. 9: The algorithm chats of the three-dimensional soft-sphere model of the DEM for computing the particle flow and heat transfer 2.1 Validation of the character of the 2 Validation of the three-dimensional particulate flow soft-sphere model of the DEM In this section, the character of the particle flow simulated The program-development process of the three- by the developed three-dimensional soft-sphere model of dimensional soft-sphere model of the DEM has been the DEM will be compared with a validation experiment of presented above in detail. However, the reasonability glass beads discharging from a transparent silo to validate and correctness of the program, especially the particle- the reasonability and correctness of the particle-mechanics mechanics model and the particle-thermodynamics model in the particle-status analysis of the three-dimensional model, should be validated against experimental data. soft-sphere model of the DEM. The experiment [37] has been In this research, one major goal was to confirm some carried out by González-Montellano et al., whose results will key parameters in the particle-mechanics model and be referenced to compare with the results calculated from the particle-thermodynamics model of the three- the developed three-dimensional soft-sphere model of the dimensional soft-sphere model of the DEM, which is DEM program in this research. The illustration of González- useful and necessary in the application of the model to Montellano’s experimental equipment appears in Fig. 10. research problems. t = t + Δt Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 154 | Clean Energy, 2021, Vol. 5, No. 2 L = 0.25 m in the silo is ~86% of the height of the right quadrangular prism, and the filling density of the glass beads in the silo –3 is 1423  kg  m [37]. The simulated results from the devel- oped three-dimensional soft-sphere model of the DEM program show that the filling height of glass beads in the silo is 87.7% of the height of the right quadrangular prism, with an relative error of 1.98% compared to the measured value in the validation experiment, and the calculated –3 filling density of glass beads in the silo is 1490.45  kg  m , with a relative error of 4.74% compared to the measured value in the validation experiment. Thus, the simulation results from the developed three-dimensional soft-sphere model of the DEM are acceptable. 2.1.2 Discharge time The discharge time of glass beads T from the silo can discharge be measured by simply opening the gate at the exit of the hopper to complete the discharge process. The measured T and the simulated T are listed in Table 4. discharge discharge Compared with the validation experiment, the T discharge calculated by the developed three-dimensional soft-sphere model of the DEM has a relative error of 3.24%. Thus, the agreement between the measured T in the check ex- discharge 62.5 periment and the simulated T in the DEM simulation discharge is satisfactory. 2.1.3 Flow pattern of glass beads in the silo The flow pattern of glass beads during the process of discharge is an important result in the validation of the mechanics model of particle-status analysis in the three- dimensional soft-sphere model of the DEM program. The simulated results by the program developed in this re- D = 57 mm search not only can be validated by the flow pattern re- corded in the experiment, but also can be checked by Fig. 10. Schematic diagram of the transparent silo for containing the TM commercial software, such as EDEM . The validation re- glass beads for validating the mechanics model in particle-status ana- lysis in the three-dimensional soft-sphere model of the DEM program sults of the flow pattern of the glass beads in discharge from the silo are shown in Fig. 11. Each column of Fig. 11 is the validation results of the As shown in Fig. 10, the transparent silo is made of flow pattern in the silo at the same reduced time, which plastic (PMMA), through which the character of par - is defined as the ratio of real time t to discharge time ticulate flow can be recorded by a camera (Genie H1400- real T . As shown in the recorded experimental graphs Monochrome) with a rate of 50 fps and resolution ratio of discharge in Fig. 11, three colours of the glass-beads bed layer are 280 × 630 pixels. The shape of the transparent silo is com- filled in the silo in the order of grey, white, black and posed of a right quadrangular prism and a right frustum, white (the order of blue, white, black and white in the whose size is illustrated in Fig. 10. The glass beads with a simulation) at the reduced time of 0T . During the diameter of 13.8 mm were chosen as particulate materials discharge discharge process of glass beads in the silo, the flat bed filling the transparent silo, whose parameter values of the layer of glass beads is curved in the right quadrangular mechanics model listed in Table 3 are the same as reported prism part of silo and is curved more seriously from a U by González-Montellano. shape to a V shape in the right-frustum part of the silo because the particle velocity in the centreline of the silo 2.1.1 Filling height and filling density in the silo is larger than that closer to the wall of the silo. It can About 14  000 particles were calculated according to the be concluded from Fig. 11 that the flow pattern of the mass and the mean particle diameter of the glass beads. glass beads in the silo simulated by the developed three- Similarly, 14000 glass-bead units filling the same size dimensional soft-sphere model of the DEM program is of silo freely were simulated by the developed three- similar to that recorded in the experiment and the re- TM dimensional soft-sphere model of the DEM program. In sults simulated by the commercial EDEM Academic 2.3 the validation experiment, the filling height of glass beads version software. L = 0.2 m D = 57 mm T =0.19m H = 0.50 m Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 155 Table 3: The values of parameters in the mechanics model of glass beads Recovery coefficient Friction coefficient e/– μ/– Density Elastic module Poisson ratio –3 ρ /kg m E/Pa v/– e e μ μ p pw pp pw pp 2516 4.1 × 10 0.22 0.62 0.75 0.3 0.3 Table 4: The discharge time of glass beads in a silo measured (a) Observed in the validation experiment in the check experiment and that calculated by the developed three-dimensional soft-sphere model of the DEM program discharge, simulation No. T (program developed in this research) discharge, experiment 1 29.32 sec 30.25 sec 2 29.28 sec 3 29.20 sec 29.27 sec 30.25 sec TM (b) Simulated by the EDEM Academic 2.3 2.1.4 Particle-velocity distribution in the silo The mechanics model of particle-status analysis in the three-dimensional soft-sphere model of the DEM program also can be validated through the distribution of the vel- ocity of the glass beads in the silo, which appears in Fig. 12. As shown in Fig. 12a, the radial-velocity distribution from the exit of the silo 0.02 m at the time of 0.2 sec after opening the gate of the silo is similar to but a little less TM than the simulated results from the commercial EDEM Academic 2.3 version software. In addition, MFI, the ratio of the glass-bead velocity near the wall of the silo to that (c) Simulated by the DEM program developed by this research in the centreline of the silo along the height of the silo at the time of opening the gate of the silo, has been compared TM with the simulated results from the commercial EDEM Academic 2.3 version software, as illustrated in Fig. 12b . Therefore, the particulate flow simulated by the developed three-dimensional soft-sphere model of the DEM can be satisfactorily validated by experimentation and commer - cial software. 2.2 Validation of the character of particulate heat transfer 1/2 T 3/4 T 0 T 1/4 T discharge discharge discharge discharge The particulate-heat-transfer model has been coupled into Fig. 11: Comparison of the flow pattern of the glass beads in the silo the developed three-dimensional DEM based on a soft- during the discharging process observed in the check experiment, the sphere model. However, the new coupled particulate heat- TM calculation from EDEM Academic 2.3 and the simulation by the three- transfer model and the values of its parameters can be dimensional soft-sphere model of the DEM program validated by some laboratory-scale experiments. In this re- search, a rotary calciner built by Chaudhuri [38], as shown 2.2.1 Alumina particles heated in a rotating calciner in Fig. 13, is adopted to validate the correctness of the About half the volume of the calciner was filled with the particulate-heat-transfer model embedded in the devel- alumina particles at a room temperature of 25°C and the oped 3D DEM program and estimate the effect of various wall of the calciner was uniformly heated using an indus- materials on the heat transfer. trial heat gun maintained at 100°C. Meanwhile, the calciner –1 The particulate materials consist of two types of spher - with a rotation speed of 20 r min was controlled by a step ical particles: alumina particles and copper-alloy particles, motor . The calciner was stopped at a certain time interval whose parameter values in the mechanics model [39] and and the temperature of the particulate bed was tested the thermodynamic model [38] are summarized in Table 5. using 10 inserted thermocouples, as shown in Fig. 4. After Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 156 | Clean Energy, 2021, Vol. 5, No. 2 TM TM Simulated in the EDEM Academic 2.3 Simulated in the EDEM Academic 2.3 Simulated in the developed 3D-DEM Simulated in the develped 3D-DEM 1.0 0.45 0.9 0.40 0.8 0.35 0.7 0.30 0.6 0.25 0.5 0.20 0.4 0.15 0.3 0.10 0.2 0.05 0.1 0.0 –0.15 –0.10 –0.05 –0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Radius position x / m Height position of silo z / m Fig. 12: The comparison of the particle-velocity profile of the bed layer along the radius direction at different height positions (0.2 sec) between the TM simulated results by EDEM Academic 2.3 and the simulated results by the three-dimensional soft-sphere model of the DEM program As shown in Fig. 15a, the average temperatures of the (a) (b) particulate bed at the positions of the thermocouples are increased with the rotating process turbulently because the relative standard deviation of the alumina particles in the calciner will be increased at the beginning of the heating process. The heat balance of the particulate bed layer in the calciner can be carried out and the coefficient of heat transfer can be calculated as follows: dT m c = α e CL (T − T )(30) s p,s s s wall s dt Å ã Å ã T − T α e CL s s s wall ln = − t(31) T − T m c D = L = 0.0762 m wall s,0 s p,s Fig. 13: The experiment built up by Chaudhuri for validating the par - where m is the whole mass of the aluminium-particulate ticulate heat  transfer model coupled in the three-dimensional soft- sphere model of the DEM program bed layer in the calciner, 0.875 kg; c is the heat capacity p,s –1 –1 of the alumina particle, 875 J  kg K ; T and T are the s s, 0 the temperatures of 10 thermocouples were recorded, the particulate layer temperature and the its initial value, K; thermocouples were extracted and the calciner was ro- T is wall temperature of the calciner, 398.15 K; stands C wall tated again. The temperature profile of the particulate bed for the circle length of the calciner, 0.203 m; π e represents in the calciner is simulated and deduced by the new im- the ratio of the flank area of the calciner occupied by the proved 3D DEM program coupled to the thermodynamics aluminium-particulate bed layer, ~0.5; α describes the co- model from 0 to 12 sec, which is illustrated in Fig. 14. efficient of heat transfer of the calciner heating process, It is not difficult to observe that the heat is transferred –2 –1 W m  K . The calculated value of the heat-transfer coeffi- from the boundary to the centre of the particulate bed –2 –1 cient is ~170.751 W m  K , a relative error of –27.07% with gradually. During the uniform rotation of the calciner, –2 –1 the measured value by Chaudhuri of 124.527 W m K . the alumina particles near the hot wall are heated dir - According to the results simulated by the new improved ectly and rise up to a higher position on the incline of 3D DEM program coupled with the heat-transfer model the dynamic angle of repose. Furthermore, the alumina compared with the measurement in the laboratory experi- particles at the free surface of the inclined particulate ment, the correctness and agreement of the new coupled bed are warmed up indirectly by the higher-temperature heat-transfer model are acceptable. particles heated by the calciner wall rolling down from the top of the inclined bed layer to the bottom. The curve 2.2.2 Copper-alloy particles heated in a rotating calciner of the average temperature at the positions of the 10 Similarly, the calciner heating process of copper-alloy par - thermocouples is simulated by the new improved 3D ticles, as another type of particulate material for validating DEM program and compared with that measured by the accuracy of the new improved 3D DEM program, is Chaudhuri, which are illustrated in Fig. 15a. The coeffi- also simulated in this research. About half the volume of cient of heat transfer of this process also can be found as the calciner was filled with the copper-alloy particles at shown in Fig. 15b. a room temperature of 25°C and the wall of the calciner Particle vertical velocity –1 V / m.s p,z MFI Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 157 Table 5: The parameter values of alumina and copper alloy in a mechanics model and a thermodynamic model in the developed three-dimensional soft-sphere model of the DEM program Thermal- Poisson Recovery conduction Particle Density Particle Elastic module ratio coefficient Heat capacity coefficient –3 –1 –1 –1 –1 material ρ /kg m radius/mm E/Pa v/– e/– c /J kg  K k /W m  K p p p Alumina 3900 1.0 7.17 × 10 0.33 0.80 875 36 Copper 8900 2.0 9.6~11 × 10 0.34 0.80 172 385 (a) 0.0 sec (b) 2.4 sec (c) 4.8 sec (d) 7.2 sec (e) 9.6 sec (f) 12.0 sec Particle temperature/°C 60 55 50 45 40 35 30 25 20 Fig. 14: The temperature distribution prediction of particulate material of the alumina spherical particles heated in the calciner simulated by the three-dimensional soft-sphere model of the DEM program was uniformly heated using an industrial heat gun main- temperature difference between the copper-alloy par - tained at 1025°C. Meanwhile, the calciner with a rotation ticles and the calciner wall. Fig. 16b shows that the relative –1 speed of 20 r min was controlled using a step motor. The standard deviation of the temperatures of the copper-alloy average-temperature curve of the copper-alloy particles particles increases with heating time from 0 to 5 sec, and and the relative standard deviation curve of all copper- is decreased with heating time after ~5 sec. The uniformity alloy temperatures can be calculated and compared with of the original copper-alloy-particle temperature is des- those simulated by Chaudhuri, as shown in Fig. 16. troyed by the higher temperature of the particles near the It can be concluded from Fig. 16a that the average tem- calciner wall as compared to that of the particles in centre perature of the copper-alloy particulate bed in the calciner of the calciner particulate bed layer. Since the average increased with the heating time. However, the amplifi- temperature of the copper-alloy particles is limited to the cation of that declined gradually because of the reduced calciner-wall temperature, the temperature difference Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 158 | Clean Energy, 2021, Vol. 5, No. 2 (a) (b) 0.10 0.00 Measured in experiement by B. Chaudhuri (a) (b) 0.09 Simulated by authors of this research –0.01 0.08 -0.02 0.07 -0.03 0.06 -0.04 0.05 -0.05 0.04 -0.06 0.03 -0.07 0.02 -0.08 Measured in experiement by B. Chaudhuri Simulated by author of this research 0.01 -0.09 Linear fit of measured value Linear fit of simulated value 0.00 -0.10 02468 10 12 02468 10 12 Time t / sec Time t / sec Fig. 15: Comparison of the average temperature curve of the alumina spherical particles between that measured by Chaudhuri and the results cal- culated by the three-dimensional soft-sphere model of the DEM program (a) (b) 1000 300 Simulated by B.Chaudhuri et al. Simulated by B.Chaudhuri et al. 900 Simulated by the authors of this research Simulated by the authors of this research 500 150 0 0 012345678 910 0123456789 10 Time t / sec Time t / sec Fig. 16: The comparison of the average  temperature curve and standard deviation of copper-alloy spherical particle temperatures between the simulated results by Chaudhuri and those by the three-dimensional soft-sphere model of the DEM program among all particles in the particulate layer is reduced and 3 Prediction of particulate behaviour of a new uniformity of copper-alloy-particle temperature coal and sand appears at the end of the heating process in the rotating The reasonability and correctness of the developed three- calciner. dimensional soft-sphere model of the DEM program, The temperature profile of the copper-alloy particulate especially the particle-mechanics model and the particle- bed in the calciner is simulated and deduced by the new thermodynamics model, have been validated through a improved 3D DEM program coupled to the thermodynamics series of validation experiments. However, to realize the model from 0 to 20 sec, which is illustrated in Fig. 17. ultimate goal of the simulation of particulate behaviour of The temperature profile of copper-alloy particles in flow and heat transfer in the process of coal pyrolysis in a the rotating calciner also can be simulated by the new downer reactor by the developed three-dimensional soft- improved 3D DEM program and compared with the sphere model of the DEM program, the physical-property simulation results from Chaudhuri, which are shown in parameters of coal particles as fuel and sand particles as Fig. 17. According to the results simulated by the three- heat carriers should be clearly validated and confirmed. dimensional soft-sphere model of the DEM program The parameter values of the mechanics model of coal coupled with the heat-transfer model compared with the and sand in the particle-status analysis of the three- measurement in the laboratory experiment, the correct- dimensional soft-sphere model of the DEM program for ness and the agreement of the new coupled heat-transfer simulating the particulate behaviour of flow are listed in model are acceptable. Table 6 [32, 40]. Average temperature Dimensionless temperature T /K copper, avg (T -T )/(T -T ) avg o w o Standard deviation of temperature ln[(T -T )/(T -T )] wall avg wall avg,0 S / K copper Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 159 Simulated by B. Chaudhuri Simulated by author 0.0 sec 3.0 sec 9.0 sec Particle temperature/K 950750 550350 Fig. 17: Comparison of copper-alloy-particle temperature distribution in the calciner at different times between the calculated results by Chaudhuri and those by the three-dimensional soft-sphere model of the DEM program Table 6: The values of the parameters of particle materials in the filling and heaping test Particle-size Elastic Poisson Sliding-friction Rolling-friction Recovery Density distribution modulus ratio coefficient coefficient coefficient –3 Particle typesρ /kg m d /mm E/Pa v/– μ /– μ /– e/– p p s r 9 –5 Coal 1250 1~2 5.08 × 10 0.28 0.51 5 × 10 0.85 10 –5 Sand 2450 1~2 4.10 × 10 0.22 0.66 5 × 10 0.90 For convenience, the wall is assumed to have the same properties as particles with infinite diameter. Table 7: The relationship between the particle diameter and the particle number with a constant mass for coal and sand (1) (2) (3) (4) (5) Coal-filling mass m /g 3.7468 7.5285 11.3451 15.1965 19.0828 coal Coal-particle number 1665 3328 5069 6804 8576 Sand-filling mass m /g 6.5266 13.0199 19.6314 26.3589 33.1530 sand Sand-particle number 1467 2951 4462 6041 7578 particles of coal or sand in the simulation are generated 3.1 Filling tests for coal and sand by an initialization program of the three-dimensional soft- The filling of a vertical pipe with coal and sand separately sphere model of the DEM simulation with the uniform for testing the filling height and filling density of coal and distribution of particle diameters ranging from 1 to 2 mm, sand individually has been simulated by the developed whose particle densities are considered as the values listed three-dimensional soft-sphere model of the DEM program in Table 6, and the generated particle numbers of coal and and compared with the practical experiment. Experimental sand are listed in Table 7, separately. results for the filling height and the filling density are com- The particle-filling state of coal and sand in the vertical pared with the model simulation. The particle diameter quartz pipe are photographed, as shown in Fig. 18a and with uniform distribution is confined to between 1 and b, separately. Meanwhile, the simulations of the particle- 2  mm through screening. The vertical pipe with an inner filling state for coal and sand using the three-dimensional diameter of 1.78  cm is made of quartz. Five different ini- soft-sphere model of the DEM are illustrated in Fig. 18c tial masses of coal particles and sand particles, listed in and d. The particle-filling height shown in Fig. 18e and Table 7, are weighed using an electronic balance and filled the particle-filling density shown in Fig. 18f have been in the same type of vertical quartz pipe, respectively. The Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 160 | Clean Energy, 2021, Vol. 5, No. 2 (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) 0.11 2000 0.11 1000 The measured h The measured h Coal Sand fill fill 0.10 0.10 The simulated h The simulated h fill fill 0.09 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 Themeasured Themeasured fill 650 fill 0.01 0.01 ρ ρ (e) Thesimulated (f) Thesimulated fill fill 0.00 600 0.00 1000 2468 10 12 14 16 18 20 48 12 16 20 24 28 32 36 Filling mass m / g Filling mass m / g fill fill Fig. 18: The simulation and experiment of the filling height and the filling density in the vertical pipe with an inner diameter of 1.78 cm for coal and sand, respectively experimentally measured and calculated in the simulation, further increasing the filling density. (iii) The average rela- the distinction of which is also shown in Fig. 18e and . f tive errors of the filling density in the vertical pipe between The conclusions from Fig. 18e and f can be summarized the simulated and measured values are ~6.7% for coal par - as follows. (i) The maximum relative errors of the fill height ticles and 6.5% for sand particles. The agreement between in the vertical pipe between the simulation and the meas- the simulation and the experiment is satisfactory. urements are –9.5% for coal particles and –5.7% for sand particles, respectively. (ii) The effect of the initial filling 3.2 Voidage distribution along the bed height mass on the particle-filling density in the vertical pipe is not remarkable, although the filling density is slightly in- The voidage distribution along the bed height can be creased with the initial filling mass or filling height, which used to check the reliability of the parameters in the can be interpreted as the pressure acting on the lower bed mechanics model of coal and sand in the particle- layer from the upper zone to reduce the bed layer voidage, status analysis of the three-dimensional soft-sphere Fillingheight h /m fill –3 Filling density / kg•m fill Filling height h / m fill –3 Filling density / kg•m fill Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 161 model of the DEM program. At the same time, the effect The bed-voidage profiles along the bed height for par- of the particle diameter on the profile of voidage along ticles with a uniform diameter from 1.0 to 2.0 mm filling in the bed height can be simulated using the developed the vertical glass tube with an inner diameter of 1.78 cm three-dimensional soft-sphere model of the DEM pro- for coal and sand separately are shown in Fig. 19c and . f gram. In this section, 19.083  g of coal particles and It can be observed that the average bed voidage ranged 38.013 g of sand particles with five different diameters from 0.3 to 0.4 and slightly increased with the bed height. are respectively placed into five different tubes with an The reason is that the lower-layer particles are compelled inner diameter of 1.78  cm. The particle numbers cor - by the pressure from the upper bed layer, and the voidage responding to the different particle diameters at one in the lower bed is slight smaller than that in upper bed constant mass are listed in Table 8 for coal and sand layer. It is not difficult to confirm that the voidage of separately. spherical particles under the cubic arrangement is 0.48 The simulation results for particles of coal and sand and the voidage of spherical particles under the closest with five diameters but with the same constant mass arrangement is 0.26. It can be concluded that the param- filling in the five vertical pipes are shown in Fig. 19. eter values of the mechanics model of coal and sand in Table 8: The relationship between the particle diameter and the particle number with a constant mass of materials No. (1) (2) (3) (4) (5) (6) Particle diameter/mm 2.0 1.8 1.6 1.4 1.2 1.0 Particle number of 19.0828 g coal 3644 4999 7118 10 625 16 872 29 156 Particle number of 38.0125 g sand 3702 5079 7232 10 795 17 142 29 622 (1) (2) (3) (4) (5) (6) (1)(2) (3)(4) (5)(6) CD 0.10 0.10 Particle diameter Particle diameter 0.08 0.08 2.0mm 2.0mm 1.0mm 1.0mm 0.06 0.06 0.04 0.04 0.02 0.02 0.00 0.00 0.00.1 0.20.3 0.40.5 0.60.7 0.80.9 1.0 0.00.1 0.20.3 0.40.5 0.60.7 0.80.9 1.0 Bed voidage ε Bed voidage ε / – / – bed bed Fig. 19: Calculation results of the filling experiment of coal and sand under the condition of the same filling mass but different particle diameters and the description of bed-voidage distribution along the height Bedheight h /m bed Bedheight h /m bed Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 162 | Clean Energy, 2021, Vol. 5, No. 2 the particle-status analysis of the three-dimensional soft- sand in the simulation are generated by an initialization sphere model of the DEM program are suitable. program of the three-dimensional soft-sphere model of the DEM simulation with a uniform distribution of par - ticle diameter ranging from 1 to 2 mm, and the generated 3.3 Heaping test for coal and sand particle numbers of coal and sand that are listed in Table 7, separately. The purpose of the particle-heaping test is to check the The experimental results and the simulation results reasonability of the sliding-friction coefficient μ and the for particles of coal and sand heaping on the platform are rolling-friction coefficient μ in the mechanics model of shown in Fig. 20. coal and sand in the particle-status analysis of the three- Fig. 20a shows the front views of the heaping state in dimensional soft-sphere model of the DEM program. The the experiment and simulation for observing the angles of pile height and pile-base radius of coal and sand are repose of coal and sand. Meanwhile, the simulations of the measured and simulated in this case. The particle diam- particle-heaping state for coal and sand using the three- eter with uniform distribution is confined to between 1 dimensional soft-sphere model of the DEM are illustrated and 2  mm through screening. The particles of coal and Front view Sand Coal Coal Sand Top view Coal Sand The linear relationship between the heaping height and heaping diameter 0.026 0.026 Measured value in test Measured value in test Calculated value in DEM Calculated in DEM 0.024 0.024 Linear fit of measured value Linear fit of measured value 0.022 0.022 Linear fit of calculated value Linear fit of calculated value 0.020 0.020 0.018 0.018 0.016 0.016 0.014 0.014 0.012 0.012 0.010 0.010 0.0150.020 0.0250.030 0.0350.040 0.0150.020 0.0250.030 0.0350.040 r /m r /m coalheap sandheap Fig. 20: The particle-heaping-state comparison of the experiments and the simulations for coal and sand. (a) The front view of the heaping states in the experiments and simulations for coal and sand, separately. (b) The top view of the heaping states for five kinds of heaping mass in the simula- tions for coal and sand, separately. (c) The linear relationship between the heaping height and the heaping diameter in experiments and simulations for coal and sand, separately. h /m coalheap h /m sandheap Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 163 in Fig. 20b. The heaping height and heaping diameter of than that of the sand hourglass. The relative errors of the the coal and sand particles are chosen to describe the simulated results of coal hourglass and sand hourglass comparison of the simulation and the experimental re- are 13.59% and 12.68% compared with the experimental sults, which are presented as Fig. 20c. results. The heaping height and heaping diameter are measured According to the conclusions from the discussion con- during coal and sand heaping on the platform with five ini- tent in the research, the developed three-dimensional tial masses as listed in Table 7 and plotted with filled points soft-sphere model of the DEM program can be capable of in Fig. 20c. The relationship between the heaping height and predicting the particulate behaviour of static state and the heaping diameter can be fixed through linear regression, flow for coal and sand. which is also plotted with the solid line in Fig. 20c. The filling height is related to the filling diameter as follows: 4 Conclusions h ≈ 0.51 · r (32) coal,measured coal,measured A three-dimensional soft-sphere model of the DEM for h ≈ 0.66 · r (33) simulating the particulate behaviour of flow and heat sand,measured sand,measured transfer is developed, the reasonable hypothesis and the The slopes of Equations (32) and (33) are presented as the ingenious algorithm of which have been presented in as tangent values of the angles of repose of coal and sand, much detail as possible. The organizational structure of separately. Therefore, the angles of repose are 27.07° for the developed three-dimensional particulate soft-sphere coal and 33.35° for sand. model of the DEM contains algorithms for determining The heaping height and heaping diameter are simu- particle collisions, for the status analysis of particles and lated, and have been plotted using open symbols in Fig. for the kinematics analysis of particles. The search algo- 20c. The relationship between the heaping height and the rithm for particle collisions is based on cubic grids and heaping diameter can be fixed through linear regression, the splicing of cubic grids can originally solve the problem which is also plotted with a dotted line in Fig. 20c. The of searching for particle collisions in a particulate system. filling height is related to the filling diameter as follows: h ≈ 0.45 · r (34) coal,simulated coal,simulated The particulate-status analysis is used for solving the causes of particle motion and heat transfer, which in- h ≈ 0.68 · r (35) sand,simulated sand,simulated volves the mechanics model and the thermodynamics Similarly, the simulated angle of repose of the coal pile is model. The particulate-kinematics analysis is described 24.17° with a relative error of –10.71% compared with the by some basic laws of physics containing Newton’s second experimental value, and the angle of repose of the sand law for particle translational motion, the rigid-body law of pile in the simulation is 34.17° with a relative error of 2.46% rotation for rotating particles and the heat-transfer law compared with the experimental value. for two particles with different temperatures. The correctness and reasonability of the developed three-dimensional soft-sphere model of the DEM program 3.4 Hourglass using coal and sand for timing has been validated and checked through simulating the Particles can be filled into an hourglass for timing. conditions compared with those in the experiment. In this However, the discharge time of the same volume of par - research, the mechanics model in particle-status analysis ticles with the same uniform diameter distribution is and the heat-transfer model in particle-kinematics ana- determined by the physical properties of the solid par - lysis of the three-dimensional soft-sphere model of the ticles. Therefore, the discharge times of the same volume DEM have been validated by the experiment of glass-bead of coal and sand with the same particle size from the discharge from a silo and metal particles heating up in a same hourglass must be different. 13500 coal particles calciner. The agreement between the experimental and (about 70.65 g) and 13 500 sand particles (about 138.47 g) simulated results is consistent and satisfactory, which with the same diameter of 2  mm have been filled into can indicate that the developed three-dimensional soft- the hopper of the hourglass to measure the time scale sphere model of the DEM program is successful and cap- of the two hourglasses, which has been also simulated able of predicting the flow and heat transfer in particulate by the developed three-dimensional soft-sphere model systems. of the DEM program. The simulation results of the dis- In addition, to prepare the simulation of the coal- charge process of coal and sand from the hopper of the pyrolysis process in a downer reactor using the developed hourglass are shown in Fig. 21. three-dimensional soft-sphere model of the DEM program The time scales of the two hourglasses containing coal in the next step, some necessary physical-property param- and sand separately have been compared between the eters of the fuel particles of coal and the heat-carrier par - experiment and simulation, respectively, which has been ticles of sand have been validated and confirmed through listed in Table 9. comparing the experiments of particle filling in the tube It can be concluded from Fig. 21 and Table 9 that the and particle heaping on the platform. The agreement be- discharge time of the coal hourglass is a little shorter tween the simulated results and tested data is satisfactory. Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 164 | Clean Energy, 2021, Vol. 5, No. 2 Coal Sand A B 1/4 T 1/2 T discharge, coal discharge, coal CD 3/4 T 1 T discharge, coal discharge, coal Fig. 21: The simulated results of the sand clock using coal and sand by the developed three-dimensional soft-sphere model of the DEM program Table 9: The discharge times of coal and sand in the hourglass measured in the validation experiment and those calculated in the DEM simulation Coal Sand No. T /sec T /sec T /sec T /sec discharge, tested discharge, simulated discharge, tested discharge, simulated 1 10.54 12.2 11.59 12.8 2 10.88 11.52 3 10.89 11.12 4 10.76 11.32 5 10.61 11.27 Average 10.74 12.2 11.36 12.8 Therefore, it can be concluded that the parameter values Nomenclature of the status-analysis model and the kinematics-analysis Scalar: model for coal and sand particles are acceptable for the A = particle surface area, m (p) next step of simulation of the coal-pyrolysis process in a A = particle contact area, m (p),c downer reactor using the developed three-dimensional A = area of tiny gas interval between two contact sur - gap soft-sphere model of the DEM program. faces, m Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 165 –1 –1 –1 c = specific heat capacity, J kg  K v = translational-movement-velocity vector, m s –1 d = width of tiny gas interval between two contact v = particle normal translational velocity, m s gap –1 surfaces, m v = particle tangential translational velocity, m s d = particle diameter, m x = space-position vector, m I = particle rotational inertia, kg m ϕ = phase-position vector, rad (p) –1 l = distance of two particles, m ω = rotating-movement-velocity vector, rad s l = length of side a of a cubic grid unit, m l = length of side b of a cubic grid unit, m l = length of side c of a cubic grid unit, m l = distance between particle centre and wall, m Conflict of Interest pw E = elastic modulus, Pa None declared. e = recovery coefficient, − r = particle radius, m (p) m = particle mass, kg (p) References m* = reduced mass of two particles, kg k = thermal-conduction coefficient of gas phase, [1] Zhu HP, Zhou ZY, Yang RY, et  al. 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Algorithm design and the development of a discrete element method for simulating particulate flow and heat transfer

Clean Energy , Volume 5 (2) – Jun 1, 2021

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Copyright © 2021 National Institute of Clean-and-Low-Carbon Energy
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Abstract

Keywords: DEM; algorithm; particulate flow; solid mechanics; heat-transfer model algorithm was originated by Cundall in 1971 [26] and ap- Introduction plied to simulate the behaviour of soil particles under dy- Granular materials in nature and particulate technology in namic loading conditions by Cundall and Strack in 1979 process industries are globally researched and developed [27]. The DEM algorithm is more efficient in quasi-static because the interactions among individual units in par - systems because it is capable of handling multiple particle ticulate systems are so complicated that the behaviours contacts. The particle deformation can be explicitly deter - of particulate-laden gases are quite different from those of mined by the interaction force of collisions. The transient conventional fluids. Thus, it is important for real interdis- contacting surface of particle-to-particle collisions can be ciplinary research on particulate systems to understand confirmed by the particle positions determined by the well- the mechanism at the microscopic level. established Newton’s laws of motion. That is to say, the With the rapid development of computer science and DEM algorithm is a powerful numerical method, in which technology, the discrete element method (DEM) [1 2, ] has the motion of each individual particle is determined by the been applied globally in recent years to gain knowledge of net force acting upon it. Another advantage of the DEM al- the microscopic mechanisms in particulate systems [3–6]. gorithm is that it can present more detailed information Thus, the soft-sphere model of the DEM has been conveni- on each particle, such as its position, trajectory, velocity ently applied to simulate grain processing in many fields and contact force, which are usually not easy to gain from of the processing industry, such as particle mixing/seg- experiments. In addition, the transient contacting surface regation [7–15] and screw transportation [16–18], and has can be modified to consider the heat-transfer area of two even been used to simulate the gas-solid two-phase flow colliding particles with different temperatures. Thus, the system in fluidization engineering coupled with computa- DEM algorithm coupled with thermodynamics and heat- tional fluid dynamics (CFD) [19–25]. transfer models can simulate the transient heat-transfer As a well-documented numerical tool and a prime or phenomena in a particulate system. Furthermore, the outstanding example of a dynamic algorithm, the DEM Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 143 Net force → Translation motion Total moment → Rotating motion The status analysis The kinematics analysis (Mechanics model) (Kinematical model) (Thermodynamic model) (Heat transfer model) Time driven: t = t + d t Particles attached in grids Searching technology Fig. 1: The foundational principle of the particulate three-dimensional soft-sphere model of the DEM for computing particle flow and heat transfer particulate macro-chemical-reaction kinetics also can be and metal particles heating in a calciner. Finally, to prepare introduced to the kinetics-analysis program to describe the the simulation of the coal-pyrolysis process in a downer variance of density or size at the particle scale. Therefore, reactor using the developed DEM algorithm in the next the soft-sphere model of the DEM algorithm can provide step, some necessary physical-property parameters of the an effective numerical method to simulate the particle be- fuel particles of coal and the heat-carrier particles of sand haviour of flow and heat transfer in a particulate system. have been validated and confirmed through comparing the To utilize the DEM algorithm to serve the simulation of experiments of particle filling in the tube, particle heaping particulate processes extensively and openly, for example, on the platform and particle flowing through an hourglass the mixing and heat transfer of coal and sand during the for timing. coal-pyrolysis process in a downer reactor or screw reactor, an integrated DEM program package has been developed 1 Development of the DEM algorithm and presented in detail. For one thing, the detailed struc- 1.1 Hypothesis and principle ture of the DEM algorithm, such as the searching algorithm for determining the particle collisions, has not been intro- In the DEM algorithm, each element can be considered as duced in previous publications. For another, the parameter one rigid body with a fixed shape, such as a sphere, a billet values are meaningless if the contact-mechanics model or a pyramid. A  little overlap is allowed to happen when or heat-transfer model is unknown or undocumented in two elements contact each other, which is calculated by some simulation tools. The contact-mechanics model and a force-interaction model. As a dynamic algorithm, the heat-transfer model should be stated clearly and the phys- status of all elements is unchanged in each time step, such ical meaning and values of the parameters should be well as force analysis and thermodynamics analysis. The foun- understood. dational principle or big picture of the DEM algorithm is During the algorithm design of the DEM, a new method described in Fig. 1. for cubic-grid formation and connection was developed, As shown in Fig. 1, the DEM algorithm, as a Lagrangian especially for application to irregularly shaped simula- method for modelling the individual trajectory of each ele- tion domains, which can save the overhead and computa- ment, can be described as a dynamic algorithm, whose tion time of systems. The particle-dynamics model in the structure consists of three parts: the status analysis, the kinematics analysis of the DEM program has been supple- kinematics analysis and an algorithm for the prediction mented by the particle-rotation model controlled by the of particle collisions. In the particle-status analysis, the rigid-body rotation’s law to simulate the angle of repose particle-mechanics model calculates the particle-force of a particle pile accurately. Most importantly, to simu- analysis and the particle-torque analysis, and the particle- late coal pyrolysis in a downer reactor, the particle-heat- thermodynamics model confirms the particle temperature transfer model and particle-thermodynamics model have according to the heat balance of one single particle. In the been coupled to the developed DEM algorithm. Second, the particle-kinematics analysis, the particle-dynamics model correctness and reasonability of the developed DEM al- takes charge of simulating the particle position and the gorithm have been validated and checked through simu- velocity of the particle in the next time step, and the heat- lating the conditions compared with experiments. In this transfer model can calculate the quantity of heat transfer research, the mechanics model for particle-status analysis between particles or from a particle to its surroundings. For and the heat-transfer model for particle-kinematics ana- a large number of particles in the system, it is a major task lysis of the developed DEM algorithm have been validated to detect the particle collisions and to calculate the contact by an experiment with glass beads discharging from a silo forces acting on the particle. A  search algorithm based on Searching Association Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 144 | Clean Energy, 2021, Vol. 5, No. 2 cubic grids for particle collisions is adopted in the whole In the Hertz theory, the normal stiffness coefficient k is program to realize the particulate three-dimensional soft- associated with the physical parameters of particles, such sphere model of the DEM on the computer more efficiently. as elastic modulus E, Poisson’s ratio ν and the particle ra- The whole calculation process of the dynamic algorithm of dius of two colliding spherical particles. e is the normal- the three-dimensional soft-sphere model of the DEM will not unit vector of the contact surface, which can be obtained be finished until the calculation step approaches the end. by the position vectors of two colliding particles as x − x j i 1.2 Particle-status analysis e = (3) x − x¯ j i Each particle has its own status parameters, such as The calculation method of the normal-unit vector can be the space-position vector x =(x , x , x ), phase-position 0 1 2 obtained from Fig. 2. The normal damping coefficient η vector ϕ =(ϕ , ϕ , ϕ ), translational-movement-velocity 0 1 2 can be calculated by [29]: vector v =(v , v , v ), rotating-movement-velocity vector 0 1 2 ω =(ω , ω , ω ), net-force vector F =(F , F , F ), total- 0 1 2 0 1 2 F = −η · v dn,ij n,ij n,ij  √  ∗ 1 moment vector M =(M ,M ,M ), particle temperature T  m k ·ln 0 1 2 n.ij ( ) p e η = 2 · n,ij and its enthalpy value H . All the status parameters of each 2 1  π + ln p [ ( )] m m (4) particle simulated in the DEM algorithm should be deter - ∗ i j m = m +m i j mined at the end of each time step. v  −ev n n,0 v = −v = − v · e e n,ij n,ji ji n n 1.2.1 Mechanics model where the reduced mass m can be calculated by two col- Normal-contact-force model. liding particles; e represents the recovery coefficient ac- It is very important to calculate the net force and total moment cording to the initial velocity and the rebound velocity of of each particle when particles contact each other, as illustrated one particle; and v is the normal relative velocity of par - n,ij in Fig. 2, because accurate particle motion is determined by the ticle i relative to particle j. exact particle-mechanics model introduced in this section. To calculate the normal contact force acting on each Tangential-contact-force model. particle, the Hertz theory [28] is applied to obtain the rela- Vemuri theory [30] is employed to obtain the tangential tionship between the normal contact force and the particle interactions between the colliding particles, which can be deformation, which is summarized as expressed as follows: F = F + F (1) n,ij cn,ij dn,ij (5) F = F + F t,ij ct,ij dt,ij F = −k · δ · e  cn,ij n,ij n However, the calculation method of the tangential force be- n,ij  √ 4 ∗  ∗ k = E r tween two colliding particles is complicated because the n,ij 2 (2) physical parameters cannot be tested easily in experiments. 1−υ 1−υ 1 j  i  = +  E E E i j Therefore, the following methods [3132 , ] are used to calcu-  1 1 1 = + r r r late the tangential force between the colliding particles: i j  ï ò ¶ © 3/2    δ t,ij F = −μ F  1 − 1 − min ,1 e ct,ij s cn,ij t  t,max (6) δ = v · Δt t,ij t,ij 2−ν δ = μ δ t,max s n,ij 2(1−ν ) →→ i τ where μ is the coefficient of side friction; F is the same cn,ij as mentioned above as the normal contact force of two col- ji → lision particles; δ is the relative displacement of two colli- j t,ij sion particles at contact time t; and Δ e is the unit tangential ji vector of the contact surface, which can be obtained by j τ ν ji τ → v − v · e e i n n ji ji ν e = (7) →→ v − v · e e ji ji n n → X – X j i e = – – The calculating method for the normal-unit vector can be X X j i →→ → → obtained from Fig. 2. η is the tangential damping coeffi- t, ij ν – ν •e ji ji n () cient, which can be calculated by e = →→ →→ – ν •e e 1 ν Å ã ji n n ji ()  δ t,ij  ∗ F = −η 6m μ F  1 − /δ v s t,max  dt,ij t,ij cn,ij t,ij t,max (8) Fig. 2: The solutions of unit normal vector and unit tangential vector 2 η = η t,ij n,ij during the contact collision of two spherical particles with different v = −v = − v · e e = − v × e × e t,ij t,ji ji t t ji n n diameters Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 145 According to the assumption mentioned above, the accel- where v is the tangential relative velocity of particle j t,ji eration of the particle motion in a small-enough time step relative to particle i. is a constant and the angle of acceleration of the particle Conservative forces, such as gravity or electromagnetic rotation in a small-enough time step is also a constant. forces, acting on the centre of a homogeneous spherical The velocity vector of the particle translational movement particle cannot cause the particles to rotate. However, v and the angular velocity vector of the particle rotation the line of total contact force acting on the particle - sur ω at the next time step can be expressed by the first-order face does not go through the centre of a spherical particle, Taylor series expansion at t = t . The spatial-position which makes the spherical particle rotate. The total rota- vector of particle x and the phase-position vector of par - tion moment is composed of the tangential contact torque ticle ϕ at the next time step can be derived by the second- and the rotational damping moment: order Taylor series expansion at = t t : (9) dv (t ) M = M + M . .. (9) i 0 i t,i r,i v (t)= v (t )+ · Δt i i  dt (15) dx (t ) d x (t ) M = r × F i 0 1 i 0 2 t,i i t,ij x (t)= x (t )+ · Δt + · · Δt 0 2 i i dt 2! dt (10) μ F ω r| cn,ij| ij M = − r,i | | ij dω (t ) i 0 ω (t)= ω (t )+ · Δt i i dt 2(16) where r is the vector of the radius of particle i, whose dir - dϕ (t ) d ϕ (t ) i i 0 1 i 0 2 ϕ (t)= ϕ (t )+ · Δt + · · Δt 0 2 i i dt 2! dt ection is from the centre of the spherical particle to the The particle-kinematics equations of translational motion contact point of collision; F is the tangential component t,ij and rotational motion can be discretized into a Verletleap- of the contact force from particle j to particle i; and is the μ frog scheme as in Equations (16) and (17): damping coefficient of the rolling friction. −→ a (t )= F (t )  i 0 i 0 −→  Δt Δt 1.2.2 Thermodynamics model v t + = v (t )+ a (t ) ·  i 0 i 0 i 0 2 2 In this research, the enthalpy of a solid particle is assumed Δt x (t + t)= x (t )+ v t + · Δt(17) i 0 i 0 i 0 to be a function of the temperature, i.e. the enthalpy H of  −→ a (t +Δt)= F (t +Δt) 0 0 i i  m a particle at temperature T can be calculated from that at  −→ Δt Δt v (t +Δt)= v t + + a (t +Δt) · i 0 i 0 i 0 2 2 the specified temperature T , i.e.: −→ j (t )= M (t )  i 0 i 0  i H (T)= H (T )+ m · c dT(11) 0 p p   −→  Δt Δt ω t + = ω (t )+ j (t ) ·  i 0 i 0 i 0 2 2 Δt (18) ϕ (t + t)= ϕ (t )+ ω t + · Δt i 0 i 0 i 0 where m is the mass of the particle; and is the heat cap c -  −→ p p   1  j (t +Δt)= M (t +Δt)  i 0 i 0 acity of the particle at constant pressure, which can be cal-  i −→  Δt Δt ω (t +Δt)= ω t + + j (t +Δt) · culated as c = a + bT + cT . Therefore, the enthalpy change i 0 i 0 i 0 2 2 ΔH of the particle can be calculated as Å ã Ä ä Ä ä 1 1 3 3 2 2 (12) ΔH = m · c T − T + b T − T + a (T − T ) p 0 1.3.2 Heat-transfer model 0 0 3 2 The calculation method for the heat transfer between two –1 –1 –1 –2 –1 –3 where parameters a/kJ· kg  ·K , b/kJ· kg  ·K and c/kJ ·kg  ·K colliding particles or between a particle and a wall with are constants collected in the thermodynamics hand- different temperatures can be obtained by the theoretical books. Therefore, the change in particle temperature can T analysis of Fourier’s law, which can be described as follows: be calculated from the enthalpy difference Δ thr H ough the heat balance of particle i: ΔT ij Q = (19) ΔH i α A c,ij c,ij T = T + (13) i i,0 m · c i p,i where |ΔT | is the absolute temperature difference be- ij tween two colliding particles or between a particle and a wall standing for the impetus of heat transfer through 1.3 Particle-kinematics analysis thermal conduction; A is the area of the thermal conduc- ij 1.3.1 Dynamics model tion between two colliding particles or between a particle The translational movement of the particle is driven by the and a wall; α is the contact coefficient of thermal con- c, ij net force acting on it, and accords with Newton’s Second duction; and the factor of stands for the resistance α A c,ij c,ij Law of Motion. The rotating movement of the particle is of heat transfer. It is worth noting that the contact coeffi- driven by the total moment acting upon it and is described cient of thermal conduction is affected by the roughness of by Newton’s Law of Rotation: the contacting surface, the acting force on the contacting 2 surface, the gas pressure in the gap between two colliding −→ dv d x i i F = F + F + F + F + m g = m a = m = m i cn,ji dn,ji ct,ji dt,ji i i i i i dt dt contact surfaces, etc. It cannot be tested by experimenta- −→ dω d ϕ i i M = M + M = I j = I = I i t,i r,i i i i i dt dt tion for most engineering materials except for some highly polished metals. Therefore, the calculation method for (14) Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 146 | Clean Energy, 2021, Vol. 5, No. 2 in Fig. 3a–c. At this time, the heat transfer through thermal conduction is only controlled by the overlap of the gas r r r 1 2 2 layers of the two particles, whose thermal resistance R can be calculated as follows: = dS l =+ r r + l =+ r r 12 1 2 12 1 2 R Δ l g Π (a) (d) 2πr sin θ = k d (r sin θ) g i l − 2r cos θ a a 12 i a 1 2 β β 1 2 sin θ cos θ = 2k πr dθ l − 2r cos θ r ++ r l r r + δ l r + r << < 1 2 12 1 2 12 1 2 − cos α l i 2r (b) (e) i = k πr (cos α − 1)+ ln (i = 1, 2) i i R 2r i − 1 g1 2r R R R R R (20) s1 g s2 s1 s2 where k is the thermal-conduction coefficient of the gas g2 layer; and Δl is the distance between two relevant particle (c) (f) surfaces along the direction of the centre line in metres. Fig. 3: Illustration of the calculation of the thermal resistance of α is illustrated in Fig. 3b and can be calculated as follows: thermal conduction between two colliding particles Ç å 2 2 l + r − (r + δ) −1 12 i α = cos (i = 1, 2)(21) 2r l the contact coefficient of thermal conduction is the core A similar calculation method can be applied to the thermal of the modelling of particulate thermal conduction. The resistance between a particle and a wall with different equivalent thermal-resistance model of the heat transfer temperatures through overlapped gas layers, as follows: between contact particles with different temperatures is  Ñ é illustrated in Fig. 3. l pw − cos α 1 l i pw   As shown in Fig. 3, the particle can be assumed to = 2k πr (cos α − 1)+ ln (22) i i pw R r g i − 1 be surrounded by a thin gas membrane with thick- i ness δ, which is much less than the particle diameter where l presents the distance between the particle centre pw d . According to the conclusion of Delvosalle from the and the wall surface along the direction of the normal measurement in experiment [33], the thickness of the vector of the wall surface. assumed gas layer can be considered as 0.1 times the (ii) The thermal conduction will be enhanced when two particle diameter, i.e.δ = 0.1d . Thus, the thermal con- particles with different temperatures collide with each duction between two colliding particles with different other, i.e. l ≤ r + r . The heat can be conducted through 12 1 2 temperatures is affected by the thin gas layer around the not only the overlapped gas layers, but also the tiny gas particle. Further assumption should be supplemented to interval d between the contacting surfaces of two par - gap build up the model of particulate thermal conduction. ticles with different temperatures. Thus, the thermal re- First, the contacting surface is smooth, whose normal sistance is composed of two parts: the thermal resistance vector is parallel with the centre line of the two colliding of the tiny gas interval R and the thermal resistance of g1 particles, the direction of the heat transfer through the annular gas layer R. Therefore, the thermal resistance g2 thermal conduction. Second, the area of the contacting under this condition can be calculated as follows: surface is equal to that of the overlap circle based on 1 1 d k gap g + = + dS the assumption of DEM modelling. Third, there is a min- R R k A Δ l g g g gap Π 1 2 i –10 ˆ i imum tiny gap interval with a thickness of 4 × 10 m −10 4.0 × 10 2πr sin θ = + k d (r sin θ) g i between the contacting surfaces. Last, the thermal con- l − 2r cos θ k π(r sin β ) 12 i g i i duction can be transferred through the internal min- ˆ i −10 4.0 × 10 sin θ cos θ = + 2k πr dθ imum tiny gap and the annular overlap of the gas layer g l − 2r cos θ k π(r sin β ) 12 i g i i around it. Therefore, the heat transfer through thermal −10 12 − cos α 4.0 × 10 l 2r 12 i conduction between two particles with different temper - = + k πr (cos α − cos β )+ ln (i = 1, 2) g i i i l12 2r − cos β k π(r sin β ) i g i i i 2r atures can be discussed from two conditions, as follows. (23) (i) The heat transfer based on thermal conduction will where α and β (illustrated in Fig. 3e) can be calculated as follows: i i not occur when two particles are far away from each other, 2 2 2 l +r −(r +δ) −1 12 i i.e. l ≥ r + r + δ. l is the distance between two particle α = cos 12 1 2 12 i 2r l i 12 Ä ä (i = 1, 2)(24) centres; r and, r are the radii of two particles separately. −1 l  12 1 2 β = cos 2r However, thermal conduction will not begin until the A similar calculation method can be applied to the overlap of the gas layers of two particles with different tem- thermal resistance between a particle and a wall with peratures appears, i.e. r + r < l < r + r + δ, as illustrated 1 2 12 1 2 Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 147 different temperatures through the gap interval and In a cuboids space with a constant volume, the scale the annular overlapped gas layers during the contact of one cubic unit determines the distribution density of of a particle with a wall with different temperatures as the cubic grids. The scale of one cubic unit is determined follows: by its side length. The distribution density of cubic grids  Ñ é in a constant volume can be determined by the particle- lpw −10 − cos α 1 1 4.0 × 10 pw i   + = + 2k πr (cos α − cos β )+ ln size distribution and the density of the particulate phase. i i i 2 l pw R R r g g i 1 2 k π(r sin β ) − cos β g i i i Generally, a small scale for the cubic unit is suitable for (25) the dense phase of particle flow and a larger side length of the cubic unit is acceptable with the dilute particle flow. where l presents the distance between the particle centre pw According to the search algorithm based on cubic grids, and the wall surface along the direction of the normal the side length of the cubic unit is usually larger, but not vector of the wall surface. too much larger, than the maximum particle diameter, i.e. As known to all, the thermal flux can be also affected by l > d . box p, max the solid resistance of the particle’s interior, which can be Each cubic unit has its own serial number (ID). As pre- calculated as follows: sented in Fig. 4, each current cubic unit coloured in pink has 1 1 1 its own 26 neighbours including the 8 units in the same layer R = − s (26) 2πk r r p,i i,m i (NW, N, NE, E, SE, S, SW, W), the 9 units in the upper layer (U, UNW, UN, UNE, UE, USE, US, USW, UW) and the 9 units in where k is the thermal-conduction coefficient of the par - the lower layer (D, DNW, DN, DNE, DE, DSE, DS, DSW, DW). ticulate material; r presents the radius of the particle; and –1/3 However, it is obvious that half of the neighbours of one cubic r = r . The spherical surface a radius of r divides the i, m i m unit should be recorded in order to save computer-memory internal energy of the particle equally. space, because the neighbour relationship is mutual. For ex- Above all, the total thermal resistance of heat transfer ample, cubic unit U is the neighbour of the current box unit through thermal conduction between two particles with coloured in pink and the current box unit is also the neigh- different temperatures can be calculated as follows: bour of cubic unit U at the same time. When the neighbours +∞l > r + r + δ||l > r + δ 12 1 2 pw i of cubic unit U are searched, the current box unit coloured in R + R + R r + r ≤ l < r + r + δ||r ≤ l < r + δ s g s 1 2 12 1 2 pw pink can be omitted, because cubic unit U has been searched R = 1 2 i i total  R R g g 1 2  when the cubic unit coloured in pink is treated as the cur - R + + R l < s s 12 r + r ||l < r 1 2 1 2 pw i R +R g g 1 2 rent one. Therefore, those in the 26 neighbour units whose (27) ID numbers are larger than (or less than) that of the current cubic unit can be considered as the neighbours recorded in the data structure of the current one. In this research, the 1.4 Search algorithm based on cubic grids uncoloured neighbours whose ID numbers are less than that of the current cubic unit coloured in pink (D, DNW, DN, DNE, 1.4.1 Formation of cubic grids DE, DSE, DS, DSW, DW, W, SW, S, SE) are omitted to save calcu- Detecting colliding particles is an important component lation time. In other words, the spatial structure consisting of the algorithm of the three-dimensional soft-sphere of cubic units can be built up and described by the 13 neigh- model of the DEM, which makes up a large percentage of bours of the current unit coloured in pink (NW, N, NE, E, U, the computation time. In the particle-status analysis, cal- UNW, UN, UNE, UE, USE, US, USW, UW) in the memory of the culation of the total contact force acting on one particle computer and the criteria for the formation rule mentioned requires calculation of the contact forces from all the par - above have been listed in Table 1. ticles colliding with it. Therefore, it is essential to deter - In Table 1, ID_BOX indicates the series number of the mine all particles that collide with the current one at each cubic unit; COLUMN shows the column number of the time step. To save calculation time, a series of search algo- cubic unit in one layer of the whole cuboids space; LAYER rithms have been developed and applied in the soft-sphere represents the number of the cubic number in one layer of model of the DEM, such as the Neighbour List Method [34], the whole cuboids space. If one of the recorded neighbours Bounding Box Method [35, 36] and Boxing Method [36]. In does not exist, the linker field of the corresponding neigh- this research, a search algorithm based on cubic grids has bour of the current cubic will be set as NULL. been developed to execute the task of determining particle collisions in the three-dimensional soft-sphere model of Splicing cubic grids between two cuboids spaces. the DEM. The number of cubic units in the cubic grids seriously af- fects the memory overhead and the calculation speed of Cubic-grid formation in a cuboids space. the computer. The spatial domain of the particle motion Before the calculation program of the three-dimensional is usually an irregular one, i.e. the space of particle mo- soft-sphere model of the DEM is executed, the entire cu- tion is not usually a cuboids space. If the cubic grids are boids space of the particle motion will be meshed into a meshed all at once in a larger cuboids space involving an series of cubic units, as shown in Fig. 4. Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 148 | Clean Energy, 2021, Vol. 5, No. 2 (k + 1) layer (k+1) layer k layer (k–1) layer k layer UNW UN UNE UW UE (k–1) layer USW US USE NW N NE W E SW SSE DNW DN DNE DW DDE DSW DS DSE Fig. 4: The structure of cubic grids and the illustration of 26 neighbours of one cubic unit irregular shape space of particle motion, a large number subspaces of particle motion. Each cubic unit in the frac- of redundant cubic units will be generated in the memory ture surface of segment part W is connected with UNE, of the computer and a large quantity of computational UE, E, NE and USE as neighbours from segment part E, time will be wasted. Therefore, an irregularly shaped space and the one in the fracture of segment part E is needed can be divided into several regularly or quasi-regularly to link to USW, UW, UNW and NW as neighbours from shaped subspaces, in which the cubic grids can be meshed. part W. The third cutting direction can be perpendicular Therefore, the whole irregularly shaped space of particle to S–N and the subspaces of segment part S and seg- motion can be recombined through splicing the cubic grids ment part N are separated from the irregularly shaped in the regularly or quasi-regularly shaped space of particle space of particle motion. All the cubic units in the frac- motion together, as described below. ture surface of segment part S are attached to UNW, UN, The design of the splicing algorithm for cubic grids UNE, NW, N and NE as neighbours from segment part is completely determined by the partitioning of the ir - N, and those of segment part N are needed to adjoin regularly shaped space of particle motion that can be USW, US and USE as neighbours from segment part S. It segmented along the direction vertical to D–U into two can be concluded from these three segmentation cases parts: the segment part U and the segment part D. Each that nine neighbour relationships should be connected. cubic unit in the fracture surface of segment part D is However, the first segmentation case is beneficial for the spliced with UNW, UN, UNE, UW, U, UE, USW, US and design of the splice algorithm, because the cubic units USE as neighbours from the segment part U, and the on the fracture surface of only one segment part D are one in the fracture surface of segment part U does not matched with the neighbours from segment part U. The need to be spliced with any cubic unit in segment part algorithm can be described as below. D according to the cubic-grid-formation rule mentioned In the fracture surface, the series number of the first in the ‘Cubic grids formation in a cuboids space’ section. If cubic unit in segment part D is begin ; step and step are 0 E0 N0 the segmentation direction vertical to W–E is adopted, count steps along the direction of E and N in segment the segment part W and segment part E are the two part D. Similarly, the series number of the first cubic unit Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 149 Table 1: The criteria for the existing 13 neighbours of the cur - Table 2: The criteria condition of the splice cubic unit be- rent cubic grid in the cuboids space of particle motion tween two segment parts of an irregularly shaped space of particle motion Neighbours Criteria Neighbours Criteria E ①[(ID_box + 1)/column]==[ID_box/column] N ①[(ID_box + column)/layer]==[ID_box/layer] UE ①[(obj1 + stepE1)/stepN1]==[obj1/stepN1] NE ①[(ID_box + 1)/column]==[ID_box/column] UW ①[(obj1-stepE1)/stepN1]==[obj1/stepN1] ②[(ID_box + column)/layer]==[ID_box/layer] ②(obj1-stepE1)>=begin1 + i × stepN1 NW ①[(ID_box-1)/column]==[ID_box/column] UN ①[(obj1 + stepN1)/ ②[(ID_box + column)/layer]==[ID_box/layer] (lengthE× lengthN)]==[obj1/ ③(ID_box-1)>=0 (lengthE× lengthN)] U ①(ID_box + layer)<=end US ①[(obj1-stepN1)/(lengthE× lengthN)]==[obj1/ UE ①(ID_box + layer)<=end (lengthE× lengthN)] ②[(ID_box + layer + 1)/ ②(obj1-stepN1)>=begin1 + i × stepN1 column]==[(ID_box + layer)/column] UNE ①[(obj1 + stepE1)/stepN1]==[obj1/stepN1] UW ①(ID_box + layer)<=end ②[(obj1 + stepN1)/ ②[(ID_box + layer-1)/ (lengthE× lengthN)]==[obj1/ column]==[(ID_box + layer)/column] (lengthE× lengthN)] UN ①(ID_box + layer)<=end UNW ①[(obj1-stepE1)/stepN1]==[obj1/stepN1] ②[(ID_box + layer + column)/ ②(obj1-stepE1)>=begin1 + i × stepN1 layer]==[(ID_box + layer)/layer] ③[(obj1 + stepN1)/ US ①(ID_box + layer)<=end (lengthE× lengthN)]==[obj1/ ②[(ID_box + layer-column)/ (lengthE× lengthN)] layer]==[(ID_box + layer)/layer] USE ①[(obj1-stepN1)/(lengthE× lengthN)]==[obj1/ UNE ①(ID_box + layer)<=end (lengthE× lengthN)] ②[(ID_box + layer + 1)/ ②(obj1-stepN1)>=begin1 + i × stepN1 column]==[(ID_box + layer)/column] ③[(obj1 + stepE1)/stepN1]==[obj1/stepN1] ③[(ID_box + layer + column)/ USW ①[(obj1-stepE1)/stepN1]==[obj1/stepN1] layer]==[(ID_box + layer)/layer] ②(obj1-stepN1)>=begin1 + i × stepN1 UNW ①(ID_box + layer)<=end ③[(obj1-stepN1)/(lengthE× lengthN)]==[obj1/ ②[(ID_box + layer-1)/ (lengthE× lengthN)] column]==[(ID_box + layer)/column] ③[(ID_box + layer + column)/ [] is floor operation. layer]==[(ID_box + layer)/layer] USE ①(ID_box + layer)<=end segmentation direction. Additionally, eight neighbours in ②[(ID_box + layer + 1)/ segment part N of the current cubic unit in segment part column]==[(ID_box + layer)/column] U can be gained according to the criteria list in Table 2. The ③[(ID_box + layer-column)/ neighbours will be NULL if the conditions are not satisfied layer]==[(ID_box + layer)/layer] with the criteria listed in Table 2. USW ①(ID_box + layer)<=end ②[(ID_box + layer-1)/ column]==[(ID_box + layer)/column] 1.4.2 Association between particles and cubic grids ③[(ID_box + layer-column)/ The program of the three-dimensional soft-sphere model layer]==[(ID_box + layer)/layer] of the DEM will launch the dynamic mechanism after the formation of cubic grids. In each time step, all particles are [] is floor operation. associated with the corresponding cubic units according to their spatial-position vectors, in which particle collisions in segment part U is begin1; stepE1 and stepN1 are count will be determined. In this section, two problems are ad- steps along the direction of E and N in segment part U. If dressed: (i) how to find the series number of the cubic unit the fracture is a rectangle with row lengthN and column with which the current particle is associated; (ii) how to lengthE, the splice units on the two fracture segment parts record or remark the associations between particles and are obj and obj , which can be calculated as: cubic units. 0 1 obj = begin + j × step + i × step (i = 0, 1, . .. , length ) 0 0 E0 N0 N−1 Mapping from the position of a particle to the series number of obj = begin + j × step + i × step ( j = 0, 1, . .. , length ) 1 1 E1 N1 E−1 the cubic unit. (28) It is not difficult to imagine that a particle can be assigned For the same value of i and j for the cubic grids of two frac- to the corresponding cubic unit according to the pos- ture surfaces, the corresponding series number of cubic ition of the particle. The mapping from the position of the units obj and obj in two segment parts are just the two particle to the series number of the corresponding cubic 0 1 splice units, also a neighbour relationship, along the D–U unit can be built up based on the cubic grids created, as Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 150 | Clean Energy, 2021, Vol. 5, No. 2 described in Section 1.4.1. In a regularly shaped space of The particles are associated with cubic grids all over particle motion, the series number of the cubic unit is in- again in each new time step because of the change in the creased from the W direction to the E direction stepped by spatial position of all particles. To avoid influence from 1, from the S direction to the N direction stepped by the the last association between particles and cubic grids, all column number, and from the D direction to the U direc- pointers of a particle type in the data structure of particles tion stepped by the layer number. Therefore, the mapping and cubic units should be set as NULL before the associ- from the position of the particle to the series number of ation mechanism of the next time step is launched. When the cubic unit can be built up as: the series number of the cubic unit is calculated from Equation (29) according to the spatial position of the par - (x − x ) (x − x ) (x − x ) 0 a 1 b 2 c ID = + n + n(29) box column layer ticle, the particle unit will be linked to the pointer of the l l l a c particle type in the cubic-unit data structure. If there are where (x , x , x ) is the position of the vertex of units D, W, a b c some other particles associated with the same cubic unit, and S of the first cubic unit in the cubic grids whose series a chain table of particle units whose heads are linked to number is 0; l , l and l are the three side lengths of a cubic a b c the associated cubic unit’s pointer of particle type should unit; n stands for the column number in each layer of column be formed in ascending order of the series number of cubic units; and n represents the number of cubic units layer the particle unit. The flow sheet of the pseudo code is in each layer of the cubic grids. presented as Fig. 6. An example has been prepared for illustrating the as- Algorithm of association between the particles and the cubic sociation between the particles and the cubic units, as grids. shown in Fig. 7. The data structures of the particle and the cubic unit are When the subroutine that records the association be- illustrated in Fig. 5. The data structure of the particle not tween particle units and cubic grids is launched, a tree struc- only records the basis information of the particle, such ture like that in Fig. 7 will be generated and updated in the as the radius, density, mass, rotational inertia, position computer memory, which is beneficial for the realization vector, translational-velocity vector, rotating-velocity and design of the algorithm for detecting particle collisions. vector, force vector and moment vector, but also in- As represented in Fig. 7a, the cubic unit of 628 contains par - cludes a pointer member of the particle type. When one # # # # # # # ticle units of 428, 432 , 620 , 656 and 697 ; those of 617 , 751 cubic unit contains more than one particle, the particle- # # # and 758 are associated with the 629 cubic unit; the 630 type pointer field will link other particle units to form cubic unit involves only the 302 particle unit. The records of a chain table in a series of increasing particle numbers. the association between the particle units and cubic grids in The structure of the cubic unit has the pointer members the computer memory can be shown as in Fig. 7b . It is clear of 13 neighbours of the cubic-unit type and a pointer that the series number of particle units in the chain table member of the particle type. The pointer member of the is increased from head to tail. It is worth noting that the particle type is linked to the particle unit of the min- # # 629 cubic box unit is the east (E) neighbouring unit of 628 , imum series number of all particles matched with this and the 628 cubic unit is the west (W) neighbouring unit of current cubic unit. 629 . The west (E) neighbouring unit is invalid according to the principle mentioned in Section 1.4.1.Therefore, the 628 cubic unit is not linked to one of the 13 neighbour pointer # # # fields of 629 . The same principle acts on 629 and 630 . ID r m 1.4.3 Algorithm for detecting particle collisions den I *next ID *p[0] *p[1] A particle is associated with the corresponding cubic unit according to the spatial coordinate values of its centre. The cubic grids will be beneficial for detecting particle collisions x[0] x[1] x[2] *p[2] *p[3] *p[4] efficiently. It is not difficult to imagine that two particles close to each other may be associated with the same cubic v[0] v[1] v[2] *p[5] *p[6] *p[7] units or two neighbouring cubic units separately, as shown in Fig. 7a. Therefore, the algorithm for detecting particle w[0] w[1] w[2] *p[8] *p[9] *p[10] collisions has two steps. The particles associated with the cubic unit containing the current particle (in the case of a cubic unit containing more than one particle) should be as- F[0] F[1] F[2] *p[11] *p[12] *next sessed to determine whether they will collide with the cur - rent particle. In addition, the particles associated with the (a) particle (b) cubic unit M[0] M[1] M[2] neighbours of the cubic unit containing the current particle should be also assessed as to whether they will collide with the current particle or not. The flow sheet of a pseudo code Fig. 5: The data structure of a particle and cubic unit in the three- dimensional soft-sphere model of the DEM for detecting particle collisions is presented as Fig. 8. Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 151 Sub begin boxno=boxnof(ball[balli].x0); //calculate the series number of the cubic box unit matched with the current particle head=&box[boxno]; //get the head pointer of the chain table of particle unit node no head->next==NULL yes head->next =&ball[balli]; _pball=head->next; //link the current particle to the pointer field of //get the pointer of the first particle node of the the corresponding cubic box unit directly chain table yes _pball->next==NULL no _pball->next =&ball[balli]; _pball=_pball->next; Sub end //link the current particle to the pointer field of //move the pointer to the next particle node the end of the chain table of particle unit Fig. 6: The algorithm for recording the association between particles and cubic grids It will be helpful to take the case as shown in Fig. 7b as an ball example. When the 428 particle is the current particle, the 428 432 620 656 697 box # particle units associated with the same cubic unit of 628 628 ĂĂ ĂĂ ĂĂ ĂĂ ĂĂ will be searched along the chain table after the current par - *next *next *next *next *next *next # # # ticle unit of 428 first, which are the particle units 432 , 620 , *p[0] ĂĂ # # ball 656 and 697 . Afterwards, the particle units associated with *p[1] ĂĂ 617 751 758 # # box the 629 cubic unit, the east neighbour of 628 containing *p[2] ĂĂ 629 ĂĂ ĂĂ ĂĂ # the current particle unit of 428 , will be searched along the *p[3] ĂĂ *next *next *next *next chain table to assess whether they collide with the 428 par- # # # *p[4] *p[0] ĂĂ ĂĂ ticle or not, which are particle units 617 , 751 and 758 . ball *p[5] *p[1] ĂĂ 302 However, it is not necessary to search the particles asso- box *p[6] ĂĂ *p[2] ĂĂ 630 ĂĂ ciated with the same cubic unit whose series numbers are *p[7] ĂĂ *p[3] ĂĂ *next *next less than that of the current particle, or whose positions in the chain table are in front of the current particle. As shown *p[8] ĂĂ *p[4] ĂĂ *p[0] ĂĂ # # # in Fig. 7b, particle units 620 , 656 and 697 will be searched *p[9] *p[5] *p[1] ĂĂ ĂĂ when particle 432 is the current one. Thus, the particle unit *p[10] ĂĂ *p[6] ĂĂ *p[2] ĂĂ # # of 428 will not be searched because particle 432 has al- *p[11] *p[7] *p[3] ĂĂ ĂĂ ĂĂ ready been searched when 428 was the current particle. *p[12] ĂĂ *p[8] ĂĂ *p[4] ĂĂ The conclusion is that the current particle does not have *p[9] ĂĂ *p[5] ĂĂ to be judged as to whether or not it will collide with all *p[10] ĂĂ *p[6] ĂĂ other particles in the space of particle motion. Therefore, *p[11] *p[7] ĂĂ ĂĂ the computation time spent on detecting particle colli- *p[12] *p[8] ĂĂ ĂĂ sions can be saved through introducing the search algo- (a) *p[9] ĂĂ (b) rithm based on cubic grids. *p[10] ĂĂ *p[11] ĂĂ 628 629 630 1.5 Algorithm of the three-dimensional soft- 656 617 302 *p[12] ĂĂ sphere model of the DEM As mentioned in the discussion of the hypothesis Fig. 7: An example for the explanation of the association between par - ticles and cubic grids recorded in the computer memory and principle, the algorithm of the three-dimensional Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 152 | Clean Energy, 2021, Vol. 5, No. 2 Sub begin boxno=boxnof(ball[balli].x0); _pball=&ball[balli]; //calculate the series number of the cubic box //get the pointer of the current particle node of unit matched with the current particle the chain table yes ip=0; _pball->next==NULL //the first neighboring unit yes no ip<13 _pball=_pball->next; no //the 13 neighboring units //move the pointer to the next particle node yes no box[boxno].p[ip]==NULL Collision yes no yes head=box[boxno].p[ip]; Status analysis //get the pointer of the head particle node of (Mechanical model) the chain table (Thermal dynamics model) ip++; //the next neighboring unit Sub end Fig. 8: The algorithm of determining particle collisions in cubic grids soft-sphere model of the DEM contains three key parts: from the particles to the cubic grids should be found. The the particle-status analysis, the particle-kinematics tree structure describing the association between particle analysis and the search algorithm for particle collisions units and cubic units will be constructed and it will be up- based on cubic grids. The algorithm flow chart is illus- dated at each time step according to the new mapping be- trated in Fig. 9, the calculation process of which can be tween the particles and the cubic grids. presented as follows. (iv) The particle collisions will be searched based on the (i) Before the main program of the three-dimensional constructed tree structure describing the association be- soft-sphere model of the DEM is launched, the initializa- tween the particles and the cubic grids. The particle unit tion information of all particles will be scanned from an linked to the current cubic unit will be first searched and original data file, which is formatted by an initialization judged whether or not collisions with current particles will program or a backup of the results of the main program for occur along the chain table of particle units. The particle the continuing calculation. The initialization information units linked to the neighbours of the one containing the involves all the detailed records of each particle, such as the current particle unit will be searched along the chain table radius r , density ρ , temperature T, the spatial-position of particle units. p p p coordinates x =(x , x , x ), the translational-velocity (v) If the current particle does not collide with any other 0 1 2 vector v =(v , v , v ) and the rotating angular velocity particle in the space of particle motion, the status ana- 0 1 2 ω =(ω , ω , ω ). lysis of the current particle will not be changed, i.e. the 0 1 2 (ii) The space of particle motion will be meshed to cubic current particle-kinematics analysis will remain the same grids. As preparative work, the formation of cubic grids in the as in the last time step. Otherwise, the new status analysis space of particle motion is indispensable to the whole algo- will be launched and the results of dynamic analysis of the rithm of the three-dimensional soft-sphere model of the DEM. current particle will be changed according to the current (iii) All particles at any time step should be associated new status analysis. with the corresponding cubic unit of the cubic grids ac- (vi) The iteration will be continued until the number of cording to the particle spatial position, i.e. the mapping time steps exceeds the maximum value. Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 153 Begin Particle initialization …… Grids formation t = t …… no t ≤ t max yes Associating particles with grids Neighbor searching no Particle collision yes Status analysis (Mechanical model) (Thermodynamics model) Kinematics analysis t = t (Kinematical model) t = t 0 0 (Heat transfer model) Particle info Particle info Particles initialization End t = t programme Particle info Results calculation Results analysis Analysis 1 Analysis 2 Analysis 3 …… Analysis n …… Result 1 Result 2 Result 3 Result n Fig. 9: The algorithm chats of the three-dimensional soft-sphere model of the DEM for computing the particle flow and heat transfer 2.1 Validation of the character of the 2 Validation of the three-dimensional particulate flow soft-sphere model of the DEM In this section, the character of the particle flow simulated The program-development process of the three- by the developed three-dimensional soft-sphere model of dimensional soft-sphere model of the DEM has been the DEM will be compared with a validation experiment of presented above in detail. However, the reasonability glass beads discharging from a transparent silo to validate and correctness of the program, especially the particle- the reasonability and correctness of the particle-mechanics mechanics model and the particle-thermodynamics model in the particle-status analysis of the three-dimensional model, should be validated against experimental data. soft-sphere model of the DEM. The experiment [37] has been In this research, one major goal was to confirm some carried out by González-Montellano et al., whose results will key parameters in the particle-mechanics model and be referenced to compare with the results calculated from the particle-thermodynamics model of the three- the developed three-dimensional soft-sphere model of the dimensional soft-sphere model of the DEM, which is DEM program in this research. The illustration of González- useful and necessary in the application of the model to Montellano’s experimental equipment appears in Fig. 10. research problems. t = t + Δt Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 154 | Clean Energy, 2021, Vol. 5, No. 2 L = 0.25 m in the silo is ~86% of the height of the right quadrangular prism, and the filling density of the glass beads in the silo –3 is 1423  kg  m [37]. The simulated results from the devel- oped three-dimensional soft-sphere model of the DEM program show that the filling height of glass beads in the silo is 87.7% of the height of the right quadrangular prism, with an relative error of 1.98% compared to the measured value in the validation experiment, and the calculated –3 filling density of glass beads in the silo is 1490.45  kg  m , with a relative error of 4.74% compared to the measured value in the validation experiment. Thus, the simulation results from the developed three-dimensional soft-sphere model of the DEM are acceptable. 2.1.2 Discharge time The discharge time of glass beads T from the silo can discharge be measured by simply opening the gate at the exit of the hopper to complete the discharge process. The measured T and the simulated T are listed in Table 4. discharge discharge Compared with the validation experiment, the T discharge calculated by the developed three-dimensional soft-sphere model of the DEM has a relative error of 3.24%. Thus, the agreement between the measured T in the check ex- discharge 62.5 periment and the simulated T in the DEM simulation discharge is satisfactory. 2.1.3 Flow pattern of glass beads in the silo The flow pattern of glass beads during the process of discharge is an important result in the validation of the mechanics model of particle-status analysis in the three- dimensional soft-sphere model of the DEM program. The simulated results by the program developed in this re- D = 57 mm search not only can be validated by the flow pattern re- corded in the experiment, but also can be checked by Fig. 10. Schematic diagram of the transparent silo for containing the TM commercial software, such as EDEM . The validation re- glass beads for validating the mechanics model in particle-status ana- lysis in the three-dimensional soft-sphere model of the DEM program sults of the flow pattern of the glass beads in discharge from the silo are shown in Fig. 11. Each column of Fig. 11 is the validation results of the As shown in Fig. 10, the transparent silo is made of flow pattern in the silo at the same reduced time, which plastic (PMMA), through which the character of par - is defined as the ratio of real time t to discharge time ticulate flow can be recorded by a camera (Genie H1400- real T . As shown in the recorded experimental graphs Monochrome) with a rate of 50 fps and resolution ratio of discharge in Fig. 11, three colours of the glass-beads bed layer are 280 × 630 pixels. The shape of the transparent silo is com- filled in the silo in the order of grey, white, black and posed of a right quadrangular prism and a right frustum, white (the order of blue, white, black and white in the whose size is illustrated in Fig. 10. The glass beads with a simulation) at the reduced time of 0T . During the diameter of 13.8 mm were chosen as particulate materials discharge discharge process of glass beads in the silo, the flat bed filling the transparent silo, whose parameter values of the layer of glass beads is curved in the right quadrangular mechanics model listed in Table 3 are the same as reported prism part of silo and is curved more seriously from a U by González-Montellano. shape to a V shape in the right-frustum part of the silo because the particle velocity in the centreline of the silo 2.1.1 Filling height and filling density in the silo is larger than that closer to the wall of the silo. It can About 14  000 particles were calculated according to the be concluded from Fig. 11 that the flow pattern of the mass and the mean particle diameter of the glass beads. glass beads in the silo simulated by the developed three- Similarly, 14000 glass-bead units filling the same size dimensional soft-sphere model of the DEM program is of silo freely were simulated by the developed three- similar to that recorded in the experiment and the re- TM dimensional soft-sphere model of the DEM program. In sults simulated by the commercial EDEM Academic 2.3 the validation experiment, the filling height of glass beads version software. L = 0.2 m D = 57 mm T =0.19m H = 0.50 m Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 155 Table 3: The values of parameters in the mechanics model of glass beads Recovery coefficient Friction coefficient e/– μ/– Density Elastic module Poisson ratio –3 ρ /kg m E/Pa v/– e e μ μ p pw pp pw pp 2516 4.1 × 10 0.22 0.62 0.75 0.3 0.3 Table 4: The discharge time of glass beads in a silo measured (a) Observed in the validation experiment in the check experiment and that calculated by the developed three-dimensional soft-sphere model of the DEM program discharge, simulation No. T (program developed in this research) discharge, experiment 1 29.32 sec 30.25 sec 2 29.28 sec 3 29.20 sec 29.27 sec 30.25 sec TM (b) Simulated by the EDEM Academic 2.3 2.1.4 Particle-velocity distribution in the silo The mechanics model of particle-status analysis in the three-dimensional soft-sphere model of the DEM program also can be validated through the distribution of the vel- ocity of the glass beads in the silo, which appears in Fig. 12. As shown in Fig. 12a, the radial-velocity distribution from the exit of the silo 0.02 m at the time of 0.2 sec after opening the gate of the silo is similar to but a little less TM than the simulated results from the commercial EDEM Academic 2.3 version software. In addition, MFI, the ratio of the glass-bead velocity near the wall of the silo to that (c) Simulated by the DEM program developed by this research in the centreline of the silo along the height of the silo at the time of opening the gate of the silo, has been compared TM with the simulated results from the commercial EDEM Academic 2.3 version software, as illustrated in Fig. 12b . Therefore, the particulate flow simulated by the developed three-dimensional soft-sphere model of the DEM can be satisfactorily validated by experimentation and commer - cial software. 2.2 Validation of the character of particulate heat transfer 1/2 T 3/4 T 0 T 1/4 T discharge discharge discharge discharge The particulate-heat-transfer model has been coupled into Fig. 11: Comparison of the flow pattern of the glass beads in the silo the developed three-dimensional DEM based on a soft- during the discharging process observed in the check experiment, the sphere model. However, the new coupled particulate heat- TM calculation from EDEM Academic 2.3 and the simulation by the three- transfer model and the values of its parameters can be dimensional soft-sphere model of the DEM program validated by some laboratory-scale experiments. In this re- search, a rotary calciner built by Chaudhuri [38], as shown 2.2.1 Alumina particles heated in a rotating calciner in Fig. 13, is adopted to validate the correctness of the About half the volume of the calciner was filled with the particulate-heat-transfer model embedded in the devel- alumina particles at a room temperature of 25°C and the oped 3D DEM program and estimate the effect of various wall of the calciner was uniformly heated using an indus- materials on the heat transfer. trial heat gun maintained at 100°C. Meanwhile, the calciner –1 The particulate materials consist of two types of spher - with a rotation speed of 20 r min was controlled by a step ical particles: alumina particles and copper-alloy particles, motor . The calciner was stopped at a certain time interval whose parameter values in the mechanics model [39] and and the temperature of the particulate bed was tested the thermodynamic model [38] are summarized in Table 5. using 10 inserted thermocouples, as shown in Fig. 4. After Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 156 | Clean Energy, 2021, Vol. 5, No. 2 TM TM Simulated in the EDEM Academic 2.3 Simulated in the EDEM Academic 2.3 Simulated in the developed 3D-DEM Simulated in the develped 3D-DEM 1.0 0.45 0.9 0.40 0.8 0.35 0.7 0.30 0.6 0.25 0.5 0.20 0.4 0.15 0.3 0.10 0.2 0.05 0.1 0.0 –0.15 –0.10 –0.05 –0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Radius position x / m Height position of silo z / m Fig. 12: The comparison of the particle-velocity profile of the bed layer along the radius direction at different height positions (0.2 sec) between the TM simulated results by EDEM Academic 2.3 and the simulated results by the three-dimensional soft-sphere model of the DEM program As shown in Fig. 15a, the average temperatures of the (a) (b) particulate bed at the positions of the thermocouples are increased with the rotating process turbulently because the relative standard deviation of the alumina particles in the calciner will be increased at the beginning of the heating process. The heat balance of the particulate bed layer in the calciner can be carried out and the coefficient of heat transfer can be calculated as follows: dT m c = α e CL (T − T )(30) s p,s s s wall s dt Å ã Å ã T − T α e CL s s s wall ln = − t(31) T − T m c D = L = 0.0762 m wall s,0 s p,s Fig. 13: The experiment built up by Chaudhuri for validating the par - where m is the whole mass of the aluminium-particulate ticulate heat  transfer model coupled in the three-dimensional soft- sphere model of the DEM program bed layer in the calciner, 0.875 kg; c is the heat capacity p,s –1 –1 of the alumina particle, 875 J  kg K ; T and T are the s s, 0 the temperatures of 10 thermocouples were recorded, the particulate layer temperature and the its initial value, K; thermocouples were extracted and the calciner was ro- T is wall temperature of the calciner, 398.15 K; stands C wall tated again. The temperature profile of the particulate bed for the circle length of the calciner, 0.203 m; π e represents in the calciner is simulated and deduced by the new im- the ratio of the flank area of the calciner occupied by the proved 3D DEM program coupled to the thermodynamics aluminium-particulate bed layer, ~0.5; α describes the co- model from 0 to 12 sec, which is illustrated in Fig. 14. efficient of heat transfer of the calciner heating process, It is not difficult to observe that the heat is transferred –2 –1 W m  K . The calculated value of the heat-transfer coeffi- from the boundary to the centre of the particulate bed –2 –1 cient is ~170.751 W m  K , a relative error of –27.07% with gradually. During the uniform rotation of the calciner, –2 –1 the measured value by Chaudhuri of 124.527 W m K . the alumina particles near the hot wall are heated dir - According to the results simulated by the new improved ectly and rise up to a higher position on the incline of 3D DEM program coupled with the heat-transfer model the dynamic angle of repose. Furthermore, the alumina compared with the measurement in the laboratory experi- particles at the free surface of the inclined particulate ment, the correctness and agreement of the new coupled bed are warmed up indirectly by the higher-temperature heat-transfer model are acceptable. particles heated by the calciner wall rolling down from the top of the inclined bed layer to the bottom. The curve 2.2.2 Copper-alloy particles heated in a rotating calciner of the average temperature at the positions of the 10 Similarly, the calciner heating process of copper-alloy par - thermocouples is simulated by the new improved 3D ticles, as another type of particulate material for validating DEM program and compared with that measured by the accuracy of the new improved 3D DEM program, is Chaudhuri, which are illustrated in Fig. 15a. The coeffi- also simulated in this research. About half the volume of cient of heat transfer of this process also can be found as the calciner was filled with the copper-alloy particles at shown in Fig. 15b. a room temperature of 25°C and the wall of the calciner Particle vertical velocity –1 V / m.s p,z MFI Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 157 Table 5: The parameter values of alumina and copper alloy in a mechanics model and a thermodynamic model in the developed three-dimensional soft-sphere model of the DEM program Thermal- Poisson Recovery conduction Particle Density Particle Elastic module ratio coefficient Heat capacity coefficient –3 –1 –1 –1 –1 material ρ /kg m radius/mm E/Pa v/– e/– c /J kg  K k /W m  K p p p Alumina 3900 1.0 7.17 × 10 0.33 0.80 875 36 Copper 8900 2.0 9.6~11 × 10 0.34 0.80 172 385 (a) 0.0 sec (b) 2.4 sec (c) 4.8 sec (d) 7.2 sec (e) 9.6 sec (f) 12.0 sec Particle temperature/°C 60 55 50 45 40 35 30 25 20 Fig. 14: The temperature distribution prediction of particulate material of the alumina spherical particles heated in the calciner simulated by the three-dimensional soft-sphere model of the DEM program was uniformly heated using an industrial heat gun main- temperature difference between the copper-alloy par - tained at 1025°C. Meanwhile, the calciner with a rotation ticles and the calciner wall. Fig. 16b shows that the relative –1 speed of 20 r min was controlled using a step motor. The standard deviation of the temperatures of the copper-alloy average-temperature curve of the copper-alloy particles particles increases with heating time from 0 to 5 sec, and and the relative standard deviation curve of all copper- is decreased with heating time after ~5 sec. The uniformity alloy temperatures can be calculated and compared with of the original copper-alloy-particle temperature is des- those simulated by Chaudhuri, as shown in Fig. 16. troyed by the higher temperature of the particles near the It can be concluded from Fig. 16a that the average tem- calciner wall as compared to that of the particles in centre perature of the copper-alloy particulate bed in the calciner of the calciner particulate bed layer. Since the average increased with the heating time. However, the amplifi- temperature of the copper-alloy particles is limited to the cation of that declined gradually because of the reduced calciner-wall temperature, the temperature difference Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 158 | Clean Energy, 2021, Vol. 5, No. 2 (a) (b) 0.10 0.00 Measured in experiement by B. Chaudhuri (a) (b) 0.09 Simulated by authors of this research –0.01 0.08 -0.02 0.07 -0.03 0.06 -0.04 0.05 -0.05 0.04 -0.06 0.03 -0.07 0.02 -0.08 Measured in experiement by B. Chaudhuri Simulated by author of this research 0.01 -0.09 Linear fit of measured value Linear fit of simulated value 0.00 -0.10 02468 10 12 02468 10 12 Time t / sec Time t / sec Fig. 15: Comparison of the average temperature curve of the alumina spherical particles between that measured by Chaudhuri and the results cal- culated by the three-dimensional soft-sphere model of the DEM program (a) (b) 1000 300 Simulated by B.Chaudhuri et al. Simulated by B.Chaudhuri et al. 900 Simulated by the authors of this research Simulated by the authors of this research 500 150 0 0 012345678 910 0123456789 10 Time t / sec Time t / sec Fig. 16: The comparison of the average  temperature curve and standard deviation of copper-alloy spherical particle temperatures between the simulated results by Chaudhuri and those by the three-dimensional soft-sphere model of the DEM program among all particles in the particulate layer is reduced and 3 Prediction of particulate behaviour of a new uniformity of copper-alloy-particle temperature coal and sand appears at the end of the heating process in the rotating The reasonability and correctness of the developed three- calciner. dimensional soft-sphere model of the DEM program, The temperature profile of the copper-alloy particulate especially the particle-mechanics model and the particle- bed in the calciner is simulated and deduced by the new thermodynamics model, have been validated through a improved 3D DEM program coupled to the thermodynamics series of validation experiments. However, to realize the model from 0 to 20 sec, which is illustrated in Fig. 17. ultimate goal of the simulation of particulate behaviour of The temperature profile of copper-alloy particles in flow and heat transfer in the process of coal pyrolysis in a the rotating calciner also can be simulated by the new downer reactor by the developed three-dimensional soft- improved 3D DEM program and compared with the sphere model of the DEM program, the physical-property simulation results from Chaudhuri, which are shown in parameters of coal particles as fuel and sand particles as Fig. 17. According to the results simulated by the three- heat carriers should be clearly validated and confirmed. dimensional soft-sphere model of the DEM program The parameter values of the mechanics model of coal coupled with the heat-transfer model compared with the and sand in the particle-status analysis of the three- measurement in the laboratory experiment, the correct- dimensional soft-sphere model of the DEM program for ness and the agreement of the new coupled heat-transfer simulating the particulate behaviour of flow are listed in model are acceptable. Table 6 [32, 40]. Average temperature Dimensionless temperature T /K copper, avg (T -T )/(T -T ) avg o w o Standard deviation of temperature ln[(T -T )/(T -T )] wall avg wall avg,0 S / K copper Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 159 Simulated by B. Chaudhuri Simulated by author 0.0 sec 3.0 sec 9.0 sec Particle temperature/K 950750 550350 Fig. 17: Comparison of copper-alloy-particle temperature distribution in the calciner at different times between the calculated results by Chaudhuri and those by the three-dimensional soft-sphere model of the DEM program Table 6: The values of the parameters of particle materials in the filling and heaping test Particle-size Elastic Poisson Sliding-friction Rolling-friction Recovery Density distribution modulus ratio coefficient coefficient coefficient –3 Particle typesρ /kg m d /mm E/Pa v/– μ /– μ /– e/– p p s r 9 –5 Coal 1250 1~2 5.08 × 10 0.28 0.51 5 × 10 0.85 10 –5 Sand 2450 1~2 4.10 × 10 0.22 0.66 5 × 10 0.90 For convenience, the wall is assumed to have the same properties as particles with infinite diameter. Table 7: The relationship between the particle diameter and the particle number with a constant mass for coal and sand (1) (2) (3) (4) (5) Coal-filling mass m /g 3.7468 7.5285 11.3451 15.1965 19.0828 coal Coal-particle number 1665 3328 5069 6804 8576 Sand-filling mass m /g 6.5266 13.0199 19.6314 26.3589 33.1530 sand Sand-particle number 1467 2951 4462 6041 7578 particles of coal or sand in the simulation are generated 3.1 Filling tests for coal and sand by an initialization program of the three-dimensional soft- The filling of a vertical pipe with coal and sand separately sphere model of the DEM simulation with the uniform for testing the filling height and filling density of coal and distribution of particle diameters ranging from 1 to 2 mm, sand individually has been simulated by the developed whose particle densities are considered as the values listed three-dimensional soft-sphere model of the DEM program in Table 6, and the generated particle numbers of coal and and compared with the practical experiment. Experimental sand are listed in Table 7, separately. results for the filling height and the filling density are com- The particle-filling state of coal and sand in the vertical pared with the model simulation. The particle diameter quartz pipe are photographed, as shown in Fig. 18a and with uniform distribution is confined to between 1 and b, separately. Meanwhile, the simulations of the particle- 2  mm through screening. The vertical pipe with an inner filling state for coal and sand using the three-dimensional diameter of 1.78  cm is made of quartz. Five different ini- soft-sphere model of the DEM are illustrated in Fig. 18c tial masses of coal particles and sand particles, listed in and d. The particle-filling height shown in Fig. 18e and Table 7, are weighed using an electronic balance and filled the particle-filling density shown in Fig. 18f have been in the same type of vertical quartz pipe, respectively. The Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 160 | Clean Energy, 2021, Vol. 5, No. 2 (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) 0.11 2000 0.11 1000 The measured h The measured h Coal Sand fill fill 0.10 0.10 The simulated h The simulated h fill fill 0.09 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 Themeasured Themeasured fill 650 fill 0.01 0.01 ρ ρ (e) Thesimulated (f) Thesimulated fill fill 0.00 600 0.00 1000 2468 10 12 14 16 18 20 48 12 16 20 24 28 32 36 Filling mass m / g Filling mass m / g fill fill Fig. 18: The simulation and experiment of the filling height and the filling density in the vertical pipe with an inner diameter of 1.78 cm for coal and sand, respectively experimentally measured and calculated in the simulation, further increasing the filling density. (iii) The average rela- the distinction of which is also shown in Fig. 18e and . f tive errors of the filling density in the vertical pipe between The conclusions from Fig. 18e and f can be summarized the simulated and measured values are ~6.7% for coal par - as follows. (i) The maximum relative errors of the fill height ticles and 6.5% for sand particles. The agreement between in the vertical pipe between the simulation and the meas- the simulation and the experiment is satisfactory. urements are –9.5% for coal particles and –5.7% for sand particles, respectively. (ii) The effect of the initial filling 3.2 Voidage distribution along the bed height mass on the particle-filling density in the vertical pipe is not remarkable, although the filling density is slightly in- The voidage distribution along the bed height can be creased with the initial filling mass or filling height, which used to check the reliability of the parameters in the can be interpreted as the pressure acting on the lower bed mechanics model of coal and sand in the particle- layer from the upper zone to reduce the bed layer voidage, status analysis of the three-dimensional soft-sphere Fillingheight h /m fill –3 Filling density / kg•m fill Filling height h / m fill –3 Filling density / kg•m fill Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 161 model of the DEM program. At the same time, the effect The bed-voidage profiles along the bed height for par- of the particle diameter on the profile of voidage along ticles with a uniform diameter from 1.0 to 2.0 mm filling in the bed height can be simulated using the developed the vertical glass tube with an inner diameter of 1.78 cm three-dimensional soft-sphere model of the DEM pro- for coal and sand separately are shown in Fig. 19c and . f gram. In this section, 19.083  g of coal particles and It can be observed that the average bed voidage ranged 38.013 g of sand particles with five different diameters from 0.3 to 0.4 and slightly increased with the bed height. are respectively placed into five different tubes with an The reason is that the lower-layer particles are compelled inner diameter of 1.78  cm. The particle numbers cor - by the pressure from the upper bed layer, and the voidage responding to the different particle diameters at one in the lower bed is slight smaller than that in upper bed constant mass are listed in Table 8 for coal and sand layer. It is not difficult to confirm that the voidage of separately. spherical particles under the cubic arrangement is 0.48 The simulation results for particles of coal and sand and the voidage of spherical particles under the closest with five diameters but with the same constant mass arrangement is 0.26. It can be concluded that the param- filling in the five vertical pipes are shown in Fig. 19. eter values of the mechanics model of coal and sand in Table 8: The relationship between the particle diameter and the particle number with a constant mass of materials No. (1) (2) (3) (4) (5) (6) Particle diameter/mm 2.0 1.8 1.6 1.4 1.2 1.0 Particle number of 19.0828 g coal 3644 4999 7118 10 625 16 872 29 156 Particle number of 38.0125 g sand 3702 5079 7232 10 795 17 142 29 622 (1) (2) (3) (4) (5) (6) (1)(2) (3)(4) (5)(6) CD 0.10 0.10 Particle diameter Particle diameter 0.08 0.08 2.0mm 2.0mm 1.0mm 1.0mm 0.06 0.06 0.04 0.04 0.02 0.02 0.00 0.00 0.00.1 0.20.3 0.40.5 0.60.7 0.80.9 1.0 0.00.1 0.20.3 0.40.5 0.60.7 0.80.9 1.0 Bed voidage ε Bed voidage ε / – / – bed bed Fig. 19: Calculation results of the filling experiment of coal and sand under the condition of the same filling mass but different particle diameters and the description of bed-voidage distribution along the height Bedheight h /m bed Bedheight h /m bed Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 162 | Clean Energy, 2021, Vol. 5, No. 2 the particle-status analysis of the three-dimensional soft- sand in the simulation are generated by an initialization sphere model of the DEM program are suitable. program of the three-dimensional soft-sphere model of the DEM simulation with a uniform distribution of par - ticle diameter ranging from 1 to 2 mm, and the generated 3.3 Heaping test for coal and sand particle numbers of coal and sand that are listed in Table 7, separately. The purpose of the particle-heaping test is to check the The experimental results and the simulation results reasonability of the sliding-friction coefficient μ and the for particles of coal and sand heaping on the platform are rolling-friction coefficient μ in the mechanics model of shown in Fig. 20. coal and sand in the particle-status analysis of the three- Fig. 20a shows the front views of the heaping state in dimensional soft-sphere model of the DEM program. The the experiment and simulation for observing the angles of pile height and pile-base radius of coal and sand are repose of coal and sand. Meanwhile, the simulations of the measured and simulated in this case. The particle diam- particle-heaping state for coal and sand using the three- eter with uniform distribution is confined to between 1 dimensional soft-sphere model of the DEM are illustrated and 2  mm through screening. The particles of coal and Front view Sand Coal Coal Sand Top view Coal Sand The linear relationship between the heaping height and heaping diameter 0.026 0.026 Measured value in test Measured value in test Calculated value in DEM Calculated in DEM 0.024 0.024 Linear fit of measured value Linear fit of measured value 0.022 0.022 Linear fit of calculated value Linear fit of calculated value 0.020 0.020 0.018 0.018 0.016 0.016 0.014 0.014 0.012 0.012 0.010 0.010 0.0150.020 0.0250.030 0.0350.040 0.0150.020 0.0250.030 0.0350.040 r /m r /m coalheap sandheap Fig. 20: The particle-heaping-state comparison of the experiments and the simulations for coal and sand. (a) The front view of the heaping states in the experiments and simulations for coal and sand, separately. (b) The top view of the heaping states for five kinds of heaping mass in the simula- tions for coal and sand, separately. (c) The linear relationship between the heaping height and the heaping diameter in experiments and simulations for coal and sand, separately. h /m coalheap h /m sandheap Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 163 in Fig. 20b. The heaping height and heaping diameter of than that of the sand hourglass. The relative errors of the the coal and sand particles are chosen to describe the simulated results of coal hourglass and sand hourglass comparison of the simulation and the experimental re- are 13.59% and 12.68% compared with the experimental sults, which are presented as Fig. 20c. results. The heaping height and heaping diameter are measured According to the conclusions from the discussion con- during coal and sand heaping on the platform with five ini- tent in the research, the developed three-dimensional tial masses as listed in Table 7 and plotted with filled points soft-sphere model of the DEM program can be capable of in Fig. 20c. The relationship between the heaping height and predicting the particulate behaviour of static state and the heaping diameter can be fixed through linear regression, flow for coal and sand. which is also plotted with the solid line in Fig. 20c. The filling height is related to the filling diameter as follows: 4 Conclusions h ≈ 0.51 · r (32) coal,measured coal,measured A three-dimensional soft-sphere model of the DEM for h ≈ 0.66 · r (33) simulating the particulate behaviour of flow and heat sand,measured sand,measured transfer is developed, the reasonable hypothesis and the The slopes of Equations (32) and (33) are presented as the ingenious algorithm of which have been presented in as tangent values of the angles of repose of coal and sand, much detail as possible. The organizational structure of separately. Therefore, the angles of repose are 27.07° for the developed three-dimensional particulate soft-sphere coal and 33.35° for sand. model of the DEM contains algorithms for determining The heaping height and heaping diameter are simu- particle collisions, for the status analysis of particles and lated, and have been plotted using open symbols in Fig. for the kinematics analysis of particles. The search algo- 20c. The relationship between the heaping height and the rithm for particle collisions is based on cubic grids and heaping diameter can be fixed through linear regression, the splicing of cubic grids can originally solve the problem which is also plotted with a dotted line in Fig. 20c. The of searching for particle collisions in a particulate system. filling height is related to the filling diameter as follows: h ≈ 0.45 · r (34) coal,simulated coal,simulated The particulate-status analysis is used for solving the causes of particle motion and heat transfer, which in- h ≈ 0.68 · r (35) sand,simulated sand,simulated volves the mechanics model and the thermodynamics Similarly, the simulated angle of repose of the coal pile is model. The particulate-kinematics analysis is described 24.17° with a relative error of –10.71% compared with the by some basic laws of physics containing Newton’s second experimental value, and the angle of repose of the sand law for particle translational motion, the rigid-body law of pile in the simulation is 34.17° with a relative error of 2.46% rotation for rotating particles and the heat-transfer law compared with the experimental value. for two particles with different temperatures. The correctness and reasonability of the developed three-dimensional soft-sphere model of the DEM program 3.4 Hourglass using coal and sand for timing has been validated and checked through simulating the Particles can be filled into an hourglass for timing. conditions compared with those in the experiment. In this However, the discharge time of the same volume of par - research, the mechanics model in particle-status analysis ticles with the same uniform diameter distribution is and the heat-transfer model in particle-kinematics ana- determined by the physical properties of the solid par - lysis of the three-dimensional soft-sphere model of the ticles. Therefore, the discharge times of the same volume DEM have been validated by the experiment of glass-bead of coal and sand with the same particle size from the discharge from a silo and metal particles heating up in a same hourglass must be different. 13500 coal particles calciner. The agreement between the experimental and (about 70.65 g) and 13 500 sand particles (about 138.47 g) simulated results is consistent and satisfactory, which with the same diameter of 2  mm have been filled into can indicate that the developed three-dimensional soft- the hopper of the hourglass to measure the time scale sphere model of the DEM program is successful and cap- of the two hourglasses, which has been also simulated able of predicting the flow and heat transfer in particulate by the developed three-dimensional soft-sphere model systems. of the DEM program. The simulation results of the dis- In addition, to prepare the simulation of the coal- charge process of coal and sand from the hopper of the pyrolysis process in a downer reactor using the developed hourglass are shown in Fig. 21. three-dimensional soft-sphere model of the DEM program The time scales of the two hourglasses containing coal in the next step, some necessary physical-property param- and sand separately have been compared between the eters of the fuel particles of coal and the heat-carrier par - experiment and simulation, respectively, which has been ticles of sand have been validated and confirmed through listed in Table 9. comparing the experiments of particle filling in the tube It can be concluded from Fig. 21 and Table 9 that the and particle heaping on the platform. The agreement be- discharge time of the coal hourglass is a little shorter tween the simulated results and tested data is satisfactory. Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 164 | Clean Energy, 2021, Vol. 5, No. 2 Coal Sand A B 1/4 T 1/2 T discharge, coal discharge, coal CD 3/4 T 1 T discharge, coal discharge, coal Fig. 21: The simulated results of the sand clock using coal and sand by the developed three-dimensional soft-sphere model of the DEM program Table 9: The discharge times of coal and sand in the hourglass measured in the validation experiment and those calculated in the DEM simulation Coal Sand No. T /sec T /sec T /sec T /sec discharge, tested discharge, simulated discharge, tested discharge, simulated 1 10.54 12.2 11.59 12.8 2 10.88 11.52 3 10.89 11.12 4 10.76 11.32 5 10.61 11.27 Average 10.74 12.2 11.36 12.8 Therefore, it can be concluded that the parameter values Nomenclature of the status-analysis model and the kinematics-analysis Scalar: model for coal and sand particles are acceptable for the A = particle surface area, m (p) next step of simulation of the coal-pyrolysis process in a A = particle contact area, m (p),c downer reactor using the developed three-dimensional A = area of tiny gas interval between two contact sur - gap soft-sphere model of the DEM program. faces, m Downloaded from https://academic.oup.com/ce/article/5/2/141/6220074 by DeepDyve user on 13 April 2021 Bing Liu | 165 –1 –1 –1 c = specific heat capacity, J kg  K v = translational-movement-velocity vector, m s –1 d = width of tiny gas interval between two contact v = particle normal translational velocity, m s gap –1 surfaces, m v = particle tangential translational velocity, m s d = particle diameter, m x = space-position vector, m I = particle rotational inertia, kg m ϕ = phase-position vector, rad (p) –1 l = distance of two particles, m ω = rotating-movement-velocity vector, rad s l = length of side a of a cubic grid unit, m l = length of side b of a cubic grid unit, m l = length of side c of a cubic grid unit, m l = distance between particle centre and wall, m Conflict of Interest pw E = elastic modulus, Pa None declared. e = recovery coefficient, − r = particle radius, m (p) m = particle mass, kg (p) References m* = reduced mass of two particles, kg k = thermal-conduction coefficient of gas phase, [1] Zhu HP, Zhou ZY, Yang RY, et  al. 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Journal

Clean EnergyOxford University Press

Published: Jun 1, 2021

References