American Journal of Mathematical and Computational Sciences 08; (): 0-6 http://www.aascit.or/journal/ajmcs Pneumatic Conveyin in Horizontal Pipes: Eulerian Modelin and Pressure Drop Characteristics Pandaba Patro Department of Mechanical Enineerin, Veer Surendra Sai University of Technoloy, Odisha, India Email address ppatro_me@vssut.ac.in Keywords Pneumatic Conveyin, Pressure Drop, Eulerian Modelin, KTGF, CFD Received: July, 07 Accepted: November, 07 Published: January, 08 Citation Pandaba Patro. Pneumatic Conveyin in Horizontal Pipes: Eulerian Modelin and Pressure Drop Characteristics. American Journal of Mathematical and Computational Sciences. Vol., No., 08, pp. 0-6. Abstract In the present paper, as solid flow (i.e. pneumatic conveyin) in a horizontal pipe has been investiated numerically usin the Eulerian or two-fluid model to predict pressure drop. Consideration of inter-particle collisions ive rise to solid phase stresses and are modeled usin kinetic theory of ranular flow (KTGF). It was observed that consideration of inter-particle collisions, particle-wall collision and lift help in the radial dispersion of the solid particles in a horizontal pipe. The value of the numerical parameter specularity coefficient stronly affects pressure drop and hence, has to be chosen correctly. The effect of flow parameters as well as particle properties on pressure drop was investiated in detail. The conclusions are (a) pressure drop increases with as velocity (b) pressure drop increases with solids loadin (c) pressure drop first increases, reach a peak and then decreases with the increase in particle diameter in the rane 5 to 00 micron.. Introduction In pneumatic conveyin, solid particles are movin inside a pipe with the help of a hih speed as stream. The most widely used industrial application of as-solid flows is pneumatic conveyin. Pneumatic conveyin may occur in a horizontal pipe or alon a vertical pipe. Generally, most of the pipin layout is horizontal in nature. Because, pumpin power requirement in horizontal flow is less compared to vertical flow. However, ravity induced settlin on the bottom wall has always been a reat challene in horizontal flows. If the as velocity in horizontal flows is below the saltation velocity, it is insufficient to maintain the solids in suspension and solids bein to settle out and slide or roll alon the bottom of the pipe and hence, as velocity must be more than the saltation velocity to have the particles in suspended mode. For the desin of pneumatic transport systems, knowlede of pressure drop is required. A minimum value of the pressure drop for effective transport without particles settlin is preferred. Many researchers [-4] predicted pressure drop in as-solid flows under low solid loadin conditions (volume fraction in the order of 0 - ). But, in industrial practice, solids loadin are much more than this. It is very difficult to perform experiments under different operatin conditions takin different particle sizes. The availability of hih speed computers and CFD packaes make it comparatively easy to perform numerical experiments under any flow conditions. There are enerally two computational approaches used to investiate as-particle flows: Eulerian-Eulerian approach and Eulerian-Laranian approach. This approach is enerally used in very dilute phase as solid flows. The Euler-Euler method treats the solid phase as a continuum which
American Journal of Mathematical and Computational Sciences 08; (): 0-6 interacts with the fluid continuum. Basically, this approach has been developed for relatively hih solids laodin. This method has been used in the present study. Many researchers [5-0] investiated as-solid flows usin this approach. They had shown that Eulerian model is capable of predictin the flow physics of as-solid flows qualitatively as well as quantitatively. In the present investiation, Eulerian-Eulerian approach has been used for the flow of a as-solid flow in a horizontal pipe accountin for four-way couplin i.e. particle-wall interaction and inter-particle collisions. Lift forces are also considered which is caused by particle rotation due to collisions with the bottom wall and a nonuniform as velocity field. Numerical prediction for pressure drop and velocity profiles were made with the numerical settins validated aainst bench mark experimental data by Tsuji et al. []. An extensive study was also performed to investiate the effect of important flow parameters like inlet as velocity, particle properties and solid volume fraction on the pressure drop prediction. Particle concentrations in the rane % to 0% by solid volume fraction (α ) and particle sizes from 5 micron to 00 micron were considered for the pressure drop prediction. The as used is air with density, ρ =.5 k/m and dynamic viscocity, µ = 5.79 0 k/m.s. Solids loadin ratio (SLR) is defined as the ratio of mass flow rate of solid phase and mass flow rate of as phase. αρs SLR = ( α) ρ () The as Reynold s number is defined as Re ρud = () µ Where D is the pipe diameter, ρ and µ are the density and dynamics viscosity respectively of as phase, U is the inlet as velocity and D is the pipe diameter.. Governin Equations in Eulerian Modelin In Eulerian Modelin, both as and solid phases are treated as continuum and hence, continuity and momentum equations are written for both the phases. The as phase momentum equation is closed usin k-ε turbulence model. Solid phase stresses are modeled usin kinetic theory []. The conservation equation of the mass of phase i (i=as or solid) is ( i i).( i i i) 0 t α ρ + α ρ u = () α i = (4) The conservation equation of the momentum of the as phase is ' ' ( α ρu) +.( α ρuu) = α p +. τ.( α ρuu) t + α ρ + K ( u u ) s s (5) The conservation equation of the momentum of the solid phase is ( α ρ ).( α ρ ) α. τ.( α ρ t + α ρ + K ( u u ) ' ' s sus + s susus = s p ps + s s susus s s s s (6) Ks =? Ks is the as-solid momentum exchane coefficient Solids stress, τ s accounts for the interaction within solid phase, derived from ranular kinetic theory. The as phase stress is T τ = α µ ( u + u ) + α( λ µ ). ui (7) The Reynolds stresses of phase i ( i = as or solid), ' ' i i ρuu employ the Boussinesq hypothesis to relate to the Reynolds stresses to the mean velocity radients. The kinetic turbulent enery and dissipation enery employ the standard k- ε model. Forα > 0.8: K For α <0.8: K s α α ρ u u = (8) s s.65 s CD α 4 dp α ( α ) µ ρ 50.75 α s u s u = + (9) d s αdp In the as-solid flow, particle motion is dominated by the collision interactions (inter-particles as well as particle-wall). This ives rise to solids pressure and solid stresses. Kinetic theory of ranular flows (KTGF) [] can be applied to p
Pandaba Patro: Pneumatic Conveyin in Horizontal Pipes: Eulerian Modelin and Pressure Drop Characteristics describe the effective stresses in solid phase to close the momentum balance equation. Granular temperature is defined as the kinetic enery associated with the random motion of the particles. The ranular temperature ( θ s ) equation for the solid phase is ( ρ α θ ).( ρ α u θ ) ( p I τ ): u.( kθ θ ) γθ φ s + = + + + t s s s s s s s s s s s s s (0) Where ( psi + τs): us is the enery eneration by the solid stress tensor k θ θ s s is the diffusion of enery ( k θ s is the diffusion coefficient) γθ is the collisional dissipation of enery s φ s enery exchane between the solid and as phase The solid phase stress is (KTGF) was used to close the momentum balance equation in solid phase. The computational domain with the cross sectional mesh is shown in fiure. T τs = α sµ s( us + us ) + αs( λ s µ s). usi () Here λ is the bulk viscosity and I is the unit tensor. Solid Pressure: s αsρsθs ρs( ss) αs o, ssθs p = + + e () Where e ss is the restitution coefficient between the solid particles, θ is the ranular temperature and 0 is the radial distribution function for solid phase. Radial distribution: Bulk Viscosity: o, ss s,max αs = ( ( ) ) () α 4 θs λs = αs ρsdpo, ss( + ess)( ) (4) π Granular Shear viscosity due to kinetic motion and collisional interaction between particles µ = µ + µ s s, coll s, kin 0 d pρs( θsπ) 4 s, kin = o, ss s ess 96 αs( ess) + + o, ss 5 µ α ( ) + (5) 4 θs µ s, coll = αs ρsdpo, ss( + ess)( ) (6) 5 π. Computational Procedure Numerical simulations of turbulent and unsteady as-solid flow are performed on a horizontal D pipe of diameter 0 mm and lenth equal to 00 times the diameter. The standard k ε epsilon turbulent model [] with standard wall function was used for as phase and kinetic theory of ranular flow Fiure. Computational domain with cross-sectional mesh. Grid independence test was carried out usin rids of mesh sizes 800, 700 and 70600 cells, respectively. It has been that an increase in the number of rid points had a neliible effect on the computed profiles for velocity and pressure. We choose the second rid with 700 cells for the numerical prediction of pressure drop. The detailed model parameters and boundary conditions used in our simulation are listed in the Table. The numerical parameter values are selected based on the findins of Patro et al. [4, 5]. Description Coefficient of restitution for particle-particle collisions Coefficient of restitution for particle-wall collisions Inlet boundary condition Outlet boundary condition Wall boundary condition Table. Simulation model parameters. Value 0.9 0.95 Maximum iterations 0 converence criteria 0 - time step 0 - s ravity enabled Velocity inlet (Fully developed) Outflow (Fully developed) No slip (Gas) Partial slip (solid) Gas-solid flow is always unsteady due to the presence of the particles, which modulates the as flow field and turbulence. So, it is important to monitor the important flow parameters like solid velocity and volume fraction at outlet. After a lot of iterations, deree of unsteadiness reduces and flow variables have a periodic variation with respect to time and finally, the variation becomes neliible.
American Journal of Mathematical and Computational Sciences 08; (): 0-6 steady state or statistical steady state reime. 4. Results and Discussion Fiure. Behavior of any parameter as predicted from a two-fluid transient simulation. Fiure illustrates the behavior of any parameter as predicted from a two-fluid transient simulation of the as solid flow [6]. From a iven initial condition, the simulation oes throuh an early stae, and finally reaches the so-called statistical steady state reime. When flow parameters start to oscillate around well defined means, statistical steady state reime has been reached. If the behavior of the flow variable becomes a straiht line, it is said to be reached steady state reime. In the present study, measurements were taken in the statistical steady state reime, when the variables reach First of all, validation of the numerical model was done by comparin the predicted results for velocity and pressure drop in horizontal as-solid flows with the benchmark experimental findins of Tsuji et al. []. Experiments were conducted in a 0 mm diameter pipe; particle diameter is 00 micron and density 00 k/m for different solid loadin ratios. The mean velocity of as was varied from 6 to 0 m/s. As the loadin ratio is very low and as velocity is more than the saltation velocity, particle settlin do not happen as predicted in the experiments. First, we used the simplest model nelectin the effect of lift and convection and diffusion terms in the ranular temperature equation. The velocity profiles for solid phase are plotted radially at different mean velocity and solid loadin ratio (SLR) as shown in fiure. It is clearly showin the particle settlin towards the exit portion of the pipe, which is not expected. Solids are supposed to have suspended throuhout the cross section in fully developed conditions as observed in the experiments. So, the simplified model did not yield the required outcome. Fiure. Velocity profiles of solid phase at the pipe exit (radial variation). In confined as-solid flows, particles experience lift force (a phenomena called Manus lift effect) that arises due to the rotation of the particle [7]. Inter-particle collisions as well as particle-wall collisions are responsible for the rotation of the particles. Lift force as well as collisions help in the radial dispersion of the particles preventin particle settlin. Hence, we considered the effect of lift (lift coefficient=0.) and particle-wall collision (restitution coefficient= 0.95). Also, the convection and diffusion terms are considered in ranular temperature equation.
4 Pandaba Patro: Pneumatic Conveyin in Horizontal Pipes: Eulerian Modelin and Pressure Drop Characteristics Fiure 4. Comparison of normalized velocity curves at SLR =. and Um = 0 m/s for (a) as phase and (b) solid phase. Now, the numerical results for velocity profiles are in ood areement with the experimental results as shown in fiure 4. The profiles are not symmetric as the particles have the tendency to settle down due to ravity. The ravitational force makes the flow more complicated in the horizontal pipe than in the vertical one, as was mentioned by Owen [8]. As the main objective of the present research is the prediction of pressure drop, validation with experimental data for pressure drop is important. One factor contributin to pressure drop in as-solid flows is the particle-wall collision momentum loss defined by a numerical parameter known as specularity coeffcient. The correct value for this parameter was chosen by matchin the numerical predictions with the experimental data. Fiure 5 shows ood areement between the experimental and numerical data occurred at specularity coefficient equal to 0.08 with maximum error of 5%. This value alon with other numerical parameter values are used for the subsequent measurement of pressure drop 5. Pressure Drop Prediction Under variable solids loadin and operatin conditions, pressure drop in horizontal as-solid flows has been computed. The important factors like as inlet velocity, particle sizes and particle loadin are investiated here. In as-solid flows, the dra exerted by the as flow on the solid particles is mainly responsible for the transportation alon the pipe. The dra force arises due to the slip velocity and hence, the inlet as velocity is an important parameter in as-solid flows. Gas inlet velocity was varied from 0 to 5 m/s and its effect on pressure drop is presented in fiure 6. We observed that pressure drop increases with increase in as velocity. Fiure 6. Pressure drop variation with mean as velocity for 00 micron particles of density 500 k/m at α = 0.04. Fiure 5. Comparison of pressure drop prediction. Two pipes of different diameter (0mm and 50mm) were used to study the influence of inlet as velocity. Pressure drop reduces with increasin the diameter of pipe similar to sinle phase flows. Most of the pressure loss in as-solid flow comes out of enery lost due to particle-particle collision and particle-wall collision. A bier diameter pipe provides more space and hence decreases the number of collisions causin less pressure drop. Solid volume fraction is defined as the ratio of volume of solid phase and total volume of the mixture. It is a measure of solid loadin in assolid flows. Particle loadin, solids loadin ratio (SLR) and volume fraction are synonymous terms in pneumatic conveyin. Volume fraction in the rane % to 0% was considered to study its effect on pressure drop.
American Journal of Mathematical and Computational Sciences 08; (): 0-6 5 Fiure 7. Effect of solid volume fraction on Pressure drop in a 0mm diameter pipe, ρ = 500 k / m, Um = 5 m/s. s As the particle loadin or volume fraction increases, the pressure drop alon the pipe increases (fiure 7). Increasin volume fraction of solids increases the number of inertparticle as well as particle-wall collisions in the pipe and in turn increasin the pressure drop. Sinh and Simon [9] performed DEM simulation and had shown that the increase of number of particle-particle collisions is reater than the increase of wall-particle collisions. number of particles for a constant solid volume fraction. Hence, the frequency of particle-particle collision and particle-wall collision decreases. So the contributions by E pp and E pw reduce. At the critical value of particle diameter, these three parameters are optimized, and hence maximum pressure drop occurs. However, it is really a difficult task to correlate them with the particle diameter quantitatively. The variation of pressure drop with particle density is shown in fiure 9. The pressure drop increases with particle density at hiher solids loadin. The rate of increase is more as we o on increasin the solids loadin. At lower loadins ( α = 0.0 ), the increase in pressure drop is almost neliible. Fiure 9. Pressure profiles with particle density at different volume fractions for 0mm diameter pipe at Um = 5 m/s. 6. Conclusions Fiure 8. Effect of particle diameter on pressure drop for 0 mm pipe at α = 0.0, Um = 5 m/s. In industrial pneumatic conveyin systems, the same type of material or various materials of different sizes are commonly transported. So, particle size and particle density affects the flow behavior. In the present study, particles of size 5 to 50 micron and density in the rane 500 to 000 k/m are investiated in a 0 mm diameter pipe as well as in a 50 mm diameter pipe. It is observed that pressure drop increases rapidly with increase in particle diameter, reach the peak value and then start decreasin for both pipe sizes (fiure 8). Peak is different for different pipe sizes. The main contributions to the pressure drop in as-solid flows in horizontal pipes are: (a) Enery required to impart dra force on the particles ( E D ) (b) Momentum and enery lost by particle-particle collisions ( E pp) (c) Momentum and enery lost by particle-wall collisions ( E pw) An increase in particle diameter causes an increase in slip velocity and superficial area of the particle. So enery required (dra force) for the solids transport increases. It is obvious that more enery is required to convey larer particles for the same conveyin conditions. At the same time, an increase in particle diameter causes a decrease in the The pneumatic conveyin in a horizontal pipe has been numerically solved usin Euler -Euler approach. The numerical results for velocity profiles and pressure drop are validated aainst the experimental data of Tsuji et al. []. Excellent areement was found by the numerical simulation accountin for four-way couplin i.e. particle-wall and inter-particle collisions as well as considerin the effect of Manus lift. The lift force is enerally very less in as-solid flows in comparison to the dra force, but can not be nelected in the numerical simulation. The particle collisions and lift force helps in the radial dispersion of solid particles preventin settlin due to ravity. Pressure drop depends on the value of numerical parameter known as specularity coefficient. In the present study, numerical pressure drop was in ood areement with the experimental values for specularity coefficient equal to 0.08. It is also observed that pressure loss increases with inlet as velocity, and solids loadin. However, with respect to particle diameter, pressure drop first increases, reaches a peak and then decreases. References [] Molerus, O. (98), Prediction of pressure drop with steady state pneumatic conveyin of solids in horizontal pipes, Chem. En. Sc, Vol. 6, No., pp. 977-984. [] Yan, W. C. (974), Correlations for solid friction factors in vertical and horizontal pneumatic conveyin, AIChE Journal, Vol. 0, No., pp. 605-607.
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