Effect of geometry on flow structure and pressure drop in pneumatic conveying of solids along horizontal ducts

Transcription

1 Journal of Scientific LAÍN & Industrial SOMMERFELD Research: PNEUMATIC CONVEYING OF SOLIDS ALONG HORIZONTAL DUCTS Vol. 70, February 011, Effect of geometry on flow structure and ressure dro in neumatic conveying of solids along horizontal ducts S Laín 1* and M Sommerfeld 1 Energetics and Mechanics Dt., FMRG, Universidad Autónoma de Occidente, Cali (Colombia) Zentrum für Ingenieurwissenschaften, Martin-Luther-Universität Halle-Wittenberg. D Halle (Saale), Germany Received 16 July 010; revised 7 December 010; acceted 9 December 010 This study resents further develoments in Euler/Lagrange aroach to calculate confined article-laden flows in neumatic conveying lines. Secial emhasis is laced on influence of article-wall collisions and wall roughness as well as interarticle collisions with ossible agglomeration on develoing two-hase flow structure and resulting rocess arameters. Models and numerical method were validated based on ressure dro measured along a 6 m horizontal channel, and agreement was found to be excellent for different articles sizes, mass loading and wall roughness. In a horizontal ie flow, due to wall roughnessinduced focussing of article trajectories towards the core of ie, a secondary flow in ie cross-section develos. Additional ressure dro due to articles in ie flow was higher than that in channel due to different wall collision behaviour. Keywords-Gas-solid flow, Inter-article collisions, Pneumatic conveying, Turbulence, Wall roughness Introduction For numerical treatment of two or multile interenetrating hases, two aroaches commonly used are Euler/Euler (EE) and Euler/Lagrange (EL) aroach 1. First aroach is referred by industry, since considered roblem can be numerically solved rather fast. Several grous have considerably advanced two-fluid aroach for calculation of turbulent disersed two-hase flows over the last 0 years,3. Second aroach is only alicable to disersed two-hase flows and is based on a couled comutation of fluid flow (Eulerian) and article hase (Lagrangian). Disersed hase is modelled by tracking a large number of reresentative oint-articles through reviously comuted flow field by solving equations of motion accounting for all relevant forces acting on articles and other imortant elementary rocesses 4. An essential advantage of this method is that discrete nature of disersed hase articles is retained. Moreover, hysical effects influencing article motion (article-turbulence interaction, article-wall collisions, inter-article collisions or agglomeration), can be modelled on the basis of hysical rinciles 5,6. Essential in the alication of both aroaches is modelling of elementary rocesses *Author for corresondence slain@uao.edu.co affecting article motion, which considerably influences temoral and satial distributions of disersed hase in the rocess of consideration. For oerational conditions, where influence of article hase on fluid flow (so-called two-way couling) becomes imortant, also fluid flow is strongly affected by elementary rocesses. As a consequence, integral arameters are influenced, as ressure dros in neumatic conveying 7. EL-aroach has been alied to neumatic conveying with more or less sohisticated modelling aroaches for disersed hase 8. A very detailed numerical calculation for a article-laden horizontal channel flow on the basis of EL-aroach in combination with k e turbulence model and full two-way couling has been resented 9. One of the first fully couled EL calculations of horizontal article-laden channel flow by accounting for wall roughness effects has been introduced by Lain et al 10 in comarison with exerimental data 11. Recently, numerical redictions for different degree of wall roughness were erformed and article diameter was in excellent agreement with measurements 1. This study resents EL calculations of confined horizontal gas-article flows (channel and ie flows) emhasising imortance of elementary rocesses (article collisions with rough walls and inter-article

2 130 J SCI IND RES VOL 70 FEBRUARY 011 collisions) on redicted overall erformance of these rocesses. Exerimental Section Euler-Lagrange (EL) Aroach Numerical scheme adoted to calculate different configurations of confined article-laden two-hase flows was fully couled EL aroach 1. Fluid flow was calculated based on Euler aroach by solving Reynolds-averaged conservation equations in connection with a full Reynolds-stress or standard k-e turbulence model. Conservation equations were extended in order to account for the effects of disersed hase (two-way couling). Time-deendent three-dimensional conservation equations for fluid may be written using tensorial notation as ( ρφ ), t + ( ρui φ),i = ( Γikφ,k ),i + Sφ + Sφ (1) where ρ is gas density, U i are Reynolds-averaged velocity comonents, and Γ ik is an effective transort tensor. Usual source terms within continuous hase equations are summarised in S φ, while S φ reresents additional source term due to hase interaction 1. m du dt i dx = dt i + m = u 3 ρ 4 D i m D g i 1 dωi I = Ti dt c ( u u ρ ρ i i + F ) u u lsi + F lri P () where x i, u i and u i are comonents of article location vector, article and fluid velocity, resectively. m is article mass, I moment of inertia, D article diameter, g i gravity vector and ρ and ρ are fluid and article material densities. Different forces acting on articles and resective resistance coefficients allowing extension of equation of motion to higher article Reynolds numbers are reorted 1. Two-way couling considers momentum transfer from disersed hase to continuous hase through aroriate source terms in the momentum, Reynolds-stress or turbulent kinetic energy and dissiation rate conservation equations. These source terms are samled for each control volume during Lagrangian tracking rocedure. An under-relaxation aroach is used when introducing the source terms in conservation equations of fluid flow 1,13. Hence, a sequential calculation of fluid flow and article hase is erformed until couled system has converged. Modelling Of Elementary Processes Instantaneous fluid velocity comonents along article trajectory are determined from local mean fluid velocity interolated from neighbouring grid oints and a fluctuating comonent generated by a single-ste Langevin model 14. In this model, fluctuation velocity is comosed of a correlated art from revious time ste and a random comonent samled from a Gaussian distribution function. Correlated art deends on article resonse characteristics (turbulent Stokes number) and is calculated using aroriate time and length scales of turbulence estimated from turbulence models. In confined flows, change of linear and angular article velocity during a wall collision rocess is calculated based on the solution of imulse equations couled with Coulomb s law of friction. Searating the rocess in a comression and rebound eriod and introducing a restitution coefficient as ratio of normal imulse of rebound to that of comression yields two sets of equations for a sliding and sticking collision. Distinction between these two tyes of collisions is made on the basis of friction coefficient. Hence, two model arameters are needed to calculate wall collision rocess, deending on imact velocity, article size and imact angle, which generally have to be extracted from measurements 15. Wall roughness has a substantial influence on article-wall collision rocess, deending on article size and wall roughness structure. Since a detailed consideration of wall roughness rofile in Lagrangian calculations is not feasible, a stochastic model was develoed, assuming that instantaneous wall collision angle is comosed of article trajectory angle and a stochastic contribution from wall roughness as αw = α0 + γ ξ (3) where ξ is a random number with zero mean andvariance one and γ is standard deviation of roughness angle distribution deending on article size and roughness height 11,15.

3 LAÍN & SOMMERFELD : PNEUMATIC CONVEYING OF SOLIDS ALONG HORIZONTAL DUCTS 131 γ [degree] Particle diameter, µm From thorough exeriments and numerical calculations, correlations between γ and article diameter with degree of roughness as a arameter have been elaborated (Fig. 1); these curves also may be fitted by aroriate correlation functions 8. PDF of roughness angle with standard deviation γ is described with a normal distribution function as 1 γ P (4) ( γ) = ex π γ γ mean roughness.3 µm 4.3 µm 6.8 µm 17.0 µm Fig. 1 Correlation between standard deviation of roughness angle distribution and article diameter for different mean roughness height (data extracted from exeriments and numerical calculations) However, due to shadow effect, this distribution function is effectively skewed to ositive values of roughness angle, esecially for smaller wall imact angles 15. For such cases, there is a ronounced transfer of horizontal to transverse article momentum associated with a strong increase of article fluctuation velocity and as a consequence wall collision frequency in ie and channel flows. Incororating these details in wall collision model, article velocities and ressure dro 1 can be correctly redicted. Inter-article collisions are modelled by stochastic aroach 16, which relies on generation of a fictitious collision artner in each Lagrangian time ste of a considered article. Fictitious articles reresent local article oulation and hence their size and velocities are samled from reviously determined local distribution functions. Velocity distributions are assumed to be normal distributed and ossibility of a correlation between velocities of colliding articles is accounted for through a deendence on article turbulent Stokes number. Once all instantaneous roerties of real and fictitious article are known, collision robability is calculated as ( Dr + Df ) u r uf n t L π P = (5) 6 where subscrits r and f refer to real and fictitious article, resectively, n is article hase number concentration and t L reresents Lagrangian time ste of article tracking. A collision will occur, in case a random number in the range [0, 1] becomes smaller than collision robability. As it is very sohisticated to determine oint of imact on the surface of articles in an Eulerian frame of reference, velocities (linear and rotational) are transformed in a coordinate system where fictitious article is stationary. In this configuration, relative velocity vector is aligned with the axis of collision cylinder and lateral dislacement of centres of both articles as well as aroriate collision angles are samled by a random rocess 16. By solving again the imulse equations in connection with Coulomb s law of friction, one obtains two sets of equations for a sliding and sticking collision, whereby new velocities of real article are calculated. These velocities are then re-transformed in fixed Eulerian frame of reference. Velocities of fictitious article are not of interest subsequently. Results and discussion Numerical Prediction of Pneumatic Conveying in Pies and Channels EL aroach was alied to two flow configurations (article-laden horizontal channel and ie flow), which are very common in owder technology. Using code FASTEST/LAG 3D, three-dimensional steady-state comutations were done, as summarised in connection with standard k-ε turbulence model. Horizontal channel has: length, 6 m; height, 35 mm; and width, 350 mm. Comutational domain was discretised by a single block structured grid (40,000 hexahedral control volumes). Pie had hydraulic diameter identical with that of the channel, yielding a diameter of 63 mm. Pie length was 6 m. Comutational domain was discretised by 5 blocks (80,000 control volumes). For both configurations, conveying gas velocity was 0 m/s and articles were sherical glass beads (mono-diserse; diameter, 130 µm; and solids density, ρ =,450 kg/m 3 ). Particle mass loading (article mass flow rate/gas mass flow rate) was set as 1.0. In each of the couling

4 13 J SCI IND RES VOL 70 FEBRUARY 011 coefficient was taken as a constant (0.9) and static and dynamic friction coefficients were assumed to be 0.4. Cross-sectional distributions of different gas and article hase roerties were considered at 5.8 m downa) b) Fig. Comarison of calculated flow structure in a article-laden develoed channel and ie flow for different cases [left column: twoway couling and without inter-article collisions, γ = 0.8 ; middle column: four-way couling, γ = 0.8 ; right column: four-way couling with γ = 5 ; a) article concentration distribution; b) distribution of stream-wise article velocity; c) distribution of stream-wise gas velocity and white lines: streamline of gas-hase cross-sectional velocity (average conveying velocity 0 m/s; D P = 130 µm, article mass loading 1.0) c) iterations, article hase was simulated by tracking 40,000 arcels through flow field in order to yield statistically reliable article hase roerties and source terms. For article article collisions, restitution

5 LAÍN & SOMMERFELD : PNEUMATIC CONVEYING OF SOLIDS ALONG HORIZONTAL DUCTS 133 stream of inlet, where flow is fully develoed. Wall collision rocess was modelled, accounting for a certain degree of roughness. A detailed comarison of -d calculations with Reynolds stress turbulence model and exeriments was erformed for channel flow 1. Agreement of calculation and exeriment 11 for vertical rofiles of gas and article hase roerties was found to be very good for different article size, mass loading and also different degree of wall roughness. Esecially, consideration of ressure dro along the channel revealed a drastic increase of additional ressure dro due to article hase when wall roughness height is increased. This is because wall roughness enhances wall collision frequency 8, whereby average energy dissiation due to article hase increases and accordingly ressure dro due to required article reacceleration. For mass loading of 1.0 and a low roughness of.3 µm, total ressure dro at the channel end was 500 Pa, whereas a mean roughness height of 17 µm yielded more than 1000 Pa total ressure dro. Calculated ressure dro for different article sizes and mass loading was in excellent agreement with measurements. A comarison of cross-sectional distributions of gas and article hase roerties at the end of channel and ie (Fig. ) demonstrated influence of inter-article collisions and degree of wall roughness at constant mass loading (1.0). Neglecting inter-article collisions results in an unrealistic accumulation of articles near the bottom of ie or channel (Fig. a). Collisions between articles yield a somewhat better disersion of articles, which are however still concentrated in lower half of cross-sections. An increase of wall roughness to γ = 5 causes a better disersion of articles in cross-section. Stream-wise, mean velocity of articles has a maximum in the core of ie, but the region of highest velocity has a different shae for three calculations. Due to strong gravitational settling of articles, when neglecting inter-article collisions, the highest article velocities are found above the region of high article mass flux where flow resistance is larger (Fig. a). For low roughness case with inter-article collisions, article velocity distribution is almost symmetric with resect to the ie axis. In high roughness case, region of highest article velocities has an ellisoidal shae, flattened in the horizontal direction (Fig. b). Similar cross-sectional distributions are found for stream-wise mean gas velocity, since articles are of course conveyed by gas flow (Fig. c). Pressure dro P, a ressure dro P [Pa] Pressure dro P, a ressure dro P [Pa] Channel Flow single hase two-hase, 4-way, γ=0.8 o two-hase, 4-way, γ=5.0 o Length (l), m Pie Flow single hase two-hase, 4-way, γ=0.8 o two-hase, 4-way, γ=5.0 o Length (l), m Fig. 3 Calculated ressure dro along the channel (To) and the ie (bottom), comaring single-hase flow and article-laden flow (mass loading 1.0) for different wall roughness (four-way couled calculation, D P = 130 µm, article mass loading 1.0, conveying velocity of 0 m/s) In grahs for stream-wise gas velocity also streamlines of cross-sectional comonent of gas velocity are shown (Fig. c). A secondary flow develos in cross-section of ie for all cases. Without inter-article collisions and for low roughness, two circulation cells are visible, whereas other two conditions yield four circulation cells. This henomenon is only observed in circular ies and not in channels. Such a secondary flow is originating from a so-called focussing effect. Particles sedimenting in a horizontal ie will collide with the bottom art of ie wall and then are rebound towards the core of ie. This is associated with a concentrated momentum transfer to the fluid (if mass loading is high enough) inducing such a secondary flow. In case of strong gravitational settling (without inter-article collisions and at low roughness), articles referably collide with the bottom wall. Almost no articles are colliding with to wall. Hence, cross-sectional flow is ushed uwards by the article focussing forming only two circulation cells. If vertical disersion of articles is enhanced (due to roughness or

6 134 J SCI IND RES VOL 70 FEBRUARY 011 inter-article collisions), collision frequency with the uer ie wall increases. As a consequence, these articles are also rebound from uer wall towards the core of ie and due to momentum transfer to the fluid, four circulation cells develo in ie cross-section (Fig. c) Finally, a comarison of ressure dro along channel and ie is shown (Fig. 3) for ure gas flow and article-laden flow. Single-hase ressure dro is almost identical for channel and ie since hydraulic diameter was chosen to be identical. In article-laden flow, an additional ressure dro arises mainly due to article-wall collisions (article wall friction). For a low roughness situation, this additional ressure dro is considerably smaller than for higher roughness case, caused by enhanced wall collision frequency 8. Remarkable is the considerably higher ressure dro in ie comared to channel, which is again the result of larger wall collision frequency of articles in a ie. Thus, a channel used for neumatic conveying requires less energy than a ie. Conclusions Numerical redictions of integral oerational arameters for horizontal neumatic conveying were erformed using EL aroach. A detailed modelling of elementary rocesses occurring on the scale of articles was required for comuting flow structure and ressure dro. In confined flows, effect of wall roughness needs to be accounted for, since thereby wall collision frequency is considerably altered. In neumatic conveying, an increase of wall roughness enhances wall collision frequency and due to the associated article hase momentum loss, ressure dro is also remarkably increased. Inter-article collisions have already a ronounced effect on flow structure even at rather low overall mass loading. For horizontal ie, due to a article focusing effect and two-way couling, a secondary flow develos in ie cross-section, which might have two or four circulation cells deending on the degree of article disersion. Pressure dro was remarkably higher in ie flow comared to channel, which is again associated with wall collision frequency. References 1 Lain S, Modeling and Simulation of Bubble Induced Flows : [Universidad Autónoma de Occidente, Cali (Colombia)] 007 (in sanish) Simonin O, 000, Statistical and continuum modelling of turbulent reactive articulate flows: Part I. Theoretical derivation of disersed hase Eulerian modelling from robability density function kinetic equation, Von Karman Institute for Fluid Mechanics Lecture Series, No (Brussels, Belgium) Zaichik L I, Pershukov V A, Kozelev M V & Vinberg A A, Modeling of dynamics, heat transfer, and combustion in twohase turbulent flows: 1. Isothermal flows, Ex Therm Fluid Sci, 15 (1997) Laín S, On Modeling and Numerical Comutation of Industrial Diserse Two-Phase Flow with the Euler-Lagrange Aroach (Shaker Verlag, Aachen, Germany) 010, Sommerfeld M, Modellierung und numerische Berechnung von artikelbeladenen turbulenten Strömungen mit Hilfe des Euler/ Lagrange-Verfahrens, Habilitationsschrift (Universität Erlangen-Nürnberg, Shaker Verlag, Aachen) 1996, Gouesbet G & Berlemont A, Eulerian and Lagrangian aroaches for redicting the behaviour of discrete articles in turbulent flows, Progr Energy & Combust Sci, 5 (1999) Lain S & Sommerfeld M, Exerimental and Numerical Study of the Motion of Non-Sherical Particles in Wall-Bounded Turbulent Flows [Universidad Autónoma de Occidente, Cali (Colombia)] 008a, Sommerfeld M, Analysis of collision effects for turbulent gasarticle flow in a horizontal channel: Part I. Particle transort, Int J Multihase Flow, 9 (003) Lun C K K & Liu H S, Numerical simulation of dilute turbulent gas solid flows in horizontal channels, Int J Multihase Flow, 3 (1997) Laín S, Sommerfeld M & Kussin J, Exerimental studies and modelling of four-way couling in article-laden horizontal channel flow, Int J Heat Fluid Flow, 3 (00) Sommerfeld M & J. Kussin J, Wall roughness effects on neumatic conveying of sherical articles in a narrow horizontal channel, Powder Technol, 14 (004) Lain S & Sommerfeld M, Euler/Lagrange comutations of neumatic conveying in a horizontal channel with different wall roughness, Powder Technol, 184 (008b) Kohnen G, Rüger M & Sommerfeld M, Convergence behaviour for numerical calculations by the Euler/Lagrange method for strongly couled hases, in Numerical Methods in Multihase Flows 1, FED-Vol 185, edited by C T Crowe et al (ASME Fluids Engineering Division Summer Meeting, Lake Tahoe, USA) 1994, Sommerfeld M, Kohnen G & Rüger M, 1993, Some oen questions and inconsistencies of Lagrangian Particle disersion models, in 9th Sym on Turbulent Shear Flows (Kyoto, Jaan) 1993, Paer 15.1, Sommerfeld M & Huber N, Exerimental analysis and modelling of article-wall collisions, Int J Multihase Flow, 5 (1999) Sommerfeld M, Validation of a stochastic Lagrangian modelling aroach for inter-article collisions in homogeneous isotroic turbulence, Int J Multihase Flows, 7 (001)