Selçuk J. Appl. Math. Selçuk Journal of Vol. 7. No.2. pp. 55-68, 2006 Applied Mathematics
Thermal Turbulent Simulation Of Gas-Solid Flow In A Riser
Zohreh Mansoori1, Payam Ramezani12, Majid Saffar-Avval12,Hassan Basirat Tabrizi2
1Energy Research Center,Amirkabir University of Technology, P.O. Box: 15875-4413,
Tehran, Iran
e-mail: z.m anso ori@ aut.ac.ir, pay_ ram ezani@ yaho o.com
2Mechanical Eng. Dep.,Amirkabir University of Technology, P.O. Box: 15875-4413,
Tehran, Iran
e-mail: m avval@ aut.ac.ir ,hbasirat@aut.ac.ir
Summary.A new dynamic and thermal turbulent Gas- solid flow simulation is introduced.Gas- solid flow simulation by the new dynamic and thermal turbulent time scale transport model (− −) shows the advantage of calculating the turbulent Prandtl number directly from the turbulent field calculations. Also, this model eliminates the difficulties of determination of wall boundary condition in well- known − model. An upward turbulent gas-solid flow in a vertical pipe is simulated using Eulerian- Lagrangian approach. Particle-particle and particle-wall collisions simulation are done considering deterministic approach. The results of the dynamic and thermal turbulent model are compared with the ones of − model for gas and particle velocity and temperature fields. The results showed that new model can predict the experimental results better than the − model.
Key words: Modeling and Simulation Methods, Gas-solid flow, Turbulence and Particle collisions
1. Introduction
Fundamental understanding of the complex behavior of particles in gas-solid turbulent flow and their interaction with other particles and surrounding fluid is important in a number of industrial processes such as vaporization of liquid droplets, coal gasification, and combustion and environment pollution.
Three main modes of heat transfer may arise in gas-solid beds ; bed -to- surface, gas -to- particle, inter-particle heat transfer [1].The inter-particle heat transfer
Heat transfer due to particle - to - particle can result from three mechanism; heat transfer by radiation, heat conduction through the contact points between the particles and unsteady -state conductive heat exchange through the gas layer separating particles. The first mechanism is only significant at the temperatures higher than 600$\QTR{group}{ }^\circ C^$. The second one occurs when moving particles impact each other and heat conduction through the impact area would be a portion of particle contribution to heat transfer. Sun and Chen [2] pointed out that this phenomenon can be important in beds with large and metallic particles that have high velocities. Also Ben Ammar [3] in an experimental effort pointed out that this term is not important in moderate velocities.
Schlunder [4] pointed out that heat conduction could constitute the major con-tribution when the particle size is small. Delvosalle el al. [5] developed an inter-particle heat transfer model due to conduction through the gas between a hot and a cold particle. They concluded that particle to particle heat transfer is nearly independent on particle thermal properties and is more significant for smaller particles. The simulation of Mansoori et al. [6] for injecting hot and cold particles to a gas- solid vertical pipe flow indicates that although the heat transferred to each group of hot and cold particles through other particles are significant, but its effect on mean values of gas and particle temperatures and suspension heat transfer coefficient is negligible in dilute flows.
The effect of particle collision on hydrodynamic field is studied later. Berlemont, Simonin and Sommerfeld [7] presented different formulations for particle-particle collision in hydrodynamic modeling approach. Inter-particle collisions can be computed either deterministically or stochastically. A deterministic method without coupling between the phases was used by Tanaka and Tsuji [8] in a hor-izontal pipe and effect of inter-particle collisions was reported. Sommerfeld [9] performed a stochastic method without coupling between phases in a horizontal channel and proved that the particle collisions have a significant effect on the particle velocity fluctuation field. Sundaram and Collins [10] considered a four way modeling using (DNS) for an isotropic turbulence including the effect of inter-particle collisions to study the turbulence modulation. They found that the rate of viscous dissipation of turbulent energy is enhanced by particles. Yamamoto et al. [11] performed four way coupled simulation of downward gas-solid flow in a vertical channel considering inter-particle collisions simulated by a deterministic method. They investigated the effect of inter-particle collisions on the flow field and showed that inter-particle collisions make profile of the particle concentration much flatter.
Sommerfeld [12] investigated the motion of particles in a horizontal channel flow with the effect of wall collisions, wall roughness and inter-particle colli-sions. Experimental and numerical investigation of gas-solid flow with collision of large spherical particles in a vertical convergent channel was carried out by
Fahanno and Oesterle [13]. The effect of the particle collisions on the particle characteristics has been observed.
There are only a few works on studying the temperature fluctuation field and the thermal interactions between the two phases are not as yet well understood. Soo [1] formulated the statistical properties of the temperature fluctuations. Shraiber et al. [14] showed heat transfer from fluid to particles decreases as the size and heat capacity of particles increase. Han et al. [15] using a two-fluid model with thermal eddy diffusivity concept studied heat transfer in a duct. Using an Eulerian-Lagrangian model and considering gas k−equations, Avila et al. [16] studied heat transfer coefficients in a pipe with constant wall temperature. Rizk et al. [17] modeled the source term due to the solid phase in the fluid k− transport equations in an Eulerian approach in terms of particle relaxation time and turbulence time scale. Andreux et al. [18] used an Eulerian-Lagrangian model solving the momentum and energy equations for each phase. The one—way and two—way coupling simulations of Jaberi [19] showed that the thermal coupling is quite important.
Chagras [20] studied heat transfer in gas-solid pipe flow and they observed that the overall heat transfer increases with particle-particle collisions.
Schwab and Lakshminarayana [21] showed that using the transport equations for dynamic and thermal turbulence time scales and kinetic energy, eliminates the difficulties of numerical problems in single-phase flows and leads to simpler wall boundary conditions. Mansoori et al. [22, 23] used a new four way interac-tion Eulerian-Lagrangian model for predicting turbulent heat transfer in a fully developed flow and showed that the level of thermal turbulence intensity and the heat transfer are strongly affected by the particle collisions. They introduced the source term due to solid phase in the k− transport equations.
The purpose of this research is to extend the analysis of previous studies [22, 23] to compare the new turbulent model results with the well-known − model ones. The influence of the particle collisions on the thermal turbulent field characteristics of an upward gas-particle flow in a developing vertical pipe under a uniform heat flux condition is considered. Four- way interaction model of two phase flows within the framework of the Eulerian-Lagrangian approach is used in the numerical simulation.
2. Mathematical Modeling
Classical simulations of gas - solid flows can be classified as following:
Eulerian - Eulerian formulation ; Gas and solid flows are assumed as two con-tinuum media.
Eulerian - Lagrangian approach; Gas flow is simulated by an Eulerian model, while individual particles or ensembles of particles are tracked through the flow field.
In one - way coupling Lagrangian formulation particle trajectories are computed in a known fluid flow. When the influence of particles on gas is assumed by introducing source terms in gas continuum equation, and is known as Two - Way simulation. In Four- way approach, the effect of particle collision is included. In this article, the mathematical modeling based on Four- way simulation has been established.
2.1. Particle Equation of Motion
The motion of each individual particle is subject to Newton’s equation of motion. The drag and the gravity forces are considered in this equation and the effects of particle rotation and Magnus lift force are neglected. All particles are considered spherical with the smooth wall. The final form of particle equation can be shown by Crow, Sommerfeld and Tsuji [24]:
(1) = 3
4
(2) =
Where, , are the diameter, and the particle velocity component
respec-tively, and the drag coefficient is assumed to be independent of voidage in dilute flow and is chosen by simple model of Schiller & Nauman [25]:
(3) = 24−1 (1 + 0667 6 )
Where, is the instantaneous gas velocity and is given by = + 0, and, 0are the fluid mean velocity and the fluctuation component generated by Kraichnan [26] model respectively.
2.2. Particle Thermal Energy Equation
The temperature equation of each particle is calculated by:
(4) () = (− )
is the heat transfer coefficient between gas and particle and is the fluid temperature at the particle location. Note that = ¯+ 0 here 0is generated
using the extended Kraichnan [26] model developed by Mansoori et al. [22]. Particle Nusselt number is modeled by the equation:
(5) = = 2 + 06 05 03 Pr 2.3. Mutual Collision Of Solid Particle
Simulation of collision model by deterministic approach is done [23]. Hard sphere particle-wall and particle-particle collision model described by Crow, Sommerfeld and Tsuji [24] are used. Considering mutual collisions between two particles, the decision on whether a particle collides with other particles is based on the centroid distance between the particles. The rebound velocities after inter-particle collisions are evaluated by the hard sphere model. It is assumed that the particles will slide over one another during the collisions and the Coulomb friction law is used to relate the tangential and normal collision impulses. For simplicity, however, the particle rotation and deformation are ignored
2.4. The Turbulent Gas Velocity and Temperature Field
− model for dynamic field and − model for thermal field is applied in the present simulation. The general form of transport equations for axi-symmetric developing two phase flow in a vertical pipe is given in previous works of Mansoori et al. [22, 23].
The expressions of the fluctuational components of gas velocity are obtained by a modified Gaussian random field model proposed by Kraichnan [26]. This approach was extended to nonhomogeneous flows by Li and Ahmadi [27]. Here the CGRF model is used to generate the instantaneous turbulence fluctuation of gas velocity and gas temperature as described by Mansoori et al. [23].The well- known − model is described in Saffar- Avval et al. [28].
The two models are the same for the continuity, velocity, and temperature and turbulence variance. The − model simulates the turbulent field through the turbulence dissipation equation, while in the − − model the equations are considered too.
3. Numerical Procedures
The time dependent computations are carried out for a turbulent axisymmetric developing pipe flow. An iterative procedure between Eulerian mean flow eval-uation and the Lagrangian particle tracking is used to account for the Four-way interactions as described in the followings:
• To start the solution procedure, the experimentally available data for fully developed single-phase gas flow is used as the initial gas flow condition. • The Lagrangian particle trajectory and heat transfer equations are solved
in the known gas velocity and temperature fields, with particle-particle collisions being included. In this way the first estimate of individual par-ticle locations, velocities and temperatures after each Lagrangian time step are evaluated.
• At the next step the Eulerian field equations are modified by addition of the source terms due to presence of particles in the gas and then solved. • After obtaining the converged solution for gas phase mean velocity and
temperature and turbulence mechanical and thermal intensities, as well as time scales, all particles are tracked again in order to evaluate their corrected positions.
• This iterative procedure is repeated until the convergence is achieved.
The governing equations for are solved in the core region (be-tween the pipe center and a node located at+ = 30), while the gas mean temperature equation is solved in entire region (up to the wall).
4.Results and Dıscussion
To compare the two turbulent models in a gas- solid flow the experimental data of Tsuji et al. [29] are chosen to validate the dynamic characteristics. The model and experimental results of gas mean velocity profiles are presented in Fig. 1. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 K-e k- t ex p R r C U U ` k k e xp . T s uji a)
0 0 .2 0 .4 0 .6 0 .8 1 1 .2 0 0 .2 0 .4 0 .6 0 .8 1 K-eps il K-teitl ex p C U U R r k k e xp . T suji
Fig. 1: Comparison the model results for gas mean velocity with experimental data of Tsuji et al. [29]. Solid lines: the ( − − )model, dashed lines: − model, Symbols: the experimental data.(a)Pure gas, (b) gas- solid flow
The velocity is non- dimensioned with the centerline velocity. The case of gas-solid pipe flow with a Reynolds number of 23000, loaded with polystyrene par-ticles 243 with density of 1030 3 is addressed. The symbols show the experimental data. While, the solid lines present the results of ( − − ) model and the dashed lines indicate the results of − model. Fig. 1(a) shows the pure gas mean velocity data and Fig. 1(b) illustrates the results for two-phase gas- solid flow with mass loading ratio (the solid to gas mass flux) of 1.3. It can be seen that the ( − − )model results show better agreement with the experimental data especially in the region near the wall for pure gas and gas- solid flow in the range of study.
0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 0 .1 0 .1 2 0 0 .2 0 .4 0 .6 0 .8 1 K-ep K-tei Ts uji ex p. C rms U u R r k k e xp . Tsuji a)
0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 K-e K-te Tsuji exp. C rms U u R r k k exp. Tsuji b)
Fig. 2: streamwise gas rms velocity profiles. Solid lines: the
( − − )model. dashed lines: − model. Symbols: the experimental data. (a) Pure gas, (b) gas- solid flow
The two model results of gas turbulence intensities are compared with the ex-perimental data of Tsuji et al. [29] in Fig.2.
Fig. 2(a) shows the results for pure gas with Reynolds number of 23000 and Fig. 2(b) illustrates the profiles for two- phase gas- solid flow with loading ratio of 1.3. Fig. 2 indicates that considering the variations of thermal turbulent field cause the better prediction of the results of hydrodynamic field.
To illustrate the results of two models, computations are carried out for the experimental conditions of Depew and Farbar [30] with 200 spherical glass and Reynolds number of 13500.
In Fig. 3, the dimensionless wall temperature for the pure gas flow versus axial distance is shown. The figure shows that ( − − )model prediction is in good agreement with the experimental results for pure gas temperature.
0 0.02 0.04 0.06 0.08 0.1 0.12 0 10 20 30 40 50 cal. tei exp R r k r q T Tw in 0 k k k , Experiment
Fig. 3: Dimensionless wall temperature for the pure gas flow versus axial distance ( = 13500)
Local heat transfer coefficient ratio versus axial distance is illustrated in Fig.4. The results indicate that the ( − −)model can estimate the heat transfer coefficient better than the − model for both pure gas flow and two- phase gas-solid flow. (a) 0 .8 1 .2 1 .6 2 0 1 0 2 0 3 0 4 0 5 0 ca l. i te i e xp
46.5
h D x h D x k k k , E xp e rim e nt a)(b) 0 .8 1 .2 1 .6 2 0 1 0 2 0 3 0 4 0 5 0 ca l. f te i e xp
46.5
h D x h D x k k , k E xp e rim e nt b)Fig.4: Local heat transfer coefficient ratio versus axial distance; (a) pure gas flow( = 13500), (b) gas-solid flow ( = 13500 = 200 = 052)
The previous figures indicate the ability of the new model of better predicting the experimental results. In Fig. 5 the two model results are compared for gas and particle mean temperature at the fully developed region for gas- solid flow with = 13500and 200 micron particles and z=0.52. It can be seen that the result for two models are different and the new ( − − ) model predicts lower temperature ratios.
0 0 .2 0 .4 0 .6 0 .8 1 0 0 .2 0 .4 0 .6 0 .8 1 1 Series 2 W T T k k k , R r a)
0 0 .2 0 .4 0 .6 0 .8 1 0 0 .2 0 .4 0 .6 0 .8 1 1 Series 2 W P T T k k k , R r b)
Fig. 5: (a) gas temperature profiles, (b) particle temperature profile at the fully developed region.
5.Conclusions
Simulation of gas-solid turbulent upward flow in a vertical pipe was performed by using two turbulent model of ( − − ) and the − model.
Gas- solid flow simulation by the new dynamic and thermal turbulent time scale transport model ( − −) cause better results due to two advantages of the model. The one is that new model is capable of the calculating the turbulent Prandtl number directly from the turbulent field calculations. The other is that this model eliminates the difficulties of determination of wall boundary condition in well- known − model.
6. Nomenclature particle surface specific heat, −1◦ −1 drag coefficient pipe diameter, particle diameter, gravity acceleration, −2
heat transfer coefficient, −2 ◦−1 turbulent kinetic energy, 2−2 temperature variance, ◦2
gas heat conductivity, −1 ◦−1 particle mass,
Nusselt number = Pr Prandtl =
radial coordinate
Reynolds number = mean temperature, ◦ mean axial gas velocity, −1 0 velocity fluctuation, −1 ∗ friction velocity, −1 vertical coordinates, distance from wall, solid loading ratio Greek Letters
thermal eddy diffusivity, 2−1 turbulence time scale,
thermal turbulence time scale, viscosity, −1−1
kinematic viscosity, 2−1 eddy viscosity, 2−1 density, kg/m3
Subscripts and Superscripts fluid gas particle velocity average quantity 0 fluctuating quantity References
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