Object-oriented simulation of pneumatic conveying-application to a turbulent flow

Object-oriented simulation of pneumatic conveying-application to a turbulent flow

S. J. M. CARTAXO and S. C. S. ROCHA

Departamento de Termofluidodinâmica, Faculdade de Engenharia Química, UNICAMP,

C. P. 6066, 13083-970, Campinas - São Paulo, SP, Brazil,

E - mail [e-mail]

(Received: July 21, 1999; Accepted: August 22, 1999)

Abstract - In this work a model was developed to describe the dynamic behavior of fluid and particles in pneumatic conveying. The object-oriented simulation (OOS) technique was applied, which allowed a discrete, deterministic and transient treatment for the particulate phase. The fluid phase was modeled assuming a pseudo two-dimensional flow, separated from the particulate phase. The dynamic effects coupling both phases were taken into account in the interaction terms as the drag force acting on the particles and the momentum generation in the fluid phase. The model presented in this work considers bidirectional coupling of the particulate and fluid phases and tridimensional movement of the particles. The properties of the object-oriented technique allow the simulation of polydispersed mixtures, permitting segregation studies in pneumatic conveying.

Keywords: pneumatic conveying, two-phase flow, modeling of gas-solid flow, object-oriented modeling and simulation.

INTRODUCTION

Pneumatic conveying is defined as the transport of solid particles by a gaseous stream through a pipe. This operation has been used in chemical industries for a long time, aiming at the transport of several kinds of materials, such as lime, coal, polimer pellets, soda ash and granular chemicals (Sadler, 1949). However, the operation of pneumatic conveying is not limited to carrying particulate solids, but is also applied in other processes such as drying, biological and catalytic reactions, as for example, the catalytic cracking of gas-oil and nafta residues (Pratt, 1974; Yerushalmi & Cankurt, 1978).

An empirical approach to pneumatic conveying has been used, to obtain useful and relevant results and conclusions, due to the complexity of physical phenomena occurring in particulate two-phase flows. However, increasing computational power and the development of efficient numerical methods allows the modeling and simulation of complex physical problems not previously possible.

Several studies have proved that the Navier-Stokes equations can be used to describe the flow of pure fluids – as the continuum approach is valid – but to date no method to incorporate the particulate phase in a two-phase or gas-solid flow has gained worldwide acceptance. The most used method consists of considering the particulate phase, which is discrete, as a continuum, applying a space leveling on the movement equations, which results in differential equations very similar to the Navier-Stokes equations. The differential equations obtained involve additional terms of generation and flux of energy and momentum on the particulate phase, which are modeled by phenomenological relations. As an alternative technique to incorporate the particulate phase in the gas flow, one can find in the literature the trajectory models, which are based on a modified form of Boltzmann equation associated with the Monte Carlo method (Kriton et al, 1990).

METHODOLOGY

The main characteristic of the object-oriented simulation (OOS) method is the deterministic treatment of particles and fluid as real and individual entities in the model abstraction. This means that dynamic phenomena and neighborhood interactions are considered for each particle in the model. As a consequence, the dynamic state of the particles is updated according to their interactions during the simulation and since particle dynamics are individually treated, detailed information on position, velocity and acceleration can be obtained in each time step. In this work, we are not including the heat and mass transfer phenomena. However, the OOS method permits the determination of particle temperature, moisture content or concentrations of chemical species if adequate transport equations are added to the model. At the present stage, the model developed here can be applied to polydispersed mixtures of particles. The use of tridimensional equations of motion and the bidirectional coupling of gas and solid phases allow that description of the dynamic flow pattern be described and that important characteristics of a real gas-solid transport system be captured.

Modeling of the Particulate Phase

The physical basis for particles movement is Newton´s second law without any simplifying assumption. Thus, the translational movement of the particles is realistic, with the centrifugal force acting in relation to any reference axis and tridimensional characteristics. The equation for the movement of each particle is

(1)

and the particle displacement is described by

(2)

In the absence of magnetic and electric field effects, the only field force to be considered is due to acceleration of gravity and we can write

as

(3)

One can observe that it is not necessary to model the centrifugal force, or the Coriolis force, in a explicit manner in this equation, as Newton`s second law, written in the general form of equation (2), already includes these effects.

The contributions of buoyancy and field forces are given by

(4)

and the drag force exerted by the fluid is calculated by the following equation:

(5)

The drag coefficient can be obtained for two ranges of the Reynolds number by the following correlations:

; (6)

; (7)

where Re_p stands for the particle Reynolds number, defined by the following equation:

(8)

The use of a correlation to calculate the drag coefficient is not mandatory, since any other feasible technique to obtain the force exerted by the fluid on the particles can be included in the proposed model. There are some CFD numerical models available in the literature that are able to calculate the force exerted by the fluid flowing around objects. These numerical schemes can be more precise than an empirical correlation; however, the computational time required to evaluate the drag force for each particle and at each time step would be prohibitive. Therefore, the utilization of techniques that will lead to more accurate results for the model parameters is limited only by the computational power available.

The numerical solution of Newton´s second law can be obtained by several discretization algorithms, using a finite time step, Dt. These methods will reduce to a differential equation in the limit as Dt ® 0.

For reasons of stability and precision, we decided to apply an explicit central difference discretization, obtaining the following equations:

(9)

(10)

These equations provide second-order precision values for position and first-order for velocity. The numerical scheme is explicit as particle position and velocity at time k+1 are calculated from the values of these variables at the previous time k. With this method, the solution of the differential equations follows a progressive step-by-step process that advances in time.

It can be verified that evaluation of particle trajectory requires knowing the fluid velocity at each time step. Thus, it is necessary to evaluate the transient velocity field of the gas phase before determination of position and velocity for each particle in the following time step.

In the modeling of the fluid phase presented below, the fluid flowing in the transport tube is divided in cuboids, which move in axial direction. These cuboids are, in fact, moving fluid volumes or elements, and the model for the fluid phase is transient and pseudo two-dimensional. The approach used in the fluid phase modeling is similar to the particles modeling, since the fluid volumes are also treated as objects in the model abstraction.

Tangential forces acting on each fluid element in the axial direction are generated by shear stresses caused by adjacent fluid elements, by normal forces due to the pressure gradient and by the weight. Further on, if there are particles transported by the fluid, an additional force will appear, which corresponds to the reaction of the drag force exerted by the fluid itself. This reaction force corresponds to the fluid-particle interaction term, and its consideration provides the model with the fundamental mechanism necessary to simulate the effects that particle movement causes on the fluid flow profile.

The force exerted on the fluid element (i,j), assuming the effect of the presence of the particles, is given by

(11)

The term Fp_z represents the sum of the drag forces acting on each particle within the fluid element, and each fluid element must have its interaction term calculated at each time step in the model simulation.

The resulting force on a column of j fluid elements, including the pressure forces is written as

(12)

and the acceleration on a fluid column is evaluated by the following equation:

(13)

The shear stress acting on the fluid cuboid faces is calculated by the discretized equations below :

(14)

(15)

The movement and the velocity changes of the fluid elements are obtained by the same numerical scheme applied to particle dynamics, but in this case only the axial direction is taken into account. The position and velocity of the fluid elements are updated at each interaction by

(16)

(17)

Dissipative work on a fluid element corresponds to the mechanical energy exchange with the particles within and with adjacent fluid elements and is given by

(18)

The correction of pressure is obtained from the first law of thermodynamics written for an isothermic flow, in the following form:

(19)

Utilization of discrete elements to represent the fluid phase makes it necessary to discretize the flow field. Thus, the equipment where fluid flow occurs must be divided in spatial cells of defined sizes and dimensions. These cells are responsible for movement of fluid elements, which flow through them from cell to cell from the inlet to the outlet of the transport tube. The fluid flow domain was discretized using the functional nonuniform segmentation technique, developed in this work to combine the high computational efficiency obtained with an uniform grid and the high stability and accuracy offered by a nonuniform grid.

Functional nonuniform segmentation divides the fluid flow field into variable intervals, according to the following equation:

, (20)

where I is the number of intervals and P is the discretization step for each interval length. In this case, a uniform discretization is obtained considering P(i) constant. Assuming that P(i) a function of i, one can obtain a variety of nonuniform grids. P(i) can assume any functional form, but in order to increase computational efficiency, simple functions are indicated.

Turbulence Modeling

The inclusion of turbulent effects in the fluid flow model was achieved using the k-e turbulence model, modified for low Reynolds numbers (Jones and Launder, 1973). The objective of including a turbulence model was to obtain a fluid velocity profile that is more realistic and able to represent the additional mechanical energy dissipation verified in turbulent flows.

Solid Fluid Bidirectional Coupling

The proposed object-oriented model allows consideration of the dynamic interaction effects between the disperse and continuous phases. The influence of the fluid flow on the particles is included by the fluid-dynamic force, estimated by the drag coefficient, while the effect of particle movement on the fluid flow pattern is included by the interaction term, Fp_z, which comprises the change in the particles linear momentum in each fluid element. In the next time steps, this interaction term is included in the transport equations for the fluid, incorporating the momentum exchange between the particulate and fluid phases.

RESULTS

The object-oriented simulation technique was applied to some case studies of pneumatic conveying to validate the model and realize the phenomenological analysis of fluid-dynamic behavior.

The case studies are presented below. In all the simulated cases, the fluid used was air with a density of 1,205 Kg/m³ and a viscosity of 1.819x10^-5 Pa.s. The particles with a density of 7,700 Kg/m³ and a diameter of 3mm were considered spherical. Pneumatic conveying was carried out in a vertical tube with an internal diameter of 7.62 cm of and a length of 9.14 m. The injection of solids was set to 23.869 kg/sm² and the pressure drop along the tube was set to 2,200 Pa. These values correspond to experimental conditions used in Capes and Nakamura´s experiments. (Capes and Nakamura, 1973).

Two-phase Turbulent Flow

This case study is the simulation of a gas-solid flow with one-directional coupling, where the particles do not influence fluid dynamic behavior. The fluid radial velocity profile obtained is shown in figure 1, where we can verify a typical flat profile of a homogeneous turbulent flow. The velocity profile obtained is in perfect agreement with the physical expectation, validating the model and the techniques employed in the simulation, as this case implies that the fluid flow is not influenced by the particles being transported.

Figure 1
: Radial velocity profile for fluid phase in two-phase turbulent flow.

Figure 2 presents the change in solids concentration inside the tube over time. We can verify that after about 2 seconds the solids holdup in the tube is very close to the final steady state value, of approximately 2.9 Kg/m³.

Figure 2:
Temporary mass concentration profile ofparticles inside the pipe in two-phase turbulent flow.

The local mean particle velocities are represented in the diagram of figure 3. The radial variation in solids axial velocity is clearly shown in the diagram. A slower particle flow is found in the region near the tube wall, and the faster particles move in the central region of the tube section. This profile verified for the particles is a reflection of the fluid velocity profile.

Figure 3:
Axial velocity distribution for particles in two-phase turbulent flow.

The diagram in figure 4 permits the instantaneous observation of the positions of individual particles inside the tube, and the lower diagram corresponds to the vertical projection, while the upper is the horizontal projection of the tube. It can be observed that the solids concentration is higher at the base of the tube, in the region which coincides with the acceleration zone. The upper view reveals the existence of an annular region, without solids, near the tube wall.

Figure 4:
Modeling of particle placement inside a pipe undergoing turbulent two-phase flow.

The annular region is due to the low fluid velocity near the tube wall, which does not attain the terminal velocity of the particles. Thus, solid particles injected into this region are not carried by the gas and fall. This fact can be experimentally verified in two-phase flows at low velocities, where the collisions between particles are not significant and the dispersion of the discrete phase does not have a relevant influence on bed dynamics.

Two-Phase Turbulent Flow with Bi-directional Coupling

Bidirectional coupling is a fundamental characteristic of solid-fluid systems, and its inclusion is essential to making the model results more realistic. Having this in mind, the present case study refers to the simulation of the same system and conditions as those in the previous section, but now considering bidirectional coupling. Both fluid and particle dynamic profiles are influenced by the presence of the other in the flow.

The fluid radial velocity profile is shown in figure 5. The profile shows a general reduction in the value of the velocities, with the average velocity being equal to 41.7 m/s in the case of Figure 1 and equal to 32.3 m/s in this case. Another effect which can be observed in Figure 5 is a decrease in the value of the fluid velocity at the center of the tube, resulting in a profile with the shape of a crown. This flow pattern is experimentally verified and results from the transfer of momentum from the fluid to the particles, which reaches its highest value at the center of the tube, due to the highest slip velocity.

Figure 5:
Fluid radial velocity profile in two-phase flow assuming bidirectional coupling.

The curve in figure 6 describes the change in solids concentration over time. Comparing this curve with the one in figure 2, we see that there is a tendency towards stabilization at about 6 seconds of flow at an approximate value of 4.8 kg/m3, while in figure 2 the solids concentration reaches a maximum value of 2.9 Kg/m³. Thus, we can conclude that solid-fluid coupling tends to increase the holdup of solids in the pneumatic conveying tube, which was, in this case, 65.5 % higher than in the one-directional coupling case considered before.

Figure 6:
Temporary mass concentration profile of particles inside pipe in a two-phase flow modeled with bidirectional coupling.

The low values for particle velocities near the tube wall and at the tube base can be seen in the diagram in figure 7. The acceleration zone is detailed, after which the changes in shade occur only for bigger intervals.

Figure 7:
Particle axial velocities in turbulent two-phase flow assuming bidirectional coupling.

The existence of regions of the same shades in figure 7 indicates that particles having the same velocities move together and are accelerated at the same time. Also, there is not an effective mixture of particles with very different dynamic states.

The diagram in figure 8 shows the local voidage along the tube. There is a higher solids concentration at the center of the tube, according to the diagram. The existence of this dense nucleus contributes to the formation of the "crown" profile obtained in figure 5, since the high number of particles at the center of the tube makes the loss of momentum by the fluid more intense.

Figure 8:
Local voidage in turbulent two-phase flow assuming bidirectional coupling.

CONCLUSIONS

Object-oriented simulation has proved to be efficient in the study of the operation of pneumatic conveying. Detailed information about the flow pattern of both fluid and particulate phases, such as local velocities and voidage, could be obtained.

The case studies conducted by simulation allowed the verification of an annular region without particles very close to the tube wall, and the inclusion of bidirectional coupling in the model showed an increase in solids holdup, when compared with the simpler case of one-directional coupling. The influence of bidirectional coupling was also verified by the crown-shaped profile obtained in the fluid velocity profile.

The results presented in this work permit one to evaluate the type of information obtained using the OOS method, as well as its potential in modeling two-phase particulate flows.

NOMENCLATURE

A interfacial area of the fluid element, m²

a_z axial fluid acceleration, m/s²

C_D particle drag coefficient,

d_p particle diameter, m

field force on the particle, N

drag force on the particle, N

solid-fluid interaction force, N

resulting force on the particle, N

axial force on the fluid element, N

gravity acceleration, Kg/m³

particle mass, Kg

Reynolds number of the particle,

resulting axial force on the fluid, N

particle velocity, m/s

V volume of the fluid element, m³

W fluid dissipative work, N.m

particle position

Greek Symbols

DP inlet-to-outlet pressure drop, N/m²;

m_ef effective fluid viscosity, Kg/ms;

r_f fluid density, Kg/m³;

r_p particle density, Kg/m³;

s_rz component of shear stress , N/m²

• Capes, C.E. and Nakamura, K., Vertical Pneumatic Conveying An Experimental Study with Particles in the Intermediate and Turbulent Flow Regimes, The Canadian Journal of Chemical Engineering, (51), 2, pp. 31-38 (1973).
• Crowe, C.T.; Sharma, M. P. and Stock, D. E., The Particle-Source-in-Cell (PSI-cell) Model for Gas-Droplet Flows, ASME, J. Fluids Eng., 99, pp.325-332 (1977).
• Jones, W.P. and Launder, B.E., Prediction of Low-Reynolds-Number Phenomena with a 2-Equation Model of Turbulence, International Journal of Heat & Mass Transfer, (16), p. 1119 (1973).
• Kiton, A.; Elperin, T. and Tamir, A., Monte Carlo Simulation of Gas-Solid Suspension Flows in Impinging- Reactors, Int. J. Multiphase Flow, (16), 1, pp. 1-17 (1990).
• Pratt, K.C., Chem. Eng. Sci., (29), p. 747 (1974).
• Sadler, A.M., Chem. Eng., (56), p. 110 (1949).
• Yerushalmi, J. and Cankurt, N.T., Chem. Technol., p. 564 (1978).

Publication Dates

• Publication in this collection
  01 Mar 2000
• Date of issue
  Dec 1999

History

• Accepted
  22 Aug 1999
• Received
  21 July 1999