Extension of kinetic theory to cohesive particle flow
Introduction
The dynamics of inter-particle collisions in non-cohesive particle flow systems have been reconsidered based on the concept of the kinetic theory [4], and further modified and applied to the particulate phase by Jenkins and Savage [6], Savage and Jeffrey [14], Savage [15], Savage and Sayed [16], Lun et al. [11], Johnson and Jackson [9], Jenkins and Richman [7], [8], Ding and Gidaspow [5], Sinclair and Jackson [17], Louge et al. [10], and Arastoopour and Kim [2]. The simulations of gas/non-cohesive particle flow systems using the kinetic theory model by Ding and Gidaspow [5] and Sinclair and Jackson [17], and the experimental verification of the Maxwellian type distribution of cohesive solid particles by Zhang et al. [19] showed that the adoption of the gas kinetic theory for gas/particle systems is a promising approach. Furthermore, experimental data on particle fluctuation velocity using a laser doppler anemometer (Yang and Arastoopour [1] and Zhang and Arastoopour [18]) showed the significance of the fluctuation velocity in identifying the particulate phase flow patterns in gas/agglomerating particle flow systems. This also revealed the need for a model such as the kinetic theory that considers fluctuation of the particle as a major parameter. However, in the literature, models developed for non-cohesive particles do not have the capability of describing flow behavior for cohesive particle systems because the most significant characteristic of cohesive particle flow is the formation of agglomerates, which considerably affect flow patterns. The formation of agglomerates results in a reduction in the number and an increase in the size of particles, both of which directly affect the inter-particle collisions and, in turn, the particle phase properties such as viscosity and pressure, as well as gas/particle drag force in gas/particle flow systems. Because the number of particles is not conserved upon agglomeration, the Maxwellian–Boltzmann integral–differential equation needs to be evaluated each time particle agglomeration occurs. To overcome this problem, we defined a new distribution function of the instantaneous velocity of the particles relative to the average velocity based on the volume fraction of the particulate phase, which is conserved upon agglomeration. Furthermore, to account for the effect of diameter growth on agglomeration, we considered a new conservation of the number of particles equation. Based on our distribution function, governing equations were derived; namely, conservation of mass, momentum, fluctuation energy, and number of particles equations. This set of equations is capable of describing particle flow behavior as well as the particle diameter variation due to agglomeration.
Section snippets
Agglomeration process
The agglomeration of two particles upon collision is a phenomenon which is directly related to surface forces and interaction of boundaries of two particles. In the agglomeration process, not only is the kinetic energy of the colliding particles dissipated or stored, but also significant surface cohesion forces result in the binding of particles. Clustering may also enhance the agglomeration process. Clustering is a phenomenon, which is associated with energy dissipation due to the collision of
Conservation equations
Due to agglomeration, the number of particles per unit volume is no longer conserved, so the Maxwell–Boltzmann integral–differential equation needs to be evaluated at each agglomeration occurrence. In order to overcome this problem and to modify the distribution function for particles with finite size, we defined a new distribution function of the instantaneous particle velocity based on the volume fraction of the particulate phase which was conserved throughout the agglomeration with no phase
Governing equations and constitutive equations
The following are the conservation of mass, momentum, fluctuating energy, and number of particles equations, and the constitutive equations for cohesive particle systems.Conservation of massConservation of momentumConservation of fluctuation energyConservation of number of particlesConstitutive EquationsSolid phase stress
Shear flow analysis
In this section, we focus our attention on the case of a homogeneous simple shear flow with no spatial gradients of solid volume fraction, fluctuation velocity, and particle diameter under agglomerating conditions. The present shear flow analysis is based upon the assumption that the rate of energy dissipation is in the moderate range; thus, the dimensionless parameter , defined by Savage and Jeffrey [14] as the ratio of the characteristic mean shear velocity to the r.m.s. of
Conclusions
(1) Mathematical equations were developed for cohesive particle collision based on the criterion of agglomeration in order to account for the effect of cohesiveness of particles and particle agglomeration during collision. This resulted in an effective restitution coefficient, which is a function of contact bonding energy and relative velocity of particles before collision.
(2) Kinetic theory for the particulate phase was modified. The modification included the development of the distribution
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Current address: Department of Robot System Engineering, Tongmyong University of Information Technology, Namgu, Pusan 608-080, Republic of Korea.