Fully Developed Dilute Pipe Flows

To analyze the dynamic behavior of gas-solid pipe flows, the most common and easiest system to consider is a dilute gas-solid pipe flow which is fully developed and is subject to the effects of electrostatic force and gravitational force. The fully developed flow here refers to the situation where the velocity profiles of both gas and particles are unchanged along the axial direction. The system of this nature was analyzed by Soo and Tung (1971). In this section, the analysis of Soo and Tung (1971) is presented. It is assumed that no particles are deposited on the wall surface of the pipe (or the particle deposition rate is zero). Moreover, the pipe flow is considered to be turbulent, as is true for most flow conditions. 

For dilute suspensions, particle-particle interactions can be neglected. The extent of transfer of particles by the gradient in the particle phase density or volume fraction of particles is proportional to the diffusivity of particles Dp. Here Dp accounts for the random motion of particles in the flow field induced by various factors, including the diffusivity of the fluid whether laminar or turbulent, the wake of the particles in their relative motion to the fluid, the Brownian motion of particles, the particle-wall interaction, and the perturbation of the flow field by the particles. 

The flow system consists of a circular pipe with an inclined angle of ft to the direction of gravitational acceleration g, as shown in Fig. 11.11. Thus, for f) of 90° the flow is horizontal and for of 0° or 180° the flow is vertical. In Fig. 11.11, z, r, and 9 are the axial, radial, and azimuthal coordinates U, V, W are the conjugate components of velocity of the fluid and Up, Vp, and Wp are those of the particle phase. The particles are assumed to be monodispersed and spherical. 

---

Figure 11.11. General coordinates for the analysis of fully developed dilute pipe flows.

It is recognized that within the range where drag reduction occurs, the solids concentration is so dilute that the averaged distance between particles is usually 10 or more particle diameters. Therefore, under this flow condition, interparticle effects can be neglected. Consider the case of a fully developed horizontal pipe flow with negligible electrostatic effects. From Eq. (11.6), the pressure drop depends only on the wall friction, as given by... 

Consider a fully developed dilute gas-solid flow in a vertical pipe in which solid particles carry significant electrostatic charges. The particle charges vary radially. It is assumed that (a) the flow and the electrostatic field are axisymmetric and (b) the radial charge distribution C(r), defined as q(r)af(r), is known. Derive an expression for the radial volume fraction distribution of the particles. 

Example 11.4 Consider a dilute gas-solid flow in a horizontal pipe made of electrically conducting materials. The pipe is well grounded. The flow is fully developed. Show that (1) the particle concentration is exponentially distributed and is a function of the vertical distance only and (2) the ratio of particle volume fraction at the top to that at the bottom depends on the density ratio of gas to particle, particle diffusivity, and pipe diameter. 

---