The work performed by our group during the previous IFPRI contract period focussed on fully developed, turbulent flow of gas-particle mixtures in vertical pipes, A K-E model for the fully developed flow was formulated and used to explain the mechanism responsible for the nonuniform distribution of particles over the pipe cross section (please refer to last year’s annual report, FRR 09-15, for a detailed description). In the present contract period is work will be extended to developing flow problems in an effort to understand (1994-97), th’ how the flow evolves in risers and standpipes, and get an appreciation for the length scales associated with various regimes of flow development.

We first derive the time-smoothed equations for a 2D, developing, turbulent gas- particle flow. The coordinate system used for this purpose is shown in Figure 1: x and y denote the axial and transverse coordinates, respectively. A Cartesian coordinate system is used rather than a cylindrical one so that asymmetric inlet conditions can eventually be analyzed. The vertically upward flow in a slit (Figure la) should have characteristics similar to those of flow in a riser so, henceforth, this geometry will be referred to as riser flow. By a similar token, the flow corresponding to Figure lb will be called a standpipe flow. The equations take the form of eight coupled partial differential equations ‘representing two mass balances, four momentum balances (two axial, two transverse) and balances for turbulent kinetic energy and its dissipation rate per unit volume of suspension. As in the case of the fully developed flow equations derived in FRR 09-15, separate transport equations for K and E are not required for the two phases, since, for the solid mass fluxes being considered, the inertia associated with the gas is negligible when compared to that of the solids. A full fledged analysis of developing flow in vertical ducts on the basis of this K-E model, taking into account the entrance and exit effects and the possibility of internal recirculation of particles and gas, is a very large-scale computational problem.

The axial development of the flow of gas-particle suspension in vertical ducts is char- acterized by a number of distinct zones. The pariticles are initially accelerated by a high- velocity gas stream (acceleration zone), where large changes in the particle concentration, pressure gradient, etc. take place over very small axial distances. Accurate numerical computations in this zone will necessarily require a very fibe axial mesh. Following the acceleration, the particle concentration and velocity fields slowly evolve to their fully developed states over a relatively larger distances (transition zone). This is followed by a fully-developed zone, after which is an end zone where the flow patterns change again to conform to the exit geometry. If the riser is not “sufficiently tall,” the fully developed zone may be absent and exit and entrance zones will interact strongly. A general purpose computational code that allows for all these regions will require large computer resources. It is therefore of interest to identify simplifications which will make the problem computationally more tractable.

In the present annual report, we have derived the time-averaged equations of motion for steady developing flow and simplified the resulting system of equations through a scaling 1 analysis (see Section 2). This analysis led to a system of equations containing only the first derivatives of dependent variables in the axial (vertical) direction, and the first and second derivatives in the transverse direction, This form allows us to view the developing flow problem as an initial value problem (as opposed to a boundary value problem) and compute the solution by marching form the inlet to the exit. It should be noted that the existence of a solution for such an initial value problem is not guaranteed. For example, when the flow develops an internal recirculation, such an approach will necessarily fail; however, in case no recirculation develops, this approach will yield the entire solution. Nevertheless, it is useful to solve the initial value problem and get a feel for the developing flow, largely because of the tremendous computational advantages it offers over the boundary value problem. With this in mind, we performed a number of calculations and these are described in Sections 3 and 4.

Section 3, which is devoted to a laminar flow model, that neglects the effect of tur-bulent fluctuations, shows that the entrance length in two-phase flow is considerably larger than that in a comparable single-phase flow, and this is a consequence of the particle seg- regation in the acceleration zone. In Section 4, results based on laminar and turbulent flow models are compared to demonstrate that the initial segregation of particles to the tube wall in the acceleration zone in purely a continuity effect, and the turbulent fluctuations, which come into play only in the transition zone, are solely responsible for causing segregation of particles in fully developed flow The main findings of the study are summarized in Section 5.