(267c) Method of Estimating the Solids Mass Flow Rate in a Gas-Solids Riser Using the Integrated Mixture Momentum Equation and the Dynamic Pressure Gradient Distribution

The determination of the solids mass flow rate (G_p) is important for efficient control of conversion processes run in risers. For this reason a significant amount of effort has been dedicated to the development of solids mass flow rate determination methods which often focus on the closure of a one-dimensional, two-fluid model using a differential approach. This requires accurate prediction of the effective drag coefficient (C_d), which is not possible at this time because of the complexity of relating the effects of turbulence in a riser and particle acceleration to C_d. In this work, a novel approach to closing a one-dimensional, two-fluid model is developed. The result is a means of estimating the G_p from riser conditions using three readily measurable and physically relevant quantities; the superficial riser gas velocity (U_r) and the dynamic pressure drop across (ΔP_ra) and gradient at the end of (P'_ra(1)) the acceleration section of a riser.

The method of analysis is made possible by augmenting a one-dimensional, two-fluid model by the inclusion of a theoretically derived expression for the dynamic pressure gradient distribution that was obtained from a solution to an isoperimetric problem of the calculus of variations. The resulting expression for the dynamic pressure gradient requires two parameters to define it; the dynamic pressure drop across (ΔP_ra) and gradient (P'_ra(1)) at the end of the acceleration section. The ratio of these two quantities provides the acceleration parameter a_p ≡ΔP_ra/(-P'_ra(1)) which reflects the extent to which particles increase in velocity and reduce in volume fraction in the acceleration section of a riser.

A series of approximations made to the one-dimensional, two-fluid model provide a means of integrating the mixture momentum equation. The resulting expression allows relationships between integrated terms to be quantified providing additional expressions used for closure and prediction of the solids mass flow rate (G_p) from measured values of the gas velocity (U_r) and the defining parameters of the dynamic pressure gradient distribution. The model prediction of G_p is tested using literature data obtained transport experiments using several types of particles transported through risers of varying diameters. The agreement between the predicted and experimentally measured values of G_p is good to excellent.