Method of Estimating the Solids Circulation Rate in a Riser Using the Theoretically Predicted Pressure Profile from the Variational Calculus

Analysis of the flow in a gas-solid riser demonstrates that
specifying two variables in the non-acceleration section determines the
conditions there including the solids circulation rate. In order to predict the
conditions there necessitates closing the momentum equations which requires the
accurate prediction of the wall friction and the drag coefficient (*C _{D}*).
The wall friction for flows that do not include a core-annulus configuration
was well described by using a model that includes a turbulence response
parameter which allows the prediction of the particle-wall friction using the
gas and particle characteristics, riser diameter and the gas velocity. The
accurate prediction of the drag coefficient, however, remains intractable due
to the variety of variables that affect the value of C

_{D}such as turbulence scale, intensity, particle and riser geometry, and ability for the particle trajectory to be influenced by the turbulence. Therefore, in order to estimate the solids circulation rate (

*G*) in a riser an alternative method musts be developed that does not depend on using the value of the drag coefficient.

_{p}In previous work, it was shown how the variational calculus
provides a means of accurately predicting the dynamic pressure gradient in the
acceleration section of a riser using two physically relevant and readily
measured parameters which are the pressure drop across and the . The shape of
dynamic pressure gradient function was shown to be determined by the
acceleration parameter (*a _{p}*) which is the ratio of the dynamic
pressure drop across and the pressure gradient () at the end of the acceleration
section of a riser where .

In this work, a consistency relationship between the dynamic
pressure gradient and solids fraction profiles will be developed which provides
a means of approximating the solids fraction over the particle acceleration
length, *H _{ra}*. The consistency relationship provides a means of
obtaining an approximation to the analytical integration of the mixture
momentum equation to relate the dynamic pressure drop across and the pressure
gradient at the end of the acceleration section with the solids fractions at
the inlet to and the end of the acceleration section where

Further analysis shows a direct relationship between the particle inertial term of the momentum equation and the acceleration parameter that is a result of the theoretically predicted dynamic pressure profile where

and

The specification of at least two conditions in a riser and these relationships provide a means of estimating the solids circulation rate given the pressure profile and the specifications of two variables in the riser. This work will use data obtained for the transport of 1-mm glass spheres undergoing pneumatic transport in a 28.45-mm stainless steel grounded riser. The results of this analysis will provide additional insight for future modeling practices as well as potential methods for the operation and control of gas-solid riser systems.

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