One-Dimensional Thermal Model for Bed and Freeboard

In the absence of any chemical reaction or phase transformations these equations simplify to 

The system of equations. Equations (8.30a,b,c), can be solved for successive axial positions by any of a variety of techniques (e.g., Runge Kutta) provided that the various heat transfer terms are characterized in terms of the local gas, bed, and wall temperatures. Thus, by starting at either end of the kiln, a complete solution of the thermal problem can be developed. It is chiefly the methodology employed in evaluating the heat transfer terms that distinguishes the various one-dimensional models. 

Chapter 8 Heat Transfer Processes in the Rotary Kiln Bed 

In applying these expressions the hydraulic diameter and transverse area of the freeboard must be utilized in evaluating the dimensionless groupings and mass flux terms. At the covered wall/covered bed interface the model described earlier is applied. 

Since there is no intent to restrict the work to nonreactive conditions in the bed and freeboard, Equation (8.30) can be expanded to include the reactive terms, which are originally present in Equation (8.29), to yield the system 

---