Side view of a rotary kiln with counter-flow gas, bed temperature profile, calcination progress and an energy balance.
The science
A rotary cement kiln is a slightly inclined steel tube that turns slowly. Raw meal enters the cold end and slides to the hot end against a stream of hot gas from the flame. Along its length the bed is heated, limestone calcines (CaCO3 -> CaO + CO2, endothermic, about 178 kJ/mol) and finally, above roughly 1450 C, calcium silicates (alite) form in a partial melt.
The model is a steady one-dimensional counter-flow heat exchanger. Per kg of clinker: bed m_s c_s dT_s/dx = U (T_g - T_s) - H_rx dalpha/dx and gas m_g c_g dT_g/dx = U (T_g - T_s) + q_loss, with the end gas temperature set by the fuel heat. Calcination follows Arrhenius kinetics above the 900 C decomposition temperature. The bed residence time uses the classical formula t = 1.77 sqrt(theta) L / (s D N).
Process CO2 is fixed by chemistry: 65 % CaO in clinker means 0.51 kg CO2 per kg clinker regardless of the fuel. Simplifications: constant heat capacities, one lumped heat-transfer coefficient, no dust or coating, no cooler, the clinker-phase index is a smooth heuristic in bed temperature and not a phase-diagram calculation, and the precalciner is represented only by its calcined fraction.