Rotary Kiln Residence Time

What is modelled

A rotary kiln is a slightly inclined rotating cylinder. The solids bed slides and tumbles down the slope, so the time it spends inside depends on slope, rotation speed and geometry. The bed is heated by hot gas, treated here as a constant-temperature stream, and the model follows the bed in plug flow.

Equations

• Residence time (Sullivan) τ = 1.77 √θ (L / D) / (S N) with θ the dynamic angle of repose in degrees, S the slope in degrees and N in rpm; Perry's form 0.19 (L / D) / (tan S · N) is shown for comparison.
• Bed segment (β − sinβ) / 2π = fill, so the bed angle β follows from the fill fraction; hold-up m = ρb A L and throughput mdot = ρb A L / τ.
• Heating dTs/dt = (Tg − Ts) / τth, τth = ρb cp A / (h A'), giving Ts = Tg − (Tg − T0) e−t/τth.
• Heat transfer area per metre A' = D (sin(β/2) + β/2): exposed bed surface plus covered wall.

Checks run in the page

• The two residence-time correlations against each other, and v τ = L.
• Bed segment area by numerical quadrature, and hold-up equals throughput times residence time.
• The heating ODE integrated numerically against the closed form.
• The heat passed from the gas along the kiln against the enthalpy rise of the solids.

Honest simplifications

The gas is held at one temperature (no counter-flow gas profile, no burner model), the solids have constant properties, there are no reactions, no dust, no radiation from the flame and no mixing or back-flow. The correlations are empirical with a typical scatter of tens of percent. This is not a kiln design tool and ignores all burner safety logic.

• Choose Fast and steep: the residence time drops and the outlet solids leave colder.
• Choose Slow and full: a long residence time brings the solids close to the gas temperature.
• Raise the heat transfer coefficient to see the heating time constant shrink.