Indeed, the answer to your question depends both on phase and conventions used. Assuming plug flow, an average residence time for the solids can be derived from the filled volume and the feeding rate or (better) it can determined experimentally by introducing an inert tracer and note the time between introduction and emerging in the ash.
The right answer is almost impossible to give for the gas phase, since it is subject to various rather undefined factors and considerable heterogeneity and stratification, so much so that the presence of a separate after combustion chamber is unavoidable. A reasonable answer is to compute the internal volume and divide it by the total flue gas flow at typical conditions.
As order of magnitude, I suppose that 10 to 20 minutes would be a reasonable order of magnitude estimate for the solids, and about 1 second for the gases. The influencing factors are kiln length and volume, inclination, and speed of rotation. Some patents tried to lengthen or shorten residence times, such as scrolls in the refractory, chains in (now defunct) wet cement kilns, etc.
Answer to Alfons Buekens, but to calculate the filled volume you need to know the residence time, using the formula given in other text and also Perry, that is to used the residence time to calculate the diameter and length , speed, l believed we can assume the residence time, calculate the diameters and length, then model the kiln and then optimize or do experiments.
I guess your question is a bit general. First of all, the residence time in a rotary kiln can be referred both to the gas residence time and solid residence time, however, considering your process I suppose you should be interested in solid residence time in the rotary kiln. Again, solid residence time in rotary kiln depends on your process aims, feed characteristics, reactor size, feed rate, etc. Assuming that in a waste incineration process you want to obtain as less solid residue as possible, then you should increase the residence time by adjusting the feed rate, kiln slope, rotational rate, etc to the extent that you get as less solid residue as possible. It should be also mentioned that energy consumption and economics of the process may restrict your residence time as well.