Hank: Shipyard Sam is onta sumpin'. However, first, you need to know the size of the grates required to release enough heat to provide the horsepower to give the speed desired. I do not have the numbers near at hand but using 1.3 X the square root of the water line length is the formula to determine what the natural "hull speed" of a displacement hull might be. Anything over hull speed is gotten at an increasingly severe tax on horsepower required. Usually only the military and ships desiring high speed to attract pax can afford such an extravagance. All of the info that I have relates to small steam launches...not big steamboats. Having determined the horspower required, there are tables giving square footage of grate required to produce enough heat release to provide that horsepower (depending upon fuel). Then there is the expansion due to the oxygen combining with the carbon in the fuel to produce carbonic acid gas plus the thermal expansion due to the increased temperature of the gasses. Stack temperature has to be greater than the temperature of the water at the pressure of the boiler and/or superheat of the steam desired...likely over 600 degrees F. in an old fashioned steamer. If the temperature is too low, draught suffers and you are likely to get condensation on the inside of the stack of sulfurous acid (from the sulfur in the coal when coal is burned) which will destroy the stack in short order. On the other hand, high stack temp is waste of heat. I'd say that the stack height was dictated by the bridges that must be negotiated; the diameter determined by the aggregate area of the flues plus a bit of a fudge factor. Not mentioned is the effect of air passing over the tops of the chimneys, inducing further draft when under way. I would suppose that the fancy "feathers" on the stacks would tend to negate this effect. I really don't know, but I would speculate that designing chimneys for steamboats is akin to designing propellors for displacement hull boats in that it is really hard to design a really bad propellor. I suspect that the reason that you observe so many ratios is that each yard had their own ideas about it and exercised their ideas accordingly. Since it was hard to measure the real efficiency, they did'nt and were able to live with whatever they produced. The Ohio/Mississippi river steamers that we all know and love are pretty much a study in inefficiency from stem to stern, but profits were great enough in their day to ignore efficiency (and often safety, years ago) in favor of convenience, cheap first cost and expediency. Freely translated: I don't know the answer either!