I'm looking for an answer concerning a steam powered airship. The steam is generated via coal burning. The ship is propelled by an X amount of steam engine propellers (think Besler steam plane). The craft would have to have a coal reserve and an X amount of water tanks which would make for a good amount of weight. Would the ship be able to get off the ground, and if so, would it be able to fly at a useful speed?
You could theoretically make a steam-powered airship, but it wouldn't be worth the hassle. Moreover, by the time that we had the technology to make the airship frame itself we already had very good diesel engines.
The efficiency of a plain old non-compound Watt-type reciprocating steam engine is about 2.5%. For an airship you want a condensing engine, because you absolutely want to minimise water loss. Water is heavy.
Stephenson's Rocket locomotive of 1829 weighed 4300 kg (complete with water in the boiler). You need at least two engines, unless you plan to have some fantastically complicated transmission chains. Let's say that with a little better engineering you can reduce the weight to 3 tons per engine.
The first commercially sort-of successful airship, the Zeppelin LZ10, was 140 meters long, had 17800 cubic meters of hydrogen and could carry 13 crew and 20 passengers; it was powered by 3 engines of 100 kW each, with a maximum speed of 77 km/h.
Say that your airship will use 2 light-Rocket style engines, souped up to give 100 kW each, and you want to have enough fuel for 5 hours; suppose that you burn some kind of coke or anthracite giving 30 MJ/kg. The two engines will weigh 6 tons, and you need to carry about 5 tons of fuel. You need 12 tons of lift only for the engines, water and fuel!
12 tons of lift require 12000 cubic meters of hydrogen. The ship structure will also require lifting, plus any passengers. Overall, your ship needs to be much bigger than LZ10; say, 180 meters long, 34000 cubic meters of hydrogen.
And from here comes the fundamental problem: what material are you going to use for the ship structure? The airship cannot be a blimp, because if needs to lift those heavy engines, water and coal: it needs a rigid structure. The structure cannot be made of iron — too heavy. Cannot be made of wood — too weak. You need aluminum. But if you have aluminum, why are you playing around with steam engines?
The second big problem of using inefficient steam engines in an airship is the loss of weight. Every ton of coal you take on board at take off and burn requires that you vent 1000 cubic meters of hydrogen in order to remain in equilibrium with the displaced air — airships are aerostats. When you land, you need to refill the gas cells, otherwise you won't be able to take off again.
The third problem is the nature of steam engines, which burn their fuel in the open, which is a big huge enormous hazard for a hydrogen-filled airship. (Cannot be filled with helium, for reason of the need to vent lifting gas to compensate for the stupendous loss of weight given by burning coal with a very low energetic efficiency.) Real life zeppelins used diesel engines, because they don't need sparks...
P.S. The picture in the question is fundamentally wrong, in that the gondola is way too big. Look for example at LZ120 (better pictures at airships.net) for a clear view of the relationship between the body filled with lifting gas and the gondola.
Since we have Steam engines, you might sidestep the problem with the complimentary technology of Stirling Engines. Rev Stirling developed the engine because of the safety issues of contemporary steam engines (especially boiler explosions and steam leaks), and inadvertently also developed perhaps the most efficient heat engine possible (Stirling Engines can operate at close to the Carnot Limit).
Simple Stirling Engine. Plans here
A Stirling engined airship would not have a boiler, water, condensers or much of the other paraphernalia associated with steam engines, making the airship much lighter and also making trim changes due to the consumption of fuel and water much easier (no water is being used and the amount of coal fuel needed can also be reduced). While contemporary Stirling engines had low power to weight ratios compared to steam engines, modern Stirling engines can have comparatively high power to weight ratios. If there is a strong demand for this type of engine in your world, it can be easily assumed the engineering community would have worked hard to discover ways to make Stirling engines lighter and have higher power outputs.
The only real issue would be to develop some sort of "firebox" or combustion chamber which can provide heat to the engine without endangering the rest of the ship.
If you had a fantasy steampunk world where the air pressure is comparable to what it is on the planet Venus (90 atmospheres) but breathable like Earth then such a vehicle might be possible. However you'd have to ignore all of the other negative effects that such high pressure would cause. Or maybe a thicker atmosphere combined with lighter gravity. The only other approach would be some steampunk fantasy material like Cavorite (H.G. Wells First men in the Moon) or Liftwood (RPG game Space 1889). Otherwise such a vehicle is not possible. Too little attention to weight reduction and not enough gas volume. Oh by the way the Besler aeroplane used a coil-tube flash boiler that burned oil producing steam at 1130 psi and 430° C. It could NOT run on coal.
Given you've front ended the tags with magic I'm going to go that road; you build a closed cycle steam engine using material reinforcement spells so it's basically made of foil, a locomotive engine we based on the Flying Scotsman built this way weighed about 20 kilograms (we made ours using a couple of other permanent spells; a heater that kept the "combustion" chamber white hot and a cooling spell that kept the condenser ice rimed, to make it into a perpetual motion device). You use further material reinforcements to built vacuum cells instead of using lifting gas and armourplate the shit out of the whole thing because you've got to overcome all that excess buoyancy somehow. Speed is limited only by story.
Depending how far you want to go you can use any configuration and materials you like really because "a wizard did it" so you can mess with the standard densities etc... or you can create materials that don't exist that make the whole exercise realistic; have a look at The Edge Chronicles by Paul Stewart and Chris Riddell, Wiki here, where lighter-than-air wood and stone both feature as methods to facilitate skyship building.
Ridiculous, huh? Check out Wikipedia Henri Giffard. I don't disagree, but the problem is the exhaust, imho. Your pictures seem to neglect the enormous problem of smoke, soot, embers, and sparks. Take a look at any photo of a steam powered coal fueled locomotive to "get real" here. Some reason the steam couldn't run a turbine with very small (negligible) losses? Even a very 'clean' combustion machine - like modern internal combustion engines, can 'back-fire' and start fires. But, seems to me that coal power isn't out of the question, with proper exhaust (and exhaust pre-processing).
One alternative to steam could be the use of gears via manual labor. For example, 16th century galley ships relied on rowing as a primary means of propulsion. They achieved this via men (often hundreds) rowing large oars below deck. What if, instead of oars, the men cranked gears. The gears would in turn power propellers that would propel it forward.
This would solve the problem of burning coal on a wooden craft and would eliminate soot, steam and smoke.
Lets do some math
To create 1 kilowatt hour of energy, it takes about 95 Liters of water in a steam turbine. Now these things are about 1 ton each let's say.
Now a typical jet engine (can be used on an air ship) uses about 5 megawatts of energy. 1 Megawatt is 1000 kilowatts. So for this jet engine to power your airship you will need 5,000 kilowatts of energy. Now multiply that by 95.
You would need 475,000 liters of water for a 1 hour flight. This is carrying about 220 tons.
1 liter of water is 2.20 pounds.
You will carry about 1,045,000 pounds of water. Which is about 500 tons. This is a lot more than what your airship can carry.. This is excluding the amount of coal you will be carrying. You will also need multiple steam engines to process enough water into steam. If this airship does fly, it will be quite slow.... An attack on this airship would be way too easy by the other nations.
PS. Please correct me on any mistake in MATHS
As Li Zhi has said these steam engines arn't always 100% efficient, actually they usually are not.
According to Wikipedia Henri Giffard flew a steam powered airship in 1852.
The Giffard dirigible or Giffard airship was an airship built in France in 1852 by Henri Giffard, the first powered and steerable (French: dirigeable - "directable") airship to fly. The craft featured an elongated hydrogen-filled envelope that tapered to a point at each end. From this was suspended a long beam with a triangular, sail-like rudder at its aft end, and beneath the beam a platform for the pilot and steam engine. Due to the highly flammable nature of the lift gas, special precautions were taken to minimise the potential for the envelope to be ignited by the engine beneath it. The engine's exhaust was diverted downwards to a long pipe projecting below the platform, and the area surrounding the boiler's stoke hole was surrounded by wire gauze. On 24 September 1852, Giffard flew the airship from the hippodrome at Place de l'Etoile to Élancourt, covering the 27 km (17 mi) in around 3 hours, demonstrating maneuvering along the way. The engine, however, was not sufficiently powerful to allow Giffard to fly against the wind to make a return journey.
Obviously more advanced steam engines could have been used in later dirigibles if more efficient internal combustion engines had not become available.
Would steam turbines have had a better power/eight ratio?
Would using oil or kerosene as fuel for a steam engine have been more efficient?
How far could steam engines for airships be developed? Could they have competed with airplanes as well as airships with internal combustion engines did?
How could they avoid being replaced by airplanes?
If anyone has good answers to those questions you might be able to have reasonably plausible sort of steam powerered airships.
When looking a your picture, and thinking about history, all you would need is the technology to store compressed hydrogen. Use it both to replenish the envelope for elevation changes and also to burn for fuel, perhaps in a stirling engine like @Thucydides suggested. You would use the cold compressed hydrogen on one side and the burning hydrogen on the other.
It would all depend of the level of technology your world has and how wedded you are to the idea of lifting a wooden ship complete with a hole in the side and useless bowsprit along with all the other heavy junk a sea ship needs to weigh it down so winds don't make it capsize. If so I'd consider making it out of Martian liftwood (Space 1889) with a varnish of H. G. Welles Cavorite (First Men In The Moon). Weight vs Buoyancy is the primary issue. However lets assume that your civilization has advanced material science but just never got around to developing internal combustion engines (piston or jet) or lightweight electric motors and fuel cells. The reasons I'll leave up to you. In that case it is definitely possible. Consider the Magesterium airship seen in the movie The Golden Compass. Discard the fantasy 'anbaric' thrusters and the oversized tail fin and you have a basic dirigible/blimp with one important addition. The gas bag filled with lift gas (helium or hydrogen) has a secondary chamber surrounding the front end. See the picture. I would suggest that a secondary liftgas fills this chamber. One that can be adjusted in buoyancy, isn't flammable, and doesn't need the heavy complication of high pressure cylinders and compressors. That liftgas would be steam. Steam has over twice the lifting power of hot air and you will have plenty of it in the engine exhaust. Modern day polyester mylar, teflon and a new recent flexible aerogel would make a lightweight and durable laminate balloon material with good thermal retention properties. As steam condensed it would drain down to a reservoir for return to the boiler. This will eliminate the need for a separate condenser further lightening your airship. A closed steam cycle would also reduce the amount of heavy water you would have to carry. If you have access to lightweight low voltage high amperage DC generators you could carry a small one to make fresh hydrogen from your water supply in an emergency. A win-win design. Since you are insisting on coal as a fuel I would suggest a fluidized bed combuster with bed of a lightweight refractory sand, possibly alumina. Over it would be a flash boiler similar to that used in the 1910 White Steam Car. The steam engine would be a smaller simpler version of the radial steam turbine used in the 1894 Turbinia steamship. Of course everything would be constructed of lightweight polymers, and aluminum and magnesium alloys. Heavier metals would be used only where absolutely necessary and kept to a minimum. Aerogel would be used for insulation where possible. If you designed it properly I think you could get a pretty respectable performance out of a vehicle like that.