3) specify the desired ground speed for each segment, which, together with the segment length, will determine the expected travel time;
4) calculate the resulting air speed for each segment,
5) calculate, for each segment, its power and energy requirements.
The winds obviously, are educated guesswork, but the rest of steps 1-4 are straightforward spherical geometry and vector trigonometry.
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To calculate engine power, fuel energy, and fuel weight requirements, we need some additional numbers.
First, we need to calculate the drag force based on the airspeed. That force equals one-half times the air density times a dimensionless drag coefficient times the reference area of the object times the square of the airspeed; this formula is likely to be known in Grantville; see McGHEST/Wind Stress, Aerodynamic Force, etc.
Determining the dimensionless drag coefficient would have to be determined by wind tunnel experiments if the data wasn't in some book in Grantville. Airship designers typically calculate it based on a reference area defined as the 2/3rd power of the volume, in which case it's called a "volumetric drag coefficient" but the computation could just as easily be based on the cross-sectional area or the total surface area. A particular drag coefficient is only good for a particular shape, anyway. Determining the reference area of course requires additional calculations but the formulae are well known to the mathematically-trained up-timers.
To calculate fuel requirements, you would need to know the energy content of the fuel, and the efficiency with which the engines burn that fuel and use it to generate a propulsive force.
There is probably data in Grantville on the energy content of individual hydrocarbons, and of some typical up-time fuels, that can be used for estimation of the energy content of down-time fuels. For more precise information, you would ideally measure the "heat of combustion" using a constant volume "bomb" calorimeter (OTL, the first one was built by Berthelot in 1881), with combustion occurring inside the calorimeter. Energy content of fuels is measured in undergraduate chemistry labs, but more approximately, using a constant pressure calorimeter and external combustion.
As long as we are using up-time engines, there is reasonable chance that one of the up-timers will have a car manual that provides a performance curve (power versus engine speed) for that engine. It may be possible to determine the power of an engine made down-time by some sort of "tug-of-war" test against an up-time one of known power. If not, then we will need a dynamometer. OTL, the first dynamometer was invented by Regnier in the 1780s (Horne), for measuring the strength of men and animals, but improved versions were commercially important from the 1820s on, when they were used to measure the tractive power of locomotives.
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The characters will have to do these calculations the hard way-unless they have a computer or calculator. You may use my spreadsheet to do the work for you.
Sample Route Analysis: Cadiz to Havana
Cadiz to Havana-the Spanish treasure fleet route-wouldn't be one of my first choices for an airship route, but hey, for enough Spanish reales, I'm happy to oblige.
The great circle distance from Cadiz (latitude 36.5361o, longitude -6.29917o) to Havana (23.133, -82.3833) is 7300 km (4536 mi.), and the initial course is 281.65o . In contrast, the rhumb line distance is 7456 km (4633 mi.), and the constant course is 258.48o. (All of the numbers in this section come from my spreadsheet, and I will sometimes allude to spreadsheet results that are not included in the tables quoted below; putting all the numbers in the tables would have made them unwieldy.)
Here are the assumptions I made in creating tables 3A-3D:
Airship Volume: 1,008,300 ft3 or 28,552 m3 (thus, volumetric area of 10,657 ft2 or 934 m2)
(while my spreadsheet no longer allows volume as an input, this volume can be achieved with an ellipsoid having a length of 345.1 feet and a diameter of 74.7 feet, yielding a length/diameter ratio of 4.62 (which Zahn said had minimum drag).
This assumed volume was based on one of the many iterations of Kerryn's airship design. However his design has changed since then so we will have different results for required power and fuel consumption. Note that he postulates different envelope volumes depending on the type of engine, because they have different fuel weight requirements for the route, and his goal is to carry a fixed amount of cargo.
Efficiency: engine is hot bulb with efficiency of 0.14, overall efficiency is 0.1, so the assumed propulsive efficiency is about 0.71.
Energy Density of Fueclass="underline" 40,000 KJ/kg.
Cruising Altitude: 3000 feet (914 meters).
Drag Force: drag force nominally proportional to square of air speed, but the drag coefficient is itself a function of airspeed and length/diameter ratio, according to Konstantinov equations 1.19 and 1.24. Note that Kerryn calculates drag differently.
If we ignore the wind (pretend that you are traveling in still air), the ship's air speed will equal the ground speed. If we assume a ground speed of 30 mph (13.41 m/s), the drag force is 3,642 newtons, the propulsive power is 66 hp, and the required engine power output (given the propulsive efficiency) is 92 hp. Note that because the air density is 92% that at sea level, you have to use a higher engine setting (in terms of rpm) to achieve the required power output than you would at sea level.
Fuel consumption is at a rate of 96.92 pounds/hour or 3.23 pounds/mile. The specific fuel consumption is 1.06 pounds/hp-hr-about double that reported for gasoline aero engines, consistent with hot bulb having about half the efficiency of a gasoline engine.
Total fuel consumption would depend on the route; on the great circle, it's 7.4 tons, one-way. The rhumb line route is only about 2% longer.
It's easy enough to compute the (still air) effect of a different ground (and thus air) speed; just remember that the drag force is roughly proportional to the square of the speed, and the power and fuel consumption rate to the cube of the speed.
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Now, let's consider winds. Let's assume that the northern limit of the northeast trades at 30oN and that these winds are from the NE (duh!) at 14 mph at the surface (10 m height), and that the southern limit of the westerlies is at 35oN and that these come from the west at 21 mph, surface. Finally, we are going to assume (for now) that the variables, in-between, are on average without wind.
Suppose we aim to fly at a ground speed of 30 mph (unless the wind will let us fly faster for "free"), and at an altitude of 3000 feet (unless otherwise indicated). The winds are stronger at that height; given a typical "wind shear exponent" of 0.2, the trade winds are 34.54 mph, and the westerlies 51.81 mph!
Let's begin by assuming that the airship ignores the wind; it flies the rhumb line back and forth from Cadiz to Havana. The rhumb line crosses the 35oN line at 15.5869oW and the 30oN line at 44.6861oW. Because we are in three different wind zones (westerlies, variables, trades) on each of the two passages, we have a six segment route (Table 3A).
The required engine power is the propulsive power divided by the propulsive efficiency (0.71). The required fuel energy is the propulsive work divided by the overall efficiency (0.1).
The total travel time is 301 hours. Assuming an energy content of 40 kJ/kg fuel, we would need to carry about 49.5 tons of fuel to fly this route (without any allowances for mishaps). That's a lot of fuel, more than three times the still air requirement!
Why did we end up in this strait? We face unfavorable winds in segments 1 and 4 (their effect could be muted by flying at a lower altitude). And we spend relatively little time in the favorable winds of segments 3 and 6.