I have not attempted to make the exact calculation for a great circle route. Why not? Since the course is constantly changing, the effect of the wind (even a constant wind) is also constantly changing along the course of the route. We are talking about solving the integral of a very complex nonlinear function, and it is not a standard integral, so it has to be approximated by numerical methods.
But if they're curious about what would be the benefit of a great circle route crossing particular wind zones, there is a way to obtain an approximate answer. In essence, you calculate intermediate points on the great circle route, and calculate the power and energy requirements for rhumb line segments connecting those points.
The more segments there are, the more computational work you are inflicting on yourself, but the closer you come to approximating a great circle route (if that's what you want).
In any event, in order to quantify what fuel is needed for different routes we need to
1) break the route down into segments, each segment expected to experience a "uniform" wind (that is a wind that doesn't change mid-segment) and calculate the length of each segment
2) specify what the wind is for each segment (this could be an "average," "worst case" or "best case" prevailing wind, or the wind forecasted to occur on that segment on a particular flight by the time we traverse it)
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 Fuel: 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 ai
rship 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.
It's also wise to look at the engine power required column (in the spreadsheet). For the table 2A route, the highest engine power required is 1487 hp on segment 1. If your engines can't put out that cruising power at cruising altitude, then you can't fly the route with the conditions given. (And of course you actually need more power, because winds could be worse than the average values placed in the spreadsheet.) If your power is inadequate, you need more powerful engines, more efficient transmission, or a less power-demanding route.
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Now let's examine a wind-friendly route Table 3B). The simplest assumption is that we fly directly south from Cadiz through the westerlies and variables to 30oN (this is the shortest route to the trade winds zone), then fly directly (rhumb line, although great circle would be shorter) to Havana (completely within the trade winds zone), then directly north through the NE trades and variables to 35oN, and finally directly (rhumb line) to Cadiz (completely within the westerlies zone).
If the wind is unfriendly (segments 1 and 4), then we fly low (300 feet). If the wind is favorable enough so that the down-course component is greater than 30 mph (segment 6), we take advantage of this and set the ground speed accordingly. On segment 6 it's disadvantageous to fly a ground speed less than 52 mph, we want to "free balloon."
This time, the fuel requirement is 7.9 tons, only one-seventh that for the "brute force" strategy. While the route is longer, the total travel time is only 282 hours.
I experimented with the effect of flying NW, rather than N, in segment 4 (out of the NE trades). While this reduced the power requirement, it increased the distance even more so, and the net result was that it was less energy efficient.
Segments 1 and 4 have the least favorable winds, and we can reduce fuel consumption even further by reducing ground speed for those segments. If they were both reduced by 50%, the travel time would increase to 301 hours, but the fuel requirement would drop to 9.6 tons.
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Is it helpful to fly a great circle route? I replaced (Table 3C) segment 3 of the last example with a great circle approximation, four smaller segments (3-6), with intermediate waypoints at 25%, 50%, and 75% of the great circle route between where we entered the NE trades and Havana. This reduces the total distance to 10,178 miles.
For segments 1 and 7 (old 4), the wind direction is unfavorable, so it's advantageous to reduce the strength of the wind aloft, and hence we fly low. And for segments 6 and 9 (old 6), winds are favorable enough to mandate a ground speed higher than our 30 mph default.
The route is 75 miles shorter, 4 hours quicker, but less energy-efficient (8.4 tons fuel). Why? On segment 3 in table 3B, the CWV angle was a constant 39o. On the corresponding segments 3-6 in table 2C, it was 54, 44, 33 and 25o. Because of the nonlinear nature of drag, the higher angles on segments 3 and 4 hurt more than the lower angles on segments 5 and 6 helped.
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What if we replaced the return passage with a great circle approximation? Segments 1-5 are the same as for table 2B, whereas segment 6 is replaced with segments 6-9. On those segments, the ground speed is increased (48, 51, 51, 49, respectively) in view of the high down-course winds. Segments 1 and 4 are still flown low to minimize unfavorable winds.
The travel distance is 10,141 miles, and the travel time is 283 hours. The least efficient segment has a propulsive power of 363 hp and the best a mere 0.5. The total propulsive work done is 11,340 hp-hr. With the assumed overall efficiency, fuel consumption is 8.4 tons.
Why? The CWV angle for segment 6 on the table 2B route was a mere 1o! So the great circle return route, while shorter, will certainly experience more drag. The CWV angles aren't bad-5 to 20o-but they can't beat 1o.
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If those fuel requirements are still too high, then you need to bring down the speeds, increase the altitude, and/or shorten the route.
If you can't find a workable combination of route, set air or ground speed, and altitude, then you have to reexamine your airship design. In essence, use a more efficient engine (diesel instead of hot bulb, hot bulb instead of steam) or find ways to increase the propulsive efficiency.
Remember, the calculations above assume overall efficiency of 10%, so the fuel requirements are ten times what they would be with an ideal (100%) system. (Of course, an ideal system is impossible, but you can do better than 10%.)
Other Routes
There isn't space to discuss alternative routes in the same detail that I did Cadiz-Havana.
What I can do is give some idea of the magnitude of the task they present.
Spain-Peru. This is quite tricky. I imagine that the outward flight would feature a refueling stop at a Spanish holding in the Caribbean, possibly Hispaniola or Puerto Rico.
The obvious continuation is to take the northeast trades over the Amazon and on to Peru. There are two considerations here. The first is timing. Northern South America has a monsoon and the winds are from the northeast in January but from the east or southeast in July. The other problem is, how to you get over the Andes?
So, you say, let's cut across Central America. Fine. Now what? All along the South American coast, the winds blow north up the coast. Sailing ships had to beat down, but an airship will have to pour on the power (and consume a lot of fuel).
The return isn't much easier, because you're fighting across the entire north-south extent of the northeast trades.
Whether this airship route is worth it, to avoid the long watership haul around Cape Horn, remains to be seen. The Spanish didn't try; they shipped gold and silver up the coast to Portobello and then moved it overland across the isthmus for pickup by an element of the flota.
Europe-India. By way of example, the great circle route from Amsterdam to Chennai is almost entirely overland, although it does cross the Caspian Sea. Unfortunately, while it avoids the Himalayas, the Elburz Range and even the Plateau of Iran, not to mention the southward extension of the Hindu Kush, are quite high enough to cause problems. Hence, it's likely to be necessary to head south first, skirting the Alps, then follow the Mediterranean eastward, cross the Saudi Arabian Peninsula to the Persian Gulf, and then follow the coast to Gulf of Gambay. If the airship has sufficient cruising altitude, it can cross the Deccan Plateau, otherwise it must work its way around India. Timing will be important because of the Indian monsoon; the winter is the best time to be crossing the Arabian Sea if you need to make southing; however, if you must round the southern tip of India, the winds of the eastern (Coromandel Coast) are more favorable in April on. Winds aren't favorable for a return until November, and then you want to work your way over to the Red Sea and back to the Mediterranean.
India-China. Unfortunately, the great circle crosses quite a few mountain ranges. So we'd need to take a mostly oceanic route, which subjects us to several monsoon belts.
Europe-China. We probably would need several refueling stops for this to be feasible, but I can envision an overland route, more or less the great circle route from Amsterdam to Beijing. It passes north of the major Russian and Chinese mountain ranges. The winds on this route are mostly light,
on the order of 4 m/s. Fly high on the way eastward (to maximize the westerly wind) and low on the return (to minimize it).
Mexico-Philippines. This would follow the standard Manila galleon route.
Pressure Pattern Navigation
Up until now, we have assumed that you will play the odds, that is, plan your route with the expectation that the actual winds you encounter will more or less correspond to the winds that prevail during the current month over the stretch of land or water you are flying over.
However, with the right resources, you can make ad hoc adjustments to your flight plan to take advantage of the winds that are actually blowing at the time of your flight.
The lower atmosphere is characterized areas of high and low pressure; these appear, intensify, move around, stay in one place for a while, fade, or disappear altogether. In low pressure areas, the air rises, and clouds are formed. In high pressure areas, the air subsides, and is dried out. Air flows (wind blows) out from highs and into lows. However, because of the coriolis force caused by the rotation of the earth, the near-surface wind is deflected. In the northern hemisphere, the winds spiral clockwise out of highs and counter-clockwise into lows.
So, that means that if a storm system is crossing the North Atlantic at the time of your flight, you can take advantage of the winds around it by staying south of the storm when flying to the east, and north of the storm when flying to the west.
At a minimum, this pressure pattern flying requires that you have reliable meteorological information for the area you are crossing. The reports could come from ground stations, or from ships or aircraft. Preferably, you have access to reliable meteorological forecasts, because, by the time you fly from point A to point B, the winds at point B have probably changed as a result of the movement of that storm. On the Graf Zeppelin (1928-37), the navigator picked route segments in one hour flight time increments, based on weather reports and forecasts.
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