The Effect of Altitude
We want to minimize airspeed, obviously. The wind speed is partially under our control-we can choose our course and we can choose our altitude. Once we fix our course, the CWV angle is fixed, but how do we determine the optimum altitude?
If the CWV is at least 90 degrees, the wind is definitely unfavorable, pick and altitude at which the wind is weak.
For smaller angles, there is an optimum ratio of wind speed to ground speed for minimum drag, and you can adjust your altitude up or down to obtain it.
If we rearrange equation 1 to solve for (AS/GS)2, differentiate (AS/GS) with respect to (WS/GS), set the derivative equal to zero, and solve for WS/GS, we get the marvelously simple result:
(WS/GS)=cos (CWVang) [equation 2]
****
In general, wind speeds increase with altitude. If the wind shear exponent is 0.2, a ten-fold change in altitude results in a 1.58 fold change in wind speed. So that gives you an idea of how much of a wind speed change you can effectuate by flipping between 100 and 1000 meters altitude.
However, there are exceptions. For the Graf Zeppelin, returning to Germany from South America, it had to fight through the northeast trades. It found it advantageous to ascend to 4000-5000 feet, where the trade winds were weaker. (Dick 58).
****
Increasing altitude also reduces drag. Drag force is proportional to the density of the air, which decreases linearly (temperature lapse rate of 6.5oK/kilometer) as altitudes increase, up to the tropopause at 11 kilometers. The physics-trained up-timers will realize that they can calculate the pressure and density using the temperature, the Ideal Gas Law and the hydrostatic equation. The necessary constants should be in CRC.
Air Speed versus Ground Speed
As a practical matter, an airship pilot is going to have a much better knowledge of the air speed (displayed on the dashboard) than the ground speed (calculated from observation of time between positional fixes). As Munk was perhaps the first to point out, the power requirement for an airship is proportional to the cube of the air speed, and the consumption of fuel over a segment is proportional to the power divided by the air speed. Given the wind velocity and course, it's possible to calculate (Munk 4) the airspeed and heading that the airship should be holding. On the other hand, a passenger or freight shipper cares a lot more about the ground speed.
In my route planning spreadsheet, I allow the user to specify either the ground speed or the air speed for a route segment. An experienced pilot has told me that I need only worry about setting the air speed, since aircraft engines are designed to work efficiently only in a narrow power band, which will in turn determine the air speed the pilot will seek to maintain while in flight. That will, in turn, limit what missions a given aircraft, whose engine has a given power band, will fly.
I understand his reasoning, but I think the 1632 writing community needs a more flexible tool. Airships, especially large airships, will be extraordinarily expensive by seventeenth-century standards. While a pilot may be more concerned with air speed, the airship customers are more interested in ground speed-how soon will passengers or cargo be delivered to a particular destination. Being able to meet particular ground speed requirements may determine whether an airship even gets built.
It's also worth remembering that there are no pre-RoF aircraft engines in Grantville. All of the engines used in the first decade of the 1632 universe will be re-purposed auto and truck engines, or down-timer built first generation steam and gasoline engines. Their performance characteristics will be different from those of a modern general aviation aircraft engine. In particular, I would expect that they will have a broader but lower power band.
Still, it's worth taking a closer look at the issues of internal combustion engine and propeller performance. I will be doing just that, in a future article.
A Look at the Hindenburg (Still Air Conditions)
Table 2A presents the dimensions of the Hindenburg, which can be used (with air density) to calculate the drag force upon it for a given air speed, and the propulsive power required to overcome that drag.
With a few assumptions, I have calculated (Table 2B) the required engine power and required fuel for the reported cruising speed, and required engine power for the reported maximum speed. I assumed energy density of 40,000 MJ/kg; and the following efficiencies: diesel powerplant 40%, propulsive 85% (but bear in mind that that propeller efficiencies are dependent on airspeed and most likely optimized for cruising speed), and thus overall 34%. The notes are to sources/explanations given in Appendix 3.
At the stated altitude, air density is 98% that at the surface. For the cruising speed, the required power almost exactly matched the published cruising power and the implied range was only 5% greater than the published range. But bear in mind that the assumed efficiencies are educated guesses, they aren't known values for the specific diesel engine and propeller used on the Hindenburg. I was actually surprised by how close the published and calculated numbers were.
Flight 14 demonstrated just how vulnerable airship performance is to adverse winds. It was "one of the longest flights to Lakehurst of the entire 1936 season:" (Dick 126). The Hindenburg encountered a front, and its ground speed dropped as low as 30 mph. (127). Later, "head winds, some as high as force 9 [47-58 mph] . . . were encountered until the ship was almost five hours out of Lakehurst" (130).
Wind-Adjusted Power and Fuel Requirements for Different Routes
Once we try to take wind into account, the calculations get hairy quickly. For this reason, I constructed a spreadsheet to do the heavy lifting. In Appendix 1, I will describe how to use the spreadsheet.
Please note that none of the calculations are actually beyond the down-timers; we know that they can do trigonometry and they can certainly learn spherical geometry and vector arithmetic. It will just take them longer, and the calculations will be more prone to error if they don't have access to one of the up-timer's computers or calculators.
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)