Grantville Gazette 36 gg-36

by Paula Goodlett

  For another, it's inconvenient to make all the necessary course changes. A modern sailor might approximate a great circle route by a series of rhumb lines, changed daily. An airship might make hourly changes but the principle is the same.

  Winds, of course, offer another reason for deviating from the great circle route. In general, you want to take the shortest path through a region of unfavorable winds, and keep the route as much as possible where the winds are favorable.

  If the great circle route is overland, then it may pass over mountains. You have three choices: (1) fly above them, but at the cost of having to carry more hydrogen and less cargo in order to achieve the necessary buoyancy (and there are some mountains you still won't be able to fly over), (2) skirt them, at the cost of increased travel distance, or (3) thread through the same passes that the mule trains do, but at risk of encountering turbulence and mountain storms.

  Limits on Route Length

  The length of the route is limited by the amount of fuel that the airship can carry, the energy content of that fuel, the efficiency with which the airship transforms fuel into propulsion, and the availability of refueling stops en route. The more fuel the airship carries, the greater its range, but the less its cargo capacity.

  The loss of hydrogen, whether through leakage or deliberate venting for altitude correction, can also limit the route. The less hydrogen, the less buoyant the airship is. It eventually needs to stop at a depot with a supply of iron, fuel and water so that it can make more hydrogen by the steam-iron process. (Hydrogen may also be made by the acid-iron process, but sulfuric acid is likely to be harder to find, especially outside Europe.)

  Prevailing Winds Navigation

  The minimum distance route (always a great circle) is not necessarily the minimum time or minimum energy route. That's because winds affect how quickly an airship can travel and how energy it must expend to move a particular distance. Ornithologists tell us that "birds will wait to embark on a migration until they can fly with a tail wind and minimize the energy they must spend." (Deblieu 77). Airships can take advantage of the wind, too.

  Initially, the best that the characters will be able to do is to plan their routes to take advantage of prevailing winds; later, "pressure pattern" navigation, which takes advantage of chance "highs" and "lows," will be possible. Prevailing winds are "typical" winds; on a day-to-day basis, the wind varies in speed and direction. Indeed, the average wind velocity distribution itself varies, at a single location, on a seasonal basis.

  While sailors have taken advantage of prevailing winds for millennia (since a sailing ship cannot sail directly upwind, and can only beat obliquely upwind with difficulty, they had no choice), the formal mathematical theory of planning a minimum time path for a sailing ship was developed by Francis Galton in the 1860s and 1870s. Maurice Giblett, in 1924, proposed a similar scheme for use by airships. Unlike sailing ships, airships need fuel, and therefore there has also been interest in identifying the minimum energy route given a particular wind distribution (Munk; Zhao).

  In my article, "Untying the Wind," (Grantville Gazette 35), I explain what the characters might reasonably be expected to know, or find out, about the prevailing winds, and guide prospective 1632 universe authors to sources of more detailed information.

  Speed Variation

  Just as with a sailing ship, an airship can expect to experience both poor and good passages, depending on the vagaries of the wind. In December, 1934, over the Mediterranean, the Graf Zeppelin encountered northwest winds of 45-56 mph, which increased its ground speed to 122 mph. (Dick 52).

  For the Hindenburg, the Frankfurt-Lakehurst passage varied from 52h49m to 78h57m, while the return was usually faster, ranging from 43h02m to 60h58m. The latter no doubt resulted from the advantage of flying with the westerlies. For the passage from Frankfurt to Rio, the Hindenburg's times ranged from 85h13m to 111h41m, and the return was 93h17m to 105h57m. (airships.net). There was a fairly wide variation in westbound routes-as far south as the Azores and as far north as the Orkneys-but the return flights were, at mid-ocean, between the latitudes of Bordeaux and Aberdeen.

  Not only did airships pick their routes to benefit from favorable winds, they chose their cruising altitudes with the same consideration in mind. The normal cruising altitude of the Graf Zeppelin was 575-820 feet, but it went higher if the upper winds were better. (Dick 67).

  What a drag . . .

  A balloon rises until the buoyant force lifting it and the gravitational force pulling it back toward the surface are equal. Its horizontal movement is dictated by the wind, which exerts a drag force on it, pushing it downwind. A force, by definition, causes an object to accelerate (gain speed); the less massive the object, the faster it accelerates.

  The drag force on the free balloon is proportional to the square of its airspeed, that is, its speed relative to the air mass. Its airspeed is thus the difference between its ground speed and the wind speed.

  When the balloon is launched, it has no ground speed, and so its air speed is equal to the wind speed (but opposite in sign). The drag force on it is high, and it accelerates quickly. As it accelerates, its ground speed increases. This causes the air speed to decrease, and thus the drag force on it to decrease. Thus, it continues to gain speed, but more slowly. If the wind remained constant, its ground speed would approach, more and more closely, the wind speed. A gust could cause it to temporarily travel faster than the "normal" wind speed, in which case the drag force would cause it to decelerate.

  While only one horizontal force (the wind) acts on free balloon, an airship is subject to such forces, the wind and the propulsive force of its propellers or jets. For the purpose of this article, we assume axial propulsion, that is the engine drives the airship forward.

  It's time to talk about velocity. To a physicist, velocity isn't the same as speed, velocity is a vector which has both a magnitude (the speed) and a direction.

  The basic equation of airship motion is:

  velocity (ship relative to ground) = velocity (ship relative to air mass) + velocity (air mass relative to ground).

  This equation may be rearranged to solve for the velocity (ship relative to air mass).

  The ground velocity is defined by a ground speed (ship relative to ground) and a course (the geographic direction toward which the ship is moving). The air velocity is defined by an air speed (ship relative to air mass) and a heading (the geographic direction toward which the nose is pointing). And the air mass (true wind) velocity is defined by a wind speed and direction.

  Course, heading and wind direction are all defined so that north is zero degrees, and the angle increases clockwise. (This is not, by the way, the same convention that mathematicians used to express angles, so some mathematical conversions are necessary in order to apply trigonometric functions properly.)

  A further complication is that for the vector mathematics to work, the wind must be expressed as the direction the air is moving toward, whereas meteorologists define the wind direction as the direction the wind is coming from. If I refer to the direction of the wind vector, I mean the TO direction.

  The drag force is based on the apparent wind, that is, the velocity of the air mass relative to the ship. That's the opposite of the velocity of the ship relative to the air mass. The ship's heading is chosen so that this apparent wind is coming over the nose (headwind) or over the tail (tail), i.e., so that there's no crosswind. (The ship, aided by its fins, acts like a weathervane, turning into or away from the wind to make this happen).

  If the ship is unable to quite make this heading (because the wind keeps shifting faster then the ship can turn), then it will feel an apparent crosswind, creating a side force that causes it to "crab" or "sideslip," a movement sidewise in the direction the apparent wind is blowing (the equivalent of leeway for a watership). For the purpose of this article, we will be ignoring sideslip and side drag.

  If there's no wind, the air speed equals the ground speed. A wind that's a headwind (di
rectly opposing movement down-course) increases the airspeed (and thus the drag), and a tailwind decreases it, by the amount of the wind speed.

  Vector mathematics is necessary to calculate the effects of in-between winds. To add (or subtract) vectors, we "resolve" the vector into two mutually perpendicular components, for example, a north-south and an east-west component. If you are heading 20 mph northwest, that resolves to 14.1 mph north and 14.1 mph west.

  If vectors are to be added, we separately add up their north-south components, and their east-west components, and then recombine the components to get the combined vector (resultant). For example, motion 40 mph west and 30 mph north corresponds to movement of 50 mph in a direction about 37o north of west. Trigonometry, which is known to the down-timers, is needed to make these calculations, but vector mathematics is new to them.

  Sometimes, it's informative to resolve a vector into components other than geographic. For example, if we resolve the wind into a component in the direction ("down") the ground course and one perpendicular ("cross") to it, then we can readily see how much a favorable wind is helping us along and how much it's trying to blow us off course.

  The CWV angle in the table below is the unsigned angle between the course (C) and the true wind vector (WV). Thus, if your course is due west, and the wind is from the northeast, the wind vector is to the southwest and the CWV angle is 45o. The CWV angle is 0 for a down-course (tail) wind and 180 for an up-course (head) wind.

  The underlying equation, if you're wondering, is rather simple:

  (AS/GS)=sqrt((WS/GS)^2 -2*(WS/GS)*cos(CWVang)+1) [equation 1].

  It shouldn't be surprising that even an oblique headwind increases airspeed as a percentage of ground speed.

  However, notice that even a pure crosswind is bad. Why? because to keep the crosswind from pushing you off course, you have to point the nose a little bit upwind to compensate, and then you have to increase power so you maintain the required ground speed.

  If you're familiar with sailing ships, that may seem strange. Sailing ships do quite well with a wind off the beam. However, sailing ships capture the wind mostly on their sails, not their superstructure. The sails can be braced about to face the wind more directly. The more directly it faces the wind, the greater the percentage of the wind force that is felt by the ship. However, the greater the bracing angle, the greater the percentage of that felt force that is driving the ship sidewise rather than forward. But a watership isn't forced directly downwind like a free balloon because the lateral resistance is proportional to the density of water (much higher than air) and the resistance is increased by the keel, centerboard, etc. The bracing angle chosen compromises between increasing driving force and increasing leeway. For a wind off the beam, it would be 45o.

  For an airship, which doesn't have sails, only the component of the wind in the down-course direction is helpful, the cross-course component must be fought.

  Note that if wind speed exceeds the desired ground speed, drag increases even when the wind is from a favorable direction. Drag is the result of the relative difference in speed between the ship and the air, and it doesn't matter which is moving faster. Of course, it's likely that if you are in an area of strong wind of favorable direction, you will happily allow your airship to behave like a free balloon, and let its ground speed equal the wind speed and its airspeed (and drag) drop to zero. But if there was a reason you couldn't do this-perhaps you're escorting a surface ship-then you must pay the piper.

  ****

  Now, let's look at the consequences of the numbers in the table. Drag is proportional to the square of the airspeed, and the power to overcome the drag is proportional to the cube of the airspeed. The energy required for the journey is proportional to the square of the airspeed (and the distance to be traveled).

  So, if the airspeed is 50% of the ground speed, then the drag is 25% of what you'd experience with the same ground speed in still air, and the power requirement just 12.5%.

  Plainly, if you don't have to go very far out of the way to take advantage of a favorable wind, you should do so.

  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.