Highways of the Sky


In late twentieth-century America and Europe, freight and passengers are transported by autos and trucks, trains, ships and aircraft. However, there was once a dream that lighter-than-air airships, capable of powered flight, could play an important role in the transport network. Both Kevin Evans and Kerryn Offord have explored the possibility that airships could be built at a relatively early date in the 1632 universe, and carve out a niche for themselves.

In this article, I will consider what routes an airship of given capabilities can fly, and how the choice of route can compensate for the deficiencies that can be expected in early airships, i.e., that they are under-powered and have limited fuel capacity.

I’ll take the High Road, You Take the Low Road . . .

At the time of the Ring of Fire, trade routes linked together Europe, Africa, Asia and the Americas. Some routes were by land, others by sea, and these could be in competition. The sea route between Europe and India required circumnavigating Africa, and thus was much longer than the overland route across the Middle East. However, it was also safer (despite navigation errors, storms and pirates), faster (a ship in a good wind was several times faster than a train of pack mules), and able to accommodate larger volumes of cargo (it takes a lot of pack mules to carry as much as a single large merchant ship).

An airship can travel at least as fast as a watership and can take an overland route (at least if there are no mountains in the way). It is not vulnerable to piracy (except when it lands) and it can safely sail at night (except in mountainous regions).

Since airships can travel over land just as easily as over sea, they are likely to first make their commercial mark on those trade routes for which the land route is much shorter.

The shortest distance between two points on the Earth’s surface is a great circle route, and I’ll explain how to calculate that distance in the next section. Unless a mountain barrier blocks the way, an airship can fly a great circle route (although there are reasons that we will reveal that it might want to deviate from such a route). Waterships are more constrained, since any intervening land forces them to find a way around.

To calculate the length of a sea route, use the Portworld calculator:

http://www.portworld.com/map/

Be sure to disallow the Panama and Suez Canals!

For example, the sea distance from Amsterdam to Chennai is 12,634 miles (and that’s for a modern ship, that isn’t worrying about catching the right winds). That’s more than twice the great circle distance (4,899 miles)!

There are similar advantages to flying from Europe to China, or from Europe to the west coast of the Americas (e.g., Spain to Peru).

That said, airships can compete on the normal oceanic shipping routes, too. If they have enough fuel and engine power, they can behave like steamships, more or less ignoring the wind (although the wind will have more effect on any airship than it would on a water-bound steamship, because the airship doesn’t have an underwater section to confer lateral resistance if the wind is coming cross-course). If the airship needs to conserve fuel, then it behaves like a hybrid sail-steam watership, powering through regions of neutral or unfavorable wind (horse latitudes, doldrums) and taking advantage of favorable winds.

Route Selection: Great Circle, Rhumb Line and Composite Routes

Since the Earth is (more or less) spherical, the shortest path between any two points on the surface is a great circle route. The great circle is defined by the intersection of that surface with a plane containing the origin, the destination, and the center of the earth.

If you have the longitude and latitude, you may calculate the great circle distance using the Great Circle worksheet on my spreadsheet.

However, it’s nice to be able to actually see the route displayed graphically. If you have Google Earth, use that. Draw a line with the ruler between your origin and destination; that line is a great circle route. You can see where it crosses coastlines, or particular latitudes or longitudes. In addition, if you save the line as a path, you can right-click on the pathname in the sidebar and pick “show elevation profile.” This is by far the easiest way to figure out what is the minimum altitude you need to fly at to avoid an embarrassing tête-à-tête with a mountain (the mountain always wins).

If for some reason you can’t use Google Earth, you may try use Great Circle Mapper

http://www.gcmap.com/.

Prior to the Ring of Fire, it was impractical to sail a great circle route. On such a route, the course (the angle of the ship’s track relative to true north) is continuously changing. You must adjust your heading depending on where you are. However, pre-RoF navigation capabilities were gravely limited. Latitude was determined by sightings of the sun at noon, or stars at night, and the accuracy was no better than a quarter-degree. Longitude was determined by dead reckoning and could be wildly in error (tens of degrees!) if you had spent a long time at sea.

I discussed the possible improvements in the art of marine navigation in my two part article, “Soundings and Sextants” (Part I, “Navigational Instruments Old and New” in Grantville Gazette 14; Part II, “Celestial Navigation Methods,” in 15). Ultimately, of course, we will have sophisticated sextants and accurate chronometers, but it will take years, if not decades, for these to become commonplace.

Moreover, since an airship travels substantially faster than a watership, the time between celestial observations is more of a factor, and there are obvious problems with measuring the elevation of a celestial object above the horizon when you are in the air.

The Hindenburg didn’t in fact rely much on celestial navigation. Rather, it used the combination of a gyroscopic compass and dead reckoning. If it were traveling in still air, its position could be accurately calculated from its airspeed and heading. Wind could be assessed by flying a special pattern; every hour, head 45 degrees off course, first to port and then to starboard, and take drift readings. (Dick 60). The zeppelin was equipped with a searchlight and a telescope; water ripples or landmarks below were studied to determine the ships angle of drift (the angle between its heading and its course). (Grossman).

Prior to RoF, the standard navigational practice was either to follow a coastline (or other landmarks), or to sail a rhumb line (loxodrome). The latter means to sail with a constant compass heading. Even that had its difficulties, as it was difficult to correct compasses for magnetic deviation or variation, but at least at night the Pole Star provided a check on the accuracy of your compass reading. Even so, navigators preferred whenever possible to “run down a line of latitude,” that is, sail directly east or west, as that way the noon sun sightings could be used to verify that they were still on course.

Even after the introduction of the sextant and chronometer, mariners didn’t follow a perfect great circle route even when the winds were not an issue. For one thing, taking a great circle path could force the ship into high latitudes with stormy weather. Hence, even modern sailors sometimes follow a “composite” route in which they truncate the great circle at a particular maximum latitude, thus following a great circle route at the ends and a rhumb line (of constant latitude) in the middle.

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 (directly 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.

Table 1: Airspeed (AS) as Percentage of Ground Speed (GS) for Different Relative Wind Speeds and Directions

Optimal (Lowest AS/GS) WS/GS% for Given CWV

Wind Speed (WS) as Percentage of Ground Speed (GS)

CWV Angle

25

50

75

100

150

200

WS/GS%

AS/GS%

0

75

50

25

0

50

100

100

0

22.5

77.50

57.11

42.03

39.02

69.16

114.21

92.39

38.27

45

84.20

73.68

70.84

76.54

106.24

147.36

70.71

70.71

67.5

93.34

93.13

99.42

111.11

144.98

186.26

38.27

92.39

90

103.08

111.80

125

141.42

180.28

223.61

0

100

135

119.00%

139.90%

161.96%

184.78%

231.76%

279.79%

0

180

125%

150%

175%

200%

250%

300%

0

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.

Table 2A: Hindenburg (LZ129) Dimensions

Parameter

Value

Note

Length (ft)

803.8

1

Diameter (ft)

135.1

1

L/D ratio

5.95

Volume (ft3)

7,681,700

2

Volumetric Area (ft2)

38,932

2

Cross-Sect Area (ft2)

14,335

2

Surface Area (ft2)

271,258

2

Maximum Fuel Capacity, lb (tons)

143,200 (71.6)

3c

italicized dimensions assume ellipsoid shape.

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.

Table 2B: Hindenburg, Predicted vs. Actual Performance

Parameter

Predicted

Actual

Source

Still Air Performance at Cruising Speed. mph

76

4

and Cruising Altitude, feet

650

5

Drag Force, newtons

63,918

?

Propulsive Power, hp

2,912

?

Required Engine Power, hp,

3,426

4×850=3400

3b

Fuel Consumption time rate, lb/hr

1,267

1166

@78.3 mph

3g

Fuel Consumption dist rate, lb/mi

16.67

Still Air Range, mi

8588

8400

3c

Specific Fuel Consumption, lb/hp-hr

0.3699

0.37

3e

Fuel Needed, short tons

Frankfurt->NYC, 3859 mi

(great circle route)

32.2

Performance at Maximum Speed, mph

84

3f

Required Engine Power, hp

4586

4×1025=4200

3a

4×1320=5260

3b

Expected Fuel Consumption Rates, Various Speeds

lb/hr @82.7 mph (1450 rpm)

1604

1430

Dick 123

“@78.3 mph (1350 rpm)

1377

1166

“@69.3 mph (1250 rpm)

980

880

“@66.0 mph (1150 rpm)

855

660

Individual Flight Performance (actual include wind effects)

Flight #9, Friedrichshafen to Rio, 3/31-4/4/1936, 5948 nmi (6845 mi) in 100:40, average speed 68 mph

Fuel consumption

93,535 (46.8)

102,581 (51.3)

Dick 112

Fuel consumption time rate lb/hr

929

1019

Fuel consumption dist rate lb/mi

13.66

14.99

Flight #14, Frankfurt to Lakehurst, 5/17-20/1936, 3920 nmi (4511 mi) in 78:30, average speed 57.5 mph

Fuel consumption

45,655 (22.8)

90,632 (45.2)

Dick 126

Fuel consumption time rate lb/hr

582

1151

Fuel consumption dist rate lb/mi

10.12

20.03

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)

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.

****

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.

****

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.

****

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).

Table 3A: Cadiz to Havana, 6 segments, simple rhumb line both ways with no concession to prevailing winds (except taking advantage of tailwind on segment 6)

#

Segment

Distance

(mi)

Set Ground Speed (mph)

Wind @ Alt (mph)

Calc

Air Speed (mph)

Propulsive Power Output

(hp)

Propulsive

Work Output

(hp-hr)

1

RL to variables

531

30

51.8

81

1062

18,802

2

RL to NE trades

1728

30

0

30

67

3,774

3

RL to Havana

2374

30

34.5

19

19

1,469

4

RL out of NE trades

2374

30

34.5

62

493

38,970

5

RL to westerlies

1728

30

0

30

92

3,774

6

RL to Cadiz

531

52

51.8

10

4.8

35

Total

9,266

66,823

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.

****

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.”

Table 3B: Cadiz to Havana, 6 segments, “rectangular” route that uses prevailing winds

#

Segment

Distance

(mi)

Set Ground Speed (mph)

Wind @ Alt (mph)

Calc

Air Speed (mph)

PropulsivePower

(hp)

PropulsiveEnergy

(hp-hr)

1

S to variables

106

30

33

44

292

738

2

S to NE trades

345

30

0

30

92

754

3

RL to Havana

4,718

30

35

22

39

4,385

4

N out of NE trades

474

30

22

48

363

4,095

5

N to westerlies

345

30

0

30

92

754

6

RL to Havana

4,264

52

52

1.34

0.01

1

Total

10,253

20,726

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 7.3 tons.

****

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.

Table 3C: Cadiz to Havana, 9 segments (includes great circle approximation), uses prevailing winds

#

Segment

Distance

(mi)

Set Ground Speed (mph)

Wind @ Alt (mph)

Calc

Air Speed (mph)

PropulsivePower

(hp)

PropulsiveEnergy

(hp-hr)

1

S to Variables

106

30

33

44

292

738

2

S to NE trade

345

30

0

30

92

754

3

25% GC

1,161

30

35

30

90

2,486

4

50% GC

1,161

30

35

24

52

1,432

5

75% GC

1,161

30

35

19

26

713

6

Havana

1,160

31

35

14

12

312

7

N to Variable

474

30

22

48

363

4,095

8

N to Westerl.

345

30

0

30

92

754

9

RL To Cadiz

4,264

52

52

1.3

0.01

1

Total

10,178

11,285

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.

****

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.

****

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.

Unfortunately, it will be some years before there’s a good network of weather stations across the shipping routes of interest, and it will be even longer before we have reliable weather forecasting.

The next stage in the evolution of pressure pattern flying was made possible by the invention of the radar altimeter. Previously, the aircraft altimeter inferred the altitude on the basis of the barometric pressure. Since the radar altimeter measured altitude directly, that meant that a barometer could be used to measure the air pressure at the known altitude. Winds, especially at high altitude, tend to follow the curves of constant pressure (isobars) so you could use the barometer (pressure altimeter), adjusted for your altitude, to follow the wind. If you found that your pressure was dropping, it would warn you that you were getting closer to the center of the storm.

Predicting when the 1632 universe will develop radar altimeters is well beyond my area of competence, but I would not expect them until the 1640s at the earliest.

The Graf Zeppelin determined height “by firing a shotgun and calculating altitude from the time delay of the returning echo.” (Miller; Dick 61). The Hindenburg had a fancier sonic altimeter, using “a compressed air whistle at station 228, whose sound bounced off the ground and was picked up by a receiver at station 188.” (Dick 88; airships.net). Yet another trick, used on both the Graf Zeppelin and the Hindenburg, was to drop a water bottle overboard, and time its fall with a stopwatch (Dick 61-62).

The Graf Zeppelin also is said to have dropped smudge pots and observed the smoke to determine wind direction and speed (Miller), but that would only speak to surface wind velocity.

****

It’s important not to expect too much of pressure pattern navigation. Even if you know exactly where the lows and highs, and the associated winds, are, it doesn’t guarantee that it’s worth detouring to exploit them. It all depends on the size and intensity of the pressure system, its location relative to that of the airship at a given time, and its own speed and direction of movement relative to the airship’s intended course.

During World War II, pressure pattern navigation typically reduced transatlantic flying time “an average of 10% compared to a great-circle track, with occasional savings exceeding 25% . . . .” (Kayton 12).

Conclusion

“The Bozo people, who live on the southern fringes of the Sahara, believe that Wind wrestled with Water, and Water lost . . . .” (DeVilliers 12). If the new airships of the 1632 Universe, fight the wind, instead of exploiting it, they too will lose.

Author’s Note: Appendices, bibliography and the spreadsheet will be posted to www.1632.org in the Gazette Extras section.

****

Share

About Iver P. Cooper

Iver P. Cooper, an intellectual property law attorney, lives in Arlington, Virginia with his wife and two children. Two cats and a chinchilla rule the household with iron paws. Iver has received legal writing awards from the American Patent Law Association, the U.S. Trademark Association, and the American Society of Composers, Authors and Publishers, and is the sole author of Biotechnology and the Law, now in its twenty-something edition. He has frequently contributed both fiction and nonfiction to The Grantville Gazette.

 

When not writing (or trying to get an “orange blob” off his chair so he can start writing), he has been known to teach swing dancing and folk dancing, or to compete in local photo club competitions. Iver adds, “I can’t get my wife to read my fiction, but she has no trouble cashing the checks.”

Iver’s story “The Chase” is in Ring of Fire II