First Parallel: A Technical Appendix: The Thermodynamics of High Altitude Air-to-Air Combat in the Propeller Age

The "funny" way of talking about the science of aeroengines --thermodynamics-- is to say that "you can't win, you can't break even, and you can't get out of the game." That's three laws, and I only propose to talk about one today: the Second Law. It is one of those hard-to-teach rules, and I am going to throw the "ideal heat engine" formulation up and call it a day: Efficiency=1-Temperature(reaction)/T(sink). The efficiency of the best possible heat engine is less than one hundred percent, and it is less than one hundred percent by the ratio of the temperature in the engine with that of the sink into which it is rejected. You think that engines are about power? I tell you that they are about rejecting waste heat. (Or, amateurs study brake effective horsepower. Professionals study calories rejected.)

If I wanted to turn the laws of thermodynamics into cod philosophy (call me, TV! I will!) then I'd call this the diminishing returns rule. Which reminds me.

Photoshop 1939! From Flight Global Archives for 2/2/39, the first official Air Ministry publicity still of the Boulton Paul Defiant. How many guns in the turret? Ha! We're not telling!

Flight, amplifying what the Air Ministry has to say, points out that sometimes official secrecy makes people nervous. "For the past year or two," the paper tells us, people have been writing in to ask what has been happening on the two-man fighter front. The Hawker Demon has been the RAF's frontline two seat fighter since 1930, and is overdue for replacement. Now, at last, the Air Ministry responds. The Defiant, we are told, is a stressed-skin fighter of all-metal, mainly light alloy construction (so much for the patent Boulton Paul construction system) with flush rivets for an exceptionally smooth surface. The feature of main interest is the compact, power-driven turret, armed with a still-undisclosed number of guns. It is presumed that the Defiant's main mode of use will be to fly slightly ahead of, and below enemy bombers, where they are typically ill-protected, as has been demonstrated by later-model "turret" Demons. Given, the paper speculates, equipment with a Merlin I engine, the Defiant will have a top speed of slightly above 300mph, lower than the Hurricane or Spitfire due to the encumbrance of the turret and because of its slightly larger size. Fitted with a Merlin II, it should be as fast as any comparable aircraft in enemy service, while with a Merline RM2SM, giving 1145hp at 16,500 feet when fuelled with 100 octane, it would have an even higher performance. The airscrew is a 3 bladed variable pitch de Havilland, and the exhaust is of the new ejector stub (i.e. turn your exhaust into rocket fuel) type.

But the point, now that I have buried the lede so thoroughly, is this:

It is October, 1943. The Home Front is bracing for defeat in the air, an ignominious retreat to night bombing that will delay the strategic air offensive by months. Meanwhile the United States Army Eighth Air Force is getting ready to win the most extraordinary victory in the history of air warfare. It will, flying from expeditionary air fields in the United Kingdom, win complete air superiority over the home skies of the world's second largest industrial economy before the first tide of summer.

Some people will say that it did not matter, that the only bombing that counts is the kind done in visual range of a foxhole. Some people apparently cannot wait to see the United States Army hump off its entire artillery arm on the USAF (or turn the USAF into the army's artillery replacement). Despite my frustration with them, I cannot say that they deserve to live in America whose only army is light infantry. No-one deserves to live in Breaking Bad America. 

The P-51 is said to have saved, by itself, the whole of America's industrial strategy. It is the classic example of Joel Mokyr's formulation of technology as a "free lunch," an exogenous input to the economic process, unpredictable and unplanned. One can only lay out a welcome mat on the doorstep, as meritocratic America and miltarist-racial Germany do, and class-ridden Britain does not. I have taken aim at the P-51's status as a magic aeroplane before, and yet its origins were almost entirely serendipitous. North American Aviation had established a relationship with the Air Ministry by selling it the Harvard, so when Sir Henry Self of the British Purchasing Commission approached it about licensing production of the P-40, company president Kindelburger was listened to when he proposed a new fighter, instead. A four gun machine using the Allison engine was proposed. The design was commissioned in March, 1940, and the prototype flew first on 26 October 1941. This is a short development period rather than an unprecedented one, and speaks, as much as anything, to the extent of underemployment in the Los Angeles basin in the spring of 1940. Yet it is also one of the most historically important of all successful rushed technological development programmes.

I have in the past featured the P-51 as a "magic aeroplane," basically as a means of launching into the transformation of the refining industry during World War II. As the cars of 2013 crawl through Vancouver's morning fog this morning, they are carried by the invisible tides of history. But no-one knows about them because they're invisible! Today I shall take another tack, all appropriately autumny, about how we're all trapped in a cellar while the entropic level rises around us ineluctably. 

Second year physics emo is so lame. In ten billion years, we'll all be dead! In your workplace, you might have noticed that a single missed shift has the potential to raise entropy to the eleventy-millionth power. Perhaps you have thought about the war, and imagined a bomb piercing the roof, and either said to yourself, "how could they possibly cope?" Or, alternatively, "How could it have made things worse?"

Or maybe you asked yourself how Erik could possibly have got from the Defiant to the P-51.

Back up a little, to just past the mythic days of the origins of air-to-air combat, to this

And this.

On the night of 2/3 September, 1916, William Leefe Robinson went aloft in a B.E. 2C fighter, converted for night operations, as this was understood in 1916, when automatic flare droppers in the wing tips were high tech, and shot down an attacking German dirigible. He was promoted, sent to France, and captured when the flight of brand-new Bristol Fighters he was leading were massacred by the Richthofen Circus. It must have been a bit of a humiliation to be so treated in an aircraft that would soon win a formidable reputation as a plane that handled like a single-seater with a "sting in its tail." Thus did the world begin to learn that thrust-to-weight ratio is often (and, since jets,  always) better than handiness.  

Robinson's Lanze-Shutze shoot down helped drive the Germans towards a daylight raiding strategy. Soon, bombers "with stings in their tails" would appear over London by daylight. Technology had reached the point where a big bomber could carry a free-mounted machine gun, so that any fighter that tried to shoot them down would do so into the teeth of a comparable defensive fire.

Technology in 1918:

The Bristol Fighter continued tin RAF service until 1927, although as an "army cooperation" type rather than as a fighter. Insofar as the key to its tactical superiority was a large engine in spite of the extra crew, a smaller fighter with a better engine was going to have Brisfits for breakfast, the RAF concluded.

Meanwhile, the RAF seemed to be spending the decade of the 1920s converting legacy airframes from wood to metal. What did the future of air combat look like?

This:  

Entering service in 1928, the Boulton Paul Sidestrand was liked for its manoeuvrability, we are told. Or perhaps it was its quiet, since it was the first bomber with geared engines. Or the fact that it was named for the residence of the Secretary of State for Air. Or perhaps it was all three, as it did pleasantly noiseless acrobatics over chez Hoare. It would be hard to argue that it was that well loved, given that only 20 or so were ordered. It looks like a transitional aircraft. And "transition" would have been a word well-used by its crew. With now three defensive positions (ventral, dorsal, and fixed front), it only came equipped with two machine guns, the lone dorsal/ ventral gunner presumably shifting his weapon between positions depending on which angle the interceptors came in at. 

Three positions is better than two, but, still, an improvement was on order. Four Overstrands were actually converted Sidestrands. A comparison of the two pictures suggests that a great deal of converting was done. Apart from the actual lifting surfaces, the Overstrand is starting to look vaguely modern. That is an actual power turret in the front, pneumatically rotated and hydraulically trained, the first in the world, supposedly installed because at its giddy speed of 140mph, the front gun could not be handled, even on a Scarff ring, though in fact more because Farnborough wanted to play with powered turrets for a bit before going all in. 

The plane could also carry two gunners and two machine guns, allowing it to defend itself against attack from below and behind as well as from above and behind. Still not much doing if a fighter got on your tail, though, as I hear has been known to happen.

I thought that this problem was "solved" in the early 1930s by the French, but some noodling about Wikipedia is failing to produce any French bombers of the era with a "tunnel" mounted tail machine gun like the one that the B-17 ended up with. There is a pretty good argument for not extending the fighting compartment back all the way to the tail, after all. It increases drag and introduces stability issues. I want to say that the French concept of the "multiplace de combat," an aircraft bristling with gun positions in all directions gives us the first tunnel mounting, but I do not know. Italians? 

Hmm. That makes this all the more remarkable. 

No, not the daft flight angle. That's exactly what you might expect from playing with bath toys these many years ago. Someone screwed up the aerodynamics, and it flies slightly sideways. It's the four-gun powered turret in the back. Which Armstong Whitworth stressers might put forwards as an excuse for getting the dihedral wrong.

All that work for four rifle-calibre machine guns, with their very dubious "anti-materiel effect." The idea that you have to punch much bigger holes in planes in order to destroy them had a comfortable pedigree in the RAF in the late 1930s, but things got a bit . . . hypertrophied as the war loomed. The Air Ministry went out and specified that the next generation of heavy bombers would carry power turrets equipped with four 20mm cannons each, while the armaments wallahs independently evaluated various makes of same and voted for the Hispano-Suiza make, a design that originated from a moteur-canon  concept in which the barrel had to be long enough to fire througdh the propeller shaft, while the engine mounting picked up the recoil energy. To put this in perspective, the Browning 1919.303 has a list weight of 31lbs and a length of 53;" the Browning .50 a weight of 61lbs and a length of 65"; the Oerlikon 20mm FF cannon weighed only 53lb and was ... comparably short, Wikipedia failing me here. The  Hispano-Suiza weighed 94lb and was 100.6" long! 

Obviously, you get what you pay for in terms of muzzle energy, but you also have to pay the cost on the airframe side. Tolerable in the twin-engine fighters under development in 1939, the Hispano cannon entailed some design effort when it was shoehorned into the Typhoon, Hurricane, and even Spitfire. The logic, however, was inescapable. If you are on the tail of an enemy bomber that is shooting back with 4x (big!) 20mm, you want to return a comparable weight of fire.

Or was it really so inescapable? What if you abandoned the tail chase and flew "slightly ahead of, and below" the enemy bomber? Then you could use a turret, and the Defiant became a practical application, and you can also see the Air Ministry's rationale for whiting out the number of guns in the turret, which in some rarified land of pure reason might be taken as admitting that the four-gun turret, already seen on the Whitley, but not the Hampden, was to be the default installation going forward.  

Except that by now we have whiplash. A moment ago, and it really was a moment ago, bombers were defended from ventral and dorsal positions and the direct rear approach was not so important. Then, it was vital. Now, suddenly, fighters are attacking from any angle they choose. What, exactly, is going on?

The answer, I think, is that we need to start getting Second Law here. The hidden factor, of which we dare not speak, is the interception envelope. Given bombers approaching the target, on which vectors will fighters close? In the limiting case of a stationary bomber, or a fighter with infinite speed, the interception envelope is a homogenous, infinite shape. The fighter can approach from any angle it chooses. For the limiting case in which the fighter is only infinitesimally advantaged energetically over the bomber (speed, rate of climb, and the extension of the intercept window in time by early warning are all factors here), the only plausible intercept that allows enough time to aim for effect is directly from the rear. 

That said, there are a series of graze intercepts, in which, depending on warning time, the fighter will reach a minimum distance within effective weapon range at various different angles off the bomber. For a fighter with a fixed, forward firing armament, the result is a difficult deflection shot. John Lundstrom thinks that thanks to trained excellence in deflection shooting, the United States naval air arm achieved a high kill ratio against the Japanese in the Pacific War. I am a little skeptical, and would like to see the experiment replicated against less recklessly underweight aircraft flying shorter missions, but I do not think that anyone would disagree that deflection shooting is hard enough that a good technological approach is to minimise it. 

One such approach is to load armament on the fighter and straight-up win the broadside duel. Another is to install free weapons. The final one is to seek speed/rate-of-climb/early warning advantage. The fact that so much was going on with the latter, all secret, prevented people from disentangling the threads and talking clearly about the development of interception in 1939. By the spring of 1940, when North American Aviation countered a proposal to produce the P-40 under license by offering their own fighter, somewhat more clarity had been achieved.  whereas the Air Ministry had inherited the Curtiss P-40 contract from the French, and were thoroughly unimpressed. It is a little hard to divine Harry Self's intent here. March of 1940 was an exceedingly loose time; the British had a preference for making planes in California, for whatever reason. Most importantly, probably, North American was already one of the Air Ministry's preferred customers, thanks to their Harvard order. The Air Ministry was thoroughly "inside" North American, having intervened to introduce a retractile undercarriage. I am not seeing any sign of an RAF engineer officer's memoir entitled I Spent World War II in LA Liaising With the Aviation Industry And Getting A Really Good Tan, but there probably wouldn't be.

Although it is a little unfair to take away from North American's achievement. The rollout of the initial P-51 was impressively fast, only 150 days, and if the resulting aircraft was a little half-baked (armament of 4 .303s, two in gondola mounts under the engine, totally inadequate even by USAAC standards, never mind RAF), the company did an impressive job of shepherding some seriously innovative aerodynamics through the University of Washington and Caltech wind tunnels. The resulting laminar-flow wing was a bit of a sore spot strengthwise (you'll have to take my word for it that Freeman's review shows an elevated level of wing spar failure in P-51s), while the ridiculously big ramscoop and vulnerable air cooler that surely gave the plane its wink-wink nudge-nudge nickname of wild stallion, or "Mustang" (where we get the word from, I think) was a liability in the fighter-bomber role. Yet all of this is just a carping way of saying that the P-51 represented a huge success from the failing-forward point of view. 

In the middle of all of this, however, we might easily forget that bomber designers can work to narrow the interception envelope, too. The Amiot 354 is a good example of this. A strategic bomber designed to rely for self protection on its ability to penetrate enemy airspace at high speed and high altitude, the effort tends to be criticised with sideways looks at more-or-less failed "speed bombers" such as the Blenheim and Junkers Ju 88, and an embarrassed silence over the one that worked, the de Havilland Mosquito. 

The reason for the difference is, in turn, clear enough. Propeller-driven planes are inherently limited in speed by the efficiency of their airscrews. As tip speeds reach transsonic levels, there is stall and a loss of thrust, while greater engine  power must be absorbed by greater solidity, increasing parasitic drag, torque load, and weight. Meanwhile, the airframe itself is exposed to increasing drag for various aerodynamic reasons. Fighter speed climbed from under 100mph in the middle of WWI to 360+ in 1939, but the 460mph top speed of the Hawkers Sea Fury took another 7 years to achieve, and the clustering of fighter top speeds just under the Sea Fury is a textbook example of diminishing returns. 

As it happens, the United States Air Force did not try to penetrate German air space with speed. It used height and armament, instead. My point, and I should really learn not to bury it like this, is that in some sense, height and speed are the same. B-17s being big, they could carry turbosuperchargers and thus perform well at altitude. This was something that the Germans had trouble matching. Neither the Bf109 nor the FW190 was a particularly good high altitude fighter.

The P-51 was. We know the reason for that: Rolls-Royce had developed a two-speed, two-stage mechanically driven supercharger for the Merlin engine. Leaving the backstory of that development to the side for the moment as this post rapidly approaches wall of words status, I want to focus today on cooling. (See? Thermodynamics!)

We all know that air cooling was a serious issue for WWII fighters. I have already referred allusively to the P-51's oversized installation, which extracted ram thrust from the exhaust heat and thereby minimised cooling drag. What I do not think people properly appreciate is the extent of the problem. This is because designers found a way of hiding just how much cooling they were doing. Specifically, in 1946, Frank Nixon of Rolls Royce got up at a Royal Aeronautical Society salon to denounce the trend to use oil circulation as a hidden means of rejecting engine heat through the oil coolers. Here's a table that I took the trouble to type out from Frank Nixon, “Aircraft Engine Oil Cooling,” Jour. Roy. Aero. Soc. (1946): 123–198; apparently I felt no need to specify the page it comes from. Good luck finding it!)

Engine	Cooling	Power Condition	RPM	BHP	HP to oil	Oil Flow, Gall/Hour	%HP to oil	Temp of oil entering cooler	Temp going out
Lion XIA	liquid		2350	530	14	285	2.6	94	77
Condor III			1900	684	14.4	460	2.1	97	88
“R”		Schneider	3200	2300	11.6	665	0.5	120	80
Pegasus III			2000	600	17.5	100	2.9	102	87
P&W Wasp			2250	600	22.6	406	3.8	127	90
Peregrine			3000	885	71	594	8.0	117	90
Merlin 24			3000	1640	65	780	4.0	117	90
Merlin 100			3000	1710	65	780	3.8	116	90
Griffon 664			2750	2300	95	1150	4.1	116	90
Allison V-1710			3000	1525	155	1040	10.2	132	85
DB601A			2400	1000	86	640	8.6	117	75
Sabre IIA			3700	1875	309	2400	16.5	144	100
Taurus			2800	985	63	385	6.4	146	90
Hercules VI			2400	1356	86	495	6.3	142	80
Hercules 100			2400	1515	97	505	6.4	150	86
Centaurus IV			2400	2095	152	700	6.3	115	85
P&W Wasp Major			2550	2280	71	666	3.1

As Nixon tartly points out, the recent practice in air-cooled radials us to have pump jets irrigating cooling fins machined into the underside of piston heads. It is not surprising that oil cooling system failures are greater than all other mechanical failures combined in recent air-cooled radials. The igh figure of heat rejected in the Daimler Benz reflects the use of general-circulation engine oil in its hydraulic supercharger gearing, while in the Sabre it reflects the engine's heroically complicated inner workings in general, including but not limited to a hydraulic supercharger. 

What I do not add here, because it has not been studied anywhere near so intensively that I can find, is the intercooler that enabled the two-stage Merlin. It is pretty simple in concept. There is an impeller to compress the fuel-air charge, than a tube, lined on the outside with water pipes, which reduce the temperature of the charge, which has been heated through compression. Then, there is another impeller. It really does not sound that hard. Yet the Allison division of General Motors never got a proper two-stage supercharger working, for lack of proper intercooler, and there were problems with the late-war German two-stage projects, too. 

Again, the actual cooling does not sound challenging. The water pipes just have to be led to a radiator somewhere. Aviation's article on the P-51 hangs a big exclamation mark over the existence of three different cooling systems all working to reject heat from the Merlin installation, but, then, we have already accepted that the P-51's shrouded ram cooler is going to be big. The ever-ramifying array of doodads on the underside of the Spitfire's wings are uglier, but do the job as well. You can see them, and read the rest of R. F. Gordon's article online at the FlightGlobal archives, and do I feel like an idiot for taking the time to photocopy this article and not the Aviation design study of the P-51. What we are not getting any closer to understanding is the failure of the Germans to field an fighter that could compete with the P-51.

Here is another synoptic table that I am probably reproducing for the second time on this blog, lifted from Enzo Angelucci and Paolo Matricardi, Hunter of the Sky: A Visual Guide to World War II Aircraft abridged ed. (New York: GT Merchandising and Licensing, n.d.; original edition Milan: A. Mondavi, 1990), 42–3

Engine	Displacement	Diameter	Dry Weight	Max power	lb/hp
Mercury 9cyl	24.9L	130cm	1066lb	995hp	1.07
Allison V-1710	29L	--	1386lb–1622lb	1200-1475hp -takeoff	1.155
P&W R-1830 14 cyl	30L	122.2cm	1437lb–1460lb	1200hp	1.19ish
Merlin	27L	--	1375lb–1650lb	1030–2000hp	up to 0.825
Nakajima Sakae 14cyl	27.8L	114.4cm	1175lb	1100–1200hp	0.97
Jumo 211 V-12	35L	--	640kg/1408lb	950–1500hp	up to 0.94
P&W R-2800 18 cyl	45.9L	Not given	--	--	--
BMW 801 14 cyl	41.8L	129–144cm	1055kg/2321lb–1800kg/3960lb	1600–2300hp	1.45
Wright R-3350 18cyl	54.56L	142cm	1200kg(!)/2640lb	2200hp at 2800rpm	1.2
Griffon	36.7L	--	816kg/1796.5lb–898kg/1975lb	1730–2300hp	up to 0.86
DB605 12 cyl	--		c. 765kg/1683lb	up to 2000hp	up to 0.84

The basic reason that the P-51 was a better high altitude fighter than the Bf109 or the FW190 was not magic. It was not an exogenous intervention from the ideal world of pure innovation that lies orthogonal to our own, plus one dimension. It is that Kurt Tank put a thick wing on his plane so that it would contain a widetrack undercarriage and some cannons on a snappy little frame, and that, meanwhile, the newly-renamed Messerschmitt works were making the same kind of mess of the Bf109 succession that they had just finished making of the Me 210. Kurt Tank's upgrade of the FW190, the Ta152, looks like it had more potential, but it was too late to make a difference.

Here's the Wikipedia article on the engine that held up the Ta152, the Jumo 213. I read it as both supporting my basic point and as slightly apologetic. No mention is made of Focke-Wulf's dislike of the Jumo, which apparently had torquing issues. Perhaps more to the point, it seems unimpressive compared with the DB605. If we reject the idea that the P-51 is magic, we get to a very, very simple solution to its appearance in the skies over Germany. Get a fighter with a better engine up there, and shoot it down. The DB605, as much a development of the DB601 as the Merlin 61 was of the old Merlin I, is --well, unless Daimler Benz engineers just suddenly got stupider, a better engine. Messerschmitt even managed to fit it into the Bf109. This is hardly ideal from a pilot's point of view, but the Bf109G is almost 2000lbs lighter than the P-51. This is a plane that ought to be shooting the P-51 out of the sky.The only obvious problem is the lack of a two-stage supercharger, especially critical given waspish comments by Roy Yount of Wright about the DB's inability to maintain its height, and Gordon's observation that actual supercharger performance is likely to decay as the oil heats up.  

To give the DB605 a competitive performance at altitude, you just need to put another impeller in the way of the charge, and in intercooler between them. Daimler Benz engineers know this. They are not stupid. What's the problem here?

Again, a cite from Wikipedia: 

"One major design difference was the switch from ball bearings to sleeve bearings which, when combined with increasingly poor grades of lubricants, led to serious problems in service, including engine fires; initially, for example, the use of emergency power was forbidden. Although Daimler-Benz redesigned the bearings and added oil slingers and their associated coolers, the RLM considered the DB 605 to be a "sick engine" and the problems had not been fully resolved by the end of the war."

You've heard about how futile the Schweinfurt raids were, on account of the Germans just substituting slide bearings for roller bearings? This is the money quote. This is the moment when the reality of strategic air war comes into focus. This is why you bomb roller bearing factories in the first place. So that engines do not appear in the sky that use roller bearings to, for example, mount the impellers of two-stage superchargers.

We are talking here about an industrial war carried on at the limits of what advanced economies are able to produce, and we are pitting two powers against each other: the one bombing, the other receiving the bombs. And we are remarking the not at all coincidental way in which the power that is not being bombed is fielding more effective heat engines than the one not being bombed. 

Again. Not a coincidence.