Last week, the plug door panel on an Alaska Airlines Boeing 737 Max airplane detached during flight, leaving a gaping hole in the side of the fuselage. Fortunately, the plane was able to land safely and no one was seriously injured in the accident. I thought it would be fun to analyze this event from the point of view of physics in medicine and biology. Let me stress that I have no inside information about this accident, and I am not an aviation expert. I’m just an old physics professor playing around trying to make sense of information reported in the press.
Let’s calculate the pressure difference between the normal cabin pressure of a 737 Max and the outside air pressure. The typical pressure at sea level is 1 atmosphere, which is about 100,000 pascals. However, in most planes the cabin pressure is maintained somewhat lower than an atmosphere. Usually the cabin pressure corresponds to the air pressure at about 6000 feet, which is 1800 meters. The air pressure falls exponentially with height. Problem 42 in Chapter 3 of Intermediate Physics for Medicine and Biology asks the reader to calculate the length constant corresponding to this decay. If you solve that problem, you get a length constant of about 8700 meters. So, the cabin pressure in the plane should have been around exp(–1800/8700) = 0.81 atm before the door panel blew out.
The mid-air depressurization occurred at about 16,000 feet (4900 meters). I assume this means 16,000 feet above sea level. Therefore, the air pressure outside the plane just before the door panel failed was about exp(–4900/8700) = 0.57 atmosphere. Thus, the pressure difference between the inside and outside was approximately 0.81 – 0.57 = 0.24 atmospheres, or 24,000 pascals.
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| Otto von Guerick’s famous Magdeburg hemispheres experiment. |
The door looks to me like it is about 5 feet by 3 feet, or 15 square
feet, which is 1.4 square meters. So, the force acting on the door was
(24,000 pascals)×(1.4 square meters) = 34,000 newtons, or 7600 pounds
(almost 4 tons). That’s why it’s so important that the door panel be
attached securely to the fuselage; air pressure differences can produce
large forces, even if the pressure difference is only a quarter of an
atmosphere. If you don’t believe me, just ask Otto von Guericke, who in 1654 showed how two hemispheres held together by air pressure could not be pulled apart by two teams of eight horses.
How significant is an abrupt change of 0.24 atmospheres? The Empire State Building is 1250 feet tall (380 meters), which means the top and bottom of the building have a pressure difference of only about 0.04 atm. If you hop on an express elevator and zoom to the observation deck at the top of the skyscraper, you won’t cry out in pain, but you might notice your ears pop. The cabin pressure in a plane typically falls from 1 atm to about 0.8 atm as the plane rises. That’s why our ears feel uncomfortable. But that change occurs slowly, so it is not too bothersome. Normal skydivers jump at about 10,000 feet (3000 meters), so during their descent they experience a drop in pressure of about 0.3 atm. Skydivers often experience noticeable ear pressure, but any associated pain is not severe enough to keep them from jumping again. Unfortunately, the pressure decompression on the 737 Max happened much more quickly than the decompression during a parachute jump, so I would expect any ear problems would have been greater for the passengers on the plane than for a typical skydiver.
Pressures under water are much greater than those in the air, because water is more dense than air. Dive into a pool to a depth of 32 feet (10 meters) and the pressure on your eardrum increases by one atmosphere. Swimmers typically have worse ear problems than airplane passengers. It is one reason why you have to use scuba equipment if you’re diving deep. It’s also why submarine accidents are so much more severe than airplane depressurizations. Remember last year when that submersible was going down to the wreckage of the Titanic and suffered the catastrophic implosion? It was going to a depth of 13,000 feet (4000 meters), which means the pressure difference between the inside and outside of the sub was about 400 atmospheres! You can survive a hole in the wall of a 737 Max, but not one in a Titanic-visiting submersible.
The airplane’s oxygen masks dropped when the hole opened in the 737 Max. Did people really need the oxygen? The airplane altitude was 16,000 feet when the accident occurred. Mount Everest is 29,000 feet high (8800 meters). A few people have climbed to the peak of Everest without using supplemental oxygen, but most carry an oxygen tank. The Everest base camp is 17,600 feet (5300 meters). Climbers often experience mild symptoms of altitude sickness at base camp, but for most it is not debilitating. I suspect that if the passengers on that 737 Max flight hadn’t put on their mask they would have survived, but it might have had an impact on their ability to think straight. And everyone is different; some are more susceptible to mild oxygen deprivation than others. Certainly, the safe thing to do was to put on the mask.
What would have happened if the door hadn’t blow out until the plane reached its cruising altitude of 35,000 feet (11,000 meters). Now you are well above the height of Mount Everest. The outside air pressure would be about 0.28 atmospheres. You would go unconscious (and probably die) if you didn’t promptly put on your mask. The pressure difference between the outside pressure and the cabin pressure would be over half an atmosphere. The odds of being sucked out of the plane during rapid decompression would have been higher. Yikes! The passengers on that 737 Max were lucky that door was very insecurely attached, and not just modestly insecurely attached. If you are going to have a in-flight disaster, it is best to have it as soon after takeoff as possible, before your altitude gets too high.
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| Physics With Illustrative Examples From Medicine and Biology. by Benedek and Villars. |
Below 10,000 ft (3150) there is no detectable effect on performance and respiration and heart rates are unaffected. Between 10,000 and 15,000 ft (3150–4570 m) is a region of so-called "compensated hypoxia"... There is a measurable increase in heartbeat and breathing rate, but only a slight loss in efficiency in performing complex tasks. Between 15,000 and 20,000 ft (4570–6100 m), however, dramatic changes start to occur. The respiratory and heart rates increase markedly; there is a loss of critical judgment and muscular control, and a dulling of the senses. Emotional states can vary widely from lethargy to excitation with euphoria and even hallucinations... The final fatal regime is the altitude region from 20,000 to 25,000 ft (6100–7620 m).
Perhaps those few Mount Everest climbers who don’t carry an oxygen tank can only survive their ordeal by training their body to adapt to high altitudes.
Benedek and Villars also recount a fascinating story about oxygen deprivation from the early years of ballooning, based on an account written by Gaston Tissandier.
These various symptoms are shown very clearly in the tragic balloon ascent of the “Zenith” carrying the balloon pioneers Tissandier, Sivel, and Croce-Spinelli on April 15, 1875... The balloon’s maximum elevation as recorded on their instruments was 8600 m. Though gas bags containing 70% oxygen were carried by the balloonists, the rapid and insidious effect of hypoxia reduced their judgment and muscular control and prevented their use of the oxygen when it was most needed. Though these balloonists were indeed trying to establish an altitude record, their account shows clearly that their judgment was severely impaired during critical moments near the maximum tolerable altitudes. As they were on the verge of losing consciousness at 7450 m they decided to throw out the ballast and rise even higher. They lost consciousness above this altitude, but by good fortune the balloon descended rapidly after reaching 8600 m. On falling to about 6500 m the balloonists revived and—under the influence of the hypoxia did exactly the wrong thing once again—they threw out ballast! The second rise to high elevation killed Croce-Spinelli and Sivel.
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