Adrian Bejan | Respiration, from Design in Nature
In this video, Adrian Bejan explains that predicting design should include not only the evolution of design in space but also in time, and he introduces respiration as the most obvious rhythmic flow to analyze. He sets up the puzzle of why inhaling and exhaling take the same time, and why the time scale becomes shorter when running, while noting that the same exercise can be used for heartbeating. He models the lung with the trachea, thorax, and a flow resistance, writes mass conservation for inhaling and exhaling, and then computes the work and average power spent by the thorax muscles over a breathing cycle. He shows that power would seem to favor very long time intervals, but then adds the engine and metabolism requirement, where oxygen must diffuse from alveoli into tissue, and diffusion becomes inefficient as time grows. He combines the oxygen intake requirement with the power cost, optimizes the inhaling and exhaling times so they are nearly equal, and links a higher oxygen requirement during physical exercise with more frequent breathing.
Bejan starts from the idea of predicting the evolution of design and shifts from design in space to design in time. He connects this to rhythmic functioning in the body and the daily cycle.
He frames respiration as a problem of posing the right questions, especially why inhaling time t1 equals exhaling time t2. He also links the same type of solution to heartbeating.
Bejan models inhaling as atmospheric air entering the trachea and increasing the thorax volume by capital V, driven by a partial vacuum and limited by a global resistance R. He uses a pressure difference proportional to R and a velocity to an exponent n, and writes conservation of mass for the inflow and the accumulated air.
He models exhaling as the thorax volume shrinking, indicated by a capital V, with an excess pressure delta p2, flowing out through the same resistance and cross-section, with an average velocity u2. He then computes work as the integral of p dv over the two parts and expresses average power in terms of V, af, and the time intervals.
Bejan introduces alveoli and diffusion, where oxygen transfer scales with the square root of time and becomes inefficient, leading to intermittency and surface renewal through finite inhaling followed by exhaling. With a fixed oxygen requirement per unit time K, he minimizes average power and obtains the result that t1 and t2 are nearly equal, and higher K corresponds to shorter times and frequent breathing during running.
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