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The thread: A smooth picture proves nothing — page 32

Page 32 of 32, continuing through the 281 essays this motif runs through.

281 essays carry this thread — page 32 of 32.

At one specific speed a bigger pump is a better one, and by less as it grows. Best efficiency against impeller radius for geometrically similar pumps of 40 m head at three specific speeds; the shaft speed falls as the radius grows, so the tip speed is the same at every size. At Ωₛ 0.55 a 25 mm impeller reaches 0.797, a 0.2 m one 0.869 and a 1.6 m one 0.887: 7.17 points for the first factor of eight, 1.84 for the second.

Moody's fifth root is true at one size

A model pump is less efficient than the full-size machine, and since 1925 the gap has been bridged with a fifth root of the size ratio. Built from its losses, a pump's inefficiency scales three ways at once — leakage as the inverse size where a clearance cannot shrink, friction as a fifth root until the passages go rough, and a fixed share not at all — so the exponent runs from a third down to a twentieth, and the fifth root is right only near one size. The same arithmetic says a duty should be split by speed, not by width.

Out to a hundred neutral wavelengths the noisy flame is still speeding up. The front's speed in excess of a flat flame's against its width, on logarithmic axes, at noise strengths of a thousandth and a millionth, with the quiet pole front's and a power fitted from 20 to 100. At a thousandth the speed climbs from 1.05 at 12.5 to 2.71 at 100; at a millionth from 0.668 to 1.71. Both follow the 0.42 and 0.43 power of the width to within 4 per cent, and neither bends over; the quiet front stays at a half.

Flows and fields

A noisy flame is still speeding up a hundred wavelengths wide

A quiet wrinkled flame gains a fixed, bounded amount of speed over a flat one, however wide it is. A noisy flame keeps getting faster as it widens, and the open question was whether that ever stops. Out to a hundred neutral wavelengths it does not. The speed grows as about the 0.43 power of the width at two noise strengths a thousandfold apart. The noise sets only the prefactor, and the front's slope grows with the speed until it leaves the range the equation was derived for — which is where the growth stops being the equation's to decide.

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