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(#43) Turbomachinery Betz Limit to Cavitation Explosions · Mechanical Engineering Made Simple

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  Welcome back, design junkies, to mechanical engineering made
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  simple. Hello again.
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  Today we are strapping ourselves into the high speed, high
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  pressure world of rotating machinery, what engineers call
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  turbo machines. And our mission really is to
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  hunt down the fundamental principle.
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  Yeah, it's the elegant math, but also the harsh physical
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  constraints that determine why a machine, you know, a giant wind
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  turbine or a tiny gas compressor, why it works.
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  Or why it fails catastrophically sometimes, right?
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  This is a deep dive into the absolute limits of efficiency,
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  the violent physics of flow degradation, and the really
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  clever control methods engineers are using now to manage
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  instability. We've pulled from sources
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  covering everything hydraulic turbines, jet engines, you name
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  it And. What I find so fascinating is
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  that these things, they seem so different.
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  One's generating power, one's consuming it, but they're all
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  unified by the same basic fluid mechanics.
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  And we found this amazing common ground, starting with just one
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  simple, almost beautiful theoretical limit.
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  That's right, this universal limit on power extraction.
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  If we want to understand the maximum potential of, say, a
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  wind turbine, we have to start with the concept called the
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  actuator disk model. I think this is a perfect
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  example of engineering elegance because you don't start by
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  modeling every single curve on the blade.
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  No, you'd never get anywhere. You you substitute the entire
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  rotor with this theoretical thing, an actuator disk.
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  So it's a placeholder basically. Essentially, yes.
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  It's imagined as this permeable plane that just instantly
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  applies a force to the fluid, and that lets US model the whole
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  system by just looking at the flow way upstream and way
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  downstream. OK.
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  And the power of this model, it comes from one big simplifying
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  assumption about continuity, right?
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  That's the one, and it's actually quite intuitive when
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  you think. About it.
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  So let's break that down because this is where the physics gets
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  really powerful. The assumption is that the axial
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  velocity right at the disk. Which we call CX2.
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  CX2 right is the average of the velocity far upstream CX-1 and
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  the velocity far downstream CX3. Exactly.
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  It just means the airspeed right at the blades.
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  It's the perfect middle ground between what's coming in and
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  what's going out. Yeah, make that 1
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  simplification. And you can derive this hard non
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  negotiable cap on efficiency. The famous bets limit.
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  The bets limit. It defines the absolute maximum
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  power coefficient CP that a wind turbine can ever achieve even in
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  a perfect world. And the number itself is it's
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  surprisingly restrictive. The Max CP is 1620 sevenths.
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  Which is about 59.3%. 59.3 That's it.
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  You cannot physically pull more than 59.3% of the kinetic energy
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  out of the wind. And you only hit that number
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  under one very specific condition.
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  You do. It's when the axial flow
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  induction factor A, which is just a measure of how much the
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  wind slows down, equals exactly 1/3.
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  So if you. Slow it down too little, you
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  don't grab enough energy. Right.
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  And if you slow it down too much, the air just, it just
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  flows around the whole turbine so the performance drops off
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  again. This limit really shows why a
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  modern, you know, lift based turbines are the gold standard.
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  They get close to this number. They do, and it's why older drag
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  based machines like those ancient Persian windmills.
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  They're just they're not really relevant for modern power
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  generation. Their efficiency is just too
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  low. But our sources did point out
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  some interesting exceptions, like the vertical axis turbines,
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  the Darius or eggbeater type. Yes, their big selling point is
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  convenience. They work no matter which way
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  the wind is blowing. No need to.
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  Yaw exactly, but it's a classic engineering trade off.
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  You sacrifice that theoretical efficiency for operational ease.
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  They have really low productivity near the ground and
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  worse, they produce these huge torque fluctuations with every
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  single rotation. Which has to be a nightmare for
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  the grid connection A. Huge control challenge,
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  especially in high winds. That's a perfect transition,
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  actually. We've set the theoretical Max,
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  but now we get into the messy reality of well, of practical
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  efficiency and physical failure. And if we're going to talk about
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  trade-offs, we first have to agree on how we're measuring
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  success. Efficiency isn't just one
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  number. Right, the definitions matter, I
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  said. Total to total efficiency
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  mentioned a lot. Yes, you use that when the
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  kinetic energy coming out of one stage is immediately used by the
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  next one. Think inside a multi stage
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  compressor. But then there's total to static
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  efficiency. When does that come into play?
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  That's for when the kinetic energy leaving the machine is
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  just lost. It's wasted, like the exhaust
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  from a jet engine. If that exit velocity isn't
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  doing useful work, you have to count it as a loss.
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  And no matter how you define it, mechanical losses are always
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  eating away at your performance. Things like bearing friction and
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  a windage. Exactly.
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  Windage is just the energy loss churning the fluid around the
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  rotor. Yeah, and these losses can be
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  surprisingly big, especially in smaller machines.
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  We're talking up to 5% or even more of your total energy input
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  just. Just burned off his heat.
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  But you know, few failure mechanisms are as as dramatic or
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  as genuinely terrifying as cavitation and hydraulic
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  machines. Oh.
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  Absolutely. This isn't just a loss of
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  efficiency, this is physics turning into a Wrecking Ball.
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  What's happened? Cavitation happens when the
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  local pressure inside the fluid drops below its Vader pressure,
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  so the water it literally starts to boil at room temperature.
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  And it forms these little vapor bubbles.
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  Right in the low pressure zones, often on the back of an impeller
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  blade. Now the bubble forming is no big
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  deal. It's the collapse.
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  When it moves back into a high pressure area.
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  Precisely the collapse is almost instantaneous and it creates
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  this this incredibly powerful shockwave.
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  Our sources site that a single bubble collapse generates a
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  pressure wave of about 400 megapascals. 400 To put that in
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  perspective, that's like 600 times the pressure at the bottom
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  of the Mariana Trench. On a microscopic point on your
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  metal blade, over and over and. Over.
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  I can't even imagine. And that repetitive hammering
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  physically removes material. That's cavitation, erosion.
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  It creates these little pits in the metal.
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  You can even hear it. It sounds like gravel is running
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  through your pump. Engineers call it white noise
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  is. It's so destructive but you
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  mentioned there are ways to manage it.
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  Super cavitating pumps. Yes, this is where engineers get
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  really clever in applications like rocket engines, where size
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  and weight are everything. They don't try to avoid
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  cavitation, they encourage it. They encourage.
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  It they designed the blades to create a huge, stable vapor
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  cavity that extends way past the metal impeller.
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  The violent collapse point is moved downstream, away from the
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  hardware. It's managing the violence
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  instead of trying to stop it. Incredible.
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  Now switching back to aerodynamics.
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  Not all degradation needs a 400 megapascal blast.
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  Sometimes all it takes is a little dirt.
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  Blade fouling. It's shocking how big a deal
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  this is, especially for stall regulated wind turbines.
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  It's it's an often overlooked factor.
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  So how big an impact are we talking?
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  A few percent. Oh, much more.
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  The data suggests that annual energy losses in the US alone,
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  just from increased surface roughness, are somewhere between
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  20 and 30%. Wait, wait. 20 to 30% from a
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  layer of bugs and dust? That's astounding.
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  It is. It just tanks the performance.
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  The fouling trips the boundary layer, it causes premature
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  separation and your lift just disappears.
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  And the sources point out, it's even worse for small turbines.
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  Why is that? The relative roughness, the
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  thickness of the bug guts is a much larger fraction of the
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  small leading edge radius. So nature is constantly trying
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  to wreck our designs. How do engineers fight back?
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  This brings us to flow control. And to the main battleground in
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  axial compressors which is managing flow diffusion.
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  A compressor's job is to slow the air down and raise its
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  pressure. Right, but you can't do it too
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  quickly. If you try to decelerate or turn
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  the air too sharply, the flow separates and you get huge
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  pressure losses. And that's why like compressors
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  need so many stages compared to a turbine which is accelerating
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  the flow. Exactly.
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  Push those limits too far and you get instability, you get
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  stall and potentially surge. And the sources made a clear
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  distinction between two types of stall.
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  Yes, you have blade stall, which is what you'd expect flow
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  separating from the airfoil itself.
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  But then you have wall stall which is connected to the
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  boundary layer on the outer casing.
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  And knowing which one you have dictates the solution.
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  It does. If you're prone to wall stall,
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  you can use something called casing treatment.
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  Basically you cut grooves into the outer wall to re energize
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  that sluggish boundary layer and delay the stall.
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  That sounds pretty smart, but what's the trade off?
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  There's always a trade off. The penalty is efficiency.
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  The more effective the casing treatment is a pushing back
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  stall, the more it hurts your stage efficiency right at the
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  design point. You're sacrificing peak
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  performance for a bigger stability margin.
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  Which is what's pushing the industry toward active feedback
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  control. That is the cutting edge.
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  Moving beyond passive grooves, these active systems use sensors
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  like hot wires and tiny actuated veins to monitor the flow in
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  real time. So they can see an instability
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  forming. They can see the tiny
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  disturbances before they grow and they can actively dampen
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  them out, stopping a full blown stall before it even starts.
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  Does it work? The studies show it does.
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  It can increase the stable flow range of a compressor by as much
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  as 25%. It lets you operate safely much
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  closer to that theoretical limit.
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  So finally, let's talk about the designer's road map for choosing
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  the right machine in the 1st place specific speed.
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  Specific speed or NADS is a genius concept.
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  It's a non dimensional number, a shape number for the machine.
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  Shape the number, Yeah. It tells the designer what shape
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  of turbo machine will be most efficient for a given job, a
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  given head flow rate and speed. It takes the actual size right
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  out of the equation. So a low specific speed means
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  you need a very high head. Exactly.
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  For really high heads, like 2000 meters in a mountain hydro
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  project, the specific speed value will be very low and it
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  tells you that you must use a Pelton turbine, an impulse type
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  machine. And the opposite end, high
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  specific speed. That means low head, high flow
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  rate. Think a river project.
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  The number will be high and it points you directly to a
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  propeller type machine like a Kaplan turbine.
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  It's the fastest way to get into the right ballpark.
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  It's interesting that for a certain middle range of specific
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  speeds, the best radial flow turbines are just as efficient
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  as the best axial flow ones. They are between about .3 and
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  one point O Non dimensionally their peak efficiencies match
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  up. It just confirms that getting
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  that shape number right is what really matters.
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  What a journey from the elegance of the bets.
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  Limit that 59% cap all the way down to 400 Mega Pascal
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  cavitation explosions. It shows that engineering design
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  is always this this battle between the clean theoretical
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  potential and the messy physical reality.
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  Right. You're using complex models like
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  blade element momentum theory not just to design for
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  performance, but to balance everything else.
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  Noise cost, Material strength. You're actively managing these
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  unavoidable losses and fighting back against degradation and
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  instability every step of the way.
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  So here's a final thought for you to chew on throughout this
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  whole discussion. And we've mostly used models
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  that treat the flow as well as steady time averaged.
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  Right for the sake of analysis. But the reality is the flow
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  inside a turbo machine is fundamentally unsteady.
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  The blades are moving past stationary veins, creating
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  wakes, pressure fluctuations. It's a chaotic environment.
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  So our best models are still just approximations of what's
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  really happening. They are.
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  So consider this. How much higher could
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  efficiencies go? How much more stable could these
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  machines be if we could perfectly model and then
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  actively control that inherent unsteadiness instead of just
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  averaging it out? That's a huge question.
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  That, I think, is the next frontier.
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  Fascinating stuff to think about.
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  We'll leave you with that dynamic turbulence until the
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  next deep dive.