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(18) Energy Availability and Limits. · Mechanical Engineering Made Simple

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  Welcome back, design junkies, to Mechanical Engineering Made
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  Simple. Today we are undertaking, well,
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  a really critical mission. I think we're trying to find the
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  true measure of efficiency, and that means kind of challenging
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  the limits of something fundamental, The first law of
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  thermodynamics, yeah. Absolutely every engineer you
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  know starts with the first law, and you have to.
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  It's basically the universe's accountant.
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  And make sure energy quantity, the amount is conserved, tells
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  you what you started with, what you ended with.
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  No energy just vanishes. But you know, I've always felt
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  the first law was maybe a bit simplistic, like OK, energy is
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  conserved, great, but you could have a whole system full of
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  conserved energy that's completely useless for doing
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  actual work, right? If it's all just spread out low
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  quality ambient temperature stuff, it feels like it ignores
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  the quality aspect. That is the core problem.
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  That's where the second law steps in.
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  It's the quality control officer, if you will.
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  Now, traditionally we look at this through the lens of
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  entropy. S Entropy tells us you know
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  which way processes naturally go and crucially, how much energy
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  must be rejected. You can't convert it all.
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  Right. Entropy is fundamental,
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  absolutely. But you mentioned we're focusing
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  on a different property today, one that gets more directly to
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  the practical bottom line. We're talking about
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  availability, Yeah, sometimes called exergy.
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  Precisely, availability is engineered specifically to
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  answer the question what is the absolute maximum useful work a
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  system can perform as it comes into equilibrium with its
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  surroundings. Our mission today is to accept
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  that the second law forces some energy projection always, and
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  then find the shortcut to calculating the theoretical
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  maximum work we can actually extract Despite that.
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  OK, let's dive in. So defining useful work seems
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  like the first step. It is.
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  Let's think about a system doing work, maybe expanding gas in a
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  piston. The total work the system does,
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  let's call it delta W sis. But when that piston moves, not
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  all that work is doing what you actually want it to do.
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  OK, so if I'm using the piston to, say, lift a weight or turn a
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  shaft, that's the useful work. Let's call it Delta Wu, but the
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  piston itself, just to move, has to push the air around it out of
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  the way. The atmosphere resists it.
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  Exactly. That's the displacement work
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  done against the environment delta W Sir.
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  And if the environment's pressure is constant, say P
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  naughty ambient pressure, then that work is simply P naughty
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  times DV the change in volume. OK, so the useful work I get out
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  Delta Wu is the total work the system generates, Delta W says
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  minus this work wasted just pushing the atmosphere around.
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  Delta W Sir or P, not DV. It's like an unavoidable tax you
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  paid to the environment just for existing and changing volume.
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  A perfect way to put it, an environmental tax.
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  And this unavoidable interaction with the environment leads us
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  straight to the next key idea, the dead state.
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  The dead state sounds a bit grim.
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  Why that? Name.
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  Well, it's maybe a bit dramatic, but it's conceptually our zero
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  point. The dead state is where your
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  system is in complete thermal and mechanical equilibrium with
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  its surroundings. It's at the same temperature T
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  naughty and the same pressure P naughty as the environment.
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  At this point it's done. It's thermodynamically retired.
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  You could say it has absolutely 0 potential left to perform work
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  relative to its surroundings. OK, that makes sense.
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  It's the baseline. So we need some way, some
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  function to measure how far our system is from that dead state,
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  from that zero work potential. And this isn't just for Pistons
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  pushing air, right? We need something general for,
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  like shaft work from a turbine, or even the theoretical work
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  from a perfect heat engine. Exactly.
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  And that's where the non flow availability function just a
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  comes into play for closed systems.
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  It's actually a really elegant combination of the key energy
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  components all viewed relative to that dead state all.
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  Right, let's see it. What's the function?
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  The availability A is defined as A = U + P naughty V -, t naughty
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  S. OK, U plus P0 VGO S looks like
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  internal energy, the environmental tax term and
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  something involving entropy. Can you breakdown what each part
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  is doing there? Sure.
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  U is just the system's internal energy, the energy it possesses
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  internally. Straightforward.
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  Then P naughty V that's related to the work potential associated
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  with the system's volume against the constant environmental
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  pressure. P naughty.
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  It accounts for that displacement work potential.
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  Right, the environmental tax part we discussed and the last
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  term minus T naughty S that smells like the second law
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  making its entrance. That's exactly it.
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  The T naughty S term is the critical piece here.
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  It represents the energy that is inherently unavailable for
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  conversion into work. It's the portion that, according
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  to the second law, must be rejected as heat to the
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  surroundings at temperature T naughty during any process
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  bringing the system to equilibrium.
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  It's the mandatory heat rejection quantified.
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  So availability A is telling us, here's the internal energy you
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  have you plus the work potential related to displacing the
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  environment P naughty V minus the energy you absolutely cannot
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  turn into work because you have to dump it as heat T, not S.
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  Precisely. And now here's the payoff.
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  The reason this function is so powerful.
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  The maximum useful work W Max you can possibly get from a
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  closed system changing from some initial state 1 to a final state
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  2 is simply the decrease in availability.
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  Wu Max equals a 1 -, a two or minus delta a.
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  Hold on that. That seems almost too simple.
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  You're saying we calculate A at the start, calculate A at the
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  end, take the difference, and that's it.
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  We don't need to worry about the exact path, the friction, the
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  type of expansion, none of that messy intermediate stuff to find
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  the theoretical maximum. For the theoretical maximum
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  useful work, yes, it gives us path independence.
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  For potential, it's the ultimate shortcut.
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  We also often use the specific non flow availability little A,
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  which is just availability per unit mass.
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  So little A equals little U + P, not little V -, t naughty little
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  S using specific properties. OK.
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  This needs an example to really sink in.
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  How does the shortcut play out in practice?
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  Right, let's imagine that piston cylinder set up again.
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  We have air inside starting at say a high pressure in
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  temperature that state one, it then expands and cools down to
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  some final state 2. And the surroundings, the dead
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  state, are just standard atmospheric conditions, T
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  naughty and P naughty. OK, now the old fashioned way.
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  The textbook way. To find the maximum useful work
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  for that entire process, you'd have to meticulously calculate
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  the work done during each tiny step of the expansion, figure
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  out the heat transfer, account for the moving boundary, work
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  against P not. It could get really complicated
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  depending on the process path. Exactly.
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  You might have heat transfer involved, maybe work input or
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  output through different means. You'd sum up all the useful work
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  components along the entire possibly complex path, take
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  pages of calculation. You'd track the work against the
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  actual resisting force, maybe shaft work, maybe work via a
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  reversible engine connected, and you'd eventually arrive at a
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  single number for the net. Useful work.
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  But with availability. With availability, you just need
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  the properties UVS at state one and state 2, and you need the
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  dead state properties T naughty and P naughty which are
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  constants. You calculate A1 plus P0V1T0S1.
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  You calculate A2 plus P0V2T0S2. The maximum useful work is
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  simply A1A2. And when engineers actually do
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  this, compare the long path dependent calculation with the
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  simple availability difference. The results for the theoretical
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  maximum useful work match they have to.
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  It proves that this state function availability perfectly
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  captures the maximum work potential without needing any
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  information about how the system got from state one to state 2,
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  assuming the process was reversible.
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  That's the aha moment. Then availability defines the
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  ceiling. It tells you the absolute best
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  you could possibly do in terms of useful work output between
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  two states in a given environment.
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  Exactly. It's the theoretical limit.
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  OK, so we found the theoretical ceiling.
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  Now let's talk about reality. The gap between that beautiful
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  maximum number and what we actually get out of any real
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  engine or process, that gap must represent waste, right?
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  It does, and it has a very important name, irreversibility,
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  usually denoted by I. Irreversibility is precisely the
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  difference between the maximum possible useful work, the change
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  in availability, and the actual useful work Wu actual you
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  achieved. It represents a permanent loss
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  of work, potential, availability that has been destroyed forever,
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  energy quality that's been degraded.
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  And how do we quantify how much availability was destroyed?
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  How bad was the process? Does it relate back to entropy
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  again? It absolutely does.
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  The irreversibility I generated during a process is directly
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  proportional to the entropy generated during that process.
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  Specifically, I equals T naughty times Delta S universe, or often
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  simplified as T naughty times, the entropy generated within the
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  system and surroundings OK. Let's make this concrete.
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  Think about a turbine in a power plant.
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  Good example. An ideal perfect turbine would
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  expand the steam or gas isentropically, meaning constant
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  entropy. That would give you the maximum
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  possible work output, which would equal the change in flow
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  availability. But a real turbine has friction,
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  turbulence, maybe some heat loss.
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  It's not Isentropic entropy increases as the fluid flows
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  through. Let's say it has an isentropic
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  efficiency of maybe 85 percent. Right, so that 85% efficient
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  turbine produces less actual work WT actual than the ideal 1.
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  The theoretical maximum work it could have produced is given by
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  the change in flow availability minus delta in F.
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  The actual work is lower. Exactly.
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  And the difference between that theoretical maximum and the
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  actual work WT actual is the lost work due to
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  irreversibilities. Now if you separately calculate
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  the irreversibility I using T naughty times the actual entropy
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  change S2 S one across the real turbine let.
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  Me guess it perfectly matches the lost work.
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  It perfectly matches the lost work.
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  The irreversibility calculation T naughty arrays quantifies the
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  exact amount of availability that was destroyed by the
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  friction and inefficiencies within that real turbine.
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  Even though energy is conserved overall first law, the potential
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  to do work, second law availability has been
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  demonstrably reduced. So availability A measures the
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  maximum work potential stored in the system relative to the dead
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  state. How does the term exergy fit in?
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  I hear the muse almost interchangeably sometimes.
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  They are fundamentally the same concept, measuring maximum
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  useful work potential. The distinction is often subtle.
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  Xergy B is sometimes defined specifically as the availability
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  relative to the dead state itself.
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  So B = A -, a naughty where a naughty is the availability at
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  the dead state. Since a naughty represents the
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  baseline, often taken as zero potential, the change in extra G
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  between two states gives you the same maximum useful work as the
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  change in availability. So for practical purposes in
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  calculating maximum work, delta A and delta B tell you the same
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  thing. Pretty much, yes.
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  They both quantify that maximum useful work potential.
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  Extra G is perhaps more formally defined relative to that zero
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  point got. It and you mentioned flow
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  availability for the turbine, that's for open systems, right?
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  For systems where mass is flowing across the boundaries,
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  like turbines, compressors, pumps, nozzles, we use the
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  steady flow availability function.
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  If we neglect kinetic and potential energy changes, which
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  is often reasonable, the specific flow availability
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  little a sub F simplifies nicely.
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  It becomes AF equals H -, T naughty S where H is the
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  specific enthalpy. HE zeros so enthalpy replaces
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  internal energy compared to the non flow function.
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  Makes sense for flow systems. Oh yeah.
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  OK, let's tackle the process you mentioned earlier that really
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  highlights why we need this thinking beyond the first law.
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  The throttling valve. Yes, the throttle.
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  A deceptively simple device with profound implications.
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  What does a throttle do? It just reduces pressure,
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  usually across a valve or restriction.
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  It's typically assumed to be a diabetic.
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  No heat transfer and does no work.
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  So what does the first law steady flow energy equation tell
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  us? If Q is 0 and W is 0, then
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  enthalpy in must equal envelope out H1 equals H2.
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  Right. Energy is conserved, enthalpy is
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  constant according to the first law.
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  Nothing really bad happened. The energy content per unit mass
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  is the same. But we intuitively know
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  throttling is wasteful, right? You lose pressure without
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  getting any work. It's a highly irreversible
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  process due to the rapid uncontrolled expansion and
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  internal fluid friction. OK, so if it's irreversible,
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  entropy must increase. Entropy must increase.
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  S2 has to be greater than S1 across the throttle, which means
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  the irreversibility I which equals T naughty times S2S1 must
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  be a positive value. Availability has been destroyed.
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  Even though H stayed constant, the flow availability HT zeros
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  decreased because S went up. Wow.
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  So the first law sees nothing wrong, H1 equals H2, but the
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  second law through availability clearly shows that the fluids
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  ability to do useful work has been permanently degraded just
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  by passing through that valve. That's the power of availability
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  and irreversibility. Analysis fuels the hidden
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  losses, the degradation of energy quality that the first
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  law completely misses. So wrapping this up, the core
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  message for you, our listeners, especially engineers, seems to
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  be this availability or exergy derived from the second law is
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  the essential tool. It's not just about conserving
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  energy, First Law. It's about preserving its
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  quality, its ability to do work. Exactly.
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  The ultimate goal isn't just efficiency in the first law
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  sense output energy, input energy, but minimizing
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  irreversibility. Every real process involves some
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  irreversibility. Friction, heat transfer across a
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  finite temperature difference, mixing, chemical reactions,
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  these all destroy availability. Reducing them is the key to
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  better real world performance. It seems like there's always
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  this potential energy, this availability that we could tap
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  into if only our processes were perfect.
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  Which brings us to a final thought to chew on.
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  Consider a standard internal combustion engine, like the auto
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  cycle. Thermodynamic analysis show that
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  even after the expansion stroke, when the exhaust valve opens,
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  there's still a significant amount of availability left in
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  those hot exhaust gases being expelled.
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  Yeah, we try to capture some of that with turbochargers.
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  Yeah, using the exhaust pressure and some heat.
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  Right. Turbochargers recover some of
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  the pressure related and thermal availability, but to get all the
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  remaining thermal availability out you'd theoretically need
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  something more complex. You'd need a bottoming cycle, a
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  secondary heat engine designed specifically to take those hot
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  exhaust gases and extract work as they cool all the way down
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  towards the ambient temperature. T Naughty.
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  So the provocative question is, we know there's significant
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  useful work potential being literally thrown out the exhaust
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  pipe. Beyond just the cost and
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  complexity of adding a whole second engine cycle, What are
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  the specific technological hurdles?
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  Maybe material limits, heat exchanger inefficiencies, or
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  cycle integration challenges that prevent us from more
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  effectively and widely implementing vitamin cycles to
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  capture that wasted thermal availability from exhaust
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  streams? That feels like a major frontier
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  for pushing efficiency towards its theoretical limits.
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  A huge challenge indeed. Something for the next
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  generation. Until the next equation.
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  Design junkies. We are mechanical engineering
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  made simple.