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(19) Full Throttle Physics: Chasing Efficiency in Heat Engines and Power Plants · Mechanical Engineering Made Simple

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  Welcome back, design junkies, to Mechanical Engineering Made
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  Simple. We've spent, you know, a lot of
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  time on the first law conservation of energy.
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  But let's be real, that's only like half the story.
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  If you're actually designing power plants, just counting the
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  energy isn't enough. You've really got to dig into
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  the quality of that energy. How much can actually become
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  useful work? Exactly.
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  That's where the second law really bites, and it's what
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  separates the theory from practical engineering, isn't it?
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  So today we're really diving deep into the metrics that,
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  well, that actually matter for designing power systems.
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  We need to get past just standard thermal efficiency.
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  There are two key limits we need to understand.
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  First, there's rational efficiency that gives us the
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  absolute best case scenario for a fuel.
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  And 2nd, maximum power efficiency, which is, you know,
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  the best efficiency you can expect when your engine's
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  actually producing maximum power.
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  OK, let's unpack this. Our mission today is basically
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  to start ourselves and you from getting fooled by those
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  conventional efficiency numbers, that thermal efficiency, It
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  often makes the system look well, better than it really is
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  because it ignores the physics of availability.
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  We need something better, a benchmark that considers the
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  unusable energy from the get go. So let's quickly remind
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  ourselves of the standard way conventional thermal efficiency.
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  That's just the network out that divided by the energy you put in
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  right? Which for burning fuel is
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  usually the enthalpy of reaction delta H delay.
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  Right. And the big, big flaw is right
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  there in the denominator using Delta H dollars One, when you
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  use that, you're kind of implicitly assuming that all
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  that fuel energy could theoretically become work.
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  But we know from exergy, from availability concepts, that's
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  just not true. Some energy is fundamentally
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  unavailable because of the environment itself.
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  You just can't get it. And that's where rational
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  efficiency gala comes in, sometimes called second law
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  efficiency, so much more. Well, honest metric, it compares
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  the actual work you get to the maximum possible work you could
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  ever get from that fuel. And that maximum is defined by
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  the change the Gibbs function no delta G0.
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  OK. The Gibbs function negative
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  delta G0, that's the key. But just saying the name, can
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  you break down what it actually means?
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  Like conceptually? Yeah, absolutely.
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  So the Gibbs function change Delta G dollars or it basically
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  represents the absolute maximum network, the extra G you can
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  possibly squeeze out of a chemical reaction like burning
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  fuel. Think of like this.
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  It starts with the total energy released, which is the enthalpy
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  change Delta H dollars one, but then crucially, it subtracts off
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  the part of that energy that's physically impossible to convert
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  to work, no matter how good your engine is.
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  And that impossible part that relates to the surroundings, the
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  ambient conditions. Exactly that unavailable slice
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  is determined by the dead state temperature TOI and T dollars is
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  the temperature of your ultimate heat sink.
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  You know, the air or river, whatever you're dumping waste
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  heat into. The Gibbs calculation figures
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  out the unavoidable entropy increase in the reaction,
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  multiplies that by T dollars ores, and subtracts that
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  product. So Delta G Yorie war represents
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  the true available work potential.
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  OK, so Del R is actual work divided by that true maximum
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  potential work from the Gibbs function.
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  Wait, let me push back a bit here based on something I read.
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  If the actual numerical difference between using delta H
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  dollars lower and delta G dollars for efficiency is often
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  just, you know, a few percent, is it really worth the trouble
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  calculating delta for a practicing engineer?
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  That's a fair question. Look for maybe routine checks,
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  day-to-day stuff. Perhaps the difference isn't
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  huge numerically, but for fundamental design, for figuring
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  out where the real potential for improvement lies, oh it's
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  absolutely vital. The value of del R isn't just
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  the number itself, it's the fact that it tells you the
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  theoretical ceiling. If your system has a rational
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  efficiency of say 60%, you know fundamentally that 40% of the
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  fuels actual work potential was lost.
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  Maybe through irreversible combustion, maybe heat transfer
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  losses, whatever, but that 40% is your ultimate target.
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  Deller tells you how high the bar truly is.
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  OK, here's where it gets really interesting.
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  Let's apply this put our idea to some common power cycles.
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  How about we start with the Rankin cycle?
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  Steam plants? Good.
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  Place to start the ranking cycle has a pretty significant
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  theoretical edge because it uses phase change.
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  Boiling water, condensing steam. Now if you imagine a perfect
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  ideal ranking cycle, totally internally reversible, no
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  friction, perfect components, it can theoretically achieve a
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  rational efficiency for base of $1.00. 100% wow.
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  OK, 100% rational efficiency. So it gets out every single
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  available jewel of work that the Gibbs functions says is
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  possible. How?
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  It's because the condensation step heat rejection happens
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  isothermally at a constant temperature.
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  Now if that constant condensing temperature happens to be
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  exactly equal to the dead state temperature T dollar is going,
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  then you've perfectly matched your heat rejection to the
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  environment. All the available work, the
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  exergy in the steam has been extracted.
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  So yeah, the ideal ranking cycle becomes the benchmark, the
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  absolute best you could possibly do with that fuel in those
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  conditions. But the moment you have any real
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  world issue like a bit of pressure drop, or maybe the
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  turbine isn't perfectly efficient, or even if the
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  condenser runs just a tiny bit hotter than T developers, that
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  desert drops below 1.0 immediately.
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  Instantly it shows you exactly where potential is being lost.
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  Now let's contrast that with the jewel cycle gas turbines.
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  They use a single phase fluid, usually air.
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  Even an ideal jewel cycle. Again, perfect components, no
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  internal losses. It can never hit and drop of
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  $1.00 on. Never.
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  Why not? What's the built in limitation
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  there? It's fundamental thermodynamics,
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  really. The heat transfer is an
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  isothermal heat gets added over a range of temperatures as the
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  air heats up, and crucially, it gets rejected over a range of
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  temperatures as the exhaust gas cools.
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  Because that exhaust leaves hot, well hotter than $2.00 anyway.
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  Over a temperature gradient, the energy it dumps to the
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  surroundings, even in the perfect case, still contains
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  usable work potential and still has exergy.
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  OK, so even if the gas turbine itself is like, perfectly made,
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  the cycle's nature means it can't help but throw away some
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  available work, unlike the ideal ranking.
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  That's a big deal for design, isn't it?
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  It inherently limits the jewel cycles maximum rational
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  efficiency. Exactly.
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  It pushes engineers toward things like combined cycles,
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  trying to grab some of that energy from the hot exhaust.
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  OK, so we've set the absolute ceiling with rational efficiency
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  Tay Totter. That's the best the fuel in the
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  cycle could possibly do. But you know, power plants
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  aren't just about potential. They need to generate a flow of
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  power, actual watts. And getting that flow, getting
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  power out quickly introduces a whole new set of constraints.
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  Real world physics problems. Right, this is the big reality
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  check for engineers that maximum theoretical efficiency we all
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  learn first. The carno efficiency thin to Max
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  1 TCTH da. It's basically useless for
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  designing an engine that actually does something.
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  It really is because to achieve that carno efficiency you need
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  external reversibility, which means heat has to transfer
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  across an infinitesimally small temperature difference.
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  Delta T approaches 0. So if your temperature
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  difference is practically 0, what's your rate of heat
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  transfer? What's your?
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  Power output 0. Nothing.
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  It takes forever to move any heat, so the most efficient
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  engine possible produces no power.
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  That's yeah, that's peak thermodynamics irony, isn't it?
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  Precisely so. Engineers rely on a more
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  realistic model, the indoor versable heat engine.
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  This model assumes the engine's insides are perfect, internally
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  reversible, but it accepts that heat transfer to and from the
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  engine requires A finite temperature difference.
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  There has to be a real temperature gap between the hot
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  reservoir and the engine's not side deep felons, and between
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  the engine's cold side DDCQ 2 and the cold reservoir DCT ZOS.
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  That's external irreversibility. OK, so to get power out you need
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  that temperature difference. But that very difference causes
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  irreversibility, which lowers efficiency.
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  There's a tradeoff. So the engine has to compromise
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  efficiency just to speed up the energy flow and make useful
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  power. What's the sweet spot?
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  What's the condition for maximum power?
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  The analysis shows that maximum power output happens when the
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  temperature ratio across the engine itself TPC levels, which
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  is 2 to T of 11, is equal to the square root of the overall
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  reservoir temperature ratio. 2 CTT Vols.
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  That specific condition gives you the maximum possible power
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  output. The square root, that's the key
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  factor. And that leads to a different
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  efficiency limit, doesn't it? The maximum power efficiency
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  THP, which turns out to be $1.00.
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  Minus that same square root term, THP will be all one TCTH
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  12. That's the one, and let's put
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  some numbers on that, like the example given.
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  Say your hot source is TTH equals 1600 K and your cold sink
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  is TC equals 400 K The absolute theoretical carnote limit 1400
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  and 1600 dollars is CA1055 or 75 percent efficiency.
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  But the maximum power efficiency T Max is 1 dollar, 400 and
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  $600.00. That's $1.14 toll, which is
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  $1.22. So it's only dollars and five
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  cents or 50% efficiency. Wow, that's a huge drop.
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  You lose a full 25 percentage points of efficiency compared to
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  the theoretical Max just to get the engine to produce power at
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  its fastest possible rate. You're literally sacrificing 1/3
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  of your achievable efficiency just to make the engine
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  practical and powerful. That's the trade off, and it's
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  quite fundamental even if you try to game it by using multiple
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  engines in series, like in a combined cycle.
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  Stacking 2 or more indoor reversible engines between the
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  same overall T dollars and T to call doesn't let you beat that
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  maximum power efficiency limit for a single engine operating
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  between them. The best you could do is arrange
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  them so the product of their individual temperature ratios
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  still hits that overall square root condition.
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  The limit holds #outro. OK, so let's try and wrap this
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  up. The key take away for you, the
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  designer listening seems to be this.
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  You really need two different efficiency measures.
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  First, rational efficiency that tells you the true theoretical
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  ceiling, how much work is actually available in the fuel.
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  Considering the second law, it shows you the ultimate scope for
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  improvement. Then you need the maximum power
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  efficiency T Topei Mahlers That gives you the realistic peak
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  efficiency you can hope for when you're running the engine flat
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  out generating maximum useful power.
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  They measure different things, and you need both to understand
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  your design's potential and its practical limits.
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  Absolutely. It may be one final thought to
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  leave people with something provocative.
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  Think about that dead state, temperature $2.00.
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  We saw how crucial it is for calculating the Gibbs function,
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  and thus ATAR. Now imagine you're operating the
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  same power plant, burning the exact same fuel, but one day
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  it's in a cold climate, low T dollars, and the next day it's
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  somewhere hot, high T dollars. The actual maximum work
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  potential available from that fuel Delta G dollar week is
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  different in those two scenarios.
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  It's higher when T dollars is lower.
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  So how does that change your perspective on what constitutes
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  wasted energy When the very potential of the fuel shifts
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  with the environment? It's something to Mull over.
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  Until the next equation design junkies, we are mechanical
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  engineering made simple.