CHP primer: Fun with thermodynamics
Those of us who believe (as I do) that there are massive opportunities to reduce US energy costs while simultaneously lowering our greenhouse gas footprint spend a lot of time getting into arguments with bad economists. These folks remember just enough of freshman theory (supply, demand, price, blah blah blah) to assert confidently that if profitable opportunities existed of any consequence, they already would have been snatched up by our efficient markets. Therefore, any change from our perfectly-balanced status quo must be economically detrimental. If you believe this, there may be a job for you at the Cato Institute.
There is, however, another parallel set of objections that is no less pernicious, although it is a bit less public. These objections come from bad thermodynamicists.
Sadi Carnot was one smart dude, and the godfather of the 2nd law of thermodynamics. The first law says that all energy is conserved (e.g., you can’t build a perpetual motion machine), while the second law says that as a practical matter, you’ll always get less useful energy out of a system than you put in. As my own thermo professor put it, “the first law says you can’t win, and the second law says you can’t tie.”
But Carnot is perhaps best known for his articulation of the limits to any work cycle. His insight was that the maximum efficiency of any thermal work cycle (more on that term of art in a moment) reduces to 1 – Tc/Th, where Tc is the coldest temperature in the cycle and Th is the hottest temperature in the cycle. (Poetry majors: you have to convert temperatures into Kelvin or Rankine for this math to work, lest you think you can get >100% efficiency by using celsius on a cold day.)
…Had Nothing to Say About Cogeneration
So how does that square with cogeneration proponents when they claim to build power plants with 80 – 90% fuel efficiency? And who’s right when those claims are quickly countered in the name of Carnot? After all, if I run a thermal power plant with peak temperatures in the 2000F range on a 60F day, the highest efficiency I’m ever going to get out of that cycle is 1-519/2459 = 79%. (Temperatures converted to Rankine.) Factor in all the losses in gears, cooling towers, etc. and you’ll be damn lucky to break 50%. Who are these outrageous cogenerateurs who dare to question the fallability of ze french phyciseest?
The truth is, there’s no disagreement. Carnot didn’t say anything about heat cycles. His math applies to work cycles. Work – oversimplified – can be thought of as really high-value energy. Electric power plants are work cycles. They turn low value stuff (wood, oil, falling water) into really valuable electricity. They are like beef slaughterhouses, turning cows into filet mignons. Those filets taste awesome; but no matter how hard you might try, the total filet mignon you can get out of a cow is limited. Carnot simply articulated the theoretical maximum filet/cow ratio. But he never claimed that the rest of the cow wasn’t worth eating.
Getting back to the physics, a thermal energy plant isn’t a work cycle. If you’ve got a good, high efficiency furnace in your basement, you might realize a fuel efficiency of 75 – 80%; for every 100 units of fuel you burn, you get some 80 units of heat for your home. And yet – I’m taking a guess here – you are not a known violator of the laws of thermodynamics. That’s not to suggest that heat is the same as electricity, any more than a hamburger is as good as a filet. But make no mistake about it – your furnace is making lots of good, ground chuck, unconstrained by the filet/cow ratio.
Cogen plants thus become a neat hybrid trick. You squeeze the filets out of the cow first, capturing the highest value commodity, but then use as much of the rest as possible to make that chuck (heat). In some cases, where the heat is more valuable, you might even sacrifice a bit of filet to make some more chuck. But in all cases, your only physical constraint is your ability to fetchez those vaches.
So What Is the Efficiency?
So Carnot was right. So is the cogen community. Lazy thermodynamists … not so much. But what does that mean about the economics of cogen? Or more broadly – what does that mean about the actual fossil fuel impact of a fossil-fired cogen plant given environmental considerations? The short answer is that you can safely ignore Carnot – and if you’re lucky, can even find yourself getting around the first law of thermodynamics as well. To see why, let’s look at an example:
Let’s assume you have a power plant that operates at 30% fuel-to-electric efficiency. For every 100 units of fuel, 30 are recovered as electricity. Carnot smiles upon you. But those other 70 units of energy don’t just disappear – they get thrown out of your exhaust as heat. As a cogenerator, you’re going to try and recover that heat. You’re never going to get all of that heat back, but in a good system you can get as much as 80% of it back. So let’s assume this is a good plant, and 80% x 70 = 56 units of heat are recovered as steam. That suggests an overall cycle efficiency of (30 + 56) / 100, or 86%. Pretty good, huh?
In fact, it’s better.
What matters to your wallet, and to the environment isn’t how much steam and electricity you made per unit of fossil fuel, but what the net change was in fossil fuel use per unit of useful output. That’s a somewhat more subtle bit of math because the steam you recover from that cogeneration plant doesn’t displace fossil fuel on a 1:1 basis. Recall that the best thermal only plants (e.g., the furnace in your basement) are 75 – 80% efficient. So at the bottom end of that range, those 56 units of steam actually avoided your need to buy/burn 56 / 0.75, or 74.7 of fossil fuel.
Now let’s take a look at what that means to your operating economics, and the environment. Before your cogen plant was installed, you were buying 74.7 units of fossil fuel, burning them to make 56 units of steam and buying an additional 30 units of electricity. After your cogen plant was installed, you were now buying 100 units of fossil fuel to run your cogen plant, but you no longer had to buy 74.7 units of fuel for your thermal plant. So from a purchasing perspective, and from a global fuel combustion perspective your net increase in fuel use is only (100 – 74.7) = 25.3 units of fuel. From which you are generating 30 units of electricity, at an apparent fuel efficiency of 30/25.3, or 119%.
Suffice to say, this is the point where bad thermodynamicists get really pissed off. But who cares? You’re making real dollars on those economics, making real reductions in fossil fuel emissions while they’re arguing theory. They are no different from the economist who walks down the street and doesn’t pick up a $20 bill because he knows that the theory of free markets wouldn’t allow that $20 worth of economic inefficiency to exist.
Textbooks are nice. Dollars are better.