In Post #1706, I determined that, for heating my home here in Virginia, it was far cheaper to run my heat pumps than to run my natural gas furnace. That’s based on costs of $1.70 per therm of natural gas, and $0.12 per kilowatt-hour (KWH) of electricity. Like so:
Yesterday, the Washington Post had yet another article on heat pumps, saying more-or-less what I just said. Cheaper to run, for many people, compared to burning fossil fuels in the home.
As usual, the comments were a mix of people who had recently bought heat pumps and were satisfied with them, and the usual hash of misinformation and disinformation. The latter coming almost exclusively from people who had never actually used a modern heat pump.
I thought I would take this opportunity to dispel what I saw as two of the worst bits of disinformation on modern heat pumps: The Carnot limit, and declining COP. Both of these address the misinformation that “heat pumps don’t work when it’s cold.”
At the end of the day, I’m going to redo the analysis above, accounting for the fact that air-source heat pumps lose efficiency as the temperature drops. I’ll use historical weather data to figure out how cold it might plausibly get in my locality. And then I’ll check that against the temperature at which modern heat pumps lose so much efficiency that natural gas would be my cheaper heating choice. To figure out how often natural gas would be cheaper than a modern air-source electric heat pump, in my locality (Northern Virginia).
Spoiler alert: Never. At the prices I face ($1.70/therm, $0.12 per kilowatt-hour), and my likely lowest temperature annually (zero F), a modern high-efficiency air-source heat pump is always going to be cheaper to run than a 95% efficient natural gas furnace.
Details follow.
Carnot COP carnage.
Coefficient of Performance (COP) is the ratio of the heat energy that a heat pump moves, compared to the amount of energy used to run the heat pump. Good modern heat pumps will typically have a COP of around 3.5. This means that for one kilowatt-hour (KWH) of electricity used, they can move 3.5 KWH worth of heat from the exterior of the house to the interior of the house, under typical winter conditions. This high COP is what drives the low cost of using a heat pump in many circumstances.
Here’s a rule of thumb that has served me well over the years. When any Citizen Scientist cites the Second Law of Thermodynamics to prove that some technology cannot possibly work, the odds are overwhelming that they’re totally full of crap.
So, of all the disinformation in the comments to that Post article, the one that stood out was the guy who proved that heat pumps were a waste of money. He claimed that the Carnot limit meant that heat pumps could not possibly be any more than 15% efficient at -20F. Or, in other words, a COP of 0.15. You’d be better off running old-fashioned resistance heating, with a COP of 1.0. And that was not debatable because the Carnot limit is based on the laws of physics, which cannot be broken.
This, despite many personal attestations in those same comments showing that many people happily used their modern heat pumps down to that temperature. So, who is right? The people actually using modern heat pumps, or Mr. Second-Law-of-Thermodynamics?
While the Second Law of Thermodynamics can’t be broken, the Law of Getting a Simple Calculation Correct gets broken all the time. And, almost without exception, the same doofuses who can’t be bothered to check their calculations always think that they have managed to uncover some deeply hidden insight that nobody else understands. As opposed to screwing up their arithmetic.
First, the Carnot Limit is a real thing. That sets a true absolute upper limit on how efficiently a heat engine or heat pump can work. I won’t even pretend to understand how Carnot proved that, but it’s generally been accepted as correct since it was posited around 1824 (source: Wikipedia).
That said, it’s more of a theoretical novelty than a practical consideration, as real-world devices never even come close to the Carnot limit. It’s akin to saying that the maximum speed of a Tesla will always be limited by the speed of sound. That’s probably true, but it’s hardy what actually limits the speed in the real world.
Second, the formula for the limit on heat pump (or reversible heat engine) efficiency is incredibly simple. It’s just based on (in this case) the difference between the interior and exterior temperatures around the home (reference).
Carnot limit on COP = T(hot)/(T(hot) – T(cool))
T(hot) is the temperature in the house. T(cool) is the temperature outside. There are some practical engineering issues that would make it a little more complicated than that. But in terms of a theoretical limit, that’s it.
Easy-peasy.
Except. Except that, as with all of thermodynamics, you have to measure temperature on an absolute scale, where zero is absolute zero. In the metric system, you have to use degrees Kelvin (K). In English units, you’d have to use degrees Rankine (R), which I can honestly say I have never seen used for anything, anywhere.
So, if it’s 68F inside (= 20 C = 293 K), and -4F outside (= -20 C = 253K), then the theoretical maximum COP (Carnot limit) is (293/(293-253)) = 7.3. Which is vastly higher than any actual COP you’ll ever see on any home heat pump, anyway.
As always, I’d like to check my calculation from some other sources. So here’s a graph of the Carnot limit, for various temperatures. Note that the Carnot limit on COP, for the example I gave above, is just over 7. That matches what I calculated.
Source: This researchgate reference.
It’s easy enough to do the theoretical Carnot limit calculation for any two temperatures. In particular, if it’s -30F outside (= -34C = 239 K), the Carnot limit on COP for warming that 68F house is (293/(293-239) =) 5.4. Still vastly higher than almost any real-world heat pump COP you are ever likely to come across.
Source: Calculated.
The Carnot limit places no binding constraint on actual home heat pumps, because you’re never even going to come close to that theoretical maximum COP limit.
The Carnot limit does, however, illustrate one important point: The colder the exterior temperature, the less efficient a heat pump will be, and the lower the COP.
The only limits on actual use of heat pumps are the actual, as-observed COPs of those units, under varying conditions. And so, if the COP declines as it gets colder outside, in theory, there will be some temperature where it will pay me to switch off an air-source heat pump and switch on a natural gas furnace. That’s what I’ll calculate in the next section, given the prices I face for natural gas and electricity here in Virginia.
So, the next section asks, in practical terms, how cold it has to get before natural gas becomes cheaper than using a heat pump, given the prices I currently face in Virginia.
Break-even COP, and break-even temp.
It’s a simple matter to take the spreadsheet shown above and modify it to find the COP values that make gas and electricity equally expensive. Like so:
Source: Calculated
At the prices I pay ($1.70/therm, $0.12/KWH), assuming that my 95% efficient gas furnace actually functions at 95% efficiency, air-source heat pump COP would have to fall to 1.95 in order for a heat pump and a gas furnace to be equally costly sources of heat.
For a modern heat pump, how cold would it have to get for the COP to fall below (say) 2.0?
I can simplify that by saying that, for the past couple of decades, we have rarely seen even one day with a minimum temperature below zero F, here in Northern Virginia. I can take the COP at zero F to be the lowest COP I am likely to experience.
Beyond that, the observed COPS are going to depend on the make and model of heat pump. So, the question is, for which heat pumps will the COP fall below 2.0 at zero F?
For selected Japanese-made heat pumps, the answer is none. I am never going to see temperatures cold enough to make gas heat cost-effective relative to these modern Japanese-made heat pumps.
S
Source: “Northeast Energy Efficiency Partnerships.” Cold Climate Air Source Heat Pump. Northeast Energy Efficiency Partnerships. Web. 16 Apr. 2016. <http://www.neep.org/initiatives/high-efficiency-products/emerging-technologies/ashp/cold-climate-air-source-heat-pump>
Similarly, for this broad collection of high-temperature heat pumps, I’d have to have more than a 100 degree difference between interior and exterior temperatures before COP dropped below 2.0
Source Purdue University, Purdue e-Pubs, International Refrigeration and Air Conditioning, Conference School of Mechanical Engineering, 2018 High Temperature Heat Pumps: Market Overview, State of the Art, Research Status, Refrigerants, and Application Potentials, by Cordin Arpagaus et al.
Here’s an analysis from the U.S. DOE leading to the same conclusion. COP at zero F remains well above 2.0 for the system they modeled.
Source: U.S. Department of Energy, High Efficiency Cold Climate Heat Pump
2016 Building Technologies Office Peer Review, Oak Ridge National Laboratory, Bo Shen, shenb@ornl.gov
I think that’s enough for me to conclude that if I bought the right air-source heat pump, the answer to my original question is “never”. At the prices I pay, with the weather I face, it will never get cold enough to make it cost effective to heat my house with a 95% efficient natural gas furnace, rather then using an air-source heat pump.
