The problem with heating "efficiency" (CHP critique part 2)

My criticism of various claims about "Combined Heat and Power" (CHP) or cogeneration systems was something I had thought I could keep brief, since the issue was intuitively clear to me. But explaining my understanding of the subject in straightforward non-scientific language has proven trickier than I'd expected. This post constitutes part 2, and will cover problems with many claims of "efficiency" in heating systems; part 3 will be more specifically focused on the several different types of CHP, what is good about them, and what is frequently over-hyped.

Part 1 in this series reviewed the basic thermodynamic principles regarding energy and entropy. In particular, not all energy is "useful" in the thermodynamic sense of being available to perform real work. The way in which useful energy is "used up" is through conversion to low-temperature heat, primarily through inefficiencies that bring irreversibility to exchanges of energy between different useful forms. Energy in the form of heat at the temperature of your surroundings has a useful-energy fraction of zero - it is all waste at that point.

Efficiency

'Efficiency' can have several meanings. One of these, which I will refer to as "thermodynamic efficiency", is best defined in terms of this 'inefficiency' or 'waste'. "Using up" useful energy means turning it into waste heat at ambient temperature. Exchanging energy between its different useful forms, for example chemical (in fuel), electrical, gravitational (the energy stored by water in hydro-electric dams) only diminishes the "useful" portion of that energy inasmuch as "waste" heat is dissipated into the environment in these transformations. If that waste were eliminated somehow then you could keep transforming that energy back and forth between its different useful forms without limit, and at the end you'd have the same total energy you started with. It is only through dissipation of waste heat that useful energy is lost.

So, for a given process involving a starting quantity of useful energy E_in, a well-defined quantity E_waste is "used up" and dissipated to the environment. That gives us a well-defined waste fraction f = E_waste/E_in (this ratio must always be between zero and 1).

The thermodynamic efficiency is then the fraction of useful energy remaining at the end of the process: 1 - f, which must also be between 0 and 1, or expressed as a percentage, this efficiency must be between 0% and 100%.

The other primary meaning for 'efficiency' relates the end-use purpose to the input energy; I will refer to this meaning as 'nominal efficiency'. For a given quantity U of final "utility", the ratio U/E_in provides a measure of the efficiency of the system. If the system in question is an industrial facility, making steel for instance, then U would be measured in tons of steel, and the ratio U/E_in could have the dimension of tons of steel per Btu of input energy. Increasing that ratio means improving the efficiency of the system, at least as far as energy inputs are concerned. Similarly, if the output of interest is lighting, the number of lumens per watt is a way to measure this "nominal efficiency". If the output of interest is transportation distance then miles per gallon of hydrocarbon fuel is essentially the same kind of nominal efficiency measure.

In cases where the useful output U is measured in energy terms, the ratio appears to be a dimensionless number similar to that in the definition of thermodynamic efficiency. Sometimes the two terms are the same. In particular, when the output U is restricted to purely useful forms of energy such as electrical energy or the potential energy of pumped water, then the two definitions of efficiency match (as long as E_in includes all the inputs).

But if the output energy U includes any form of thermal energy, which is never entirely "useful", this match between nominal efficiency and thermodynamic efficiency breaks down, and the numbers you get from nominal efficiency calculations can be deeply misleading.

The nominal efficiency is meaningful in the sense that bigger values are better than smaller values, as for all such measures of utility relative to inputs. But for heating (or cooling) it is a fundamentally different measure than the thermodynamic efficiency. In particular, nominal efficiencies for heating and cooling can easily be greater than 100%, and so calling it an "efficiency" in the first place is a little dubious.

An example of this use is in US Government "ENERGY STAR" rating of furnaces, where the criterion is based on Annual Fuel Utilization Efficiency (AFUE). The definition of AFUE is the ratio of fuel converted to space heat to total input fuel, essentially the nominal efficiency definition above; that gives you a number less than one because of the way furnaces work, and so it's usually expressed as a percentage - 85% - 90% levels are required for ENERGY STAR ratings now.

That seems amazingly efficient. Whenever we see efficiency numbers close to 100% we are impressed - but in fact these numbers are dreadfully poor relative to the true thermodynamic limits. At high thermodynamic efficiency, nominal "efficiencies" for space heating could theoretically reach 1000% or more, as will be explained in detail below.

There are several things that suggest these nominal efficiencies are misleadingly high. First, whenever you burn something, as long as you burn it to completion, all of the chemical energy is converted into heat. If you set fire to your kitchen table, 100% of the heat produced goes into your house, it doesn't disappear. When you run electrical appliances, or an electric space heater, 100% of their energy is, one way or another, dissipated into your home and goes into space heating. There is nowhere for "waste" heat to go if it's entirely produced within your house. So getting to 100% should be easy. The main reason furnaces are a little lower is because they vent their exhaust gases outdoors, instead of inside the home, and so lose a little heat that way.

Setting a fire or turning on your oven is a dumb way to heat your home, so that 100% efficiency suddenly seems a little less impressive. But now, suppose the outside air temperature is warmer than indoors. You can heat your home with no expenditure of energy at all by just opening windows and doors and letting the warmer air circulate around. The nominal efficiency of heating a house by opening windows on warm days is essentially infinite - you've burned no fuel at all to do that, aside from the few calories consumed by your own body while running around. Infinite "efficiency" definitely beats 100% - that's impressive.

When it's too hot and you need to cool your house, our standard tool now is the air conditioner; it is removing heat energy rather than adding it, but the same sort of nominal efficiency calculation applies. ENERGY STAR ratings for air conditioners are based on a nominal efficiency measure known as the Energy Efficiency Ratio (EER), calculated as the cooling capacity (in Btu/hour) divided by power input (watts). Typical home units now have EER (or seasonally adjusted - SEER) values ranging from 10.0 to 20.0. EER is explicitly a nominal efficiency measure as discussed above, but the two energy units used are different.

What happens when you factor out the energy terms in an EER rating? 1 watt is 1 joule per second, so over an hour it comes to 3600 joules. 1 Btu is 1055 joules, so 1 Btu/hour divided by 1 watt gives a dimensionless ratio of 1055/3600, or 0.293. That is, if you multiply the EER number by 0.293, you have a dimensionless energy ratio (this is referred to somewhat cryptically as the "coefficient of performance" - COP), relating the heat energy removed from a room by the air conditioner to the input electrical energy to the unit.

So for an air conditioner with an EER of 15.0 the nominal efficiency in dimensionless terms is 440%. That's already quite a bit better than 100%!

But, you may protest, cooling and heating are different. Indeed they are - cooling is harder than heating. That same process of heat removal by the air conditioner from the inside of the house to the outside means that the air conditioner is adding heat to the outdoors from inside the house, in addition to the heat that it releases as waste heat from its internal operations. That is, viewed from the heat added to the outdoors, an air conditioner adds 100% of the input energy it consumes, like any electrical appliance, plus another 440% (at EER 15.0) from the heat it moves around, for a total of 540% nominal efficiency in heating the outdoors.

If you want to heat the indoors you turn everything around - which is essentially how a heat pump works. Heat pumps also get ENERGY STAR ratings based on the nominal efficiency (COP) value, and typical numbers are again 400% (COP of 4.0), 500% or more.

So that 85% or 90% efficient furnace suddenly doesn't seem so impressive compared to the 400% or more equivalent numbers for heat pumps (or infinite nominal efficiency for opening windows). Instead of looking at this nominal efficiency value which seems to be able to be arbitrarily large, let's look at where the real limits of heating are, represented by thermodynamic efficiency.

Heat engines

Extracting useful energy from chemical energy, the form that fossil fuels hold it in, almost always involves combustion - i.e. burning the fuel. Combustion isn't absolutely necessary; there are other chemical processes, such as those used in fuel cells or batteries, which can convert more directly between chemical and electrical energy. Those direct conversions can be highly efficient, but they have other constraints that limit their applicability, so combustion remains the primary conversion mechanism for chemical energy for now.

Combustion releases the chemical energy of the fuel into the kinetic energy of the resulting gas molecules (CO2 and H2O, for full combustion of hydrocarbon fuel); the kinetic energy of those molecules is exactly what we know as thermal energy, i.e. heat. Flame temperatures can reach a few thousand degrees C, so at least at first this is very high-temperature heat.

To turn that high-temperature heat into useful work we need a "heat engine". This is some sort of mechanical system that takes high-temperature heat from a source (at temperature T_H) and runs some sort of process cycle extracting work (W); it also must connect to a cold sink (T_L) in order to return to its original state to run the cycle over again. The following diagram (from wikipedia) illustrates the generic heat engine:

Wikipedia Heat Engine image
The heat Q_L released to the cold sink in a heat engine is the fraction of the input energy (Q_H) that is "used up", i.e. from the earlier discussion of efficiency here, the waste fraction f = Q_L/Q_H. Carnot's theorem asserts that that fraction f must be at least as large as the ratio of the two temperatures:

f ≥ T_L/T_H

(where T_L and T_H are measured as absolute temperatures, in Kelvins). So to improve the efficiency of conversion of chemical energy to other useful forms (through combustion) we need to minimize the limit on the waste fraction f - that means keeping T_L low, and T_H as high as possible.

For a sytem running here on Earth, T_L can't be lower than the ambient outdoor temperature (otherwise you are adding additional input energy for air conditioning!), or roughly 300 K. For chemical combustion of hydrocarbons, T_H can't be higher than the flame temperature, around 2000 K (a bit higher for some fuels). That means there is already a minimum waste level asociated with chemical combustion of about 15% (300/2000). So you can never get better than about 85% thermodynamic efficiency via chemical combustion, it is simply not physically possible on this planet.

Operating the ideal Carnot cycle in reverse, adding work to low-temperature heat to produce high temperature heat, gives the thermodynamic limit on how much useful energy (work) is needed to produce high-temperature heat:

W ≥ (T_H - T_L)/T_H * Q_H

which gives us the thermodynamic limit on nominal efficiency: Q_H/W ≤ T_H/(T_H - T_L). To get 2000-degree temperatures, the nominal efficiency limit becomes 2000/1700 = 1.18, or 118%, so just burning the fuel (which gets close to 100% nominal efficiency) comes pretty close to the limit.

For residential or commercial building heat we are really not interested in temperatures of thousands of degrees. Aside from the relatively small quantities of energy needed for cooking, hot water at 120 degrees F (50 C) is about the highest temperature of any interest, and for most heating purposes all we need is to get somewhere around room temperature (68 F = 20 C). For T_H = 50 C (323 K) and T_L = 0 C (273 K), the maximum possible nominal efficiency to get hot water when it's freezing outside, the limit is 323/50 = 6.46 (646%). If all we're doing is trying to heat a room to room temperature when outside it's freezing, the nominal efficiency limit goes up to 293/20 = 14.65, or 1465%. For geothermal heat pumps that use groundwater as their low-temperature heat source, at about 10 C, nominal efficiency for heating could be as high as 293/10 or almost 3000%.

That is, the thermodynamic limit for heating a building using 50-degree F groundwater is almost 30 times the heat you get from a resistive space heater.

What this really means is that the efficiency numbers, such as the AFUE number used even by the Department of Energy to describe furnaces and hot water heaters, are really not comparable to the sort of thermodynamic efficiency values that are used elsewhere, for example in generating electricity or running motors, etc. To get a comparable real efficiency number for heating, nominal efficiencies such as AFUE should really be scaled down by the inverse of the thermodynamic limit on that nominal efficiency.

I.e. the true thermodynamic efficiency of a hot-water heater is somewhere around 0.15 times the AFUE value, or perhaps 14% for a 90%-efficient gas burner. For a furnace intended to heat a house, the scale factor is 0.07 or less. That means that oil-fired furnace you spend so much money to fuel is really operating at a thermodynamic efficiency of only 6% or less. Electric resistance heaters do little better, at 7% (plus, going down the pipeline to the primary energy source, your total efficiency for heating is more like 3% for electricity). A heat pump with nominal efficiency of 500% (COP of 5) at least gets to a relatively respectable 35% thermodynamic efficiency for home heating. But the thermodynamic limit tells us that in principle technology could be developed that does even better than that.

What that all means is that, in a thermodynamic sense, close to 94% of the money we spend on fuel oil is wasted.

Now, I hope a glimmer of understanding is starting to dawn on the problem with efficiency claims for combined heat and power systems. When you add the efficiency of electrical production to the efficiency of heat production and see big effiency numbers like 80%, 90% or better, that is using the nominal efficiency of the heat component, not the true thermodynamic efficiency. Those two efficiency numbers mean very different things and cannot just be added together like that. I'll go into the details on these bogus efficiency claims in the final installment in this series.

Part 1: Useful energy can only be used once.