I always used to think (I don’t know why!) that the efficiency of human (and animal and plant) cells should be equal to or near the efficiency of a Carnot engine or at least should be the highest efficiency among all practical engines. But I wondered when I saw the answers given to this question. They are talking about an efficiency of $18-26\%$. But you can see here that the efficiency of an Otto engine is between $56\%$ and $61\%$. Is there any explanation for this? Which cycle do human cells work with? Can we compare living cells with heat engines at all?
[Physics] Why is the efficiency of human cells less than the efficiency of an Otto engine
biophysicscarnot-cycleefficient-energy-useheat-enginethermodynamics
Related Solutions
Revamped Answer. 2017-07-01
There is no contradiction because your analysis only includes what happens to the gaseous working substance in the Stirling engine, and it neglects a crucial component of the engine called the regenerator. If the regenerator is not included as a component of the engine when we perform the efficiency analysis, then we don't have a device that qualifies as a heat engine operating between two temperatures, and we therefore shouldn't expect it to abide by Carnot's Theorem as I stated in the original version of this answer.
However, if we properly take account of the regenerator, then we find that the efficiency of the engine is the Carnot efficiency.
Of course the whole analysis here is an idealized one in which we assume, for example, that there are no energy losses due to friction in the engine's components.
Details.
A stirling engine is more complex than the $P$-$V$ diagram drawn in the question statement seems to indicate. If we conceptually reduce the engine to its simplest form, it contains two fundamental components:
- A gaseous working substance. This is the part of the engine whose thermodynamic state travels along the curve in the $P$-$V$ diagram.
- A regenerator. This part of the engine absorbs and stores the energy given up by the gaseous working substance by heat transfer during the process $2\to 3$ and then returns that same energy to the gaseous working substance during the process $4\to 1$.
The crucial point is that when the regenerator is included, there is no net heat transfer into or out of the engine during the processes $2\to 3$ and $4\to 1$. The energy that leaves the gaseous working substance during the process $2\to 3$ by heat transfer is stored in the regenerator, and that heat is then given back up to the working substance during process $4\to 1$. No heat is transferred between the engine and its surroundings during these legs of the cycle.
It follows that the only heat transferred to the engine as a whole is transferred during $1\to 2$. This qualifies the device as a heat engine (see old answer below) and the efficiency of the engine is then computed as the ratio of the net work output divided by the heat input in process $1\to 2$. This yields the Carnot efficiency as it should.
My original answer claimed that the cycle drawn does not represent the operation of a heat engine operating between two temperatures, but I was neglecting the regenerator, and I believe this is what you implicitly did in the computation you originally performed as well, and this yielded the incorrect efficiency.
Original, incomplete answer.
There is no contradiction. The Stirling cycle you drew above is reversible but does not operate between two reservoirs at fixed temperatures $T_1$ and $T_2$. The isovolumetric parts of the cycle operate at continuously changing temperatures (think ideal gas law).
Old Addendum. Note that in thermodynamics, a heat engine is said to operate (or work) between (two reservoirs at) temperatures $T_1$ and $T_2$ provided all of the heat it absorbs or gives up is done so at one of those two temperatures.
To give credence to this definition (which is essentially implicit in most discussions of heat engines I have seen), here is a quote from Fermi's thermodynamics text:
In the preceding section we described a reversible cyclic engine, the Carnot engine, which performs an amount of work $L$ during each of its cycles by absorbing a quantity of heat $Q_2$ from a source at temperature $t_2$ and surrendering a quantity of heat $Q_1$ to a source at the lower temperature $t_1$. We shall say that such an engine works between the temperatures $t_1$ and $t_2$.
However, isn't any closed loop on a PV diagram reversible? The arrows can simply be drawn in the reverse way to create a refrigerator. If any closed loop is reversible then why does the specific Carnot engine (a specific loop) have the highest efficiency?
This was exactly the question I asked myself ten years ago :-) The problem is that often students do not appreciate the whole statement: Carnot's engine is operating between two temperatures (heat sources). Any circle on the PV-plane is reversible if you have many heat sources. In the case of many heat sources, you may also know that you do not talk about the efficiency of the engine, but you talk about the Clausius' equality: $$\sum_{i} \frac{Q_i}{T_i}= 0.$$ Note that $T_i$ is the temperature of the $i$th heat source (this is a very important point often missed!), which equals the temperatures of the system when they are in reversible contact. This is not true if the process is irreversible: you have heat flow from hot sources to the (colder) engine. Then one has the Clausius' inequality: $$\sum_{i}\frac{Q_i}{T_i}<0.$$
So, in short: Carnot's engine is the only reversible engine operating between two temperatures.
Best Answer
No, not really, because the living being isn't only a heat engine. There are three main points I want to make here.
1. Homeostasis Requires Constant Energy Input
This statement is especially true and obvious of homeotherms Mammals (Mammaliaformes, descended from the Therapsid Synapsid Amniotes), and Birds (Avialae / Dinosauria, descended from Dinosauriform Amniotes), which use a great deal of energy simply keeping their body temperature within strict limits, i.e. compensating for (mainly convective) heat loss from their body in cold conditions and actively expelling heat from their bodies in hot conditions. But, more generally, the phenomenon of homeostasis also requires expenditure of energy; a living organism is a highly non-equilibrium thermodynamic system, and excess entropies produced by metabolic processes must be expelled to keep it that way. Thermodynamic equilibrium is only reached when the living creature dies.
From this consideration alone, we would expect efficiencies measured when the organism does mechanical work to be considerably less than those of a heat engine.
2. Muscular Action is Not a Heat Engine
Muscular action is much more comparable to an electric motor than a heat engine. What I mean by this is an electric motor converts essentially work from one form to another with near to zero entropy change and negligible temperature change; motor proteins convert low-entropy energy stored as ATP to mechanical work through the hydrolysis of ATP with very small temperature change in the reagents as they react. In this case, the most meaningful measure of efficiency is probably expressed in two factors: (1) the ratio of the free energy $\Delta G$ of the ATP hydrolysis reaction to the total enthalpy change $\Delta H$ of the reaction (the difference $T\,\Delta S$ being the work we have to "give up" to expel the excess entropy of the reactants relative to lower entropy reaction products) and (2) the ratio of the mechanical work done to the available $\Delta G$.
In a heat engine, we take a quantity of heat from a hot reservoir, reducing the latter's entropy by $Q_i/T_i$ in the process, but find that, if we have a colder reservoir at $T_o<T_i$ we only have to "give back" $Q_o<Q_i$ to the cold reservoir to offset the entropy drop in the hot reservoir, so we get to "keep" energy $Q_i - Q_o > 0$ for doing work with. In biological reactions, the most comparable process to this is that of photosynthesis, where the "working fluid" of light at thermodynamic equilibrium at $6000{\rm K}$ is converted to "stored work" in sugars and, ultimately, ATP, dumping excess heat at ambient temperature $300{\rm K}$ in the process. Thenceforth, all living things use this low-entropy energy store rather like an electric motor converting energy stored in a capacitor, whether it be plants using it for their own life process, or herbivores accessing it through eaten plants or carnivores accessing it through eaten plant eaters.
So the plants and their solar energy fixing are the component of the biosphere most comparable to a heat engine in a power station; the plant metabolic processes and animals that eat plants and each other to get access to stored energy in plants are more like the electrical appliances that use the work extracted by the power plant, with very little temperature change.
3. Proteins Denature at Roughly $50{\rm C}$
For any animal process that could be considered to be like a heat engine, the maximum intake temperature can be at most a few or at most a few tens of degrees kelvin above ambient. This is because biological machinery is fatally damaged by temperatures much higher than $40{\rm C}$. Proteins denature and lose their vital life functions at very low temperatures. So if there are any processes in life that can be thought of as reasonably analogous to heat engines, we would foresee their efficiencies to be very low, since the theoretical efficiency is of the order of $3\%$ given this limit.
An interesting exception to my point 3 comes up in deep sea life living near hydrothermal vents. John Rennie writes:
so we have creatures dwelling in $100{\rm C}$ and over environments and the opportunity to dump heat into the surrounding sea at much lower temperatures. However, my understanding is that these creatures still use the chemical energy from what they can extract from the volcanic vents, rather than working as heat engines taking advantage of the temperature drop.