Thursday, September 1, 2022

The Basic Physics of the Climate Heat Engine - And How It May Fail In The End

The recent climate study in the journal Communications Earth & Environment  -which found a permanent spike in heat index for mid-latitudes  -  is sobering.  The research, by Harvard climatologist Lucas Zeppetello, found  a three to tenfold increase in 103 degree heat in the mid-latitudes.   This was even in the unlikely best-case scenario of global warming limited to only 3.6 degrees since preindustrial times.  But there’s only a 5% chance for warming to be that low and that infrequent, the study found. What’s more likely, according to the study, is that the 103-degree heat will steam the tropics “during most days of each typical year” by 2100.  In other words, nearly all surviving 10-12 billion humans will be immersed in a hothouse world.

This brings to mind how we got to this point, and that concerns the climate heat engine itself. Although in past posts I have touched on this in terms of entropy and other factors-   as well as submitted thermal physics tests - mainly for climate change skeptics, e.g.

Thermal Physics Test for skeptics

All of this assumed test takers at least had a basic physics background, which I found many - even in the high IQ societies Mensa and Intertel - did not.  They simply loved to bloviate based on economic arguments that carry no ballast when the climate is in catastrophe mode. See e.g.

Skewering The Spurious Global Warming Narrative Of Another "High IQ" Libertarian: Thomas Nelson

Let's begin with the question: What constitutes a heat engine?  Basically, a heat engine is a device that converts thermal energy into useful forms such as mechanical and electrical. More exactly, it's a device which carries a substance through a cycle, during in which:

i) Heat (Qh) is absorbed from a source at high temperature (Th)

ii) Work is done by the engine (W = Qh - Qc)

iii) Heat is expelled by the engine to a source at lower temperature (Tc). This expelled heat is denoted by Qc.

An illustration for such an engine, with respective sources is provided below:

Again, the net work done is equal to the net heat flowing into the engine or W = (Qh - Qc) where the net heat Qnet = (Q h Q c).   A heat engine then produces mechanical energy in the form of work W by absorbing an amount of heat Q in from a hot reservoir (the source, for h) and depositing a smaller amount Qout into a cold reservoir (the sink, for c).

Note also that since the working substance (fluid or gas) goes through a cycle, the initial and final energies are equal and hence, delta U = 0. If the working substance is a gas, the net work for a cyclic process is just the area enclosed by the curve representing the process.  As an example, there is the Carnot Engine:

We note here that no real heat engine operating between two heat reservoirs (at temperatures Th and Tc) can be more efficient than a Carnot Engine operating between the same temperatures, thus the potential efficiency 𝜂' < 𝜂carnot always, where 𝜂carnot  denotes the Carnot Engine efficiency, defined by:

𝜂 carnot =  𝜂  =   1 - T/ Th,   or   Th  - T/ Th

Determines the maximum possible work any heat engine can perform on an external body. It is achieved by a closed, reversible (ideal) engine. Real heat engines can never truly reach the Carnot efficiency because their work output is limited by irreversible processes.

And note again:

Qc / Q h = T c / T h

The four components of the cycle are as follows:

1) A
® B: isothermal expansion at T h, so absorbs Q h and work = WAB

2) B-
® C: adiabatic expansion: Q = 0, Th ® Tc and work WBC raises piston

3) C
® D: isothermal compression at Tc, expels heat Q c, work done on gas = WCD

4)D
® A: adiabatic compression: Tc  ®T h work on gas is WDA

Note the general thermodynamic efficiency
𝜂'  of a heat engine is:

𝜂 ' = W/Qh = (Qh - Qc)/ Qh =  [1 - Qc/Qh]

Or alternatively:

𝜂 ' [1 - Q in/Q out]

It can be seen that a heat engine will only be 100% efficient if Qc = 0 (i.e. no heat is expelled at a cold temperature). However, since every process known generates some kind of heat, which is inevitably left over, this is impossible. In practice, all heat engines convert only a fraction of absorbed heat to mechanical work, e.g. for the typical auto engine an efficiency of 20% is practical, maybe a bit more (25%) on newer engines. If e = 20% that means 20% goes to useful work, say in propelling the car, but 80% is expelled as waste heat. (That is, 4 out of every 5 gallons you use of gasoline goes into producing waste heat. So if you pay $5 a gallon, you are burning up$4 as waste heat.)

The climate system is essentially a giant planetary-scale heat engine, with a basic 'cartoon' depiction shown below (from a recent Physics Today article, 'Thermodynamics of the Climate System', July, p. 30):

Heat input arrives via the absorption of solar radiation and cooling by the emission of radiation to space.  The heating is largest at the warm tropical surface, while the cooling occurs primarily in the colder troposphere and is weighted toward higher latitudes. The planetary heat engine transports heat from the warm surface source to the colder tropospheric sink by the flows of the atmosphere and oceans. For the climate system, the ultimate heat source is the Sun, with outer space acting as the sink. The work is performed internally and produces winds and ocean currents. As a result, Q in = Q out

For the climate system the question often asked is:  How is work done?  After all Earth does mot push against any external body as in a classical heat engine. As I showed in one of my early basic physics posts, e.g.

A gas confined to a container and heated - provides   heat in the amount  Q in -  and can generate sufficient pressure to displace a piston upwards, changing the volume of the gas by V2 - V1.   Then from the first law of thermodynamics:

+Q = +U + D W or

D Q = D U + p(V2 – V1) = p(D V)

Where D U  is the change in internal energy, and D W  the work done.   Then we have for the work done:

D W =    p(D V) = p(V2 – V1)

But there is no analogy to such work done for the Earth as it doesn't push against any external bodies. So how to parse the application of the given law?   As the Physics Today authors point out:

"The oceans and atmosphere do, however, perform work on themselves and each other, and that work generates the familiar winds and ocean currents that scientists observe. For climate scientists, useful work is that used to drive atmospheric and oceanic circulations.  Because the work performed by the planetary heat engine is internal to the engine itself, its efficiency is not limited by the Carnot efficiency. Rather, the climate system can, in principle, recycle some of the heat produced by the frictional dissipation of winds and ocean currents and increase its maximum efficiency to a value:

𝜂 max  1  - T/ Th

Which, they point out:  "is similar to the Carnot efficiency, except that the temperature in the denominator is replaced by that of the cold sink.3 The maximum planetary efficiency occurs when all available energy is used to drive atmospheric and oceanic currents and when the dissipation of those currents is concentrated at the warm source—for instance, through friction with Earth’s surface"

The authors go on to point out the relevance to spatial cloud and temperature distributions, especially as applicable to atmospheric circulation.  These aspects of climate change I covered in an earlier blog post on feedback mechanisms, e.g.

Add in the winds and currents, driven by Earth's heat engine -  and which affects efficiency and the amount of heat transported  -  and you have the basis for the feedback that regulates the climate. In effect, "The work performed by the planetary heat engine acts to reduce the temperature gradient that drives it."

In the case of our Earth, the input and output temperatures of the planetary heat engine (T in ,  T out ) are controlled by two temperature gradients: i) surface to upper atmosphere, and ii) equator to pole.  This elicits the question - posed by the authors: For  what factor (s) might we expect the climate heat engine to become more (or less) efficient? It turns out (p 35): "If the magnitude of moist processes increases with the water vapor content we might expect the climate heat engine to become less efficient on a warmer planet."   They then add:

"A study of global climate models shows that, indeed, the mechanical efficiency of simulated future climates may go down and decrease the net energy available to drive winds."

At this point we can conceptualize the entry point for the Boltzmann entropy, defined by S = k log (W), where k is the Boltzmann constant and W is the total of microstates needed to define the macro- state for a gas, say in the atmosphere.  But as the PT authors point out, "to apply it requires researchers discard the heat engine model entirely and consider the system of interest thermodynamically isolated and in contact with  single reservoir rather than two."

In his monograph 'Thermal Physics' (1969),  Charles Kittel wrote (p. 65):  "It is not known if the total entropy of the Earth is increasing or decreasing at this moment.  The Earth receives, generates, and radiates entropy."

But at the time of Kittel's text publication, global warming had not yet become the much more violent phenomenon it has manifested today.  If in fact we can apply the PT authors' current picture - with decreased mechanical energy (and efficiency) to drive circulation (winds)-   then by also incorporating the "number of microscale arrangements of fluid particles" i.e. in the atmosphere (to maximize the Boltzmann entropy), it seems clear Earth's entropy is increasing. And radically. Counterintuitively, it appears the failure of the planet's climate heat engine - via collapsing efficiency in its 'moist processes' -  leads inexorably to an ever hotter, less inhabitable abode for humans.

And if this continues, Earth will end up as another Venus.