By Kevin Kilty, Ph.D., P.E.

CO₂ Coalition Member

Retired Mechanical Engineering faculty member, University of Wyoming

Among the many attempts to demonstrate that CO₂ has little to no effect on climate are several that purport to use an equation of state as their basis. A recent example by Lightfoot and Ratzer is entitled “The Sun and the Troposphere Control the Earth’s Temperature” (Journal of Basic & Applied Sciences, 2023, 19, 163-173, hereafter referred to as L&R).  Let’s begin by stating that we are not denying the great utility of equations of state. The Psychrometric chart or its equivalents in table form or as an Excel spreadsheet, which is the basis for claims made in L&R, has been a basic tool in HVAC for over a century. However, the Psychrometric chart is fatally incomplete as a model of the atmosphere. Also, we do not claim that the authors of L&R have used the Psychrometric chart, or its equivalent, the equation of state for moist air, inaccurately. No equation of state can fully characterize atmospheric dynamics on its own. A model requires a great deal of additional information, little to none of which is in evidence in L&R.

Since the Psychrometric chart appears in the undergraduate studies of many mechanical engineers, a reference to other concepts found in two courses in a mechanical engineer’s education, engineering thermodynamics and transport, should make the deficiency of the L&R paper, and other similar attempts, clear.

Engineering Thermodynamics

Let’s begin with claim number 10.7. “The Sun is the primary energy source, and its variation controls the Earth’s temperature. The Sun determines the Earth’s temperature, it is currently in a solar minimum.”

There is no doubt that the Sun is the primary energy source, but the Sun does not determine the Earth’s temperature alone. Rather, Earth’s temperature distribution (since there is no single temperature involved here) is determined point to point by application of the first law of thermodynamics. Simply stated in engineering thermodynamics the First Law comes in one of two forms

• Changes in internal energy of a system = sum of heat flows – sum of work done
• Changes in enthalpy = sum of heat flows – sum of flow work

For Earth’s atmosphere the internal energy, or alternatively the enthalpy, could be provided by the Psychrometric chart. Both the internal energy and enthalpy are state functions whose numerical value is independent of how the air has arrived at its state. Thus, what is missing are the various heat flows and work processes that take a parcel of air from one state to another. How does the atmosphere arrive at a particular distribution of states? The Psychrometric chart alone cannot say. A model requires more information.

After taking engineering thermodynamics, a mechanical engineer might recall that neither heat nor work are exact differentials, and the only way to calculate how much of either is involved in a change of internal energy as a parcel evolves is to provide a process path which can be integrated in some space (temperature, entropy space, T,S is most convenient heat processes). What one would like to know is a path, or possibly the many alternative paths, that takes air from its state at a place of its origin to its destination.

In HVAC work, the process path is dictated by the equipment which is specified for the system. The equipment, compressors, heat exchangers, throttles, expansion turbines, piping, fans and so forth have specified capability which an engineer can draw upon to design the needed change of state of the working fluid. In HVAC the working fluid may be moist air itself as in a “swamp cooler” or it may be a refrigerant like R-744 (CO₂ itself!) which conditions air indirectly through a heat exchanger.

By comparison, there is no similar equipment that specifies how the atmosphere operates as both a refrigerator and a heat engine. It operates through processes uncontemplated by the equation of state.[1]

Non-Equilibrium factors

The need for evaporators or heat exchangers now brings up a second deficiency of the Psychrometric chart alone for any analysis of weather or climate. Heat transfer requires a temperature difference. This takes us out of the realm of equilibrium thermodynamics (the equation of state is an equilibrium concept) and into the realm of heat and mass transport. A mechanical engineer probably would take a course in transport – one can’t say that all do because there might be different curricula that apply between foreign countries and the U.S. One transport mechanism at the heart of Earth’s climate, and about which the Psychrometric chart can say nothing at all, is radiation.

Methods of transport

There are three modes of heat transport involved in setting Earth temperatures: 1) convection or advection of air and subsequent mixing with air of differing temperature, 2) evaporation and condensing of water vapor, and 3) radiation. Just concentrate on radiation for a moment.

For any two surfaces that can “see” one another and are at different temperatures, there will be radiation transport between them of some amount no matter what the Psychrometric chart has to say about the state of air near each surface or in between the two. Placing a radiation baffle between these surfaces is an engineering method to reduce such transfer, and in the context of Earth’s climate both water vapor and CO₂ act as radiation baffles, frustrating the transport of heat through radiation from the surface to space. One way to get an idea of how much heat is transported from Earth’s surface to space by radiation is to use the program Modtran. Modtran is not a full expression of the First Law of thermodynamics because it fails to include transport mechanisms 1) and 2) in the paragraph above. It is not a full equation of transport.[2]

Moreover, some public versions of Modtran, the one at the University of Chicago for example, have a limited wrapper (a wrapper is the GUI for the underlying code) that doesn’t allow an arbitrary input. However, despite its deficiencies, a person can still learn a lot about radiation transport by using just Modtran along with the few atmospheric models it provides.

For example, a calculation using Modtran allows me to say that I agree with this conclusion that Lightfoot and Ratzer list: “10.4. The radiation profile at the top of the atmosphere is essentially the same as at the top of the Troposphere.”

I have used Modtran to calculate that the temperature of gases between the tropopause and 70km height adds only about 10 W/m² to the total radiation of over 240 W/m² from surface to space. So, if one part in 24 is negligible, then claim 10.4 is true. How the Psychrometric chart could establish this result, however, is not apparent.

What does a calculation using Modtran have to say about heat transport from Earth’s surface to space? I have examined all the models available in Modtran along with variations of a few other inputs. The example of a mid-latitude summer atmosphere without clouds and a surface temperature of 294K. Although there is no input of a value for solar radiant intensity at the top of the atmosphere (TOA) in Modtran, we might suppose something like 320W/m² in mid-summer. Modtran calculates that the net radiation transport from the surface (net being surface transport upward minus greenhouse effect downward) is 40% at the surface and increasing to 100% at the TOA. If the temperature distribution in the model remains constant, the first law demands that 60% at the surface, and declining to 0% at TOA, must be transferred by a combination of other means – horizontal advection, vertical convection and latent heat. The Psychrometric chart is no help in sorting all this out.

Figures 2 and 3 in L&R are proof of nothing

Figure 2 in the L&R paper, despite being constructed from various weather observations, is nothing more than the integration of the relationship of enthalpy to temperature, dH=Cp dT. It would look exactly the same if one were to simply use a series of equally spaced temperatures and use the Psychrometric chart to calculate enthalpy for dry air at each. The figure below shows this through a comparison of Figure 2 from L&R against a use of Excel to plot H(T) using the third-order expansion of dry air specific heat at constant pressure. This is done independently of the Psychrometric chart. They are the same straight-line plot. Figure 2 in L&R, thus, has nothing to do with CO₂ specifically and is really just a circular activity showing that enthalpy of dry air is a straight-line relationship, which is what is programmed into the Psychrometric chart in the first place.

Figure 1. Caption: Left hand panel is Figure 2 from L&R. The right panel is simply an integration of Cp(T) dT for dry air without regard to anything specific about CO₂.  The zero point of enthalpy is arbitrarily set at -37C.

Likewise, a comparison of Figure 3 from L&R against a plot of dry air enthalpy and moist air enthalpy at saturation shows that the Psychrometric chart simply uses a mixing ratio plus latent heat of water vapor and adds this to the enthalpy of dry air to produce an air state. It is a circular activity once again that proves nothing specific about the process of water vapor warming the atmosphere.

However, Figure 3 from L&R does show something unmentioned by the authors and which I have repeatedly tried to emphasize in various places. I think what it shows is very important.

People often claim that as Earth’s atmosphere warms in the future its burden of water vapor will scale like the Clausius-Clapeyron relationship – it’s a reason for their belief that relative humidity might remain constant. Yet, Figure 3 in L&R shows through sampled observations that enthalpy in Earth’s atmosphere at any temperature spans the range from the dry air line almost to the fully saturated air curve. This is indicative of the distribution of humidity depending on not just equilibrium evaporation at a source of water but also on transport and mixing processes taking that evaporated water from its source to the place of observation. As the atmosphere warms, its water vapor burden might scale in any way from no additional vapor to something like Clausius-Clapeyron. There is no simple rule that describes the complexity of the situation in a useful way.

Figure 2. Caption: Left hand panel is Figure 3 from L&R. Right hand pane is a compilation of the dry air curve using the integration of Cp(T)dT, and the relationship for moist air by addition of the dry air curve with latent heat of water vapor at the saturation mixing ratio. Measurements of the atmosphere at any temperature reveal enthalpy value across the entire range of mixing ratio,

Misunderstanding an equation of state as implying cause and effect

An equation of state is simply a relationship among the state variables, temperature, pressure, volume, and so forth, describing a substance. It does not imply a cause for the specific value the state variables taken in any instance. A common mistake begins with organizing the ideal gas law, as an example, with temperature on the left-hand side: T=PV/nR. This makes temperature seem like a dependent variable and pressure as its cause. This mistake, which is common in engineering education, once again ignores the path or process by which material arrived at its present state. Many processes can arrive at particular states. On the other hand, some states are impossible to reach by a particular process. Yet, the equation of state doesn’t provide the slightest notion of any of this. It is a mistake of seeing capability in the equation of state that it does not have.

A conclusion in the L&R paper that I disagree with most vigorously is built from the same mistake. It is: “10.1. This study confirms quantitatively that the warming effect of CO₂ is too small to measure, i.e., negligible. In contrast, warming by water vapor is 1,000 to 7,000 times greater than CO₂.”

The calculation leading to this result involves division by a factor named the “contribution to the CO₂ (BF) temperature”. This quantity or calculation is undefined, but it cannot involve anything to do with the Psychrometric chart which involves moist air alone independent of any specific property of CO₂. However, this section of L&R hints at a more disastrous misunderstanding. The discussion of dry air having a straight-line enthalpy relationship with temperature, and that for moist air being curved, also refers to the logarithmic relationship of ΔRF to CO₂ concentration in the IPCC reports by comparison. Enthalpy appears confused here with the radiant forcing function of atmospheric CO₂ The Psychrometric chart has nothing to do per se with radiation.

I agree with the authors of L&R that water vapor is the dominant greenhouse warming factor in the Earth’s atmosphere, but it is not 1,000 to 7,000 times more potent than CO₂.

 Conclusion

This brief note has laid out a number of reasons why no useful model of the atmosphere can be based on the Psychrometric chart or any other equation of state alone. The reasons stated here are:

1. It has no capacity to model the processes that lead to a distribution of the state of the air.
2. It is not a substitute for the first and second laws of thermodynamics.
3. It possesses no way to model transport processes.
4. It is not a cause-effect relationship between temperature and other state properties.

Other reasons may come to mind, but these are sufficient to make our case.

Notes:

1- For anyone interested in an effort to use a T-S diagram to show refrigeration/heat transport/available work in a couple of example weather systems, I produced this for the WUWT website last spring at https://wattsupwiththat.com/2024/05/26/of-heat-engines-and-refrigerators/. One reference I supplied in that effort made a T-S diagram of the general circulation of the atmosphere. It is, of course, a composite of uncountably many T-S diagrams of individual smaller weather events.  People can find it through this reference: F. LaLiberte , et al, SCIENCE, 30 Jan 2015, Vol 347, Issue 6221, pp. 540-543 DOI: 10.1126/science.125710

2- The equivalent of the First Law of Thermodynamics, known to geophysicists and atmospheric physicists, is the energy equation – a 3-D vector and time dependent partial differential equation.