Energy of matter depends only on the thermodynamic parameters of that matter, for any arbitrarily chosen reference state adopted for calculations. However, the beauty of the concept of the exergy of that energy, although it also depends on these thermodynamic parameters, lies in the fact that the reference state for exergy calculations is not arbitrary and is determined by the thermodynamic parameters of the environment in which the matter is considered [1]. For humans, this environment naturally defines the reference state primarily through its chemical composition and temperature. Following the environment temperature as the defining reference, the exergetic line of inquiry in the search for energy sources leads to considerations of the possibility of exploiting natural fluctuations of that temperature.

Historically, the first monograph on exergy [2] presents the exergy balance equation for a system subjected to a fluctuating environmental temperature. This equation closes only after introducing an additional term, ΔB_en, corresponding to the variability of the environment temperature, and the value of this additional term can be determined directly from the balance equation:

Equation (a) is considered over the time interval from t = 0 to t = t . In this equation, B in is the exergy of the streams supplied to the system, B out is the exergy of the streams removed, Δ B ir is the instantaneous exergy destruction due to irreversibility, and ( B t − B 0 ) denotes the difference in the system’s exergy between the initial state (at t = 0 ) and the final state (at t = t ). Equation (a) shows that a process may be organized in time such that a change in environment temperature leads to the creation of exergy ( Δ B en > 0 ).

A simple example of such a process is the following: at t = 0 , a portion of air is taken from the instantaneous environment and placed into a perfectly insulated vessel, where it remains until time t , when the environment temperature has changed. During this interval, no streams enter or leave the vessel ( B in = 0 , B out = 0 ), and no irreversible processes occur ( Δ B ir = 0 ). From equation (a) it follows that, solely due to the change in environment temperature, exergy has been created (or destroyed) in the amount

This reasoning therefore draws attention to the possibility of generating energy by exploiting fluctuations of environment temperature. The effect of environmenttemperature change on the value of ΔB_en in the example of a sphere heated by radiation is discussed in more detail in §6.9 of [3].

In general, exergy consists of several components; however, since the present discussion concerns temperature, only one component is considered here: physical exergy, that is, the part that results mainly from the temperature level, while pressure plays only a secondary role. In any case, no other exergy components are examined here, and in particular the chemical component—arising from differences between the chemical composition of the working substance and that of the environment—is omitted. The temperature of the environment T₀ at a given location is considered, varying in time, with its maximum value denoted as T_H and its minimum as T_L. The use of slow changes in environmental temperature is very old and much older than any attempts to generate electrical energy [4]. As early as around 500 BCE, the Persians used daily temperature fluctuations and thermal radiation to the sky to produce ice in a desert climate. At night, water froze due to radiative heat loss, despite positive daytime temperatures. In India, shallow water basins were used, which cooled during the night and produced ice. Natural night cooling was employed in Roman frigidaria. In the Middle Ages, European ice houses stored winter ice, which, thanks to slow temperature variations and insulation, could survive until summer. In Iran and Arabia, wind towers were used to direct cool night air into interior spaces. Psychrometric desert cooling was also known, relying on differences in temperature and humidity between day and night. These were all uses of daily fluctuations to regulate the microclimate.

The nineteenth century brought the first heat engines operating on small temperature differences (the Stirling engine). This was the first step toward “energy harvesting” from slow temperature variations [5][6]. In the twentieth century, the first practical devices appeared that were powered solely by slow daily and seasonal changes in environmental temperature. The concept of phasechange materials emerged, intended for storing heat and releasing it slowly. Materials capable of generating electricity from temperature differences (thermoelectrics) also began to be used, although the daily ΔT is usually too small to produce significant power. “Thermal clocks” were developed as devices driven by slow temperature changes; for example, the Atmos clock by JaegerLeCoultre (1928), which operated thanks to the thermal expansion of a gas.

The twentyfirst century is marked by modern concepts of “thermal harvesting”, that is, attempts to extract energy from very slow temperature variations. Examples include thermal resonators developed at the Massachusetts Institute of Technology (MIT), materials with high heat capacity and thermal conductivity that “resonate” with the daily temperature cycle and generate small amounts of electrical power, or Stirling engines driven by Earth’s thermal radiation, which use terrestrial infrared emission as a source of mechanical energy (American Association for the Advancement of Science, AAAS).

For clarification, one of the simplest and most durable methods of converting mechanical energy into electrical energy on the milliwatt scale is the use of a piezoelectric element [7]. This is a material or a small device that produces an electric voltage when compressed, bent, or deformed. It also works in reverse; when a voltage is applied, it deforms itself. Another method involves a small container in which, during the day, the gas warms up, pressure increases, and a membrane pushes a piston outward. The piston’s motion tensions a spring or deforms a piezoelectric element. At night, the gas cools, pressure drops, and the piston returns. The return stroke generates a pulse of electrical energy.

The natural phenomenon of a varying environmental temperature has also been considered many times as a potential energy source, but it has not yet become a practical energy technology. Research and conceptual proposals exist, yet their applicability is constrained by the fundamental laws of thermodynamics. To produce energy, at least two heat sources are required. Typically, one heat source is, for example, the hot exhaust gases from fuel combustion, while the second heat source is the environment. However, in some cases, two heat sources can be identified when the environmental temperature changes. Such heat sources do not occur simultaneously, and therefore, to generate energy, one would need to accumulate the thermal effect of the environment at one moment in order to use the stored heat at another moment, when the environmental temperature has changed due to natural causes. An example of such a method of obtaining useful energetic effects is the longknown practice of collecting ice blocks in winter, storing them in insulated spaces, and using this ice for cooling purposes later in the summer.

The phenomenon of daily and seasonal environmental temperature fluctuations has long appeared in the literature as a potential energy source, yet it has not led to the development of a mature energy technology. Classical thermodynamic analyses emphasize that the available exergy at small temperature amplitudes is inherently very limited, which follows directly from the dependence of ideal efficiency on the temperature difference in the Carnot cycle [5][8][9]. Under typical terrestrial conditions, where daily temperature variations are on the order of 5 - 20 K, the maximum theoretical efficiency is negligible, and the actual efficiency, after accounting for thermal and mechanical losses, usually falls below 1% [8][10]. Consequently, most of the concepts considered remain at the level of laboratory demonstrators or niche applications with very low power output.

The most frequently analysed class of devices consists of heat engines operating at very small temperature differences, in particular lowtemperaturedifferential (LTD) variants of the Stirling engine. The literature describes both theoretical models and prototypes capable of operating at ΔT values of only a few tens of kelvins, powered for example by geothermal, solar or waste heat sources. Analyses by Senft and other authors, however, indicate strict geometric and thermodynamic constraints that prevent achieving high specific power at such small temperature differences [11][12]. In practice, LTD engines reach power levels on the order of watts or tens of watts, which is sufficient to drive small generators but does not constitute competition for conventional energy sources [11]-[13].

A second widely studied group of solutions consists of systems employing phasechange materials (PCM). These materials allow heat to be stored within a narrow temperature range, which makes them attractive for solarenergy buffering or for buildingtemperature stabilization [14][15]. However, PCM do not themselves generate mechanical or electrical energy; they serve only as a storage medium that can operate in conjunction with another energy converter, such as a heat engine or a thermoelectric module [14]. In the context of exploiting environmental temperature fluctuations, this means that PCM can smooth the temperature profile and extend the duration over which a useful temperature difference ΔT is available, but they do not resolve the fundamental limitation of low exergy in the source.

A third class consists of thermoelectric devices based on the Seebeck effect. Thermoelectric modules are capable of operating at small temperature differences, which makes them natural candidates for converting energy from daily temperature fluctuations [6][16][17]. Their efficiency, however, is limited both by the fundamental laws of thermodynamics and by the finite value of the material quality factor (ZT), which for most commercial semiconductors remains moderate [16]. Numerous environmental “energy harvesting” demonstrators have been described in the literature—installed on heating pipes, building surfaces or infrastructure elements—but the power obtained is typically in the milliwatt to singlewatt range per module, restricting applications to powering sensors and lowpower electronics [6][17].

It is also worth mentioning the concepts of socalled “thermal clocks” and microdevices driven by cyclic expansion and contraction of materials. In micro-mechanics and micro-elektro mechanical systems (MEMS) research, systems have been proposed in which environmental temperature changes induce periodic deformations of bimetals, polymers or elastic structures, generating small amounts of mechanical work [18][19]. The power of such devices is inherently very low (in the milliwatt range or below), and their applications are limited to niche solutions such as autonomous triggers, indicators or clocks with extremely low energy consumption [18]. From the perspective of systemscale energy engineering, they do not constitute a meaningful power source but rather an engineering curiosity.

Analyses of the exergy available from environmental temperature fluctuations consistently show that the main barrier is the small temperature amplitude and the difficulty of maintaining a temperature difference for the time required to perform work [1]-[3]. For a typical daily ΔT ≈ 10 K at an environmental temperature of about 300 K, the ideal Carnot efficiency is only around 3.3%, and after accounting for thermal irreversibility and conduction losses, the actual conversion efficiency falls below 1% [8][10]. Moreover, achieving noticeable power output would require processing very large heat fluxes, which entails substantial heatexchanger size, integration challenges and high capital costs. As a result, the energyeconomic balance of such systems is unfavourable compared with classical renewable technologies such as photovoltaics or wind power [9][10].

An interesting direction, appearing with increasing frequency in the literature, concerns planetary environments with extreme temperature amplitudes. On the Moon, Mercury or other bodies lacking an atmosphere, daily surface-temperature variations can reach several hundred kelvins, which radically increases the exergy potential available from cyclic temperature changes [20]. Proposed concepts include cyclic heat engines coupled with massive thermal stores that accumulate energy during the day and release it at night, generating mechanical work or electrical power [21]. It was demonstrated that radiative cooling—including nocturnal cooling—can significantly increase the effective temperature difference in solar-thermal systems [22]. Although these concepts remain at the stage of feasibility studies and numerical analyses, they demonstrate that in extraterrestrial conditions the limitations known from Earth may be partially alleviated.

In summary, the literature review indicates that the energetic use of environmental temperature fluctuations has been analysed many times at various scales—from MEMS microdevices, through thermoelectric modules and LTD Stirling engines, to concepts of thermal stores and hybrid systems. The common conclusion of all these studies is that on Earth the available exergy is too small to justify the construction of large, standalone energy installations based solely on daily temperature variations [5][6][8]-[10][16][17]. These technologies find niche applications in the field of energy harvesting for lowpower electronics and in thermalstorage systems, while their role in the overall energy balance remains marginal. In planetary environments with extreme temperature amplitudes, however, there emerges a realistic opportunity to use this phenomenon as one component of powersupply systems, which represents an interesting direction for further theoretical and design research [20]-[22].

The novelty of the present work lies in the discussion of the possibilities of producing energy from environmental temperature fluctuations on the basis of the Carnot efficiency and the socalled exergy efficiency. Both efficiencies are analysed as functions of the maximum difference ΔTin the varying environmental temperature, as well as of the temperature level (T_L), at which this difference occurs. These efficiencies are presented in diagrams, which allows for a convenient discussion of their values over a wide range of conditions, encompassing both those on Earth and in the Solar System. The use of substances (understood as gases, liquids, or solids) and of radiation (“photon gas”) as working media in devices exploiting environmental temperature fluctuations is evaluated. Radiation as a working medium is particularly important in processes occurring in vacuum. For comparison, natural fuels are briefly discussed as examples of natural environmental resources for which the exergy efficiency exceeds 100%.

A temperature difference is required to produce energy, that is, two distinct heat sources at different temperatures. A heat source is understood as a medium capable of supplying or absorbing infinitely large amounts of heat without undergoing a change in its own temperature. Let us look more closely at the temperature of the human environment. We observe its fluctuations over time: hourly, daily, and yearly. These fluctuations occur in our vicinity, in a country, in a region of the world, on the planet, and beyond the planet Earth. They may be small or large, but they all share one essential feature: the temperature difference that could potentially be used for energy production does not occur at the same time. For example, on Earth the daily variation of the environment temperature may reach 20 degrees, while the annual variation may reach even 50 degrees. On the Moon, the daily temperature change is approximately from −173˚C to 127˚C, that is, about 300 degrees [23], with the lunar day lasting 29.53 Earth days. This leads to the conclusion that one should nevertheless consider the possibility of using the environment at its minimum and maximum temperatures as the two heat sources required for energy generation. For an energyproducing process of significant power, on the order of megawatts, two possibilities emerge.

One possibility is to select, for heat storage (first period), a substance that would, during the second period, exchange the stored heat at one extreme environment temperature with a second heat source also at a constant extreme environment temperature. Theoretically, a variation of this case could involve a device producing energy that is extended in space in such a way that it would simultaneously be in contact with locations having extreme environment temperatures.

The second possibility would rely on the appropriate use of insulation to preserve a substantial portion of the environmental effect captured during one extreme temperature period, so that it could be used during the opposite extreme temperature period. The weakness of this process is that the substance, once isolated, becomes a heat source whose temperature changes as the process proceeds.

The economic viability of energy generation in both cases increases with the amplitude of the environment temperature fluctuations, that is, the peaktopeak value (T_H − T_L), and also depends on the values of the extreme temperatures themselves—for example, on the lower extreme temperature T_L; the lower it is, the higher the efficiency. Naturally, when determining economic feasibility—an issue not considered in this work—many additional factors would have to be considered, such as the time scale of the environment temperature variations.

For energy generation, the ideal situation would be that locally extreme differences in environmental temperature occurred within a very short time and preferably were available simultaneously. Since this is impossible, only the two possibilities described in Section 2 are considered.

In the first case, one would need to ensure constant values of the extreme temperatures for the energygeneration process, for example for the cycle of a heat engine. This could be achieved as follows:

a) Ideally, one would thermally insulate a working substance after it has been heated by the environment at the high extreme temperature T_H. This substance could, for instance, be saturated vapor at temperature T_H, which would then release its condensation heat isothermally to the engine cycle at the same temperature T_H. The second heat source required for the process could be another suitably insulated working substance that evaporates at low extreme temperature T_L, removing heat isothermally from the cycle at constant temperature T_L, and then undergoes isothermal condensation as it is cooled by the environment at temperature T_L. These would be evaporation/condensation processes.

b) An analogous process could be implemented using a substance undergoing melting/freezing, thereby creating the two required heat sources at constant temperature for the operation of a heat engine.

c) One could also consider a sublimation/resublimation process.

d) Another possibility would be a substance undergoing chemical reactions at constant temperature; this would involve exothermic/endothermic processes.

In the above cases, the magnitude of the latent heat of phase change (a, b, c) or the reaction heat (d) plays a major role. The larger this value, the smaller the required amount of working substance. For phase changes, the heat of fusion of metals is often moderate (100 - 300 kJ/kg), the heat of sublimation of metals is enormous (several to several tens of MJ/kg), the heat of vaporization of water is only about 2257 kJ/kg, the heat of vaporization of pentane is about 360 kJ/kg, and the heat of vaporization of ammonia is about 1370 kJ/kg [24]. For typical thermalstorage materials, the reaction heat—depending on the specific system, for example for salt hydrates (dehydration/hydration reactions)—is on the order of 200 - 800 kJ/kg [25]. The heat of sorption reactions (e.g., zeolites, silica gel + water) is on the order of 500 - 2000 kJ/kg [26]. Some hightemperature chemical reactions (e.g., oxides, carbonates) have reaction heats of about 1000 - 3000 kJ/kg [27].

Thus, the heat of vaporization of water is relatively large, but chemical reactions in thermalstorage substances can yield similar or even greater heat values per kilogram. Sublimation of metals (especially highmelting ones) may have a transformation heat larger than typical thermalstorage reactions but is completely impractical technically.

All the above cases are characterized by isothermal heatexchange processes, which affects the efficiency of using the temperature difference T_H-T_L. Cases a) through d) should therefore be analyzed on the basis of the Carnot efficiency.

A less favorable, though perhaps easier to implement, possibility is:

e) simply storing a substance at the extreme temperature T_H or T_L and using it in the next period, either during cooling to T_L in a forward heatengine cycle, or during heating to T_H in a reverse cycle.

In case e), the temperature of the substance is not constant and changes during heat exchange. Instead of the Carnot efficiency, it is more appropriate to use another efficiency measure that is less strict and more realistic. It is therefore proposed to use, in this case, the exergy efficiency discussed later.

The thermodynamic assessment of the feasibility of using environmental temperature fluctuations is carried out by comparing the process with the corresponding theoretical maximum efficiency, which cannot be exceeded in practice. The following sections therefore discuss the values of Carnot efficiency or exergy efficiency for the considered cases. The values of both efficiencies depend not only on the peaktopeak difference (T_H − T_L), but also on one of the temperatures, T_L or T_H, which determines the temperature level of that difference.

Three efficiencies have been discussed. The first one, when the working fluid is a substance and both heat sources have constant temperature, can be considered as the Carnot efficiency defined by Equation (2):

The second case, when the working fluid is a substance and only one heat source has a constant temperature, allows one to consider the exergy efficiency defined by Equation (7):

The third case, when the working fluid is radiation, which serves as a heat source with a variable temperature while the second source has a constant temperature, allows one to consider the exergy efficiency η_E,r defined by Equation (16) in the form:

The characteristic variability of these three efficiencies is illustrated by the plot in Figure 5.

Figure 5. Comparison of the efficiencies η_C, η_E and η_E,r.

For relatively large values of ΔT, one can observe that η_C > η_E,r > η_E, whereas for small values of ΔT the relation becomes η_C > η_E > η_E,r. In Figure 6, example points are shown at which η_E = η_E,r.

Figure 6. Two example points, at which η_E = η_E,r.

In general, depending on the problem considered, the exergy of a substance may consist of physical, chemical, kinetic, potential, and also other components in special cases. Up to this point, in discussing the exergy efficiency of a substance, only the physical exergy has been considered, because the focus was on the temperature of the environment. However, for completeness and comparison, another component of the substance’s exergy is considered here: only the chemical exergy, assuming that the substance has the constant temperature (and pressure) of the environment. Chemical exergy arises from the difference between the chemical composition of the substance under consideration and the common constituents of the environment [2].

As an example, natural fuels from the geological sequence wood-peat-lignite-coal-anthracite are examined. It turns out that while the efficiency of physical exergy cannot exceed 100%, the efficiency of chemical exergy may be greater than 100%, which is the case, for example, for natural fuels. Such fuels are regarded as a natural asset.

The chemical exergy efficiency of a fuel η_E,ch is understood as the ratio of the chemical exergy b_ch of the fuel to the input to the process of complete combustion of that fuel, in which the full energetic value of the fuel is realized, that is, the net calorific value V_N of the fuel:

Based on the statistical method [34], the values of η_E,ch for fuels were determined. A formula was derived which, after simplification by omitting—because of their relatively small significance—the mineral matter, moisture, sulfur, and nitrogen in the fuel, takes the form:

and the net calorific value was determined using the formula [35]:

where H, C, O—gram fraction of hydrogen, carbon and oxygen, respectively, in the fuel.

Based on formulas (18) and (19), Figure 7 presents a plot of the approximate dependence of the efficiency η_E,ch on the volatilematter content VM for natural solid fuels [36]. The VM content represents the geological age of the fuel: the higher the content, the geologically younger the fuel. It follows that the carbonization of natural fuels over time causes a degradation of the value defined by exergy, which shows wood as the most valuable substance in the sequence of natural fuels. The younger the fuel, the more chemically complex the substance is, and the more valuable it becomes as a feedstock for chemical processing.

Figure 7. Exergy efficiency of chemical exergy η_E,ch of natural solid fuels as a function of the volatilematter content VM.

An exergy efficiency greater than 100% may indicate that the utilization of natural fuels through the energetic process of combustion is unjustified, because such a process can at most yield the calorific value at 100% efficiency. This therefore suggests using natural fuels by means of some process other than combustion—namely, a process or processes that would theoretically allow the full exergy efficiency to be achieved. One such solution is the appropriate chemical processing of fuels (for example, by devolatilization), as a result of which substances with already lower exergy efficiencies are obtained. Only the utilization of such products of chemical processing, with efficiencies below 100%, could then be considered as fuels for combustion.

A strong conclusion therefore follows it is justified to burn only the products of chemical processing of natural fuels whose exergy efficiency is less than 100%.

In every newly arising situation in which two heat sources are available, the possibility of obtaining energy is examined primarily on the basis of the theoretical efficiency appropriate to that situation. If both sources have constant but different temperatures, the Carnot efficiency is used. If one of the sources changes its temperature as the process proceeds, then exergy efficiency may be applied. The use of both efficiencies has been discussed in the present work.

If the value of the theoretical efficiency is considered sufficiently high, and thus one may expect that the lower real efficiency will still remain acceptable, then one can proceed to consider and select the process and the device by which energy will be produced. It should be emphasized that even the best device, with the highest possible efficiency and quality, cannot exceed the theoretical efficiency applied.

It is assumed that the aim of the present work is to analyse the situation for a specified value of the environmental temperature. The climatological data for this temperature were taken from the sources cited in the present work. However, it may occur that the value of the environmental temperature is not unambiguous, since the very notion of the environment is not straightforward to define. The method of determining this environmental temperature in such particular specific cases—including considerations involving substances or radiative media—constitutes a broad issue that warrants a separate study, and it may be necessary to introduce a suitable concept of an effective environmental temperature.

The short description of the process that exploits environmental temperature differences, together with its evaluation using the efficiencies applied in this work, is provided in the paper. For the detailed theoretical assumptions, the reader is referred to the literature—A. Bejan for the Carnot efficiency and R. Petela (Florence) for the exergy efficiency.

Using theoretical efficiencies, the possibilities of exploiting environmental temperature fluctuations on Earth and in the Solar System have been evaluated. Conditions on the Moon allow a Carnot efficiency of 75%, and on Mercury even 86.8%. On Earth, daily environmental temperature fluctuations allow only 12.1%, although annual environmental temperature fluctuations on Earth may be used at a Carnot efficiency of 33.8%.

An interesting observation is that when radiation is used as the working medium, the exergy efficiency—compared with the exergy efficiency in the case of using a substance—is higher for large temperature differences ΔT and lower for small ΔT. For example, for Mercury, η_E = 69.2% for a substance and η_E = 82.4% for radiation; whereas, conversely, the corresponding efficiencies for Lhasa are 5.2% and 1.9%. However, obtaining significant power would require handling large volumes of radiation, considering that, for example, 1 m³ of radiation at a temperature of 100 K possesses a relatively small exergy of 0.0129 mJ/m³.

For comparative purposes, it has been shown that exergy efficiency can serve as a tool for identifying natural resources in the environment, taking values greater than 100%, as is the case for natural solid fuels in the sequence wood-peat-lignite-coal-anthracite-graphite.

The issue of the scale of energy production under a given pattern of environmental temperature fluctuations is not included within the scope of the present work. This scale may depend on the quality of the device used or, for example, on the local level of energy demand, especially under unusual or exceptional circumstances. In general, however, the scale of energy generation is in the milliwatt range when efficiencies are low, and it increases to multiwatt levels as the achievable efficiency rises.

The considerations in this work concern locations on Earth and beyond. The work may be of particular relevance for Earth at present, because it discusses one of the possible energy sources that are highly sought after as nonpolluting to the environment. The work carries an element of inspiration.

*Professor Emeritus.