The radiation of the Sun can be considered as thermal radiation that comes from a body with a temperature higher than absolute zero. Solar radiation plays a large role as an energy source. Therefore, various energetic and exergetic analyzes of this radiation and the processes of its use are widely carried out. For example, it can be concluded that the practical value of solar radiation, measured by exergy, differs from the energy of this radiation and is always about 6.7% less.

The main part of solar energy is visible white light, which, for example, by means of a prism or by diffraction (Figure 1), can be split into radiation of different colors. Colored light is also obtained through filters that allow only a given color to pass. The unicolor radiation is monochromatic radiation. The spectrum of solar radiation also contains invisible radiation, which of course is also characterized by wavelength, or frequency. Invisible radiation can be also dispersed, but remains invisible. An example of a single-color light is also laser light, which, however, is not covered by these considerations. Radiation of fluorescent phenomena is also neglected. Current considerations apply only to heat radiation, also called thermal radiation, which occurs due to surface temperature. The theory of such thermal radiation was formulated by Planck [1].

The subject of this theoretical study is the exergetic effect of dispersion of thermal radiation and the results of the exergetic analysis of monochromatic radiation arising during dispersion are the original contribution of this work to the knowledge of radiation exergy.

The literature on this topic is not very developed. Badescu (1988) [2] proposed a formula for the component of the exergy spectrum per unit volume. Moreno et al. (2003) [3] considered the reduction and splitting of the quantum states of photons. The exergy of monochromatic radiation was originally mentioned by Candau (2003) [4], then also discussed by Chu and Liu (2009) [5], and a more detailed analysis of monochromatic radiation was presented by Petela [6]. Badescu [7] thought about a solar power station in space or on the surfaces of planets, based on Carnot’s efficiency, and analyzed the efficiency of monochromatic radiation converters. However, the exergetic effect of radiation dispersing, analyzed in this paper, has never been taken into account.

An extended overview of the exergy of thermal radiation can be found in [8]. The energy and entropy of thermal radiation are considered here according to Planck (1914) [1]. Data on the solar radiation spectrum come from Kondratyev (1954) [9].

We begin with radiation, which is considered arbitrary, but non-polarized,

uniform radiation propagating in a solid angle of 2π. Uniform radiation means that the energy and entropy spectra do not depend on direction. The exergy formula for such radiation b, J/m² is as follows [10]:

b = 2 π ∫ ν i b , 0 , ν d ν − 2 π T 0 ∫ ν L b , 0 , ν d ν + σ 3 T 0 4 (1)

where:

T₀ - absolute temperature of environment, K,

ν - vibration frequency, 1/s,

i_b,0,ν - directional normal monochromatic radiation intensity of non-polarized radiation, depending on frequency ν, J/(m² sr),

L_b,0,ν - entropy of this intensity, dependent on frequency ν, J/(m² K sr),

σ - Boltzmann constant for black radiation, σ = 5.6693 × 10⁻⁸ W/(m² K⁴).

To clarify, according to Planck [1], the number 2 before the sums of intensity and entropy in formula (1) takes into account that both components of polarized radiation in two perpendicular planes are equal.

Waves of heat radiation are characterized by wavelength λ, propagation velocity c and frequency ν, (λν= c). In experimental studies, it is more convenient to measure the wavelength. However, in theoretical analyses, it is often convenient to use a frequency that does not change when radiation passes from one medium to another, although it changes the propagation speed. Both integrals in Equation (1) can also be expressed as dependent on the wavelength:

∫ i b , 0 , ν d ν = ∫ i b , 0 , λ d λ (2)

∫ L b , 0 , ν d ν = ∫ L b , 0 , λ d λ (3)

where:

λ - wavelength, m,

i_b,0,λ - directional normal monochromatic radiation intensity of non-polarized radiation, depending on wavelength λ, W/(m³ sr),

L_b,0,λ - entropy of i_0,λ, dependent on wavelength λ, W/(m³ K sr).

The black radiation intensity appearing in the formula (1), is expressed [1] as follows:

i b , 0 , ν = h ν 3 c 2 1 exp ( h ν k T ) − 1 (4)

and the entropy of the black radiation intensity:

L b , 0 , ν = k ν 2 c 2 [ ( 1 + X ) ln ( 1 + X ) − X ln X ]       where     X ≡ c 2 i 0 , ν ν 3 h (5)

where:

c - is the speed of propagation of radiation in vacuum, c = 2.9979 × 10⁸ m/s,

k - Boltzmann constant, k = 1.3805 × 10⁻²³ J/K,

h - Planck’s constant, h = 6.625 × 10⁻³⁴ J s,

T - radiation temperature, K.

The black radiation intensity appearing in formula (1) can also be expressed as a function of the wavelength λ, [1]:

i b   0   λ = c 0 2 h λ 5 1 exp ( c 0 ​ h k λ T ) − 1 (6)

and correspondently the entropy of the black radiation intensity as function of wavelength λ:

L b , 0 , λ = c 0 k λ 4 [ ( 1 + Y ) ln ( 1 + Y ) − Y ln Y ]       where     Y ≡ λ 5 i b , 0 , λ c 0 2 h (7)

Monochromatic radiation occurs only in an infinitesimally small range of wavelengths from λ to λ + dλ. However, one can also consider quasi-monochromatic radiation for a small but finite range from λ to λ + Δλ. In both cases, radiation can be understood either as only the considered component of the spectrum, not separated from this polychromatic radiation, or radiation separated from the polychromatic spectrum, characterized by such a spectrum in which the radiation intensity is for all wavelengths equal to zero, except for the considered wavelength (dλ or Δλ). It can be assumed that by applying an appropriate filter, separated radiation of the required spectrum can be obtained.

If the dependence of the intensity i and entropy of L, neither on the frequency ν nor on the wavelength λ, in the form of equations is unknown, then these two quantities have to be determined on the basis of measured data for intensity i, and in Equation (1) discretization is used:

b = 2 π ∑ i 0 , ν Δ ν − 2 π T 0 ∑ L b , 0 , ν Δ ν + σ T 0 4 3 (8)

or using wavelength instead of frequency:

b = 2 π ∑ i 0 , λ Δ λ − 2 π T 0 ∑ L b , 0 , λ Δ λ + σ T 0 4 3 (9)

For the solar radiation, it is assumed that the intensity iarrives from black surface of the sun, therefore the entropy L of this intensity is calculated also as for a black radiation.

The formula for physical exergy B of a substance contains the environmental temperature T₀, the enthalpy and entropy (H, S) for the substance under consideration and (H₀, S₀) for that substance in equilibrium with the environment:

B = H − H 0 − T 0 ( S − S 0 ) (10)

Equation (10) can be rewritten in the form:

B = H − T 0 S + ( T 0 S 0 − H 0 ) (11)

which reveals the three components of exergy; energetic (H), entropic (-T₀S) connecting to the environment through the involvement of T₀ and purely environmental (T₀S₀ − H₀).

In the same way, the components of radiation exergy can be considered. To obtain the formula for the exergy of black emission, in equation (10) the enthalpy of a substance is replaced by the corresponding σT⁴ emission of black radiation, while the entropy of the substance corresponds to the entropy 4σT³/3 of the emission. The exergy b of black emission is therefore:

b = σ T 4 − σ T 0 4 − T 0 ( 4 3 σ T 3 − 4 3 σ T 0 3 ) (12)

Equation (12) can be converted into a form with three characteristic components revealing the contribution of environment (ambient) A, entropy S and emitted energy E:

b = A + S + E (13)

where:

A = σ T 0 4 3 (14)

S = − 4 3 σ T 0 T 3 (15)

E = σ T 4 (16)

The environmental exergy component A is always positive. In the considered case of non-dispersed black radiation this component depends only on the environmental temperature. The energetic component E represents the intensity of radiation and is always positive. However, the entropic component S is always negative. These components (A' + S' + E' = 100%) are shown in Figure 2 for temperatures T below 600 K and in Figure 3 for temperatures T higher than 600

K. The ambient temperature is T₀ = 300 K. In the first case, Figure 2, (for example, for T = 450 K, A = 37.2%, S = −502.3%, E = 565.1%), all components have counting values so the emission exergy of the black surface should be calculated using the known complete formula [10] resulting e.g. from the rearranged Equation (12):

b = σ 3 ( 3 T 4 − 4 T 0 T 3 + T 0 4 ) (17)

However, in the second case, in Figure 3, it is shown that the environment component can be neglected for radiation temperatures T > ~1000 K, (for example for T = 1500 K, A = 0.07%, S ≈−25%, E ≈ 125%), and the following simplified formula can be used:

b = σ 3 ( 3 T 4 − 4 T 0 T 3 ) (18)

In exergy analyses, for comparative purposes, the exergy-to-energy ratio ψ is used. For example, for blackbody radiation at temperature T, this ratio is, Petela, 1961 [10]:

ψ = 1 + 1 3 ( T 0 T ) 4 − 4 3 T 0 T (19)

For radiation temperatures T > ~1000 K, according to (18), the formula (19) can be simplified:

ψ = 1 − 4 3 T 0 T (20)

The ratio of ψ plays a similar role as Carnot's efficiency for heat engines, since this ratio indicates a certain relative potential of the maximum work available from radiation. It is worth noting that this ratio does not describe how to get maximum work, but only expresses that the search for a way is justified. This ratio was first introduced by R. Petela, [10]. Based on later articles by Landsberg and Press, who discussed the formula (19) without changing it, it was sometimes called in the literature by three names, the Petela-Landsberg-Press ratio, although there is no rule that everyone who confirmed the newly found formula becomes its co-author.

On the basis of the discussed formulas for polychromatic radiation, represented by the full spectrum of radiation, further considerations are carried out on the exergy of monochromatic radiation.

By analogy with Equation (10), you can write the following formula for the exergy of undispersed monochromatic black radiation propagating in a unit of solid angle [6]:

b λ = ( i 0 , λ ) T − ( i 0 , λ ) T 0 − T 0 [ ( L 0 , λ ) T − ( L 0 , λ ) T 0 ] (21)

By analogy with Equation (12), exergy b_λ, expressed by Equation (21), is the sum of the components presented in Equation (13):

A λ = T 0 ( L 0 , λ ) T 0 − ( i 0 , λ ) T 0 (22)

S λ = − T 0 ( L 0 , λ ) T (23)

E λ = ( i 0 , λ ) T (24)

The sum of the percentage values of these components is 100%, and some examples of such values, depending on the wavelength λ, are shown in Figure 4. Three graphs show values for three different radiation temperatures T, (6000, 1000 and 280 K), at the same environment temperature T₀ = 300 K, while the other three graphs refer to three different ambient temperatures T₀, (300, 2000 and 150 K), at the same radiation temperature T = 6000 K. The entropic component S_λ is always negative. At high radiation temperature T, the energetic component E_λ is larger than 100%. The environmental component A_λ becomes relatively relevant when the radiation temperature T is low or the environment temperature T₀ is relatively high. At constant values of T and T₀, the environmental component A_λ increases as the wavelength λ increases.

By analyzing the Equation (1), for polychromatic radiation, and the Equation (21), for monochromatic radiation, it can be deduced that the environmental component of monochromatic radiation A_λ, W/(m³ sr), presented by the formula (22), after integration in the range from λ = 0 to λ = ∞ and at T₀ = 300 K, reaches the polychromatic component value A = σ T 0 4 / 3 = 153.1   W / m 2 .

Based on Equation (21), the ratio of exergy-to-energy ψ_λ, for monochromatic unseparated radiation, is:

ψ λ = b λ ( i 0 , λ ) T (25)

Figure 5 shows the calculated values of ψ_λ as a function of λ for five temperature

values T, at environmental temperature T₀ = 300 K. The formulas (25), (6) and (7) were used in the calculations. For comparison, the value of ψ calculated from the formula (20) is also shown.

For each of all 5 temperatures considered T, Figure 5 shows that the exergy-to-energy ratio ψ_λ of undispersed monochromatic radiation increases with a decrease in wavelength λ and can reach even values greater than ψ calculated for total polychromatic black radiation. Both the ratios ψ and ψ_λ decrease as the radiation temperature decreases and the wavelength λ increases. In other words, for any given temperature, the thermodynamic value of monochromatic radiation, measured by exergy, is the closer to the energy, the smaller the wavelength, and therefore the higher the frequency. The question now arises as to how the exergy of polychromatic radiation will change when it is optically separated into monochromatic beams.

These considerations are conducted on the assumption that dispersed rays do not change the solid angle of their spread, while changing the direction of their propagation at the changed plane angle does not affect its exergy. It was assumed that the dispersion of radiation causes a change in its spectrum without changing temperature, which means that dispersed non-polychromatic beams, with the exception of exergy, have energy and entropy values such as they had when they were part of undispersed polychromatic radiation. It is also assumed that there is no loss of energy when dispersing of radiation, e.g. that the heat released can be neglected.

Radiation dispersing can generate exergy. To theoretically explain this possibility, let's apply Equation (1). Based on the relationship (2) and (3), Equation (1) can also be written using wavelengths instead of vibration frequencies. Suppose that the polychromatic radiation, the exergy of which is determined by the formula (1), has been dispersed into a count of n beams, the spectrum of which consists of the intensity of radiation only in the appropriate intervals: Δ λ 1 , Δ λ 2 , ⋯ , Δ λ n . The radiation exergy of each dispersed beam can also be calculated by the formula (1). The sum b_d of the exergies of all the dispersed beams is therefore:

b d = 2 π ( ∫ 0 Λ 1 i b , 0 , λ d λ − T 0 ∫ 0 Λ 1 L b , 0 , λ d λ ) + 2 π ( ∫ Λ 1 Λ 2 i b , 0 , λ d λ − T 0 ∫ Λ 1 Λ 2 L b , 0 , λ d λ ) + ⋯     + 2 π ( ∫ Λ n − 1 ∞ i b , 0 , λ d λ − T 0 ∫ Λ n − 1 ∞ L b , 0 , λ d λ ) + n σ 3 T 0 4 (26)

where integration limits are:

Λ 1 = Δ λ 1 Λ 2 = Δ λ 1 + Δ λ 2           ⋮ Λ n − 1 = ∑ i = 1 i = n − 1 Δ λ i

The sum of all integrals in Equation (26) is equal to the sum of two integrals in Equation (1). The difference between the exergy of dispersed beams and the non-dispersed radiation is therefore:

b d − b = ( n − 1 ) σ 3 T 0 4 (27)

By introducing the degree of ε = b_d/b as a measure of the increase in radiation exergy by dispersion and using the Equations (13) and (14), the Equation (27) may be changed as follows:

ε = 1 + ( n − 1 ) A A + E + S (28)

Since the ratio A/(A + S + E) is always positive, the formula (28) shows that for n > 1, radiation dispersion always gives an increase in exergy, ε >1. For black radiation, using the formulae (14), (15) and (16) in (28), the following is obtained:

ε = 1 + n − 1 1 + 3 ( T T 0 ) 4 − 4 ( T T 0 ) 3 (29)

The degree of ε depends on the radiation temperature T, the ambient temperature T₀ and the number of n beams obtained by dispersions. The dependence (29) for black radiation at T₀ = 300 K is illustrated by diagram in Figure 6.

For example, as shown in Figure 6, for T = 800 K, if n = 40 then ε = 1.5075. The greater the number of n separated beams from polychromatic radiation, the

greater the generation of exergy can be expected. The formula (29) also shows that at any temperatures T and T₀, when n → ∞, also to infinity aims the exergy increase degree, ε → ∞. The lower the radiation temperature T, the smaller the number of n dispersed beams causes a remarkable increase in exergy by dispersing radiation.

Solar radiation is characterized by high temperature and a significant increase in exergy by dispersion, for example ε = 1.05 at T = 6000 K and T₀ = 300 K, occurs only at the number of dispersed beams n = 22,401. However, in some exceptional surroundings with a temperature of T₀ = 2000 K, this solar radiation will reach ε = 1. 05 already at n = 8. The calculation examples later show in more detail the dispersion effect for high-temperature solar radiation and low-temperature water vapor.

Summarizing, it can be observed that if the exergy of an initially undispersed radiation flux of a given spectrum is calculated, then according to the formula (8) the environmental component A is taken only once. However, when the initial radiation is dispersed, a series of new dispersed beams are formed, each of which has its own spectrum. Therefore, by summing up the exergy of all n dispersed beams, each of which is also calculated using the formula (8), one obtains a total exergy greater by the number (n − 1) of the environment components. Therefore, the dispersion process increases total exergy.

It seems peculiar that the separation of any radiation beam can be carried out optically in a process without any driving input. For comparison, however, it is worth mentioning the analogy to the separation of gaseous components. In addition to the differential pressure required for the transport of gases, the separation of the components is carried out without any input using molecular sieves. For example, the exergy of ambient air is zero, while the sum of the exergies of the separated components from such air is greater than zero.

The effect of increasing the exergy of products in the process of gas separation, or radiation dispersion, occurs as a result of interaction with the environment. In the case of radiation, this effect arises in the fact that environmental radiation has a full black spectrum, which confronts the spectrum of dispersed radiation without radiation intensity in some wavelength ranges. For further interpretation of such increasing exergy, one of the methods of deriving the formula on the exergy of radiation can be used. The method involves calculating the work performed by radiation in a frictionless cylinder-piston system filled with gas. Such a model was used [11] for radiation considered as a photon gas (Figure 7).

The system is located in a vacuum, which is filled only with environment radiation at temperature T₀. The cylinder contains only considered radiation trapped inside, at temperature T. The system is in an adiabatic state because, thanks to its perfect insulation and perfect mirror like walls, there is no heat exchange with surroundings, neither by convection, nor conduction, nor by radiation. The radiation pressure depends only on the temperature, and the higher the temperature, the higher the pressure. Therefore, for different temperatures T ≠ T₀, there is a different pressure on both sides of the piston, which moves the piston to the left (T₀ > T) or to the right (T > T₀), in both cases performing work determining the exergy of trapped radiation.

The dispersed radiation with a spectrum with no radiation intensity in some wavelength ranges could be imagined using the model of two cylinder-piston systems (Figure 8). For example, one cylinder is filled with any quasi monochromatic radiation in a given wavelength range at T > T₀, and another cylinder represents the absence of radiation (T = 0) in all other wavelength ranges. The sum of the work done in both systems determines the exergy of the quasi monochromatic radiation under consideration. The absence of radiation in the dλ wavelength range in the spectrum can be interpreted as a kind of monochromatic

radiation “vacuum”, which allows the environmental radiation pressure to manifest its usefulness in this range. It could therefore be noted that the factor that triggers the work from the environment may be partial pressure in the case of gases and a non-polychromatic spectrum in the case of radiation.

The considerations in this theoretical work constitute a cognitive contribution to the field of radiation exergy. This exergy has been analyzed as consisting of three components that represent energy, entropy, and the environment. For high-temperature radiation, such as solar radiation, the environmental component in the exergy of polychromatic and monochromatic radiation is meaningless. However, the lower the temperatures of such radiations, the greater the impact of the environment. To illustrate this rule, solar radiation has been compared to water vapor radiation.

The spectrum presented by quasi-monochromatic or cumulative radiation was for the first time studied using exergy. It was found that for these two types of radiation, the ratio of exergy-to-energy is smaller, the greater the wavelength. That is, the lower the wavelength (or higher frequency), the more valuable the energy, (measured by exergy) of each of these considered types of radiation.

However, the main finding is that dispersion of polychromatic radiation can increase exergy, and the dispersion process does not require any driving input, which means that it occurs for free. That is, the exergy of polychromatic radiation before dispersion is less than the sum of the exergies of all dispersed beams. An interpretation of this phenomenon has been proposed and an analogy has been cited to the increase in exergy during gas separation. The lower the radiation temperature, the greater the effect of the increase in exergy.

The dispersing of visible sunlight generates color beams, all of which have a monochromatic exergy-to-energy ratio slightly greater than such a ratio for undispersed polychromatic solar radiation.

The presented analyses may be a motivation for further research of optical phenomena from a thermodynamic point of view. Exergy effects or reversibility of optical processes could be analyzed. A more detailed analysis of exergy for the process of radiation dispersion could be carried out. Exergy analysis of processes in which visible sunlight is dispersed or involved are ultraviolet, X-ray or gamma rays could also be considered. These analyses can help to make better use of monochromatic radiation, particularly dispersed solar radiation, on Earth or elsewhere.

The author declares no conflicts of interest regarding the publication of this paper.

Petela, R. (2023) Application of Exergy for Research on Increasing the Usefulness of Solar Radiation by Dispersing It into Monochromatic Beams. Energy and Power Engineering, 15, 73-103. https://doi.org/10.4236/epe.2023.151004

Aenvironment contribution component to exergy

b radiation exergy, W/m²

B exergy consisting of three components

c speed of propagation of radiation in vacuum c = 2.9979 × 10⁸ m/s

e radiation energy, W/m²

E energy contribution component to exergy

h Planck’s constant, h = 6.625 × 10⁻³⁴ J s

H enthalpy of substance, J

i directional radiation density, J/(m² sr) or W/(m³ sr)

i successive number

j successive number

k Boltzmann constant, k = 1.3805 × 10⁻²³ J/K

L normal entropy of radiation intensity, J/(m² K sr) or W/(m³ K sr)

n number of beams

n successive number

S entropy of substance, J/K

S entropy contribution component to exergy

T absolute temperature, K

ε degree of exergy increase

Λ integration limit, m

λ wavelength, m

ν vibration frequency, 1/s

ψexergy-to-energy ratio

σ Boltzmann constant for black radiation, σ = 5.6693 × 10⁻⁸ W/(m² K⁴)

bblack

viol violet beam

λwavelength

Δλ interval of wavelength

ν frequency

0 environment

0 directional normal

Columns: 1) Successive number; 2) Wavelength, λ (10⁹), m; 3) Monochromatic radiation intensity, i_0,λ(10⁻¹⁰), W/(m³ sr); 4) Assumed wavelength range, Δλ, (10⁹), m; 5) Product i_0,ν·Δν (10⁻¹), W/(m³ sr); 6) Product L_0,ν·Δν, W/(m² K sr); 7) Exergy of monochromatic radiation, b_ν, W/m², (formula 16) 8) Ratio of exergy b_ν to energy of monochromatic radiation, ψ_ν, (formula 19) 9) Additively cumulated product, i_0,ν·Δν (10⁻¹), W/(m² sr); 10) Additively cumulated product, L_0,ν·Δν, W/(m² K sr); 11) Additively cumulated exergy of monochromatic radiation intensity, b_Δν, W/m², (formula 36) 12) Ratio of exergy-to-energy of radiation, ψ_Δν, (formula 39). (Table presents the rounded values, although the calculations were carried out without rounding).