**2.1 Using model for converting radiant energy into work**

We use the model of solar energy conversion (De Vos, 1985) shown in Fig. 1. The absorber of thermal radiation is blackbody *1* with temperature *T*A. The blackbody is situated in the center of spherical cavity *2* with mirror walls and lens *3* used to achieve the highest radiation concentration on the black surface by optical methods. Heat absorber *4* with temperature *T*0<*T*A is in contact with the blackbody.

The filling of cavity *2* with solar radiation is controlled by moving mirror *5*. If the mirror is in the position shown in Fig. 1, the cavity contains two radiations with temperatures *T*A and *T*S. If the mirror prevents access by solar radiation, the cavity contains radiation from blackbody *1* only. Radiations in excess of these two are not considered. In this model, solar energy conversion occurs at *T*0= 300 and *T*S = 5800 K. The temperature of the blackbody is *T*A = 320 K.

Photons as Working Body of Solar Engines 359

Fig. 2. Entropy diagram showing isochoric cooling of radiation (line st) in the cavity *2*. The amount of evolved radiant heat is proportional to the area ststss. The amount of radiant heat converted into work is proportional to the area abcd. The work is performed by matter in a

The absorbed radiant heat is converted into work by Carnot cycles involving matter as a working body. One such cycle is the rectangle abcd in Fig. 2. Work is performed during this

It is common to say that the matter is the working body in this cycle. But radiation can be involved in the isothermic process ad in Fig. 2, because the efficiency of a Carnot cycle does not depend on kind and state of the working body. We will not discuss the properties of a matter-radiant working body. Let us simply note that a matter-radiant working body is possible. In this case upper limit of temperature *T*A can reach 5800 K. The curve AB on Fig. 3 is the efficiency of this Carnot cycle where the matter cools down and heats up between temperatures *T*A, *T*0 and radiation has temperature *T*A. In this cyclic process the matter and

Let us show the absorption of radiation on an entropy diagram (Fig. 4) as an isothermal transfer of radiation from the volume *V*2 of the cavity (state s) to the volume *V*1 of the black body (state p). One can even reduce the radiation to state p\* in Fig. 4. We will not discuss the properties of points p and p\* here. Let us simply note that radiation reaches heat equilibrium with the black body (state e) from these points either through the adiabatic

Let us represent the emission of radiation as its transfer from the volume of the black body (state e) to the volume V2 of the cavity along the isotherm *Т*A (state t). As the radiation fills the cavity, it performs a work equal to the difference between the evolved and absorbed

*η*0 = 1 - *T*0/*T*A = 0.0625 (2)

heat engine during Carnot cycle abcd.

cycle with an efficiency of

radiation are in equilibrium.

**2.2.2 Work production during the Carnot cycles** 

between the limit temperatures *T*0=300 К and *T*A=320 К.

process p\*e or through the isochoric process pe.

### **2.2 Energy exchange between radiation and matter 2.2.1 Energy conversion without work production**

It is known that the solar radiation in cavity *2* with volume *V* has energy *U*S=σ*VT*<sup>S</sup> 4 and entropy *S*s=4σ*VT*<sup>S</sup> 3 /3, where σ is the Stefan-Boltzmann constant (Bazarov, 1964). The black body absorbs the radiation and emits radiation with energy *U*А=σ*VT*<sup>А</sup> 4 in cavity *2*. If *T*А=320 К, these energies stand in a ratio of *U*S/*U*А=(*T*S/*Т*A) <sup>4</sup> ≈ 10<sup>6</sup> , while *S*S/*S*А=(*T*S/*Т*A) <sup>3</sup> ≈ 6х10<sup>3</sup> . As the volumes of radiations are equal, the amount of evolved heat Δ*Q* is proportional to the difference *T*<sup>А</sup> 4 -*T*<sup>S</sup> 4 and is equal to the area under the isochore st on the entropy diagram drawn on the plane formed by the temperature (*T*) and entropy (*S*) axes in Fig. 2. The solar energy *U*S entering the cavity and heat Δ*Q* are in ratio:

$$
\Delta \eta\_{\rm U} = \Delta Q / \,\,\mathrm{L} \mathbf{I}\_{\rm S} = (\mathrm{L} \mathbf{I}\_{\rm S} \cdot \mathrm{L} \mathbf{I}\_{\rm A}) / \,\mathrm{L} \mathbf{I}\_{\rm S} = \mathbf{1} \cdot (\mathrm{T}\_{\rm A} / \,\mathrm{T}\_{\rm S}) \,\mathrm{4}.\tag{1}
$$

Our model considers the value *η*U as an efficiency of the photon reemission for a black body if the radiation and matter do not perform work in this process.

One should note that the efficiency of the photon absorption can be defined as (Wuerfel, 2005)

$$\eta\_{\rm abs} = 1 \text{-} (\mathcal{Q}\_{\rm emit} \text{ } / \text{ } \mathcal{Q}\_{\rm abs} \text{)} (T\_{\rm A} / T\_{\rm S})^4 \text{ } $$

where *Ω* is a solid angle for the incident or emitted radiation. In our case, the ratio *Ω*emit/*Ω*abs can be ignored because value of *η*U is close to one, for (*T*A/*T*S) 4 =(320/5800)<sup>4</sup> ≈10-5. Thereafter we have to assume that *η*U = *η*abs. The consequence is that solar energy can be almost completely transmitted to the absorber as heat if no work is done. Then a part of evolved heat Δ*Q* can be tranformed into work.

Fig. 1. Model of solar energy conversion from (Landsberg, 1978). Designations: 1. black body, *2*. spherical cavity, *3*. lens, *4*. heat receiver, *5*. movable mirror added by the author.

the volumes of radiations are equal, the amount of evolved heat Δ*Q* is proportional to the

drawn on the plane formed by the temperature (*T*) and entropy (*S*) axes in Fig. 2. The solar

Our model considers the value *η*U as an efficiency of the photon reemission for a black body

One should note that the efficiency of the photon absorption can be defined as (Wuerfel,

*η*abs = 1-(*Ω*emit / *Ω*abs)(*Т*A/*T*S)4, where *Ω* is a solid angle for the incident or emitted radiation. In our case, the ratio *Ω*emit/*Ω*abs

we have to assume that *η*U = *η*abs. The consequence is that solar energy can be almost completely transmitted to the absorber as heat if no work is done. Then a part of evolved

Fig. 1. Model of solar energy conversion from (Landsberg, 1978). Designations: 1. black body, *2*. spherical cavity, *3*. lens, *4*. heat receiver, *5*. movable mirror added by the author.

/3, where σ is the Stefan-Boltzmann constant (Bazarov, 1964). The black

<sup>4</sup> ≈ 10<sup>6</sup>

and is equal to the area under the isochore st on the entropy diagram

*η*U = Δ*Q*/ *U*S = (*U*S–*U*А)/*U*S= 1-(*Т*A/*T*S)4. (1)

4

=(320/5800)<sup>4</sup>

4

, while *S*S/*S*А=(*T*S/*Т*A)

4 and

. As

in cavity *2*. If *T*А=320

<sup>3</sup> ≈ 6х10<sup>3</sup>

≈10-5. Thereafter

It is known that the solar radiation in cavity *2* with volume *V* has energy *U*S=σ*VT*<sup>S</sup>

body absorbs the radiation and emits radiation with energy *U*А=σ*VT*<sup>А</sup>

**2.2 Energy exchange between radiation and matter 2.2.1 Energy conversion without work production** 

К, these energies stand in a ratio of *U*S/*U*А=(*T*S/*Т*A)

energy *U*S entering the cavity and heat Δ*Q* are in ratio:

if the radiation and matter do not perform work in this process.

can be ignored because value of *η*U is close to one, for (*T*A/*T*S)

entropy *S*s=4σ*VT*<sup>S</sup>

difference *T*<sup>А</sup>

2005)

3

heat Δ*Q* can be tranformed into work.

4 -*T*<sup>S</sup> 4

Fig. 2. Entropy diagram showing isochoric cooling of radiation (line st) in the cavity *2*. The amount of evolved radiant heat is proportional to the area ststss. The amount of radiant heat converted into work is proportional to the area abcd. The work is performed by matter in a heat engine during Carnot cycle abcd.

### **2.2.2 Work production during the Carnot cycles**

The absorbed radiant heat is converted into work by Carnot cycles involving matter as a working body. One such cycle is the rectangle abcd in Fig. 2. Work is performed during this cycle with an efficiency of

$$
\eta\_0 = 1 - T\_0/T\_A = 0.0625 \tag{2}
$$

between the limit temperatures *T*0=300 К and *T*A=320 К.

It is common to say that the matter is the working body in this cycle. But radiation can be involved in the isothermic process ad in Fig. 2, because the efficiency of a Carnot cycle does not depend on kind and state of the working body. We will not discuss the properties of a matter-radiant working body. Let us simply note that a matter-radiant working body is possible. In this case upper limit of temperature *T*A can reach 5800 K. The curve AB on Fig. 3 is the efficiency of this Carnot cycle where the matter cools down and heats up between temperatures *T*A, *T*0 and radiation has temperature *T*A. In this cyclic process the matter and radiation are in equilibrium.

Let us show the absorption of radiation on an entropy diagram (Fig. 4) as an isothermal transfer of radiation from the volume *V*2 of the cavity (state s) to the volume *V*1 of the black body (state p). One can even reduce the radiation to state p\* in Fig. 4. We will not discuss the properties of points p and p\* here. Let us simply note that radiation reaches heat equilibrium with the black body (state e) from these points either through the adiabatic process p\*e or through the isochoric process pe.

Let us represent the emission of radiation as its transfer from the volume of the black body (state e) to the volume V2 of the cavity along the isotherm *Т*A (state t). As the radiation fills the cavity, it performs a work equal to the difference between the evolved and absorbed

Photons as Working Body of Solar Engines 361

Fig. 3 compares work efficiencies *η*0 and ηС during Carnot cycles described above. Radiation performs work during the Carnot cycle with a greater efficiency than *η*0. *η*0 and *η*C values are calculated from Eqs. (2),(3) as a function of temperature *T*A. We see that the efficiency *η*С of conversion of heat into work in process with radiation only decreases with increasing temperature of the absorber, but the work efficiency *η*0 of matter and radiation increases.

Fig. 3 is divided in two parts by an isotherm at 500 К. On the left side is the region with temperatures where solar cells are used. The efficiency there of conversion of heat into work can reach value of 0.39 for a Carnot cycle with matter and be above 0.91 during a Carnot cycle with radiation. It is important to note, that other reversible and irrevesible cycles

The efficiency of parallel work done by radiation in the Carnot cycle sp\*et\*s (Fig. 4) and the

*η*<sup>0</sup> *η*C = (1–*T*0/*T*A)(1–*Т*A/*T*S) = 0.0591.

*η*<sup>0</sup> *η*C = *η*0+*η*C -(1–*T*0/*T*S)= *η*0+*η*C-*η*OS,

is the efficiency of the Carnot cycle in which the isotherm TS corresponds to radiation and isotherm *T*0, to the matter. Efficiency *η*OS is independed from an absorber temperature *Т*<sup>A</sup> which divides adiabates in two parts. Upper parts of adiabates correspond to the change of radiation temperature, bottom parts to that of matter. It is important that such a Carnot cycle allows us to treat radiant heat absorption and emission as an isothermal and adiabatic processes performed by the matter. Еfficiency *η*OS is limiting for solar-heat engine. It is equal to 0.948 for limit temperatures *T*0=300 K and *T*S=5800 K. This Carnot cycle is not described in

Solar energy is converted as a result of a combination of different processes. Their mechanisms are mostly unknown. For this reason, one tries to establish the temperature dependence of the limit efficiency of a reversible combined process with the help of balance equations for energy

For example, *η*AS = 0.926 when *T*A=320 К. Fig. 5 compares work efficiencies *η*AS and *η*<sup>С</sup> during the cycles with radiant working body at the same limit temperatures. ηAS and η<sup>C</sup> values are calculated from Eqs. (3),(5) as a function of absorber temperature *T*A. We see that *η*AS < *η*С, that is not presenting controversy to the Carnot theorem. The efficiencies *η*AS and *η*С of conversion of radiant heat into work decreases with increasing absorber temperature. The maximum difference *η*C-*η*AS is approx. 18% when *T*А=3500 К (Landsberg, 1980; 1978). The maximal value of the efficiency if for a black body at a temperature *Т*A < *T*S were possible to absorb the radiation from the sun without creating entropy is shown in (Wuerfel,

and entropy flows. For solar engine, it takes the form (Landsberg, 1980; 1978)

*η*OS = (1–*T*0/*T*S) (4)

*η*AS = 1 – 4*Т*A/3*T*S + *Т*A4/3*T*S4. (5)

between these limit temperatures have efficiency smaller than efficiencies *η*C or *η*0.

matter in the Carnot cycle abcda (Fig. 3) is *η*0*η*C. It follows from (2) and (3) that

Efficiencies are equal to 0.77 at *Т*A = 1330К.

After mathematical operations, it takes the form

**2.2.3 Work production during unlike Carnot cycles** 

where

literature.

heat. The radiation performs a considerable work if it reaches state t\* on Fig. 3. Our calculations show that work is performed along the path sp\*et\* with an efficiency of

$$
\eta\_{\text{\textdegree C}} = 1 - T\_{\text{A}} / T\_{\text{S}} = 0.945 \,\tag{3}
$$

when *T*A=320 К. It is important to note that, when radiation returns to its initial state s along the adiabat t\*s, it constitutes a Carnot cycle with the same efficiency *η*C.

Fig. 3. Efficiencies of Carnot cycles in which the radiation takes place. The efficiency *η*0 of work of radiation and matter in a cycle with temperatures limited at *T*0 and *T*A is shown as curve AB. Line CD shows the efficiency of a cyclic process where work is performed by radiation only.

Fig. 4. Entropy diagram showing some thermodynamic cycles for conversion of solar heat into work in cavity *2* with the participation of a black body. Isotherms represent the absorption and emission of radiant energy. Lines pe, p\*e correspond to the cooling of radiation in the black body. Line st indicates the temperature and entropy of radiation in cavity.

Fig. 3 compares work efficiencies *η*0 and ηС during Carnot cycles described above. Radiation performs work during the Carnot cycle with a greater efficiency than *η*0. *η*0 and *η*C values are calculated from Eqs. (2),(3) as a function of temperature *T*A. We see that the efficiency *η*С of conversion of heat into work in process with radiation only decreases with increasing temperature of the absorber, but the work efficiency *η*0 of matter and radiation increases. Efficiencies are equal to 0.77 at *Т*A = 1330К.

Fig. 3 is divided in two parts by an isotherm at 500 К. On the left side is the region with temperatures where solar cells are used. The efficiency there of conversion of heat into work can reach value of 0.39 for a Carnot cycle with matter and be above 0.91 during a Carnot cycle with radiation. It is important to note, that other reversible and irrevesible cycles between these limit temperatures have efficiency smaller than efficiencies *η*C or *η*0.

The efficiency of parallel work done by radiation in the Carnot cycle sp\*et\*s (Fig. 4) and the matter in the Carnot cycle abcda (Fig. 3) is *η*0*η*C. It follows from (2) and (3) that

$$\eta\_0 \eta\_C = (1 - T\_0/T\_A)(1 - T\_A/T\_S) = 0.0591...$$

After mathematical operations, it takes the form

$$
\eta\_{\!0} \eta\_{\!C} = \eta\_{\!0} + \eta\_{\!C} \text{ -} (1 - T\_0 / T\_{\!S}) = \eta\_{\!0} + \eta\_{\!C} \text{-} \eta\_{\!\!\!OS}
$$

where

360 Solar Cells – New Aspects and Solutions

heat. The radiation performs a considerable work if it reaches state t\* on Fig. 3. Our

when *T*A=320 К. It is important to note that, when radiation returns to its initial state s along

Fig. 3. Efficiencies of Carnot cycles in which the radiation takes place. The efficiency *η*0 of work of radiation and matter in a cycle with temperatures limited at *T*0 and *T*A is shown as curve AB. Line CD shows the efficiency of a cyclic process where work is performed by radiation only.

Fig. 4. Entropy diagram showing some thermodynamic cycles for conversion of solar heat into work in cavity *2* with the participation of a black body. Isotherms represent the absorption and emission of radiant energy. Lines pe, p\*e correspond to the cooling of radiation in the black

body. Line st indicates the temperature and entropy of radiation in cavity.

*η*C = 1 - *Т*A/*T*S= 0.945 (3)

calculations show that work is performed along the path sp\*et\* with an efficiency of

the adiabat t\*s, it constitutes a Carnot cycle with the same efficiency *η*C.

$$
\eta\_{\rm CS} = (1 - T\_0/T\_{\rm S}) \tag{4}
$$

is the efficiency of the Carnot cycle in which the isotherm TS corresponds to radiation and isotherm *T*0, to the matter. Efficiency *η*OS is independed from an absorber temperature *Т*<sup>A</sup> which divides adiabates in two parts. Upper parts of adiabates correspond to the change of radiation temperature, bottom parts to that of matter. It is important that such a Carnot cycle allows us to treat radiant heat absorption and emission as an isothermal and adiabatic processes performed by the matter. Еfficiency *η*OS is limiting for solar-heat engine. It is equal to 0.948 for limit temperatures *T*0=300 K and *T*S=5800 K. This Carnot cycle is not described in literature.

### **2.2.3 Work production during unlike Carnot cycles**

Solar energy is converted as a result of a combination of different processes. Their mechanisms are mostly unknown. For this reason, one tries to establish the temperature dependence of the limit efficiency of a reversible combined process with the help of balance equations for energy and entropy flows. For solar engine, it takes the form (Landsberg, 1980; 1978)

$$
\eta\_{\rm AS} = 1 - 4T\_{\rm A}/3T\_{\rm S} + T\_{\rm A}4/3T\_{\rm S}4 \tag{5}
$$

For example, *η*AS = 0.926 when *T*A=320 К. Fig. 5 compares work efficiencies *η*AS and *η*<sup>С</sup> during the cycles with radiant working body at the same limit temperatures. ηAS and η<sup>C</sup> values are calculated from Eqs. (3),(5) as a function of absorber temperature *T*A. We see that *η*AS < *η*С, that is not presenting controversy to the Carnot theorem. The efficiencies *η*AS and *η*С of conversion of radiant heat into work decreases with increasing absorber temperature. The maximum difference *η*C-*η*AS is approx. 18% when *T*А=3500 К (Landsberg, 1980; 1978).

The maximal value of the efficiency if for a black body at a temperature *Т*A < *T*S were possible to absorb the radiation from the sun without creating entropy is shown in (Wuerfel,

Photons as Working Body of Solar Engines 363

For example, the maximal efficiencies of the solar energy conversion are equal 94.8% at the limit temperatures 300 K and 5800 K (*η*OS in the Table 1). Under these temperatures the efficiencies of the solar energy conversion can be equal 5.91% (*η*O*η*C in Table 1). The one belongs to a Carnot cycle, in which a matter and radiation are found as a combined working body, i.e. matter and radiation as a whole system. The other belongs to 2 cycles running parallel. A matter performs the work with a low efficiency 6.25% (*η*O in Table 1), but the radiation performs the work with a high efficiency 94.5% (*η*C in Table 1). In these cases matter and radiation are elementary working bodies. The efficiencies of these parallel

*η*0*η*C = 0.948 \* 0.0625 = 0.0591 = 5.91%. Table 1 shows that a matter performs the work with a low efficiency in solar cells. However, the efficiency of the radiant work at the same absorber temperature is considerably higher. For example, a radiation performs the work with efficiency 92.6% during a non-Carnot cycle (*η*AS in Table 1), but a matter produces work only with an efficiency 6.25% (*η*0 in Table 1) at the absorber temperature 320 К. This difference is caused by various limit temperatures of the cycles (Table 1). The efficiencies of these processes running parallelis smaller than that of

*η*0*η*AS = 0.926 \* 0.0625 = 0.0579 = 5.79%. However, at the same temperatures the efficiency of solar energy conversion achieves 94.8% (*η*OS in Table 1), if a work is performed during a cycle with the matter-radiant working body.

**Cycle Working** 

heat Carnot Elementary / *matter or* 

heat Carnot matter-radiation / *matter* 

**body**

Carnot elementary 300-320

Carnot elementary 300-320

Carnot elementary 300-320

Carnot elementary 300-320

Table 1. Classification and efficiencies of the engines with the elementary and matter-radiant

**Carnot engines** 

solar Carnot elementary 320-5800 *η*C 94.5 3

**non-Carnot engines**  solar non-Carnot elementary 320-5800 *η*AS 92.6 5 solar- heat non-Carnot matter-radiation 300-5800 *η*L 93.1 6 **combined engines**

**Efficiency at TA = 320 K and other parameters of cycles** 

**Limit tempe-**

*and radiation in one cycle* 300-5800 *η*OS 94.8 4

*radiation* 300-320 *η*0 6.25 2

**ratures,K Symbols Limit,**

320-5800 *<sup>η</sup>*0*η*c 5.91 2,3

300-5800 *<sup>η</sup>*0*η*OS 5.93 2,4

320-5800 *η*0*η*AS 5.79 2,5

300-5800 *<sup>η</sup>*0*η*L 5.82 2,6

**%** 

**Calcilated equation** 

processes is

*η*0*η*C:

**Classification of engines** 

ideal solar-

combined Carnot,

combined Carnot

combined Carnot, non-

combined Carnot, non-

working bodies

2005). It follows from a balance of absorbed and emitted energy and entropy flows under the condition of reversibility. The efficiency of a reversible process in which radiation and matter perform work is equal to

*η*L = 1-(*Т*A/*T*S)4 - 4*T*0 [1 - (*Т*A/*T*S)3]/3*T*S. (6)

Fig. 5. Consideration of Carnot efficiencies and efficiencies of reversible processes other than the Carnot cycle. Dot lines АВ,AK denote Carnot efficiencies *η*C and *η*OS at lower limit temperatures 320 and 300 K respectively. Solid curves AFB and AЕB are efficiencies *η*L and *η*AS of non-Carnot engines at the same limit temperatures. Line AK and curve AFB discribe cycles with a radiant-matter working bodies. In the same time line AB and curve AЕB describe cycles with radiant working body only.

For example, *η*L = 0.931 when *T*A=320 К and *T*0=300 К. Condition *T*0=*Т*A excludes temperature *T*0 from expression (6) which in this case is described by the Eq. (5). It means that the work can be obtained during a cycle with a radiant and matter working bodies. Dependencies (5), (6) are shown in the Fig. 5 by curves AFB and AЕB, respectively. Line AB presenting *η*C from Eq. (3) and line АК presenting *η*OS from Eq. (4) are also shown.
