**3. Elementary and matter-radiant working bodies**

Two types of working body are considered:


### **3.1 Energy conversion without irrevocable losses**

According to Carnot theorem, an efficiency of a Carnot engine does not depend on a chemical nature, physical and aggregate states of a working body. The work presents a peculiarity of applying this theorem for solar cells. The statement is that the maximal efficiency of solar cells can be achieved with help of a combined working body only. Let's consider it in detail.

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

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

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

According to Carnot theorem, an efficiency of a Carnot engine does not depend on a chemical nature, physical and aggregate states of a working body. The work presents a peculiarity of applying this theorem for solar cells. The statement is that the maximal efficiency of solar cells can be achieved with help of a combined working body only. Let's

presenting *η*C from Eq. (3) and line АК presenting *η*OS from Eq. (4) are also shown.

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

matter perform work is equal to

describe cycles with radiant working body only.

Two types of working body are considered:

consider it in detail.

**3. Elementary and matter-radiant working bodies** 

**3.1 Energy conversion without irrevocable losses** 

 Elementary working body – matter or radiation in one cycle; Matter-radiant working body – matter and radiation in one cycle. 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 processes is

 $1\eta\_{0}\eta\_{C} = 0.948$   $\* \,0.0625 = 0.0591 = 5.91\,\%.$ 

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*C:

$$1\eta\_{0}\eta\_{\text{AS}} = 0.926\,\,\text{\*}\,0.0625 = 0.0579 = 5.79\,\%.$$

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.


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

Photons as Working Body of Solar Engines 365

**non-working conversion**  - reemission 320-5800 *η*<sup>U</sup> 99.99 1 heat, solar and heat-solar endoreversible engines elementary\* Carnot 300-320 *μ*0*μ*<sup>U</sup> 6.25 1, 2 elementary Carnot 320-5800 *μ*C*μ*<sup>U</sup> 94.5 1, 3 elementary non-Carnot 320-5800 *μ*AS*μ*<sup>U</sup> 92.6 1, 5

radiation\*\* non-Carnot 300-5800 *μ*L*μ*U 93.1 6 matter-radiation Carnot 300-5800 *μ*OS*μ*<sup>U</sup> 94.8 1, 4

elementary Carnot 300-320/300-5800 *μ*0*μ*0S*μ*<sup>U</sup> 5.93 1, 2, 4

300-320/300-5800/320-

Table 2. Classification and efficiencies of the engines with solar energy conversion as a

Based on this statement one can say that every point of the line CB is (by definition) a graphical illustration of the sequence of reversible and irreversible energy transitions. First represent a Carnot cycle abcd in the Fig.2, second represent a process of cooling of radiation running according to the line st. This engine performs the work with the efficiency *μ*0*μ*U. The combination of reversible and irreversible processes allows us to call this engine an

Curves CВ и LB in Fig.6 have some dicrepancy because of the entropy production without a work production in endoreversible engine. Indeed reversible energy conversion with the efficiency *μ*L or *μ*0 occurs without entropy production. The energy conversion with efficiency *μ*<sup>0</sup> *μ*U is accompanied by the entropy production during the solar energy reemissions. The

So, the entropy is not performed during a reversible engines. Endoreversible engines perform the entropy. Thus, the author supposes that difference between efficiencies (*μ*L- *μ*<sup>0</sup> *μ*U) of these engines is proportional to the quality (or number) of irreversible transitions. In most cases the increase of number of irreversible transitions in conversion of radiant heat into irrevocably losses calls a reduce of the efficiency of engine from *μ*L down to

latter ones do not take place in the work production and cause irrevocably losses.

Combined endoreversible engines elementary Carnot 300-320/320-5800 *μ*0*μ*C*μ*<sup>U</sup> 5.91 1, 2, 3

Carnot 300-320/320-5800 *μ*0*μ*AS*μ*U 5.79 1, 2, 5

<sup>5800</sup>*μ*0*μ*C*μ*0S*μ*U 5.60 1, 2, 3, 4

<sup>5800</sup>*μ*0*μ*0S*μ*AS*μ*U 5.49 1, 2, 4, 5

**working body cycle Limit temperatures,K Symbols Limit,**

matter-

elementary and matterradiation

irrevocably losses

*μ*<sup>0</sup> *μ*U.

elementary Carnot/non-

elementary Carnot 300-320/320-5800/300-

\*\* matter-radiant working body – matter and radiation in one cycle.

endoreversible. Engines with the efficiencies *μ*L and *μ*0 are reversible.

Carnot/non-Carnot

\* elementary working body – matter or radiation;

**Cycle parameters Efficiency at TA=320 K** 

**% eq-tion** 

A Carnot cycle with the efficiency *η*OS and the matter-radiant working body has been considered by the author earlier in the chapter, its efficiency is given by eq.4. Further we will call an engine with the matter-radiant working body an ideal solar-heat one. The elementary working bodies perform the work by solar or heat engines. Their properties are listed in Table 1. The advantage of the cyclic processes in comparison with the matter-radiant working body is obvious.

So, the elementary working bodies perform the work with the efficiencies *η*0, *η*C, *η*L and *η*AS. The matter-radiant working bodies perform the work with the efficiency *η*OS. According to the Table. 1, one can confirm:


Therefore, high efficiency solar cells should be designed as solar-heat engine only.

### **3.2 Energy conversion with irrevocable losses**

The absorption of radiation precedes the conversion of solar heat into work. In our model, the black body absorbs solar radiation and generates another radiation with a smaller temperature. Heat is evolved in this process; it is either converted into work or irrevocable lost. In this work the photon absorption in solar cells is divided into processes with and without work production. For the sake of simplicity, let us assume that heat evolved during solar energy reemission is lost with an efficiency of *η*U from Eq. (1). The work of the cyclic processes is performed with the efficiencies of *η*0, *η*C, *η*AS, *η*OS and *η*L (Tabl. 1). Then the conversion of solar heat with and without work production is performed with the efficiencies, for example, *η*C*η*U or *η*0*η*C*η*U. These and other combinations of efficiencies are compared in (Laptev, 2008).

It is important to note that the irrevocable energy losses of absorber at temperature 320 K do not cause the researchers' interest in thermodynamic analysis of conversion of solar heat into work. Actually, efficiency of the solar energy reemission as the irrevocable energy losses *μ*U is close to 1 for (*T*A/*T*S)4 = (320/5800)4 ≈ 10-5. So efficiency of the solar energy reemission at 320 К has a small effect on efficiency of solar cell. The Tables 1,2 list efficiencies of the solar cells with and without irrevocable losses calculated in this work. The difference between these values does not exceed 0.01%. Values of *μ*0*μ*C and *μ*0*μ*C*μ*U may serve as examples. It might be seen that irrevocable energy losses are not to be taken into account in the thermodynamic analysis of conversion of solar heat into work. However, the detailed analysis of efficiencies of conversion of solar heat into work enabled us to reveal a correlation between the reversibility of solar energy reemission and efficiency of solar cell. The following parts of the chapter are devoted to this important aspect of conversion of solar heat into work.

#### **3.3 Combinations of reversible and irreversible energy conversion processes**

The temperature dependencies of *μ* L from Eq. (5) and *μ*0*μ*U from Eqs. (1),(2) are shown by lines LB, CB in Fig. 6. Let us also make use of the fact that every point of the line LB is (by definition) a graphical illustration of the sequence of reversible transitions from one energy state of the system to another, because each reversible process consists of the sequence of reversible transitions only.

A Carnot cycle with the efficiency *η*OS and the matter-radiant working body has been considered by the author earlier in the chapter, its efficiency is given by eq.4. Further we will call an engine with the matter-radiant working body an ideal solar-heat one. The elementary working bodies perform the work by solar or heat engines. Their properties are listed in Table 1. The advantage of the cyclic processes in comparison with the matter-radiant

So, the elementary working bodies perform the work with the efficiencies *η*0, *η*C, *η*L and *η*AS. The matter-radiant working bodies perform the work with the efficiency *η*OS. According to



The absorption of radiation precedes the conversion of solar heat into work. In our model, the black body absorbs solar radiation and generates another radiation with a smaller temperature. Heat is evolved in this process; it is either converted into work or irrevocable lost. In this work the photon absorption in solar cells is divided into processes with and without work production. For the sake of simplicity, let us assume that heat evolved during solar energy reemission is lost with an efficiency of *η*U from Eq. (1). The work of the cyclic processes is performed with the efficiencies of *η*0, *η*C, *η*AS, *η*OS and *η*L (Tabl. 1). Then the conversion of solar heat with and without work production is performed with the efficiencies, for example, *η*C*η*U or

It is important to note that the irrevocable energy losses of absorber at temperature 320 K do not cause the researchers' interest in thermodynamic analysis of conversion of solar heat into work. Actually, efficiency of the solar energy reemission as the irrevocable energy losses *μ*U is close to 1 for (*T*A/*T*S)4 = (320/5800)4 ≈ 10-5. So efficiency of the solar energy reemission at 320 К has a small effect on efficiency of solar cell. The Tables 1,2 list efficiencies of the solar cells with and without irrevocable losses calculated in this work. The difference between these values does not exceed 0.01%. Values of *μ*0*μ*C and *μ*0*μ*C*μ*U may serve as examples. It might be seen that irrevocable energy losses are not to be taken into account in the thermodynamic analysis of conversion of solar heat into work. However, the detailed analysis of efficiencies of conversion of solar heat into work enabled us to reveal a correlation between the reversibility of solar energy reemission and efficiency of solar cell. The following parts of the chapter are devoted to

*η*0*η*C*η*U. These and other combinations of efficiencies are compared in (Laptev, 2008).

**3.3 Combinations of reversible and irreversible energy conversion processes** 

The temperature dependencies of *μ* L from Eq. (5) and *μ*0*μ*U from Eqs. (1),(2) are shown by lines LB, CB in Fig. 6. Let us also make use of the fact that every point of the line LB is (by definition) a graphical illustration of the sequence of reversible transitions from one energy state of the system to another, because each reversible process consists of the sequence of

working body is obvious.

the Table. 1, one can confirm:

solar-heat engine.

reversible transitions only.

**3.2 Energy conversion with irrevocable losses** 

this important aspect of conversion of solar heat into work.


\* elementary working body – matter or radiation;

\*\* matter-radiant working body – matter and radiation in one cycle.

Table 2. Classification and efficiencies of the engines with solar energy conversion as a irrevocably losses

Based on this statement one can say that every point of the line CB is (by definition) a graphical illustration of the sequence of reversible and irreversible energy transitions. First represent a Carnot cycle abcd in the Fig.2, second represent a process of cooling of radiation running according to the line st. This engine performs the work with the efficiency *μ*0*μ*U. The combination of reversible and irreversible processes allows us to call this engine an endoreversible. Engines with the efficiencies *μ*L and *μ*0 are reversible.

Curves CВ и LB in Fig.6 have some dicrepancy because of the entropy production without a work production in endoreversible engine. Indeed reversible energy conversion with the efficiency *μ*L or *μ*0 occurs without entropy production. The energy conversion with efficiency *μ*<sup>0</sup> *μ*U is accompanied by the entropy production during the solar energy reemissions. The latter ones do not take place in the work production and cause irrevocably losses.

So, the entropy is not performed during a reversible engines. Endoreversible engines perform the entropy. Thus, the author supposes that difference between efficiencies (*μ*L- *μ*<sup>0</sup> *μ*U) of these engines is proportional to the quality (or number) of irreversible transitions. In most cases the increase of number of irreversible transitions in conversion of radiant heat into irrevocably losses calls a reduce of the efficiency of engine from *μ*L down to *μ*<sup>0</sup> *μ*U.

Photons as Working Body of Solar Engines 367

*μ*<sup>C</sup> *μ*U if the efficiency *μ*L (*μ*-coordinates of line LB points) is taken as one. For example, the efficiency *μ*<sup>0</sup> *μ*U at 320 К (point *b* in Fig.6) is 6.25% (Tabl.2). Let us draw an isotherm running through the point *b*; the intersections of this line with the line LB and the y-axis give us the

*q*ir = (*μ*L-0.0625)100/ *μ*L=93.3% of the irreversible transitions and the fraction *q*rev = 6.7% of the reversible ones in conversion solar energy with the efficiency *μ*<sup>0</sup> *μ*U=6.25%. If matter and radiation produce work with the efficiency *μ*<sup>0</sup> *μ*<sup>C</sup> *μ*U=5.91% at 320 К (Tabl.2), the fraction of irreversible transitions as

*q*ir = (*μ*L-5.91)100/ *μ* L=93.6%. The fraction of irreversible transitions *q*rev of two Carnot cycles will be equal 100-93.6=6.4%. Thus, when matter and radiation produce work simultaneously in Carnot cycles, only less then 7% of energy transitions in solar energy conversion are working cycles. Remaining 97% of energy transitions in solar energy conversion are non-working processes. So, the efficiency of solar-heat engine sufficiently depends on the reversability of non-working processes, rather than on the efficiencies of the work cycles. Let's explain in more details.

As it was mentioned above, the reason of descrepancy between curves *μ*L, *μ*0*μ*U and *μ*0*μ*C*μ*<sup>U</sup> at *T*A<*T*S is entropy production during the processes without work production. Then the *μ*coordinates of the points belonging to the plane between line *μ*L and the lines *μ*0*μ*U, *μ*0*μ*C*μ*<sup>U</sup> on Fig. 7 are proportional to the shares of the reversible and irreversible transitions without work production. We propose the following method for calculating relative contributions of

Fig. 7. The comparison of the well-known thermodynamic efficiency limitations of the solar

value *μ*L = 0.931. Then, by the lever principle, point *b* corresponds to a fraction

**3.4 Reversible and irreversible non-working processes** 

reversible and irreversible processes without work production.

energy conversion with and without entropy production.

irrevocably losses is

Then, with help of the Fig. 6 one can find a ratio of reversible and irreversible transitions in a solar cell performing the work with the efficiency *μ*<sup>0</sup> *μ*U. For example, let point *a* on Fig. 6 denote the conversion of solar energy with an efficiency of 30%. Let us draw an isotherm running through the point *a*; the intersections of this line with the line LB and the y-axis give us the values *μ*L = 0.93 and *T*A=430 К. Then, by the lever principle, the fraction of irreversible transitions *q*ir in point *a* is

$$\mu\_{\rm ir} = (\mu\_{\rm L} \text{-30})100 / \,\mu\_{\rm L} \text{=} 68\,\%\,.$$

The fraction of irreversible transitions *q*rev in point *a* will be 100–68=32%.

In other words, when in solar-heat engine (with the efficiency *μ*L) 68% energy transitions can be made irreversible then the radiant work production will be ceased. If the reversible transitions (their number is now 32% of the total transitions' number) may form a Carnot cycle with the efficiency *μ*0, then one can say that the reduce of number of reversible transitions leads to a cease of radiant work production, makes it possible to continue matter work production and reduces efficiency of engine from 93% down to 30%. The radiant heat evolved under these conditions partially takes a form of irrevocably losses as irreversible non-working transitions.

Fig. 6. Efficiency *μ*L of solar-heat reversible engine (linе LB) as a function of the absorber temperature *T*A compared with the efficiency *μ*0*μ*U of an heat endoreversible engine (line СB). Every point of the line LB correspond to 100% reversible transitions in energy conversion without entropy production. Every point of the line CB correspond to 100% irreversible transitions in energy conversion without work production. Every point of the region LBC between lines LB and СB correspond the reversible and irreversible transitions. Their parts can be found according to the lever's law (see text).

Let's consider a case of parallel work of matter and radiation. One could assume the work of radiation should increase an efficiency of solar-heat engine. However, it is not correct. The efficiencies of endoreversible processes whose elements are Carnot cycles with the efficiencies *μ*C and *μ*0 are shown in Fig. 7 by the CEB and CFB lines. They are situated below the LB line, whose every point is the efficiency *μ*L of a reversible process by definition. We can therefore calculate the contributions of the Carnot cycle to the efficiencies *μ*<sup>0</sup> *μ*U and *μ*<sup>0</sup>

Then, with help of the Fig. 6 one can find a ratio of reversible and irreversible transitions in a solar cell performing the work with the efficiency *μ*<sup>0</sup> *μ*U. For example, let point *a* on Fig. 6 denote the conversion of solar energy with an efficiency of 30%. Let us draw an isotherm running through the point *a*; the intersections of this line with the line LB and the y-axis give us the values *μ*L = 0.93 and *T*A=430 К. Then, by the lever principle, the fraction of irreversible

*q*ir = (*μ*L-30)100/ *μ*L=68%.

In other words, when in solar-heat engine (with the efficiency *μ*L) 68% energy transitions can be made irreversible then the radiant work production will be ceased. If the reversible transitions (their number is now 32% of the total transitions' number) may form a Carnot cycle with the efficiency *μ*0, then one can say that the reduce of number of reversible transitions leads to a cease of radiant work production, makes it possible to continue matter work production and reduces efficiency of engine from 93% down to 30%. The radiant heat evolved under these conditions partially takes a form of irrevocably losses as irreversible

Fig. 6. Efficiency *μ*L of solar-heat reversible engine (linе LB) as a function of the absorber temperature *T*A compared with the efficiency *μ*0*μ*U of an heat endoreversible engine (line СB). Every point of the line LB correspond to 100% reversible transitions in energy conversion without entropy production. Every point of the line CB correspond to 100% irreversible transitions in energy conversion without work production. Every point of the region LBC between lines LB and СB correspond the reversible and irreversible transitions.

Let's consider a case of parallel work of matter and radiation. One could assume the work of radiation should increase an efficiency of solar-heat engine. However, it is not correct. The efficiencies of endoreversible processes whose elements are Carnot cycles with the efficiencies *μ*C and *μ*0 are shown in Fig. 7 by the CEB and CFB lines. They are situated below the LB line, whose every point is the efficiency *μ*L of a reversible process by definition. We can therefore calculate the contributions of the Carnot cycle to the efficiencies *μ*<sup>0</sup> *μ*U and *μ*<sup>0</sup>

Their parts can be found according to the lever's law (see text).

The fraction of irreversible transitions *q*rev in point *a* will be 100–68=32%.

transitions *q*ir in point *a* is

non-working transitions.

*μ*<sup>C</sup> *μ*U if the efficiency *μ*L (*μ*-coordinates of line LB points) is taken as one. For example, the efficiency *μ*<sup>0</sup> *μ*U at 320 К (point *b* in Fig.6) is 6.25% (Tabl.2). Let us draw an isotherm running through the point *b*; the intersections of this line with the line LB and the y-axis give us the value *μ*L = 0.931. Then, by the lever principle, point *b* corresponds to a fraction

$$\mu\_{\rm ir} = (\mu\_{\rm L} \text{-0.0625})100 / \,\mu\_{\rm L} = 93.3\%$$

of the irreversible transitions and the fraction *q*rev = 6.7% of the reversible ones in conversion solar energy with the efficiency *μ*<sup>0</sup> *μ*U=6.25%. If matter and radiation produce work with the efficiency *μ*<sup>0</sup> *μ*<sup>C</sup> *μ*U=5.91% at 320 К (Tabl.2), the fraction of irreversible transitions as irrevocably losses is

$$\mu\_{\rm ir} = (\mu\_{\rm L} \text{-5.91})100 / \,\mu\_{\rm L} \text{=} 93.6\%.$$

The fraction of irreversible transitions *q*rev of two Carnot cycles will be equal 100-93.6=6.4%. Thus, when matter and radiation produce work simultaneously in Carnot cycles, only less then 7% of energy transitions in solar energy conversion are working cycles. Remaining 97% of energy transitions in solar energy conversion are non-working processes. So, the efficiency of solar-heat engine sufficiently depends on the reversability of non-working processes, rather than on the efficiencies of the work cycles. Let's explain in more details.

### **3.4 Reversible and irreversible non-working processes**

As it was mentioned above, the reason of descrepancy between curves *μ*L, *μ*0*μ*U and *μ*0*μ*C*μ*<sup>U</sup> at *T*A<*T*S is entropy production during the processes without work production. Then the *μ*coordinates of the points belonging to the plane between line *μ*L and the lines *μ*0*μ*U, *μ*0*μ*C*μ*<sup>U</sup> on Fig. 7 are proportional to the shares of the reversible and irreversible transitions without work production. We propose the following method for calculating relative contributions of reversible and irreversible processes without work production.

Fig. 7. The comparison of the well-known thermodynamic efficiency limitations of the solar energy conversion with and without entropy production.

Photons as Working Body of Solar Engines 369

According to the Eqs. (4),(6), there are two efficiency limits of solar energy conversion in reversible processes for a pair of limiting temperatures *T*0 and *T*S, namely, *μ*OS, *μ*L. In the Fig. 8 are shown their dependencies on absorber temperature *T*A. Obviously, under all values of *T*A the condition *μ*OS > *μ*L is fulfilled. The suggestion can be made that processes can occur whose efficiency is between these limits. By virtue of the second law of thermodynamics,

For instance, radiant energy conversion with the efficiency *μ*OS*μ*U is irreversible, because radiation performs an irreversible process with the efficiency *μ*U along the line st shown in Fig. 2. We see from Fig. 8 that the *μ*OS *μ*U values (line AB) at temperatures *T*A<<*T*S are indeed

In each point of the line AB the irreversible processes appeare as a sequence of transitions without work production. Because the efficiency *μ*OS *μ*U of the processes with their participation larger then *μ*L, one can suppose that the fraction of irreversible transitions along the line LB is equal to zero, and along the line AB is equal to unity. In the plane between lines AB и LB the

*q*ir = (*μ*OS*μ*U - *μ*)100/( *μ*OS*μ*U - *μ*L). In each point of the line AB the fraction of irreversible transitions *q*0i of total number of

*q*0ir = (*μ*OS*μ*U - *μ*L)100/( *μ* S*μ*U). According to the Tables 1, 2, *q*0ir=1.8% at *T*A=320 К. The processes consisting of these

So, along the line AD (Fig. 8) conversion of solar energy is perfomed during a Carnot cycle, along the line AB – during a Carnot cycle and the processes without work production. In the plane between these lines only work is perfomed. The working process is a non-Carnot cycle. Between the lines AB and LB non-working processes are reversible and irreversible. There the fraction of irrevesible transitions decreases with closing the line LB. Along the line LB all

energy transitions (with and without work production) are reversible, i.е. *q*0ir=0.

Fig. 8. Comparison of the efficiency *μ*OS*μ*U of combinations of reversible working and irreversible non-working processes (the AB line) with the efficiency *μ*OS of a reversible Carnot working process (the AD line) and the efficiency *μ*L of reversible non-Carnot

working processes (the LB line). Lines AD и AB coincide at *T*A=0 К.

fraction of irreversible transitions *q*ir may be calculated according to the lever principle:

**3.5 Ideal solar-heat engines** 

they cannot be reversible.

irrevesible transitions is

transitions, do not produce work.

larger than *μ*L (line LB) and smaller than *μ*OS (line AD).

The temperature dependencies of *μ*L, *μ*0*μ*U, *μ*0*μ*C*μ*U derived from Eqs. (1),(2),(3) and(6) are shown by lines LB, CEB and CDB in Fig. 7. Remember that the *μ*-coordinates of the points belonging to the lines *μ*0*μ*U, *μ*0*μ*C*μ*U are proportional to the shares of reversible transitions which are one or two Carnot cycles with the efficiencies *μ*0 and *μ*C. At the same time remaining 100% irreversible transitions are (form) irreversible processes without work production as irrevocably losses.

The lines CEB and CDB on Fig. 7 are not only the illustration of possible ways of converting solar energy. For example, it is known that the limiting efficiencies of solar cells on the basis of a p,n-transition are 30% (Shockley & Queisser, 1961) and 43% (Werner et al., 1994). at *T*A=320 К. They are shown by points *a* and *b* on Fig. 7 and lie above the lines *μ*0*μ*U, *μ*0*μ*C*μ*U, but below the line *μ*L. One can say that processes without work production may be the combination of reversible and irreversible processes running parallel. Then the *μ*coordinates of the points above the lines CEB, CDB are proportional to the shares *q* of reversible and irreversible transitions, which are appearing as processes without work production. According to the lever's principle, fraction of irreversible transitions *q*ir in сonversion of solar energy is

### *q*ir =( *μ*L- *μ*0*μ*C*μ*U)100/ *μ*L.

For example, according to Tabl. 1,2, *μ*L=93.1% and *μ*0*μ*C*μ*U=5.91% at *T*A=320 К. Then *q*ir = (93.1-5.91)100/93.1=93.7%. The fraction of reversible transitions *q*rev is 100-93.7=6.3%.

Point *a* in Fig.7 denotes the conversion of solar energy with an efficiency *μ*=30% at *T*A=320 К. It lies above the line *μ*0*μ*C*μ*U. Then the value

$$
\mu\_{\rm ir} = (\mu\_{\rm L} \text{-} \mu)100 / (\mu\_{\rm L} \text{-} \mu\_{\rm 0} \mu\_{\rm C} \mu\_{\rm U})
$$

is a part of irreversible transitions in the conversion of solar energy without work production. In our case it is (93.1-30)100/(93.1-5.91)=70.0%. The fraction of reversible transitions will be equal to 100-70.0=30.0%. Note that there are no reversible transitions of Carnot cycles.

The band theory proposes mechanisms of converting solar energy into work with an efficiency of 43% (Landsberg & Leff, 1989). Let us denote last value by the point *b* on the isotherm in Fig. 7. Then the fraction of irreversible transitions *q*ir without work production is (93.1-43)100/(93.1-5.91)= 57.5%, and the fraction of reversible transitions *q*rev without work production will be equal 100-57.5=42.5%. So that, to increase efficiency of endorevesible solar-heat engine from 30 up to 43% at *T*A=320 К, it's necessary to reduce a fraction of irreversible transitions without work production from 72.4% down to 57.5%.

As an example of reverse calculations let's find out efficiency of solar-heat engine with 50% irreversible transitions without work production. Denote the value is to be found as *х* and write down the equation

$$\eta\_{\rm ir} = (93.1 \text{-x})100 / (93.1 \text{-5.91}) = 50\%. \text{ :} $$

Solving it reveal that *х*=49.5%. Note that for *q*ir = 100% efficiency *μ*0*μ*C*μ*U=50% may be achieved only under *T*A=700 К (see Fig.7). This temperature is much higher than the temperatures of solar cells' exploitation (the temperatures of solar cells' working conditions) Thus, the author supposes the key for further increase of solar cells' efficiency is in the study and perfecting processes without work production.

#### **3.5 Ideal solar-heat engines**

368 Solar Cells – New Aspects and Solutions

The temperature dependencies of *μ*L, *μ*0*μ*U, *μ*0*μ*C*μ*U derived from Eqs. (1),(2),(3) and(6) are shown by lines LB, CEB and CDB in Fig. 7. Remember that the *μ*-coordinates of the points belonging to the lines *μ*0*μ*U, *μ*0*μ*C*μ*U are proportional to the shares of reversible transitions which are one or two Carnot cycles with the efficiencies *μ*0 and *μ*C. At the same time remaining 100% irreversible transitions are (form) irreversible processes without work

The lines CEB and CDB on Fig. 7 are not only the illustration of possible ways of converting solar energy. For example, it is known that the limiting efficiencies of solar cells on the basis of a p,n-transition are 30% (Shockley & Queisser, 1961) and 43% (Werner et al., 1994). at *T*A=320 К. They are shown by points *a* and *b* on Fig. 7 and lie above the lines *μ*0*μ*U, *μ*0*μ*C*μ*U, but below the line *μ*L. One can say that processes without work production may be the combination of reversible and irreversible processes running parallel. Then the *μ*coordinates of the points above the lines CEB, CDB are proportional to the shares *q* of reversible and irreversible transitions, which are appearing as processes without work production. According to the lever's principle, fraction of irreversible transitions *q*ir in

*q*ir =( *μ*L- *μ*0*μ*C*μ*U)100/ *μ*L. For example, according to Tabl. 1,2, *μ*L=93.1% and *μ*0*μ*C*μ*U=5.91% at *T*A=320 К. Then *q*ir = (93.1-5.91)100/93.1=93.7%. The fraction of reversible transitions *q*rev is 100-93.7=6.3%. Point *a* in Fig.7 denotes the conversion of solar energy with an efficiency *μ*=30% at *T*A=320

*q*ir = (*μ*L-*μ*)100/(*μ*L- *μ*0*μ*C*μ*U) is a part of irreversible transitions in the conversion of solar energy without work production. In our case it is (93.1-30)100/(93.1-5.91)=70.0%. The fraction of reversible transitions will be equal to 100-70.0=30.0%. Note that there are no reversible transitions of

The band theory proposes mechanisms of converting solar energy into work with an efficiency of 43% (Landsberg & Leff, 1989). Let us denote last value by the point *b* on the isotherm in Fig. 7. Then the fraction of irreversible transitions *q*ir without work production is (93.1-43)100/(93.1-5.91)= 57.5%, and the fraction of reversible transitions *q*rev without work production will be equal 100-57.5=42.5%. So that, to increase efficiency of endorevesible solar-heat engine from 30 up to 43% at *T*A=320 К, it's necessary to reduce a fraction of irreversible transitions without work production from 72.4% down to

As an example of reverse calculations let's find out efficiency of solar-heat engine with 50% irreversible transitions without work production. Denote the value is to be found as *х* and

*q*ir = (93.1-*х*)100/(93.1-5.91)= 50%. Solving it reveal that *х*=49.5%. Note that for *q*ir = 100% efficiency *μ*0*μ*C*μ*U=50% may be achieved only under *T*A=700 К (see Fig.7). This temperature is much higher than the temperatures of solar cells' exploitation (the temperatures of solar cells' working conditions) Thus, the author supposes the key for further increase of solar cells' efficiency is in the study

production as irrevocably losses.

сonversion of solar energy is

Carnot cycles.

57.5%.

write down the equation

К. It lies above the line *μ*0*μ*C*μ*U. Then the value

and perfecting processes without work production.

According to the Eqs. (4),(6), there are two efficiency limits of solar energy conversion in reversible processes for a pair of limiting temperatures *T*0 and *T*S, namely, *μ*OS, *μ*L. In the Fig. 8 are shown their dependencies on absorber temperature *T*A. Obviously, under all values of *T*A the condition *μ*OS > *μ*L is fulfilled. The suggestion can be made that processes can occur whose efficiency is between these limits. By virtue of the second law of thermodynamics, they cannot be reversible.

For instance, radiant energy conversion with the efficiency *μ*OS*μ*U is irreversible, because radiation performs an irreversible process with the efficiency *μ*U along the line st shown in Fig. 2. We see from Fig. 8 that the *μ*OS *μ*U values (line AB) at temperatures *T*A<<*T*S are indeed larger than *μ*L (line LB) and smaller than *μ*OS (line AD).

In each point of the line AB the irreversible processes appeare as a sequence of transitions without work production. Because the efficiency *μ*OS *μ*U of the processes with their participation larger then *μ*L, one can suppose that the fraction of irreversible transitions along the line LB is equal to zero, and along the line AB is equal to unity. In the plane between lines AB и LB the fraction of irreversible transitions *q*ir may be calculated according to the lever principle:

$$
\eta\_{\rm ir} = (\mu\_{\rm OS} \mu\_{\rm U} - \mu)100 / (\,\mu\_{\rm OS} \mu\_{\rm U} - \mu\_{\rm L}).
$$

In each point of the line AB the fraction of irreversible transitions *q*0i of total number of irrevesible transitions is

$$
\eta^{0\_{\rm ir}} = (\mu\_{\rm OS}\mu\_{\rm U} - \mu\_{\rm L})100/(\,\mu\_{\rm S}\mu\_{\rm U}).
$$

According to the Tables 1, 2, *q*<sup>0</sup> ir=1.8% at *T*A=320 К. The processes consisting of these transitions, do not produce work.

So, along the line AD (Fig. 8) conversion of solar energy is perfomed during a Carnot cycle, along the line AB – during a Carnot cycle and the processes without work production. In the plane between these lines only work is perfomed. The working process is a non-Carnot cycle. Between the lines AB and LB non-working processes are reversible and irreversible. There the fraction of irrevesible transitions decreases with closing the line LB. Along the line LB all energy transitions (with and without work production) are reversible, i.е. *q*<sup>0</sup> ir=0.

Fig. 8. Comparison of the efficiency *μ*OS*μ*U of combinations of reversible working and irreversible non-working processes (the AB line) with the efficiency *μ*OS of a reversible Carnot working process (the AD line) and the efficiency *μ*L of reversible non-Carnot working processes (the LB line). Lines AD и AB coincide at *T*A=0 К.

Photons as Working Body of Solar Engines 371

production processes. It follows that the contribution of reversible processes is smaller than the contribution of irreversible processes in devices that are currently used. According to the maximum work principle, their efficiency can be increased by increasing the fraction of reversible processes in solar energy absorbers. The following approach to solution of this

Antenna states of atomic particles or their groups are the states between them resulting from the reemission of photons without work production and heat dissipation. Below is shown that а comparison of the efficiencies of the reversible and irreversible photon reemissions occuring in parallel made it clear that it is impossible to attain very high efficiencies in the conversion of

Matter and solar radiation are never in equilibrium in solar cells and quasi-static conversion of the solar energy is not achieved. For this reason, the irreversible thermodynamic engines are described using the method of endoreversible thermodynamics of solar energy conversion (Novikov, 1958; Rubin, 1979; De Vos, 1992). Remember that endoreversible engines are irreversible engines where all irreversibilities are restricted to the coupling of engine to the external world. It is assumed that the inner reversible part of an еndoreversible engine is a Carnot cycle. We have also considered the non-Carnot cycles and found out that photon absorption in solar cells can be considered as the external reversible part of an

The photon absorption in solar cells is separated into processes with and without work production. These processes are sequencies of transitions from one energy state of the system to another. The energy transitions between particle states are called "photon

The photon reemission is divided into reversible and irreversible processes. These nonworking processes are regarded as a continuous series of equilibrium states outside the irreversible or endoreversible engine, are isolate into separate processes, and are used to obtain a higher efficiency of the generally non-quasistatic solar energy conversion. We can consider those equilibrium processes as a base of the "exoreversible"additional device for

So, we have called the processes without work production photon reemission. Let us now

Let us consider these particle states in a radiant energy absorber, transitions between them result from the absorption of photons. The states of atomic particles or their groups, as well as energy transitions between them, are called "working" if they take part in performing work. It is known that solar energy conversions are not always working processes. The states of atomic particles and the energy transitions between them are said to be "antenna" ones if they take part in the absorption and emission of radiant energy without work production, i.e. in the photon reemission. Carnot cycles are examples of working processes, while cycles described below involving the photon reemission are examples of antenna

reemission" if they do not take part in performing work and heat dissipation.

shown that they play a special role in the conversion of solar heat into work.

solar heat into work without the reversible photon reemissions as antenna process.

problem can be suggested.

еndoreversible engine.

processes.

the irreversible or endoreversible engine.

**4.2 Antenna states of the absorber particles** 

**4. Antenna states and processes** 

**4.1 Photon reemissions as non-working processes** 

Irreversible processes without work production appear under the line LB. Their fraction increase as we move off the line LB. The previous chapter gives examples of calculating their contributions to solar energy conversion.

The value *μ*L is a keyword in separation of solar energy conversion in processes with and without work production. It is referred as Landsberg efficiency (Wuerfel, 2005). According to this tradition, solar-heat engine with the efficiency *μ*L are called Landsberg engine. It has a minimal efficiency of all solar-heat engines.

Solar-heat engine with the maximal efficiency *μ*OS we call ideal solar-heat engine. Then solar-heat engine with the efficiency between *μ*OS и *μ*OS*μ*U may be noted as ideal non-Carnot solar-heat engine, and а solar-heat engine with the efficiency between *μ*OS*μ*U and *μ*L as an endoreversible solar-heat engine.

Solar-heat engine have cycles with the matter-radiant working body. We will not discuss the properties of the matter-radiant working body here. Let us simply note that all energy transitions in solar-heat engines are performed by the matter-radiant working body (Table 3). In the endoreversible solar-heat engine the matter-radiant working body makes working transitions, their fraction of total number of energy transitions is 100-1.8=98.2% because the processes without work production have only 1.8% energy transitions. Engine with the matter-radiant working body is not produced yet.


Table 3. Comparison between the engine efficiencies and the contributions of working and non-working transitions

Existent solar cells are endoreversible combined engines with the non-Carnot cycles and еlementary working bodies (matter or radiation). The working bodies perform working energy transitions, their fraction (as it is shown in the Table 3) can achieve only 7 % of transitions total number. We cannot overcome this 7% barrier but it doesn't mean that there is no possibility to increase the efficiency of such a machine. According to the Table 3, the non-working transitions are more than 93% from the total number of energy transitions. They can be either irreversible or revesible. If the first are made as the last, the engine efficiency can be increased from 6 up to 93.1%. So the aim of perfection in endoreversible solar-heat engine processes without work production is arisen. In ideal case it is necessary to obtain Landsberg engine.

So, efficiencies of the combined endoreversible engines with the non-Carnot cycles and еlementary working bodies (matter or radiation) mostly depend on reversibility of processes without work production, running in the radiation absorber rather than on the work

Irreversible processes without work production appear under the line LB. Their fraction increase as we move off the line LB. The previous chapter gives examples of calculating their

The value *μ*L is a keyword in separation of solar energy conversion in processes with and without work production. It is referred as Landsberg efficiency (Wuerfel, 2005). According to this tradition, solar-heat engine with the efficiency *μ*L are called Landsberg engine. It has a

Solar-heat engine with the maximal efficiency *μ*OS we call ideal solar-heat engine. Then solar-heat engine with the efficiency between *μ*OS и *μ*OS*μ*U may be noted as ideal non-Carnot solar-heat engine, and а solar-heat engine with the efficiency between *μ*OS*μ*U and *μ*L as an

Solar-heat engine have cycles with the matter-radiant working body. We will not discuss the properties of the matter-radiant working body here. Let us simply note that all energy transitions in solar-heat engines are performed by the matter-radiant working body (Table 3). In the endoreversible solar-heat engine the matter-radiant working body makes working transitions, their fraction of total number of energy transitions is 100-1.8=98.2% because the processes without work production have only 1.8% energy transitions. Engine with the

Idel Carnot solar-heat 94.80 100 - equilibrium

Landsberg reversible 93.10 < 7 >93 equilibrium

Table 3. Comparison between the engine efficiencies and the contributions of working and

Existent solar cells are endoreversible combined engines with the non-Carnot cycles and еlementary working bodies (matter or radiation). The working bodies perform working energy transitions, their fraction (as it is shown in the Table 3) can achieve only 7 % of transitions total number. We cannot overcome this 7% barrier but it doesn't mean that there is no possibility to increase the efficiency of such a machine. According to the Table 3, the non-working transitions are more than 93% from the total number of energy transitions. They can be either irreversible or revesible. If the first are made as the last, the engine efficiency can be increased from 6 up to 93.1%. So the aim of perfection in endoreversible solar-heat engine processes without work production is arisen. In ideal case it is necessary to

So, efficiencies of the combined endoreversible engines with the non-Carnot cycles and еlementary working bodies (matter or radiation) mostly depend on reversibility of processes without work production, running in the radiation absorber rather than on the work

endoreversible <6 < 7 >93 Non-equilibrium

heat 94.79 100 - equilibrium

heat 93.10-94.79 98.2 1.8 Non-equilibrium

*contributions of energy* 

*transitions, % matter and radiation* 

*state working non-working* 

contributions to solar energy conversion.

minimal efficiency of all solar-heat engines.

matter-radiant working body is not produced yet.

*at 320 K* 

*Engine Efficiency* 

endoreversible solar-heat engine.

Ideal non-Carnot solar-

Endoreversible solar-

Combined

non-working transitions

obtain Landsberg engine.

production processes. It follows that the contribution of reversible processes is smaller than the contribution of irreversible processes in devices that are currently used. According to the maximum work principle, their efficiency can be increased by increasing the fraction of reversible processes in solar energy absorbers. The following approach to solution of this problem can be suggested.
