Volume 1 photovoltaic solar energy 1 15 – thermodynamics of photovoltaics

Volume 1 photovoltaic solar energy 1 15 – thermodynamics of photovoltaics Volume 1 photovoltaic solar energy 1 15 – thermodynamics of photovoltaics Volume 1 photovoltaic solar energy 1 15 – thermodynamics of photovoltaics Volume 1 photovoltaic solar energy 1 15 – thermodynamics of photovoltaics Volume 1 photovoltaic solar energy 1 15 – thermodynamics of photovoltaics Volume 1 photovoltaic solar energy 1 15 – thermodynamics of photovoltaics

V Badescu, Polytechnic University of Bucharest, Bucharest, Romania

© 2012 Elsevier Ltd All rights reserved

1.15.1 Introduction

1.15.2 Thermodynamics of Thermal Radiation

1.15.2.1 Photon Gas

1.15.2.2 The Continuous Spectrum Approximation

1.15.2.3 Fluxes of Photon Properties

1.15.2.4 Spectral Property Radiances for Blackbodies and Bandgap Materials

1.15.2.5 Geometrical Factor of Radiation Sources

1.15.2.5.1 Isotropic radiation sources

1.15.2.5.2 Geometric factor of nonisotropic blackbody radiation sources

1.15.2.6 Diluted Thermal Radiation

1.15.3 Concentration of Solar Radiation

1.15.3.1 The Étendue of Beam Radiation

1.15.3.2 Upper Bounds on Beam Solar Radiation Concentration

1.15.3.3 Upper Bounds on Scattered Solar Radiation Concentration

1.15.4 Upper Bounds for Thermal Radiation Energy Conversion

1.15.4.1 Available Work of Enclosed Thermal Radiation

1.15.4.2 Available Work of Free Thermal Radiation

1.15.4.3 Available Work of Blackbody Radiation as a Particular Case

1.15.4.3.1 Upper bound for PT conversion efficiency

1.15.4.3.2 Upper bound for PV conversion efficiency

1.15.4.4 Discussion

1.15.5 Models of Monogap Solar PV Converters

1.15.5.1 Modeling Absorption and Recombination Processes

1.15.5.1.1 The solar cell equation

1.15.5.1.2 The Shockley–Queisser model

1.15.5.2 Modeling Multiple Impact Ionization

1.15.5.2.1 Solar cell efficiencies

1.15.5.2.2 Optimum voltage across solar cells

carriers

1.15.1 Introduction

short-wavelength solar radiation consists of direct and diffuse radiation The spectrum of the direct component is characterized by many dips, due to absorption by water vapor, oxygen, and other gases in the atmosphere The spectrum of the diffuse radiation contains less energy and has a narrower spread of wavelengths than that of the direct component Many calculations involving direct solar radiation can be made by using the blackbody spectra approximation rather than the correct solar spectrum Also, diffuse solar radiation is sometimes treated as diluted blackbody radiation

Solar energy transformation into other forms of energy involves interaction between the solar photons and the particles constituting the conversion devices The energy levels of these particles (e.g., electrons, holes, excitons, and phonons) are quantified and transition of particles to higher energy levels is allowed just for particular values of the energy of the incoming photons The

is practically transparent for the other incoming photons Also, transition of particles is allowed to lower quantified energy levels and the emitted photons have, accordingly, quantified energies, corresponding to the differences between the energy levels in the conversion device

Solar energy converters may be scholastically grouped into two categories: devices based on thermal processes and devices based on nonthermal processes Usually, the processes in the latter category of converters are called quantum processes, but this is rather inappropriate because both categories in fact involve quantum particles In the first category, most part of the solar energy

is transformed into internal energy of the body receiving radiation This way of dealing with solar energy is called photothermal (PT) conversion Very often, the body receiving radiation is a metal or an alloy The internal energy may be subsequently used directly (as in case of a device providing heat to an end user) may be stored or may be transformed into mechanical, electrical, or chemical work Converters based on quantum processes transform part of the energy of solar radiation directly into electrical energy (as happens in a photovoltaic cell) or store that energy in the form of chemical energy (as it happens in case of water photodissociation into oxygen and hydrogen) The first process is called photovoltaic (PV) conversion, while the second one is called photochemical (PC) conversion Most PV devices are built with energy bandgap materials like semiconductors Photosensible substances are used within PC conversion devices In this chapter, we consider mainly the thermodynamics of

PV solar energy conversion

quantities such as the continuous photon spectrum and photon fluxes are defined Also, the geometrical factors are introduced for both isotropic and nonisotropic radiation sources Solar radiation is sometimes concentrated before reaching the converter The

concentration ratio for both direct and diffuse solar radiation is treated there Upper bounds for the conversion efficiency of

for PV conversion The solar cell equation is derived from photon number fluxes arguments Models for omnicolor converters are

1 The single quantum state probabilities p(N1), p(N2),…, are independent, that is,

2 When a new photon is added to the system, the probability that this particle is in quantum state j is independent of the number

Through the normalization condition, one finds:

½2

½3

½4

1.15.2.2 The Continuous Spectrum Approximation

The distance between the photon energy levels decreases by increasing the volume V containing radiation The model of a continuous spectrum is often used when that distance is very small Integration replaces in this case the summation over quantum

A surface element is considered, which may be part of the surface of the volume V or may be placed inside that volume A flux of

the refractive index of the medium inside the volume equals unity Then, the number of photon quantum states in the frequency

cwhere c is the speed of light while l = 1 and l = 2 stand for polarized and unpolarized radiation, respectively The number of photons

lν2

~nν ≡ The internal energy of the radiation is given by

The quantities nν, uν, and sν are photon number, internal energy, and entropy densities, respectively Their units are number of photons, energy unit, or entropy unit, respectively, per unit volume, unit frequency, and unit solid angle

and [10] can be obtained by integration over photon energies In this case, the number of photon quantum states in the energy range is given by

h3c3

1.15.2.3 Fluxes of Photon Properties

Photons traveling in free space are carrying properties, such as the number of particles, energy, and entropy The thermodynamics of free thermal radiation is shortly presented here under the assumption of the continuous approximation

fluxes come to the simplest form:

which is the so-called geometric (or view) factor that captures the geometrical relation between the source and the receiver of

1.15.2.4 Spectral Property Radiances for Blackbodies and Bandgap Materials

c2

radiances, respectively, for blackbodies or bandgap materials

1.15.2.5 Geometrical Factor of Radiation Sources

PV converters may receive radiation from various sources A common case corresponds to a spherical source of radiation (e.g., the Sun) Generally, the incident radiation is nonisotropic, but the isotropic approximation is very often used as far as solar direct radiation is concerned

1.15.2.5.1 Isotropic radiation sources

We shall consider now isotropic radiation that is not necessarily blackbody The position of a typical luminous element of the

intersection of the plane (OC, OA) with the plane (x, y)

� �

subtending the sphere when viewed from the observer A common case corresponds to a spherical source of radiation (e.g., the Sun)

(Figure 1) Let n be a unit vector on the axis z0 normal to ∑ at a point O, and let c be the unit vector in the direction of the center C of

Next, note that

Thus

An expression for the geometrical factor

Otherwise, part of the emitting disc is cut off by the horizon

bifacial cell, they should add up to 2

1.15.2.5.2 Geometric factor of nonisotropic blackbody radiation sources

When nonisotropic sources of radiation are considered, strictly, a geometrical factor does not exist However, an average geometric factor can still be used, as shown below for the case of the Sun

It is known that the solar brightness falls considerably with the distance from the center of the disc This effect, which is referred

to as limb darkening, is a consequence of the fact that the Sun is not an isotropic source of radiation Several empirical correlations

5

δ ε

luminance L(ε) at zenith angle ε is included in the diagram R(θ′) is the radiance of the radiation incident under the angle θ′ on the receiver ∑

Figure 2, one obtains after some algebra

that the nonisotropically emitting Sun is equivalent to a star of about 80% smaller size, emitting isotropically

subtending the same solid angle In order that the energy flux received from both sources be equal, the temperature of the nonisotropic has therefore to exceed that of the isotropic source Finally, one should mention that a more accurate description of

which is often used in PV efficiency calculations

1.15.2.6 Diluted Thermal Radiation

Entropy and energy fluxes of diluted radiation may be written as

3

� �

subtended by the source of radiation by

with

next is used

radiation scattering is forward (in other words, there is no backscattered radiation) Then, we define a perfectly forward

Until now, we have analyzed the scattered radiation from the point of view of an observer situated on the surface of the diffuser

derived from the relation:

We obtain

the cone subtended by the Sun)

The second scattering is described next This case implies the existence of a second perfectly forward diffuser The incoming

As we see, the doubly scattered solar radiation is still anisotropic

The above procedure can be repeated for three and four scatterings with the following results:

number of three scatterings

�

Table 1 Maximum efficiency of singly or multiply scattered solar radiation

Number of scatterings, i

Te,j (K)

0.265 2.083 12.609 30.961

90

45963 189.5 5.5 1.3

1

a Value computed from eqn [51]

dilution factor

Te, the effective temperature of scattered radiation;

subtends a blackbody at temperature Te; Cmax

1.15.3 Concentration of Solar Radiation

A solution to increase the useful energy flux provided by the solar energy conversion system is to concentrate the incoming solar radiation Another solution is to increase the surface area of the absorber The former solution has the advantage that it yields an increase of the conversion efficiency and that the cost per unit surface area of the concentrating device is smaller than that of the absorber Also, radiation concentration allows obtaining a higher absorber temperature

Two types of solar concentrators are often used in practice: three-dimensional (3D) and two-dimensional (2D) concentrators In

a 3D concentrator, the transversal surface area of the incident beam is diminished on two perpendicular directions The beam radiation is concentrated into a spot, which ideally reduces to a point In a 2D concentrator, the transversal surface area is diminished on a single direction The beam radiation is concentrated into a strip, which ideally reduces to a line

There is an upper limit for solar radiation concentration In this section, we show how this limit may be theoretically derived for both 3D and 2D concentrators

1.15.3.1 The Étendue of Beam Radiation

Figure 3 shows an optical system consisting of two homogeneous media [24] A light incident ray passing through the point

changes in emitted ray direction exhibit on the emerging ray These small displacements and direction changes make the beam of emerging rays to have a certain transversal surface area and angular extent The refractive index of the two media is

The small variations introduced above allow to define the infinitesimal variation of the Lagrangean U for a bundle of rays in a medium of refractive index n:

y v

z O

one denotes by dA the area of the emitting surface Then, the following relation applies

A classical result of geometrical optics states that the Lagrangean U is conserved for those ray beams free from energy losses

1.15.3.2 Upper Bounds on Beam Solar Radiation Concentration

have generally different values When radiation passes through interfaces separating two optical media with different refractive

radiation beam is given by

[85] and [86] yield

example, the maximum concentration ratio for a 3D concentrator in case a more accurate value is adopted for the half-angle of the

concentrator may exceed 1 00 000 This yields at concentrator exit an energy flux density higher than on the surface of the Sun, which

1.15.3.3 Upper Bounds on Scattered Solar Radiation Concentration

in ¼AðΩiÞεiσTe4 ;i ¼ 0 ¼ Ω0

π

on the receiver of a concentrator is

where C is the concentration ratio

To determine the maximum concentration ratio of a i times scattered diffuse radiation, we must observe that fully concentrated

Results are shown in Table 1 [18] The maximum concentration ratio decreases by increasing the number of scatterings, as expected

Of course, fully isotropic diffuse radiation cannot be concentrated

1.15.4 Upper Bounds for Thermal Radiation Energy Conversion

The maximum conversion efficiency of thermal radiation energy into work has been often derived from available work (exergy) considerations The vast majority of authors considered the case of blackbody radiation (i.e., thermal radiation with zero chemical potential) The diversity of results generated a long-term debate in literature Three efficiency-like factors affecting the radiation

In this section, a more general theory of radiation energy conversion into work is developed by using a simple statistical

1.15.4.1 Available Work of Enclosed Thermal Radiation

A (fixed) volume V containing thermal radiation is considered and the continuum spectrum hypothesis is adopted The energy

1.15.2.2, dNe denotes the number of photons in V with energy between e and e + de Equation [12] shows that dNe is given by the

The number of energy states of enclosed thermal radiation equals the number of vibration modes in volume V and is given in

64π4

c3ℏ3

Here the extensivity assumption was adopted, which means interactions are weak In some cases, not all the energy states are occupied

The number of particles N in volume V and their (internal) energy U is obtained after integration over all energy levels Results

Table 2 In the particular case of a blackbody radiation spectrum (eg = 0, μ = 0, and em → ∞), the extensive thermodynamic functions depend just on V and T

i3ðxμ; xg ; xmÞ þ j2ðxμ; xg; xmÞ −i2ðxμ; xg; xmÞxμ

i3ðxμ; xg; xmÞ

Table 2, that is, e ¼ U=N ¼ ½i3ðxμ; xg; xmÞ=i2ðxμ; xg; xmÞðkTÞ In the particular case of blackbody radiation (eg = 0, μ = 0, and em → ∞),

The available work is generally defined as the maximum work extractable from a system by bringing it to mechanical, thermal, and chemical equilibrium with the environment The environment (subscrip 0) is defined as consisting of m given constituents of

also used

The chemical equilibrium condition is not trivial in the present case To avoid this obstacle, we neglect most systems constituting the environment, which is simply reduced to the thermal radiation it contains The same way of defining the environment has been

However, in most cases, the equilibrium radiation in real-world environments is different from blackbody radiation Indeed, the equilibrium radiation frequency distribution depends strongly on the nature of the other components of the environment, with which radiation interacts A simple example is the equilibrium radiation of an environment consisting of a gray body inside an

� �

simple example Also, common experience shows that the environment bodies emitting radiation are usually not covering the whole hemisphere This makes the environment radiation to be nonisotropic in general The academic equilibrium radiation is a particular case of the equilibrium thermal radiation in a real-world environment (i.e., it is associated with equilibrium radiation emitted by a single hemispherical blackbody)

The radiation in volume V reaches a dead state when equilibrated with the environment The chemical potential and temperature

Table 2 for T = T0, μ = μ0, eg= eg0, and em= em0 shows that the dead-state Gibbs free energy G0 ≡ U0 − T0S0+ p0V0= μ0N0, as expected

temperatures and the initial and dead-state volumes This factor is different from the usual Carnot factor appearing in the available work carried by heat

Two cases, denoted 1 and 2 in the following, may be of interest:

j2ðxμ; xg; xmÞ T0 i3ðxμ0; xg0; xm0Þ þ j2ðxμ0; xg0; xm0Þ T0

the environment radiation have the same chemical potential (a zero chemical potential is usually considered) Second, the same spectrum dependence on temperature and chemical potential applies to both the radiation and the environment

temperature may still provide (mechanical, electrical, or chemical) work A well-known example is absorption of radiation in a bandgap material followed by electric carrier separation in the valence and conduction bands, which is associated with storage of Gibbs free energy

1.15.4.2 Available Work of Free Thermal Radiation

A simple statistical model follows It is inspired by the usual nonequilibrium thermodynamics treatment of free non-interacting particles where intensive parameters such as (effective) temperature, pressure, and chemical potential are still in use (see Section

1.15.2.4) Definitions for the most important flux quantities are shown in Table 3 in the energy representation There, θ and λ are

photon:

Table 3 Quantities related to the flux of property Ξ

Spectral property radiance

Property flux density for solid angle dΩ and energy interval de

Property spectral flux density

Property flux density

m2

Property units

m2 J Property units

m2

Ξ may be the number of particles N, the energy U, and the entropy S

Z

Ω

show that the particle number flux and the energy of thermal radiation, respectively, are proportional to the third and fourth power

of the effective temperature T, respectively, as known from the case of blackbody radiation Use of the equations in column 3 of rows

A simple model is developed here to evaluate the available work of free thermal radiation It is formally similar to the model of

Particle number flux density Energy flux density Entropy flux density Helmholtz free energy flux density

1.15.4.3 Available Work of Blackbody Radiation as a Particular Case

solar radiation is often modeled as blackbody radiation

Two different particular subcases will be treated Each case is characterized by its own (unique) dead state In the first subcase, the dead state consists of radiation of zero chemical potential In the second subcase, the dead state corresponds to luminescence radiation (nonzero chemical potential) Both cases may be used to describe, more or less accurately, situations existing in natural world In the more limited world of man-made devices, the dead state of the first subcase is similar to the radiation emitted by a thermal converter, while the dead state of the second subcase may be assimilated to the radiation emitted by a common single gap solar cell Therefore, the two available work factors derived below represent upper bounds for the efficiency for PT and PV conversion of radiation energy into mechanical and electrical work, respectively

1.15.4.3.1 Upper bound for PT conversion efficiency

means that the incident blackbody radiation is finally degraded to radiation emitted by a blackbody receiver and the available work derived below corresponds to the maximum work extractable from solar radiation by using a PT (nonselective) blackbody

available work factor is given by

Figure 5 shows that the available work factor η ′

1.15.4.3.2 Upper bound for PV conversion efficiency

environment filled with radiation emitted by a bandgap material, for example Thus, the available work factor derived below

1.0

0.05

T0/T

0.1 0.3 0.5 0.7

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

(a) Case of PT conversion: η ′ JX is given by eqn [117]; (b) case of PV conversion: η ′ JX is given by eqn [118] (computations performed for xμ0 = 0.9xg0)

eqn [115], one finds

Figure 6(b) shows the dependence of η′ J X on the radiation temperature T, for a particular value of the environment temperature T0

1.15.4.4 Discussion

containing the dead-state temperature and the radiation temperature, among other factors

dead-state radiation subtending a solid angle narrower than that of the radiation source)