|
particles and processes, and the annihilation of the grand canonical potential
current density, the term corresponding to the other elements is more conventionally
written as
˙
E
others
=
T
a
˙
S
others
.
= ˙
Q
(
4
.
40
)
where
˙
Q
is defined as the rate of heat leaving the converter.
130
THEORETICAL LIMITS OF PHOTOVOLTAIC CONVERSION
The second law of thermodynamics, expressed by equations (4.8) and (4.10), inte-
grated into the whole volume of the converter can be written, for the stationary case, as
˙
S
irr
=
U
σ
irr
d
U
=
A
i
j
s,i
d
A
= ˙
S
r
− ˙
S
s
+ ˙
S
mo
− ˙
S
mi
+ ˙
S
others
(
4
.
41
)
The irreversible rate of entropy production is obtained by the elimination of the terms
with subscript
others
by multiplying equation (4.41) by
T
a
, subtracting equation (4.39)
from the result and taking into account (4.40). In this way we obtain
T
a
˙
S
irr
=
(
˙
E
s
−
T
a
˙
S
s
)
−
(
˙
E
r
−
T
a
˙
S
r
)
+
(
˙
E
mi
−
T
a
˙
S
mi
)
−
(
˙
E
mo
−
T
a
˙
S
mo
)
(
4
.
42
)
From equation I-4 (Table 4.1) and considering the annihilation of the grand canonical
potential flow for electrons,
(
˙
E
mi
−
T
a
˙
S
mi
)
=
ε
F v
˙
N
mi
and
(
˙
E
mo
−
T
a
˙
S
mo
)
=
ε
F c
˙
N
mo
, so
that we can write
T
a
˙
S
irr
=
ε
F v
˙
N
mi
−
ε
F c
˙
N
mo
+
(
˙
E
s
−
T
a
˙
S
s
)
−
(
˙
E
r
−
T
a
˙
S
r
)
(
4
.
43
)
Taking into account that
˙
N
mi
= ˙
N
mo
=
I /q
and
ε
F c
−
ε
Fv
=
qV
, equation (4.43) is now
rewritten as
T
a
˙
S
irr
= − ˙
W
+
(
˙
E
s
−
T
a
˙
S
s
)
−
(
˙
E
r
−
T
a
˙
S
r
)
(
4
.
44
)
Here, this equation has been derived from the
local
model. However, it can also be
obtained with more generality from a classical formulation of the second law of ther-
modynamics [32]. It is valid for ideal as well as for non-ideal devices. The values
of the thermodynamic variables to be used in equation (4.44) are given in Table 4.1.
Power – which in other cases will be that of the converter under study – corresponds
in this case to the power of an SQ ideal solar cell and is given by the product of
equations (4.18) and (4.19).
We have already discussed the basic ambiguity for the thermodynamic description
of any radiation concerning the choice of the temperature and the chemical potential. A
useful corollary is derived from this fact [32]. If the power rate produced by a radiation
converter depends on the radiation only through its rate of incident energy or number of
photons, then any radiation received or emitted by the converter can be changed into a
luminescent radiation at room temperature,
T
a
, and with chemical potential,
µ
x
, without
affecting the rate of power and of irreversible entropy produced. The chemical potential
µ
x
of this equivalent luminescent radiation is linked to the thermodynamic parameters of
the original radiation,
T
rad
and
µ
rad
, through the equation
ε
−
µ
rad
kT
rad
=
ε
−
µ
x
kT
a
⇒
µ
x
=
ε
1
−
T
a
T
rad
+
µ
rad
T
a
T
rad
(
4
.
45
)
Note that, in general,
µ
x
is a function of the photon energy,
ε
, as it may also be
T
rad
and
µ
rad
.
THE TECHNICAL EFFICIENCY LIMIT FOR SOLAR CONVERTERS
131
The proof of the theorem is rather simple and is brought in here because it uses
relationships of instrumental value. It is straightforward to see that
˙
n
x
= ˙
n
rad
;
˙
e
x
=
ε
˙
n
x
= ˙
e
rad
;
˙
ω
x
/T
a
= ˙
ω
rad
/T
rad
;
˙
e
rad
−
T
˙
s
rad
=
µ
x
˙
n
x
+ ˙
ω
x
(
4
.
46
)
where, again, the suffix
rad
labels the thermodynamic variables of the original radiation.
The equality of the energy and the number of photons of the original and equivalent
radiation proves that the power production is unchanged. Furthermore, the last relationship
can be introduced in equation (4.44) to prove that the calculation of the entropy production
rate also remains unchanged.
Taking all this into account, the application of equation (4.44) to the SQ solar cell
is rather simple. Using the SQ cell model for the power and using the room-temperature
equivalent radiation (of chemical potential
µ
s
) of the solar radiation, we obtain
T
a
˙
S
irr
= −
∞
ε
g
qV
[
˙
n(T
a
, µ
s
)
− ˙
n(T
a
,
qV
)
] d
ε
+
∞
ε
g
[
µ
s
˙
n(T
a
, µ
s
)
+ ˙
ω(T
a
, µ
s
)
] d
ε
−
∞
ε
g
[
qV
˙
n(T
a
,
qV
)
+ ˙
ω(T
a
,
qV
)
] d
ε
=
∞
ε
g
[
˙
ω(T
a
, µ
s
)
− ˙
ω(T
a
,
qV
)
] d
ε
+
∞
ε
g
[
(µ
s
−
qV
)
˙
n(T
a
, µ
s
)
] d
ε
(4.47)
The integrand is zero for
qV
=
µ
s
, but as
µ
s
varies with
ε
it is not zero simultaneously for
all
ε
. To prove that it is positive, we differentiate with respect to
qV
. Using d
˙
ω/
d
µ
= −˙
n
,
d
(T
a
˙
S
irr
)
d
qV
= −
∞
ε
g
[
˙
n(T
a
, µ
s
)
− ˙
n(T
a
,
qV
)
]
=
I /q
(
4
.
48
)
Thus, for each energy, the minimum of the integrand also appears for
qV
=
µ
s
(ε)
. Since
this minimum is zero, the integrand is non-negative for any
ε
and so is the integral, proving
also the thermodynamic consistence of this cell from the integral perspective. Furthermore,
the minimum of the entropy, which for the non-monochromatic cell is not zero, occurs
for open-circuit conditions. However, the ideal monochromatic cell reaches zero entropy
production, and therefore reversible operation, under open-circuit conditions, and this is
why this cell may reach the Carnot efficiency, as discussed in a preceding section.
4.4 THE TECHNICAL EFFICIENCY LIMIT FOR SOLAR
CONVERTERS
We have seen that with the technical definition of efficiency – in whose denominator the
radiation returned to the sun is not considered – the Carnot efficiency cannot be reached
with a PV converter. What then is the technical efficiency upper limit for solar converters?
We can consider that this limit would be achieved if we could build a converter
producing zero entropy [33]. In this case, the power that this converter can produce,
132
THEORETICAL LIMITS OF PHOTOVOLTAIC CONVERSION
˙
W
lim
, can be obtained from equation (4.44) by setting the irreversible entropy-generation
term to zero. Substituting the terms involving emitted radiation by their room-temperature
luminescent equivalent, equation (4.44) becomes
˙
W
lim
=
∞
ε
g
{
[
˙
e
s
−
T
a
˙
s
s
]
−
[
µ
x
(ε)
˙
n
x
+ ˙
ω
x
]
}
d
ε
=
∞
ε
g
˙
w
lim
(ε, µ
x
)
d
ε
(
4
.
49
)
The integrand should now be maximised [32] with respect to
µ
x
. For this, we calculate
the derivative
d
˙
w
lim
d
µ
x
= −˙
n
x
−
d
˙
ω
x
d
µ
x
−
µ
x
d
˙
n
x
d
µ
x
= −
µ
x
d
˙
n
x
d
µ
x
(
4
.
50
)
where we have used that the derivative of the grand potential with respect to the chemical
potential is the number of particles with a change of sign. This equation shows that the
maximum is achieved if
µ
x
=
0 for any
ε
, or, in other words, if the radiation emitted is
a room-temperature thermal radiation. This is the radiation emitted by all the bodies in
thermal equilibrium with the ambient. However, the same result will be also achieved if
the emitted radiation is any radiation whose room-temperature luminescent equivalent is
a room-temperature thermal radiation.
Now, we can determine this efficiency, according to Landsberg [33], as
η
=
H
sr
π
σ T
4
s
−
4
3
σ T
a
T
3
s
−
σ T
4
a
−
4
3
σ T
4
a
(H
sr
/π )σ T
4
s
=
1
−
4
3
T
a
T
s
+
1
3
T
a
T
s
4
(4.51)
which for
T
s
=
6000 K and
T
=
300 K gives 93.33% instead of 95% of the
Carnot efficiency.
No ideal device is known that is able to reach this efficiency. Ideal solar thermal
converters, not considered in this chapter, have a limiting efficiency of 85.4% [34, 35]
and therefore do not reach this limit. Other high-efficiency ideal devices considered in
this chapter do not reach it either. We do not know whether the Landsberg efficiency
is out of reach. At least it is certainly an upper limit of the technical efficiency of any
solar converter.
Do'stlaringiz bilan baham: |
