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While the Clausius-Clapeyron equation is very important as it determines the saturation vapour pressure, in practice it is replaced by empirical, typically Magnus type, equations which are more accurate. It is shown that the reduced accuracy reflects an inconsistent assumption that the latent heat of vaporization is constant. Not only is this assumption unnecessary and excessive, but it is also contradictory to entropy maximization. Removing this assumption and using a pure entropy maximization framework we obtain a simple closed solution, which is both theoretically consistent and accurate. Our discussion and derivation are relevant to students and specialists in statistical thermophysics and in geophysical sciences, and our results are ready for practical application in physics as well as in such disciplines as hydrology, meteorology and climatology.
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Clausius-Clapeyron equation and saturation vapour pressure:
simple theory reconciled with practice
Demetris Koutsoyiannis
Department of Water Resources and Environmental Engineering, School of Civil
Engineering, National Technical University of Athens, Greece
E-mail: dk@itia.ntua.gr; URL: http://www.itia.ntua.gr/dk
Abstract. While the Clausius-Clapeyron equation is very important as it determines
the saturation vapour pressure, in practice it is replaced by empirical, typically
Magnus type, equations which are more accurate. It is shown that the reduced
accuracy reflects an inconsistent assumption that the latent heat of vaporization is
constant. Not only is this assumption unnecessary and excessive, but it is also
contradictory to entropy maximization. Removing this assumption and using a pure
entropy maximization framework we obtain a simple closed solution, which is both
theoretically consistent and accurate. Our discussion and derivation are relevant to
students and specialists in statistical thermophysics and in geophysical sciences, and
our results are ready for practical application in physics as well as in such disciplines
as hydrology, meteorology and climatology.
Keywords: entropy, maximum entropy, saturation vapour pressure, Clausius-Clapeyron
equation, Magnus equation.
1. Introduction
The Clausius–Clapeyron relationship characterizes the transition between two phases of matter. The
importance of this equation cannot be overemphasized and transcends thermodynamics and physics.
As it determines the saturation vapour pressure for water, it provides the physical basis of the
hydrological cycle and becomes a principal equation in hydrology, meteorology, climatology, and
other geophysical sciences. Specifically, the saturation vapour pressure, also known as equilibrium
vapour pressure, is an upper limit of the quantity of vapour that the atmosphere can contain. When
this limit is reached, no additional liquid water is evaporated, while below the limit more water
evaporates. This limit is expressed in terms of the partial pressure of the vapour. At standard
temperature and pressure (STP) conditions, i.e. at a temperature of 273.15 K (0°C) the saturation
vapour pressure is 6.11 hPa, i.e. 0.611% of the total pressure of 1000 hPa. The saturation vapour
pressure increases at higher temperatures, e.g. at 25°C it is over five times higher. Conversely, when
moist air ascends and its temperature decreases, so does the saturation vapour pressure. Vapour in
excess of the lower saturation pressure starts to condense, giving rise to the formation of clouds.
The Clausius-Clapeyron equation derives from entropy maximization, which determines the
equilibrium between two phases of a substance. Mathematically it is expressed as the relationship
between temperature, T, and pressure, p, at the equilibrium. Usually it is expressed in differential form
[e.g. 1, 2; see also below]. However, in addition to the differential form, most texts [e.g. 3, p. 98; 4, p.
180; 5, p. 612; 6, p. 300] as well as electronic sources, provide an analytical equation written as
p = CT
1
RT
L
exp (1)
where CT
1
is an integration constant (generally, we use the symbol CT
i
to denote integration
constants), L is the latent heat of vaporization and R is the specific gas constant. This is supposed to
be the integration of the differential form under the assumption that L is constant. However, as will be
shown, this assumption is inconsistent with the entropy maximization framework and, as a result, (1)
is not an integration of the Clausius-Clapeyron equation.
Corrected postprint, 2012-04-16
2
Furthermore, due to the inconsistent assumption, (1) is not accurate enough as an approximation to
be used in practice. Therefore, more accurate empirical relationships are more often used to determine
the saturation vapour pressure. The so-called Magnus-type equations are the most typical in
application and the most recent version [7] for temperature range –40
o
to 50
o
C is
p = 6.1094 exp
+T
T
04.243
625.17 [T in
o
C, p in hPa] (2)
A theoretically consistent closed solution exists [8, p. 203; see also below] and is not much more
complex than (1), but it is rarely mentioned (e.g. none of the above referenced books contains it).
Even when it is mentioned, it is still presented along with (1), which may again be characterized as the
best known approximation to calculate the liquid–vapour equilibrium pressure [9].
Here we demonstrate the inconsistency inherent in the assumptions that are made to derive (1) and
we derive a theoretically consistent closed solution, which is in line with the aforementioned existing
consistent solution, removing incorrect assumptions. We also determine its numerical constants for
the phase change of water, thus providing an expression ready for use in practice, and we show that
numerically its error with respect to standard reference data is negligible, smaller than that of (2) and
spectacularly smaller than that of (1). In all this, the underpinning logic is that the principle of
maximum entropy, which is a variational principle, is economic in assumptions needed:
Mathematically, there is no limit to the number of unknowns that can be determined in a
maximization problem (as compared to one formulated in terms of equations, where the number of
unknowns should equal the number of equations). Thus, we should be able to determine any unknown
quantity without assumptions.
2. The inconsistency
It is well known [e.g. 1, p. 143] that in two systems at equilibrium, entropy maximization constrained
with the conditions that the total energy, volume and number of particles are constant, results,
respectively, in temperature, T, pressure, p, and chemical potential, µ, that are equal in the two
systems. In particular, the last equality is
µ
G
= µ
L
(3)
where the subscripts G and L denote the gaseous and liquid phase, respectively. Classical and
statistical thermodynamics texts do not use (3) directly; rather, they derive, as a consequence of (4),
and use the equality of differentials [e.g. 2. p. 71],
dµ
G
= dµ
L
(4)
The entropies per unit mass s
G
and s
L
at the gaseous and liquid phase, respectively, are given as
s
G
= CT
2
+ c
p
ln TR ln p (5)
s
L
= CT
3
+ c
L
ln T (6)
where c
p
is the specific heat at constant pressure of the vapour and c
L
is the specific heat of the liquid.
The liquid was regarded as incompressible and, for this reason, in (6) the entropy does not depend on
pressure. The differentials of the chemical potentials are given by Gibbs-Duhem equations [e.g. 2] as
dµ
G
= v
G
dps
G
dT (7)
dµ
L
= – s
L
dT (8)
where again in (8) we neglected the specific volume v
L
of the liquid phase. Combining (4), (7) and (8)
we obtain
T
p
d
d
=
G
v
ss
LG
(9)
If we express the entropy difference in terms of the latent heat as
3
s
G
s
L
=
T
L
(10)
then we obtain the following typical form of the Clausius-Clapeyron equation,
T
p
d
d
=
G
Tv
L
(11)
Furthermore, using the ideal gas law, pv
G
= RT, we obtain the expression
T
p
d
d
=
2
RT
Lp
(12)
whose integration assuming constant L results in (1).
In fact, however, assuming a constant L renders the above derivation unnecessary. Indeed, we can
express the difference of entropies from (5) and (6) as
s
G
s
L
= (CT
2
CT
3
) + (c
p
c
L
) ln TR ln p (13)
Substituting into (10) and solving for p we readily obtain
p = CT
4
Rcc
p
T
RT
L
/)(
L
exp
(14)
where CT
4
:= exp[(CT
2
CT
3
)/R]. Equations (1) and (14) describe the same relationship of p and T,
and were derived by precisely the same assumptions. However they are clearly inconsistent, which
implies that at least one of the assumptions made is excessive and incorrect.
3. Alternative derivation using classical thermodynamics
In this section we will remove the assumption for constant L. We express the difference of entropies
in (13) as an unknown function g(T, p) := s
G
s
L
and we write (13) in differential form as
(c
p
c
L
)
T
T
d
R
p
p
d
= dg (15)
Using the law of ideal gases to eliminate v
G
from (9), we obtain
T
p
d
d
=
RT
p
(s
G
s
L
) =
RT
pg
(16)
Solving (16) for dp/p and substituting it in (15) we obtain
(c
p
c
L
)
T
T
d
g
T
T
d
= dg (17)
which can be written as
(c
p
c
L
) dT = d(Tg) (18)
and can be readily solved to give
g = s
G
s
L
=
T
α
– (c
L
c
p
) (19)
where α is an integration constant. Comparing (19) with (10), we conclude that L appears to be a
linear function of temperature, rather than a constant:
L = α – (c
L
c
p
)T (20)
Now, substituting g from (19) to (16) we obtain
T
p
d
d
=
RT
p
(c
p
c
L
) +
2
RT
pα
(21)
4
This is readily solved to give
p = CT
5
Rcc
p
T
RT
α
/)(
L
exp
(22)
Comparing (21) with the earlier results, we observe that it is functionally equivalent with (14) (both
include a multiplicative factor that is a power function of T) whereas (1) proves to have an
inappropriate functional form.
To eliminate CT
5
from (22) we assume a known saturation vapour pressure p
0
at a specific
temperature T
0
. We can then write (22) as
p = p
0
T
T
RT
α
0
0
1exp
Rcc
p
T
T
/)(
0
L
(23)
which is our final closed solution of the Clausius-Clapeyron equation.
4. Alternative derivation in a statistical mechanical framework
In this section we derive the phase transition equation in a purely statistical mechanical framework
totally avoiding the assumptions about the equality of chemical potentials and temperatures; rather we
will derive them by entropy maximization. For this maximization we assume that our system contains
a total of N particles, N
G
of which in the gaseous phase and N
L
in the liquid phase, so that
N
G
+ N
L
= N (24)
If S denotes the total extensive entropy, and S
G
and S
L
denote the extensive entropy in the gaseous
and liquid phase, respectively, then,
S = S
G
+ S
L
= N
G
s
*
G
+ N
l
s
*
L
(25)
where s
*
G
and s
*
L
denote entropies per particle. From generalized Sackur-Tetrode equations [e.g. 2] we
have
s
*
G
/k = CT
6
+ (c
v
/R) ln (E
G
/N
G
) + ln (V/N
G
), s
*
L
/k = CT
7
+ (c
L
/R) ln (E
L
/N
L
) (26)
where again we neglected the volume per particle in the liquid phase, which is by several orders of
magnitude smaller than that of the gaseous phase. In (26) k is Boltzmann’s constant and c
v
is the
specific heat at constant volume of the vapour. We recall that c
v
= c
p
R and that the quantity 2c
v
/R
represents the degrees of freedom available to the molecular (thermal) motion. In addition, E
G
and E
L
in (26) are the thermal energies in the two phases. If E is the total energy, then conservation of energy
demands
E
G
+ E
L
+ N
G
ξ = E (27)
where ξ is the amount of energy per molecule required to break the bonds between molecules of the
liquid phase in order for the molecule to move to the gaseous phase, which we assume constant.
We wish to find the conditions which maximize the entropy S in (25) under constraints (24) and
(27) with unknowns E
G
, E
L
, N
G
, N
L
. We form the function Ψ incorporating the total entropy S as well
as the two constraints with Langrage multipliers κ and λ:
Ψ = (S
G
+ S
L
)/k + κ (E
G
+ E
L
+ N
G
ξE) + λ (N
G
+ N
L
N) (28)
To maximize Ψ, we first take the derivatives with respect to E
G
and E
L
and equate them to 0 to obtain
G
Ε
Ψ
=
1
G
G
Ε
S
+ κ = 0,
L
Ε
Ψ
=
1
L
L
Ε
S
+ κ = 0 (29)
We recall that in statistical thermodynamics the temperature is defined as
T
1 :=
Ε
S
(30)
5
Thus, (29) results in
κ = –
G
1
kT = –
L
1
kT = –
kT
1 (31)
In other words, it was proved that the temperatures in the two phases are equal.
Furthermore, taking the derivatives of Ψ with respect to N
G
and N
L
, and equating them to 0 we
obtain
G
N
Ψ
=
s
*
G
R
c
v
– 1 + κξ + λ = 0,
L
N
Ψ
=
s
*
L
R
c
L
+ λ = 0 (32)
Eliminating λ, substituting κ from (31) and c
v
from c
v
= c
p
R, and making algebraic manipulations,
we find:
ss
*
L
*
G
=
kT
ξ
R
cc
p
L
(33)
On the other hand, from (26), observing that E
G
/N
G
and E
L
/N
L
are both proportional to T, while V/N
G
is proportional to T/p, we also obtain the difference of entropies per particle as:
ss
*
L
*
G
= CT
8
R
cc
p
L
ln T – ln p (34)
Combining (33) and (34), eliminating s
*
G
s
*
L
, and solving for p we find
ln p = CT
9
kT
ξ
R
cc
p
L
ln T (35)
Now if we introduce α = ξR/k (= ξN
a
, where N
a
is the Avogadro constant) and take antilogarithms,
then we obtain (22) again, which was our desideratum.
To finish this analysis, we will show the equality of chemical potentials (although the chemical
potential was not involved at all in the above proof). We recall that the chemical potential is by
definition:
T
µ
:=
N
S
= s
*
+ N
N
s
*
(36)
where the partial derivative applies for constant internal energy. Applying this definition in the two
phases and using (26) we find,
T
µ
G
= s
*
G
R
kc
v
k , –
T
µ
L
= s
*
L
R
kc
L
+ ξ
Τ (37)
where the last term (ξ/T) in the second equation represents the conversion from constant thermal
energy E
L
to constant internal energy U
L
= E
L
N
L
ξ. Specifically, this term represents the quantity
(∂S
L
/∂E
L
)(∂E
L
/∂N
L
) = (1/T) ξ. This gives the difference of chemical potentials as
T
µµ
GL
= s
*
G
s
*
L
+
R
cck
p
)( L
ξ
Τ (38)
Combining (33) and (38) we find
µ
L
µ
G
= 0 (39)
5. Application to water vapour
We choose as reference point the triple point of water, for which it is known with accuracy that T
0
=
273.16 K (= 0.01
o
C) and p
0
= 6.11657 hPa [10]. The specific gas constant of water vapour is R =
461.5 J kg
–1
K
–1
. The specific heat of water vapour at constant pressure, again determined at the triple
6
point, is c
p
= 1884.4 J kg
–1
K
–1
and that of liquid water is c
L
= 4219.9 J kg
–1
K
–1
[10], so that c
L
c
p
=
2335.5 J kg
–1
K
–1
and (c
L
c
p
)/R = 5.06.
The latent heat at T
0
is L
0
= 2.501 × 10
6
J kg
–1
so that α = L
0
+ (c
L
c
p
)T
0
= 3.139 × 10
6
J kg
–1
and
ξ / kT
0
= α / RT
0
= 24.9. According to (20), this results in the functional form
L [J kg
–1
] = α – (c
L
c
p
)T = 3.139 × 10
6
– 2336 T [K] (40)
It can be readily verified that this is very close to a commonly suggested (e.g. [11]) empirical linear
equation for latent heat, i.e.,
L [J kg
–1
] = 3.146 × 10
6
– 2361 T [K] ( = 2.501 × 10
6
– 2361 T
C
[
o
C]) (41)
Figure 1 provides a graphical comparison of equation (40) with (41), as well as with tabulated data
from Smithsonian Meteorological Tables [12], which agree with the equations. Furthermore, it is
important to know that the entropic framework which gives the saturation vapour pressure is the same
framework that predicts the relationship of the latent heat of vaporization with temperature.
2.35
2.4
2.45
2.5
2.55
2.6
2.65
-40 -30 -20 -10 0 10 20 30 40 50
Temperature (°C)
Latent heat (MJ/kg)
Theoretically derived
Observed (tabulated)
Common empirical
Figure 1. Comparison of latent heat of water as given by equation (40) proposed in this
study with the empirical equation (41) and with standard tabulated data from ref. [12].
Now, according to (23), the saturation vapour pressure will be
p = p
0
T
T
0
1921.24exp
06.5
0
T
T
, with T
0
= 273.16 K, p
0
= 6.11657 hPa. (42)
where we have slightly modified the last two decimal digits of the constant α / RT
0
to optimize its fit
to the data (see below). For comparison, the inconsistent version (1) for constant L = L
0
is
p = p
0
T
T
0
184.19exp
, with T
0
= 273.16 K, p
0
= 6.11657 hPa. (43)
Equations (42) and (43), if plotted on a p vs. T graph, seem indistinguishable from each other as
well as from the Magnus-type equation (2) (Figure 2). However, because p ranges at several orders of
magnitude, the plot of Figure 2 is misleadingly hiding the differences between the different equations.
The maximum relative difference of the proposed equation (42) with respect to (43) exceeds 7%,
while that with respect to (2) is much lower, 0.29%. It is thus more informative to compare the three
equations in terms of relative differences and also to compare them to data rather than to intercompare
to each other.
7
0.1
1
10
100
1000
-40 -30 -20 -10 0 10 20 30 40 50
Temperature (°C)
Saturation vapour pressure (hPa)
Magnus type (by Alduchov & Eskridge)
Clausius-Clapeyron as proposed
Clausius-Clapeyron for constant L
Figure 2. Comparison of saturation vapour pressure obtained by the proposed equation
(42), by the Magnus-type equation (2) from ref. [7], and by the standard but inconsistent
equation (43) for constant L.
-1
0
1
2
3
4
5
6
7
8
-40 -30 -20 -10 0 10 20 30 40 50
Temperature (°C)
Relative difference (%)
Proposed vs. IAPW S Standard vs. IAPW S
Proposed vs. ASHRAE Standard vs. ASHRAE
Proposed vs. Smiths. Standard vs. Smiths.
Proposed vs. W MO Standard vs. WMO
Figure 3. Comparison of relative differences of the saturation vapour pressure obtained
by the proposed equation (42), as well as by the standard but inconsistent equation (43),
with data of different origins (see text).
8
Table 1 Maximum values of the relative differences from data of the saturation vapour
pressure obtained by three different equations (see text).
Standard (inconsistent)
equation (43)
Magnus-type
equation (2)
Proposed
equation (42)
Difference from IAPWS data 6.8% 0.27% 0.07%
Difference from all data 7.6% 0.39% 0.15%
For the comparisons four reference data sets have been used, which are given in tabulated form
from different origins: (a) the International Association for the Properties of Water and Steam
(IAPWS), (b) the Smithsonian Meteorological Tables (Smiths.), (c) the World Meteorological
Organization (WMO) meteorological tables, and (d) the American Society of Heating, Refrigerating
and Air-conditioning Engineers (ASHRAE). The temperature domain of the comparison extends from
–40
o
to 50
o
C, that is the typical range used in hydrometeorological applications. The data set (a),
taken from ref. [10], contains values of saturation vapour pressure for temperatures higher than the
water triple point (273.16 K). The other three data sets, all taken from ref. [13], contain values also
below triple point; such temperatures prevail in the upper air and the saturation vapour pressure in
such temperatures is necessary in order to estimate the relative humidity of the atmosphere. It is
clarified that the values for T < 0
o
C are for water vapour over a surface of liquid water (not over ice),
and thus are relevant to our study. Nonetheless, it is reasonable to expect that the values for T ≥ 0
o
C
are more accurate and that the IAPWS data set, which is newer, is the most accurate among the four.
The different data sets display small differences between each other for the same temperature value,
up to 0.16%.
Figure 3 provides a graphical comparison of equations (42) and (43) with the reference data.
Clearly, the common inconsistent equation (43), derived for constant L, proves to be inappropriate, as
its relative error exceeds 7%. In contrast, the derived closed solution (42) has negligible relative
errors. The maximum relative errors of the two equations, as well as those of the Magnus-type
equation (2), with respect to the data are given in Table 1. It can be seen that the differences of the
proposed equation (42) from the data is negligible, smaller that the deviations among the values of
different data sets. The error of Magnus-type equation (2) is four times larger than that of (42) for the
most accurate IAPWS data set and 2.5 times larger for all data sets. The error of the standard equation
(43) is too high, 1.5-2 orders of magnitude higher that that of the proposed equation (42).
The simplicity of (42) makes numerical calculations easy. For known T, (42) provides p directly.
The inverse problem (to calculate T, i.e. the saturation temperature, also known as dew point, for a
given partial vapour pressure p) cannot be solved algebraically. However, the Newton-Raphson
numerical method at an origin T
0
/T = 1 gives a first approximation T΄ of temperature by
T
T
0
= 1 +
06
.
5
921
.
24
1
p
p
0
ln (44)
Notably, this is virtually equivalent to solving (43) for T
0
/T. This first approximation can be improved
by re-applying (42) solved for the term T
0
/T contained within the exponentiation, to give
T
T
0
= 1 +
921
.
24
1
p
p
0
ln +
921
.
24
06.5
T
T
0
ln
(45)
A systematic numerical investigation showed that a single application of (45) suffices to provide a
value of T with a numerical error in T
0
/T less than 0.1%, while a second iteration (setting the
calculated T as T΄) reduces the error to 0.02%.
6. Summary and concluding remarks
Evidently, theoretically consistent relationships are preferable over purely empirical ones as far as the
former agree with empirical evidence and their use is convenient. The Clausius-Clapeyron equation is
a nice theoretical relationship, but the analytical solution typically contained in books, while it is
9
simple and easy to use, proves to be flawed and also a bad approximation of reality. It is, thus,
reasonable that in calculations of saturation vapour pressure, empirical, typically Magnus-type,
equations are preferred over this theoretical equation. By removing an unnecessary, excessive and
inconsistent assumption that is made within the common derivation, we obtain a closed solution that
is still very simple and combines both theoretical consistency and accuracy. With reference to water
vapour saturation, the proposed solution is by orders of magnitude more accurate than the standard
equation of the literature and also better than the more accurate Magnus-type equation. Compared to
standard tabulated data of saturation vapour pressure for temperature range –40
o
to 50
o
C, which is
relevant to hydrometeorological applications, the derived equation has negligible error. These facts
may allow recommending the use of the derived equation (42) as one combining theoretical
consistency and accuracy.
The alternative theoretical framework proposed, which is based on entropy maximization avoiding
unnecessary assumptions, provides a better understanding and intuition development for the phase
transition. Such understating and intuition development can help to recognize, particularly within the
university education, the power of variational principles and the extremization (maximiza-
tion/minimization) approach over the more common approach that the physical laws are
mathematically expressed only by equations, as well as to recognize the fundamental character of
entropy maximization as a powerful physical principle, contrary to a common perception that physical
laws are only deterministic and mechanistic.
As first indicated by Boltzmann and Gibbs, later succeeded by Shannon who used essentially the
same entropy definition to describe the information content, entropy is none other than a measure of
uncertainty [e.g. 14 , 15 , 16 ]. Thus, the interpretation of the framework proposed is that the
quantification of the phase change relies on maximization of uncertainty. In particular, the entropy in
equations (25) and (26) represents the combined uncertainty as to (a) whether a molecule is in the
liquid or gaseous phase, (b) the molecule’s position in space and (c) the molecule’s kinetic state
expressed by its velocity. This combined uncertainty is maximal at the microscopic (molecular) level.
It is amazing that the entropy maximization represents a principle so powerful as to fully explain and
accurately quantify the phase transition determining its latent heat (equations (20) and (40)) and the
resulting equilibrium vapour pressure (equations (23) and (42)). It is even more amazing that, while at
the microscopic level the uncertainty is maximized, at the macroscopic level all the resulting laws
express near certainties, verified by measurements. This is not a surprise, though, because, given the
probabilistic meaning of entropy, the negligible macroscopic uncertainty can be predicted by
application of probability theory for systems with large number of elements, as the typical
thermodynamic systems are.
References
[1] Wannier G H 1987 Statistical Physics, Dover, New York, 532 pp.
[2] Robertson H S 1993 Statistical Thermophysics, Prentice Hall, Englewood Cliffs, NJ, 582 pp.
[3] Reif F 1967 Berkeley Physics Course: Statistical physics, 1
st
edn., McGraw-Hill.
[4] Kondepudi D and Prigogine I 1998 Modern Thermodynamics, From Heat Engines to Dissipative
Structures, Wiley, Chichester.
[5] Cencel Y A and Boles M A 2002 Thermodynamics, An Engineering Approach, 4
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[6] Stowe, K., 2007 Thermodynamics and Statistical Mechanics (2nd edn., Cambridge Univ. Press, Cambridge,
556 pp.
[7] Alduchov O A and Eskridge R E 1996 Improved Magnus form approximation of saturation vapour
pressure, J. Appl. Meteor., 35, 601–609.
[8] Denbigh K 1981 The principles of chemical equilibrium with applications in chemistry and chemical
engineering, Cambridge University Press, Cambridge.
[9] Amadei A, Roccatano D, Apol M E F, Berendsen H J C and Di Nola A 1996 Prediction of the liquid–vapor
equilibrium pressure using the quasi-Gaussian entropy theory, J. Chem. Phys. 105 (16), 7022-7025.
[10] Wagner W and Pruss A 2002 The IAPWS formulation 1995 for the thermodynamic properties of ordinary
water substance for general and scientific use, J. Phys. Chem. Ref. Data 31, 387-535.
[11] Shuttleworth W J 1993 Evaporation, ch.4 in Handbook of Hydrology (D. R. Maidment ed.), McGraw-Hill,
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[12] List R J 1951 Smithsonian Meteorological Tables, Smithsonian Institution, Washington, DC 527 pp.
[13] Gueymard C 1993 Assessment of the accuracy and computing speed of simplified saturation vapor
equations using a new reference dataset, J. Appl. Meteorol., 32 1294-1300.
[14] Koutsoyiannis D 2011 Hurst-Kolmogorov dynamics as a result of extremal entropy production, Physica A,
390 (8), 1424–1432.
[15] D. Koutsoyiannis 2011 Lecture Notes on Hydrometeorology: A probability-based introduction to
atmospheric thermodynamics, 45 pp., National Technical University of Athens, Athens,
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[16] A. Ben-Naim 2008 A Farewell to Entropy: Statistical Thermodynamics Based on Information, World
Scientific Pub. Singapore, 384 pp.
... -5.06) at an origin T0/T = 1. Empirical equations based on observations are in common use; these are regarded to be the most accurate-but in fact are less accurate than (6.131) (see Koutsoyiannis, 2012). Figure 6.4 compares the two theoretical equations (6.131) and (6.132) also with one empirical. ...
... Comparison of latent heat of water as given by equation(6.130) and with standard tabulated data fromList (1951), as well as with a common empirical equation(Shuttleworth, 1993), which is /(J kg −1 ) = 2.501 × 10 6 − 2361 /°C. (Source of graph:Koutsoyiannis, 2012.) Comparison of the saturation water vapour pressure obtained by the consistent equation (6.131) as well as by the standard but inconsistent equation(6.132). ...
Preprint
Full-text available
This is a working draft of a book in preparation. Current version 0.4 – uploaded on ResearchGate on 25 January 2022. (Earlier versions: 0.3 – uploaded on ResearchGate on 17 January 2022. 0.2 – uploaded on ResearchGate on 3 January 2022. 0.1 (initial) – uploaded on ResearchGate on 1 January 2022.) Some stuff is copied from Koutsoyiannis (2021, https://www.researchgate.net/ publication/351081149). Comments and suggestions will be greatly appreciated and acknowledged.
... The saturated pressure for forming cubic bixbyite Mn 2 O 3 phase nano-particles obtained from the P-T graph and using Clausius-Clapeyron equation is P s = [ 7.3 × 10 6 × T ] Pa [54][55][56][57][58]. For the nucleation models, a magnetic field of 0-7 T and the pressure in the range of ∼10 5 − 10 7 Pa was used [59,60]. to decrease substantially on increasing the magnetic field and pressure but model B contradicts it. ...
... To carry out the nucleation modelling, the molar volume, difference in molar volume, molar mass, interfacial free energy and saturated pressure of NiO nanoparticles used for determining the size of the nanoparticles are 1.12 × 10 −5 m 3 mol −1 , 4.61 × 10 −6 m 3 mol −1 , 74.69 ×10 −3 kg mol −1 , 1.74 J m −2 , [ 2.2 × 10 5 × T ] Pa respectively [51,55,56,59,60]. ...
Article
Pulsed laser ablation at manganese (paramagnetic)-water interface led to the formation of cubic bixbyite α-Mn 2 O 3 nano-particles. The effect of external magnetic field on to the size of the nano-particles was investigated. Nucleation modelling were carried out to validate the experimental results. To study the affect of the external magnetic field on to the nucleation dynamics, two different models were employed-model A: influence of the magnetic pressure, and model B: influence of the magnetic energy, that affects the laser-induced nucleation dynamics when an external magnetic field is applied. It was observed that the nucleation modelling using model A gives more agreeable results to the experimental observation than model B. A similar investigation was also carried out using ferromagnetic: nickel target, which shows significant influence exhibiting a decrease in nano-particle sizes using both the models. The fluid dynamical counterpart: cavitation bubbles formed at laser interaction with solid targets immersed in liquids, are also probed. Cavitation bubbles formed at the manganese-water interface seem impervious to the external magnetic field; however, for targets such as nickel, energy dispensed to ferromagnetic interactions is translated to cavitation bubbles which exhibit larger bubble radius. Supplementary material for this article is available online
... For each study, the details on the spatial scale, change and methods applied are identi- where CT is an integration constant, L is the enthalpy of vaporisation, and R is the gas constant. The assumption of constant L can be removed by expanding Eq. 11.1 to Eq. 11.2 (derived in Koutsoyiannis, 2012) given by: ...
Book
Extreme rainfall events, ranging from short- to multi-day durations, have shown increasing trends in recent decades and are projected to intensify more in the future with global warming. Moreover, short-duration rainfall extremes are more sensitive to warming as compared to long-duration rainfall extremes. Understanding the changes to and mechanisms leading to extreme rainfall events is crucial in effective water management for decision making and policy planning. In this chapter, we review recent advances and challenges in studying the changes in observed sub-daily to multi-day rainfall extremes and their future changes based on climate model outputs. This is achieved by discussing the different methods used to study changes to rainfall extremes, understanding relationships between rainfall extremes and temperature, and using climate models to study future changes in extreme rainfall.
... Associated with the Clausius-Clapeyron relation (Held & Soden, 2006;Koutsoyiannis, 2012), global warming resumption produces more water vapor into the atmosphere, resulting in an increase in water vapor (mostly the intense TCs) within the TC range (Figure 1c), which may offset the decrease in water vapor caused by the increase in track density at higher latitudes and decrease in TC intensity so that the average water vapor in LN2 years shown in Table 1 remains unchanged. ...
Article
Full-text available
Plain Language Summary The impacts on various weather phenomena associated with the global surface warming slowdown (or “hiatus”: 1998–2012) and its subsequent resumption since 2013 have been extensively investigated. An unresolved issue is whether the tropical cyclone (TC) rain rate over the western North Pacific (WNP) basin has also changed, especially during the El Niño (or La Niña) years. Considering the availability of high‐resolution satellite rainfall data, we mainly focus on the period of 2001–2020 (slowdown period: 2001–2012; resumption period: 2013–2020). Our results show that, consistent with the increase in atmospheric water vapor after the global warming resumption, the average TC rain rate has increased by ∼2%. However, TC rain rate increases in El Niño years but decreases in La Niña years during the resumption period. This opposite change in TC rain rate is mainly influenced by the strengthened WNP subtropical high. Our findings suggest that it is necessary to properly consider the influence of the subtropical high on the projections in TC migration, intensity, and rain rate under global warming.
... Design of Experiment is completed using modified a Clausius Clapeyron equation [57] given by ...
Article
Full-text available
The current study addresses the role of heat and moisture emitted from anthropogenic sources on the local weather with the aid of numerical weather prediction (NWP). The heat and moisture emitted by industries to the atmosphere are considered main sources in this study. In order to understand the effect of heat and moisture on local weather, the study is conducted to capture the impact of heat with no moisture change. The results are compared against a control run case without perturbation and also against the case where both heat and moisture are perturbed with temperature as a single parameter. The Angul district in Odisha that houses over 4000 industries is considered our study region. The numerical simulations are performed using the mesoscale Weather Research and Forecasting (WRF) model for two rain events, namely a light rain case and a heavy rain case, with different physics options available in the WRF model. The WRF simulated maximum rainfall rate using various microphysics schemes are compared with the Tropical rainfall measuring mission (TRMM) observations for validation purposes. Our study shows that the WDM6 double moment microphysics scheme is better in capturing rain events. The TRMM-validated WRF simulation is considered a reference state of the atmosphere against which comparisons for the perturbed case are made. The surface temperature is perturbed by increasing it by 10 K near the industrial site and exponentially decreasing it with height up to the atmospheric boundary layer. A numerical experiment represents heating without addition of moisture by recalculating the relative humidity (RH) corresponding to the perturbed temperature. The perturbed temperature affects sensible heat (SH) and latent heat (LH) parameters in the numerical experiment. From the results of the numerical investigation, it is found that the near-surface rainfall rate increases locally in a reasonable manner with the addition of sensible heat to the atmosphere. A comparison against the case where moisture is added shows that enhanced rainfall is more sensitive to sensible and latent heat than sensible heat alone.
... Pour que la chaleur latente suive cette loi d'évolution linéaire, nous posons que la capacité calorifique de la phase liquide vaut cl et vl est négligée devant cl ∆T. Nous allons faire le calcul pour le cycle de vaporisation suivant (figure A.2) [245], [246] : ...
Thesis
L’objectif principal du dépôt d’agrégats, préformés en phase vapeur sur substrat, est l’élaboration contrôlée, reproductible et transposable à l’échelle industrielle de matériaux nanostructurés ouvrant ainsi la possibilité à une ingénierie des matériaux à l’échelle nanométrique. Dans cette approche, les briques élémentaires sont des agrégats et leur assemblage par auto-organisation détermine les propriétés finales du matériau. Cette auto-organisation étant lié étroitement à l’interaction entre les agrégats et la surface, la problématique peut être inversée et offrir ainsi au travers des agrégats de véritables sondes pour une caractérisation fine des états de surfaces.Ce travail de thèse illustre cette approche par le dépôt d’agrégats métalliques (Ag) et semi-métallique (Sb) sur différentes surfaces carbonées (carbone amorphe, graphite, graphène, nanotubes de carbone). Les agrégats sont obtenus à l’aide d’une source à condensation gazeuse, les dépôts effectués sous ultra-vide et l’analyse par imagerie aux microscopes électroniques (MEB) et champ proche (AFM). Ceci a permis de mettre en évidence, par nucléation hétérogène, les défauts ponctuels des surfaces (lacune, inclusion, défauts de Stones-Wales) ou structuraux (marche, joints de grains, plis, champ de contrainte élastique,…). Le prolongement de ces travaux serait de déposer ces agrégats sur d’autres matériaux présentant des états de surface particuliers, topographiques (surfaces vicinales, porosité, rugosité,…) et/ou physico-chimique (corrosion, polluants et contaminants, différences d’adhésion et d’adhérence, …).D’autres part, les nanostructures obtenues sont le plus souvent des structures topologiques métastables, possédant une très grande sensibilité aux variations de températures et/ou de compositions chimiques. Cela nous a permis d’étudier le vieillissement, de nanostructures fractales d’argent. Les résultats montrent la capacité de fixation de ces nanostructures, d’agents polluants (soufre, chlore) même présents uniquement sous forme de traces dans l’environnement, permettant d’envisager leur valorisation comme filtres et/ou capteurs ultra-sélectifs. Cette étude, nous a permis également de proposer un mécanisme hypothétique pour la corrosion de l’argent à l’échelle nanométrique.
... Simulated distillation curves at 3.0 and 6.0 MPa were obtained using the Clapeyron equation modified with the Redlich Kwong (RK) equation of state to consider deviations from ideality [30]. The approximation was made considering the individual behavior of saturates, aromatics, and resins from the simulated distillation experiments. ...
Article
This work investigates the thermo-oxidative behaviour of saturates, aromatics, and resins, at different pressures of 0.084, 3.0, and 6.0 MPa. Saturates (S), aromatics (Ar), and resins (R) were characterized by elemental analysis, vapor pressure osmometry, ¹H- and ¹³C-nuclear magnetic resonance, and X-ray photoelectron spectroscopy. The atmospheric evaporation of S, Ar, and R was evaluated by simulated distillation, whereas high-pressure distillation curves were created by calculation using the Clapeyron equation with the Redlich Kwong equation of state. The mass change and rate of mass change profiles were assessed in a high-pressure thermogravimetric analyzer (HP-TGA) by injecting nitrogen and air to obtain the pyrolysis and oxidation responses. Kinetic analysis was performed using a first-order kinetic model discretizing thermal oxidation profiles in four different regions, namely, oxygen chemisorption (OC), decomposition of chemisorbed oxygen-based compounds (DCO), first combustion (FC), and second combustion (SC). As main results, it was obtained that the saturates were just described by FC and SC regions at all pressures, while aromatics and resins by the four stages (OC, DCO, FC, and SC) for pressures higher than 3.0 MPa. At 6.0 MPa, aromatics exhibit a rapid consumption in the FC region, while resins present a more controlled consumption, distributed in the three decomposition regions. This implies that as pressure increases, aromatics have a greater influence on oxidation reactions at high temperatures. From pyrolysis curves, it was noted that the amount of coke generated follows the increasing order saturates < aromatics < resins for all pressures used. The kinetic parameters corroborated the results obtained. For the higher chemisorption systems, the effective activation energy values were higher, and therefore the DCO values. At the same time, the pre-exponential factor showed opposite results given slower reaction kinetics in these stages.
Article
The existence of fracture flow has an undesirable effect on the creation of the frozen wall. Brine and liquid nitrogen combined freezing technology can ensure the safety of freezing engineering, reduce the construction period and save cost. Considering the permeability of the rock matrix, fluid exchange and Darcy–Stokes coupling effect between the rock matrix and fracture, a thermo-hydraulic model of the fractured porous rock mass under water seepage is herein established. The interfacial seepage field characteristics of fractured rock mass under different fluid flow models and interface conditions are compared. The numerical simulations of the initial brine freezing and liquid nitrogen reinforcement freezing are carried out. The results show that the overall permeability of fractured rock mass computed by free flow considering the Darcy–Stokes effect is greater than that computed by the Cubic law. The limit seepage velocity of the intact rock mass in brine freezing is 2.5 m/d, and that of fractured rock mass decreases to 1 m/d. The fracture aperture and groundwater seepage velocity are directly proportional to the closure time of the frozen wall. Liquid nitrogen freezing can seal water quickly and shorten the closure time of the frozen wall when the seepage velocity of the fractured rock mass is greater than the limit seepage velocity, and the rapid cooling of the upstream region plays an important role in the formation of the frozen wall in fractured rock mass.
Article
Membrane distillation (MD) has gained significant attention for treating hypersaline brine. However, conventional MD is still constrained by technical issues such as temperature polarization (TP) caused by the feed being continuous heated. Herein, through the use of a thermal conducting layer, this study introduced a conductive heating vacuum membrane distillation (CH-VMD) system capable of transmitting external heat directly to the membrane-water interface without preheating the feed. The influence of operational conditions on distillate flux and salt rejection was investigated. In both CH-VMD and conventional VMD systems, the temperature profile at the membrane-water interface were examined and compared. In addition, we analyzed the TP differences between the two systems under different operational conditions. In CH-VMD system, with a 35 g/L NaCl feed solution, thermal conducting layer temperature of 60 °C, a flow velocity of 3.1 cm/s, and 90 kPa vacuum level, the distillate flux reached 9.8 L/m²·h and the salt rejection achieved 99.9%. The temperature polarization coefficient was greater than unity, resulting in a thermal efficiency as high as 83.2%. Heat conduction increased the temperature of the membrane-water interface, which is critical for optimizing MD performance in terms of overcoming TP, enhancing distillate flux and salt rejection, and lowering energy consumption.
Method
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The probability-based introduction to atmospheric thermodynamics emphasizes the following issues: ● Heat is not a caloric fluid, as used to be thought of up to the 19th century. ● Thermodynamic laws (e.g. the ideal gas law) are statistical laws (typically relationships of expectations of random variables). ● While these laws are derived by maximizing entropy, i.e. uncertainty, they express near certainties and are commonly misinterpreted as deterministic laws. ● The explanation of near certainty relies on these two facts: (a) Typical thermodynamic systems are composed of hugely many identical elements: N ~ 10^24 per kilogram of mass; (b) The random motion of each of the system elements is practically independent to the others’. ● As a consequence, a random variable x expressing a macroscopic state, will have a variation std[x]/E[x] ~ 1/ N ~ 10^–12 (for a kilogram of mass). ● The fact that the macroscopic variability is practically zero should not mislead us to interpret the laws in deterministic terms.
Article
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A revised saturation vapor dataset is proposed for use in meteorology. Based on new engineering data of the American Society of Heating, Refrigerating, and Air-Conditioning Engineers for temperatures above 0°C, it should supersede the older Smithsonian and World Meteorological Organization meteorological tables.Simple new equations are proposed to compute the saturation vapor pressure over water between 50° and 50°C. Their accuracy is shown to be excellent over this range, with an nns error of 3 × 103 mb and an average relative error of 0.02%. Detailed statistics descrbing the accuracy performance of 22 other equations are presented and the speed performance of all these equations is assessed. Nested polynomials are shown to provide both good accuracy and computational speed. On a modern minicomputer, a single evaluation of saturation vapor pressure may take less than 1 µs of CPU time, 15 times less than required by the Goff Gratch equations that were used to construct the meteorological tables.
Article
Full-text available
It is demonstrated that extremization of entropy production of stochastic representations of natural systems, performed at asymptotic times (zero or infinity) results in constant derivative of entropy in logarithmic time and, in turn, in Hurst-Kolmogorov processes. The constraints used include preservation of the mean, variance and lag-1 autocovariance at the observation time step, and an inequality relationship between conditional and unconditional entropy production, which is necessary to enable physical consistency. An example with real world data illustrates the plausibility of the findings.
Book
The principal message of this book is that thermodynamics and statistical mechanics will benefit from replacing the unfortunate, misleading and mysterious term "entropy" with a more familiar, meaningful and appropriate term such as information, missing information or uncertainty. This replacement would facilitate the interpretation of the "driving force" of many processes in terms of informational changes and dispel the mystery that has always enshrouded entropy. It has been 140 years since Clausius coined the term "entropy"; almost 50 years since Shannon developed the mathematical theory of "information" - subsequently renamed "entropy". In this book, the author advocates replacing "entropy" by "information", a term that has become widely used in many branches of science. The author also takes a new and bold approach to thermodynamics and statistical mechanics. Information is used not only as a tool for predicting distributions but as the fundamental cornerstone concept of thermodynamics, held until now by the term "entropy". The topics covered include the fundamentals of probability and information theory; the general concept of information as well as the particular concept of information as applied in thermodynamics; the re-derivation of the Sackur-Tetrode equation for the entropy of an ideal gas from purely informational arguments; the fundamental formalism of statistical mechanics; and many examples of simple processes the "driving force" for which is analyzed in terms of information. © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
Article
A brief remark is made about a sentence appearing in a recent article by Koutsoyiannis (2012 Eur. J. Phys. 33 295) regarding the concept of equilibrium vapour pressure and the role of the atmosphere as the 'holder' of water vapour. As explained in the comment, this sentence conveys some common misconceptions.
Article
Relative humidity is usually measured in aerological observations and dew point depression is usually reported in upper-air reports. These variables must frequently be converted to other moisture variables in meteorological analysis. If relative humidity is converted to vapor pressure, most humidity variables can then be determined. Elliott and Gaffen reviewed the practices and procedures of the US radiosonde system. In their paper, a comparison of the relative errors was made between the saturation vapor pressure formulations of Tetens (1930), Goff-Gratch (1946), Wexler (1976), and Buck (1981). In this paper, the authors will expand the analysis of Elliott and Gaffen by deriving several new saturation vapor pressure formulas, and reviewing the various errors in these formulations. They will show that two of the new formulations of vapor pressure over water and ice are superior to existing formulas. Upper air temperature data are found to vary from about +50 C to -80 C. This large variation requires a saturation vapor pressure equation to be accurate over a large temperature range. While the errors introduced by the use of relatively inaccurate conversion equations are smaller than the errors due to the instruments, dewpoint coding errors, and dewpoint conversion algorithms (Elliott and Gaffen, 1993); they introduce additional systematic errors in humidity data. The most precise formulation of vapor pressure over a plane surface of water was given by Wexler (1976). The relative errors of Tetens` (1930) formula and one due to Buck (1981) (Buck`s equation is recommended in the Federal Meteorological Handbook No. 3, 1991) are shown. The relative errors in this table are the predicted value minus the Wexler value divided by the Wexler value.
Article
1. Introduction and acknowledgements; 2. Moon reference maps; 3. International atlas of lunar exploration: the Moon at the dawn of the Space Age; 4. Chronological sequence of missions and events; Bibliography; Websites; Index.
Article
In 1995, the International Association for the Properties of Water and Steam IAPWS adopted a new formulation called ''The IAPWS Formulation 1995 for the Thermody-namic Properties of Ordinary Water Substance for General and Scientific Use'', which we abbreviate to IAPWS-95 formulation or IAPWS-95 for short. This IAPWS-95 formula-tion replaces the previous formulation adopted in 1984. This work provides information on the selected experimental data of the thermodynamic properties of water used to develop the new formulation, but information is also given on newer data. The article presents all details of the IAPWS-95 formulation, which is in the form of a fundamental equation explicit in the Helmholtz free energy. The function for the residual part of the Helmholtz free energy was fitted to selected data for the following properties: a thermal properties of the single-phase region (pT) and of the vapor–liquid phase boundary (p T), including the phase-equilibrium condition Maxwell criterion, and b the caloric properties specific isochoric heat capacity, specific isobaric heat capacity, speed of sound, differences in the specific enthalpy and in the specific internal energy, Joule– Thomson coefficient, and isothermal throttling coefficient. By applying modern strategies for optimizing the functional form of the equation of state and for the simultaneous nonlinear fitting to the data of all mentioned properties, the resulting IAPWS-95 formu-lation covers a validity range for temperatures from the melting line lowest temperature 251.2 K at 209.9 MPa to 1273 K and pressures up to 1000 MPa. In this entire range of validity, IAPWS-95 represents even the most accurate data to within their experimental uncertainty. In the most important part of the liquid region, the estimated uncertainty of IAPWS-95 ranges from 0.001% to 0.02% in density, 0.03% to 0.2% in speed of sound, and 0.1% in isobaric heat capacity. In the liquid region at ambient pressure, IAPWS-95 is extremely accurate in density uncertainty 0.0001% and in speed of sound 0.005%. In a large part of the gas region the estimated uncertainty in density ranges from 0.03% to 0.05%, in speed of sound it amounts to 0.15% and in isobaric heat capacity it is 0.2%. In the critical region, IAPWS-95 represents not only the thermal properties very well but also the caloric properties in a reasonable way. Special interest has been focused on the extrapolation behavior of the new formulation. At least for the basic properties such as pressure and enthalpy, IAPWS-95 can be extrapolated up to extremely high pressures and temperatures. In addition to the IAPWS-95 formulation, independent equations for vapor pressure, the densities, and the most important caloric properties along the vapor–liquid phase boundary, and for the pressure on the melting and sublimation curve, are given. Moreover, a so-called gas equation for densities up to 55 kg m 3 is also included. Tables of the thermodynamic properties calculated from the IAPWS-95 formulation are listed in the Appendix. © 2002 American Institute of Phys-ics.