Appendix D
Useful Equations
Below, empirical equations, parametrisations, and derivations used in the UDF are presented. The appendix is just meant as a work of reference for the author, but it has been
included with the hope that it will also be of help to somebody else.
Forms of the universal gas law
In its most general form, the universal gas law is given by
where
(D.1)
is pressure, is volume, is number of moles, is the universal gas constant,
and is temperature. The number of moles can be calculated from:
A
m
(D.2)
where is mass, is molar weight, is the number of particles in a closed system, A
is Avogadro’s constant, and m is molar volume. Furthermore,
B A
77
(D.3)
78
where
Useful Equations
B
is Boltzmann’s constant. Therefore,
m
where M
M B
(D.4)
(D.5)
M (D.6)
M (D.7)
Ê
Å is the specific gas constant, and is density.
Relations between components A and B in a binary system
Mass fraction
Mass of one component per the total mass
A
A
A
(D.8)
The “density fraction” is equivalent to the mass fraction because for volume ,
A
A
and with ÊMÔ Ì
A
B
B
A B
A B
ÑA
A
Î
A
ÑA ÑB A
B
Î
Î
A
B
ÔÊÅÌ :
A
ÔA ÅA
ÊÌ
ÔA ÅA ÔB ÅB
ÊÌ
ÊÌ
A
A
ÅB
ÅA B
(D.9)
where A and B are the partial pressures of the components.
Mole fraction
Number of moles of one component per total number of moles
A
A A
(D.10)
79
because
A
A
How to get A from
A
A A
A
A
A
A
A
A
B
B
B
A
A
A A
A
B
A
A
A
B
A B
A
A
A
A
A
A B B
A A A
B A
A
A A B
A
A
and vice versa
A
A A B
A
B
A B
A B A A
A AA B B
A B A
A
B
A
A
A
A
A
B
A
B
B
B
(D.11)
A
A A
B
A B A
A
A
B
A
A
B
B
(D.12)
Molality Number of moles of solute per kilogram of water
mol s
s w
s s
s
s
w
s
(D.13)
when the molar weight is given in kg mol .
Molar mass of the mixture
mix
is calculated as the mass-weighted average of the
molar masses of the components of the mixture:
mix v
v
v
g
(D.14)
80
Useful Equations
Molecular thermal velocity
After Hinds (1982) (p. 19)
v
mol
(D.15)
Mean free path of the gas (air)
After Kinney et al. (1991)
g g where g
m for air at and . ¾¿
(D.16)
K is the reference tem-
perature, hPa is the reference pressure, is air temperature, and is air
pressure.
Air viscosity
Also after Kinney et al. (1991)
where Pa s
(D.17)
is the reference viscosity.
Mean free path of water vapour
Jacobson (1999) (p. 458) gives the mean free path of a trace gas, in this case water vapour
in air as
v g
v
mol g v
(D.18)
81
Latent heat of vapourisation
After Bakan et al. (1988) (p. 30)
w/v
(D.19)
where K. The parametrisation is valid for K K.
Heat conductivities
After Pruppacher & Klett (1980) (p. 418), the heat conductivities for dry air,
vapour, v , and the mixture of the two, mix , are given by
g v mix g
v
v
g, water
(D.20)
(D.21)
(D.22)
g
Diffusion coefficient water vapour – air
After Bakan et al. (1988) (p. 29)
v valid for K
(D.23)
K.
Surface tension
The original relation used in the UDF is given by (Bakan et al., 1988, p. 32):
droplet (D.24)
82
Useful Equations
valid for K
K. This parametrisation has been replaced by the one endorsed
by the International Association for the Properties of Water and Steam (IAPWS, 1994):
droplet (D.25)
crit K is the critical point temperature. (D.25) is valid
between the triple point of water (triple K) and crit . The influence of solvents
where
crit
.
on the surface tension can be taken into account in the following manner (Brechtel &
Kreidenweis, 2000):
droplet where NaCl NaCl mol
N molwater m
(D.26)
kgsolute is the surface tension change per mole
salt in 1 kg of water.
Saturation vapour pressure
For a plane surface of pure water, the equilibrium water vapour pressure is given by
(Bakan et al., 1988, p. 35):
sat,w K
which is valid for
ÌÌ
¼
̼ Ì (D.27)
K.
Osmotic coefficient
The dependence of the osmotic coefficient on molality has been calculated from the values
given in Table D.1. These values have been taken from Roth (1973), they were derived
83
from measurements of the electromotive force of the solution. For the calculations carried
out within the scope of this diploma thesis, curves have been fitted to the values at each
temperature. The change in
mol
Æ
between different temperatures is assumed to be linear.
C
ÆC
C
Æ C
C
Æ C
Table D.1: Osmotic coefficient
Æ
Æ
at different temperatures as
a function of molality (Roth, 1973).
Fuchs-Sutugin
The mass and heat transfer transition functions that appear in the single particle growth
law, Equation (2.10), are given by Fuchs & Sutugin (1971):
Mass Heat Kn
Kn Kn
(D.28)