air density and its uncertainty

AIR DENSITY AND ITS
UNCERTAINTY
Manuel Salazar
Maria Vega
CONTENTS

Air and its composition
Ways to calculate the air density
Chart
CIPM Equation
Approximate equation
Uncertainty
Air and its composition
The air is a mixture of several gases
dry air, and water in steam
form. Troposphere is the inferior layer of the terrestrial atmosphere,
terrestrial surface altitude of 6 to 18 kilometers, the air we breathed is
concentrated there.
The dry air as the water steam behaves like ideal gases. They have been
developed to empirical laws that relate the macroscopic values, in ideal
gases, these values include pressure (p), volume (V) and temperature (T)
Ley de Charles
Ley de Gay-Lussac
IDEAL GAS LAW
Ley de Boyle
Air and its composition
It is constituted by a nitrogen
mixture and of oxygen like
basic element (99%) and the
rest like noble gases. The
composition is similar around
the world.

Water Steam (0-5%), Carbon
dioxide, hydrocarbons, tars,
ashes, dust and SO2.

Electrical delivery form C2H2,
H202, 03, NO3H, NH3,
NO3NH4.
COMPOSICION DEL AIRE
PURO
Elemento
Proporción
Proporción
en volumen
en peso
Nitrógeno
78,14
75,6
Oxigeno
20,92
23,1
0,94
0,3
Argón
-3
1 10
-4
0,7 10
-4
3 10
-5
0,35 10
-5
4 10
Neón
1,5 10
Helio
5 10
Criptón
1 10
Hidrogeno
5 10
Xenón
1 10
-3
-4
-4
-5
-5
Ways to calculate the air
density
Density defined in a qualitative manner as the measure of the
relative mass of objects with a constant volume
Hypothesis de Avogadro Two gases same volume (same pressure and
temperature) contain the same number of particles, or molecules
Standard Law gases
P.V = n . R . T
n = m/Mr
Ways to calculate the air
density

As function of altitude
Using a refractometer
Air buoyancy artefacts methods
Equation CIPM /81
Approximate equation
Ways to calculate the air density
AS FUNCTION OF ALTITUDE
Atmospheric pressure drops
about or about 1.1 mbar (kPa)
for each 100 meters. Density
decreases
Ways to calculate the air density
AS FUNCTION OF ALTITUDE
L = 6,5
temperature lapse rate, deg K/km
H = geopotencial altitude
Z = geometrical altitude
To = Temperature ºK
Po = Atmospheric Pressure
Ways to calculate the air density
REFRACTROMETRY
Changes in air density can be determined with good precision using
an optical method based on the high correlation between air
density and air index of refraction.
R specific refraction or the refractional invariant in unction composition of air and
the local atmospheric conditions
ν vacio
n=
ν aire
n is determined by a simple ratio of laser frequencies:
ν vacio
laser frequency locked to one transmission peak of the interferometer under vacuum
ν aire
the frequency locked to the same peak of the interferometer placed in air.
Ways to calculate the air density
AIR BUOYANCY ARTEFACTS METHODS
The method is based on the weighing of
nominal mass and the same surface
volumes. Two weightings are necessary
one in air and one in vacuum
two artefacts having the same
area but with very different
to determine the air density ρ ,
∆maire = I 1 − I 2 + ρ (Vm1 − Vm 2 )
I1 e I2 balance readings in air mass 1 and mass2
Vm1 e Vm2 volume of m 1 y m 2
ρ
air density
∆m
=I −I
vacío
3
4
I3 e I4 the balance readings in vacuum mass 1 y mass 2
∆maire = ∆mvacio + σ∆S

∆S the difference in surface area between the two artefacts and σ
mass of adsorption per unit area.
ρ=
I 3 − I 4 − ( I 1 − I 2 − σ∆S )
Vm1 − Vm 2
Ways to calculate the air density
FORMULA CIPM
From the equation of state of a non-ideal gas and the experimental
conditions the density of moist air
[
]
M a = 28,9635 + 12,011( xco2 − 0,0004) *10 −3 kgmol −1
Where
x v = hf ( p, t )
p sv (t )
p
P pressure
T thermodynamic temperature 273,15 + t
Mv molar mass of the water
Z compressibility factor
R molar gases constant
Ways to calculate the air density
APPROXIMATE EQUATION
From BIPM formula we obtain one
numerical approximate equation :
0,34848 * p − 0,009024 * hr * e 0, 0612*t
ρa =
273,15 + t
Ways to calculate the air density
PSYCHROMETRY
Thermodynamic properties of mixtures of gas with vapor. saturation
pressure and temperature of dew, Indexes of humidity, Volume, heat
and humid enthalpy, temperature of saturation adiabatic and wet
thermometer.
Some definitions:
Relative Humidity. The relative humidity is the percent of saturation
humidity, generally calculated in relation to saturated vapor density, in
(%):
HR = 100 Pv/Ps (%)
Temperature of adiabatic saturation, Th , is the ideal temperature of
equilibrium will have the air non saturated after undergoing an adiabatic
and isobaric process (iso enthalpic), that it takes it temperature to the
saturation by means of liquid evaporation of water to this.
Temperature of wet thermometer is the temperature that it reaches a
thermometer covered with a wet cloth that is exposed to an airflow
without saturating that it flows at speeds near 5 m/s
Dew point is the temperature, at which the moisture content in the air will
saturate the air , If the air is cooled further, some of the moisture will
condense .
Ways to calculate the air density
PSYCHROMETRY
To mesure the humidity :
ARTEFACT
METHOD
psychrometer
Thermodynamic
hygrometer of hair or
others materials
Hygroscopic
Hygrometer of dew
point
Condensation
Hygrometer of Chemist
absorption
Gravimetric
Hygrometer digital
Variation of
electrical
properties
Ways to calculate the air density
PSYCHROMETRY
Psychrometer and aspiro
psychrometer Consist two
thermometers, one normal
(dry) and another with their
bulb permanently humidified
thanks to a cloth or wet gauze
that it recovers it.
The humidity can be measured
between both starting from
the difference of temperature
apparatuses
Ways to calculate the air density
PSYCHROMETRY
Diagram Carrier.
- The T represents (ºC) in the
abscissa axis (axis x) and the
mixture reason or humidity (X, in
kg of water/kg of dry air) in the
axis of orderly (axis and, to the
right).
The saturation curve (HR = 100%)
it ascends toward the right and it
represents the end of the
diagram. In this curve the
temperatures of humid
thermometer and the
temperatures of dew are located.
- The curves of humidity relative
constant are similar to that of
saturation, advancing down (lying
down more) as it diminishes the
humidity of the air.
Ways to calculate the air density
CHART
.8
ρ a( 19, 40 %, p)
0.8
1.2
0.79
0.78
1.2
1.16
1.12
ρ a( 19 , 50% , p )
1.08
0.77
ρ a( 20, 40 %, p) 0.76
ρ a( 24, 40 %, p) 0.73
ρ a( 20 , 50% , p ) 1.04
1
ρ a( 21 , 50% , p )
0.96
ρ a( 24 , 50% , p ) 0.92
0.72
0.88
ρ a( 21, 40 %, p)
0.75
0.74
0.71
0.7 0.7
600 610 620 630 640 650 660 670 680 690 700
600
p
Air density evaluated with
Relative humidity 40 %,
temperature 19 ºC – 24 ºC,
pressure 600 mbar – 700 mbar
700
0.84
0.8 0.8
660 694 728 762 796 830 864 898 932 966 1000
660
p
Air density with evaluated with
Relative humidity 50 %,
temperature 19 ºC – 24 ºC,
pressure 660 mbar – 1000 mbar
1000
Ways to calculate the air density
1.185
CHART
1.2
1.15
1.1
ρ a 20 , h r , 1000
1.05
(
)
ρ a( 20 , h r , 800)
1
0.95
ρ a( 20 , h r , 700)
0.9
ρ a( 20 , h r , 600) 0.85
0.8
0.75
0.7 0.7
40
42
44
46
48
40
Air density evaluated with
50
52
hr
Relative humidity 40 % - 60 %,
temperature 20 ºC ,
pressure 600 mbar – 1000 mbar
54
56
58
60
60
Ways to calculate the air density
CHART
1.189
1.2
1.17
1.14
1.11
ρ a( t , 40 , 1000) 1.08
1.05
ρ a( t , 40 , 800)
1.02
0.99
0.96
0.93
0.929 0.9
19 19.6 20.2 20.8 21.4 22 22.6 23.2 23.8 24.4 25
19
Air density evaluated with
Relative humidity 40 %,
temperature 19 ºC – 25 ºC,
pressure 800 mbar – 1000 mbar
t
25
CALCULATION OF THE AIR DENSITY
CIPM
pM a
ZRT
Where:
p
Ma
Z
R
T
Xv
Mv


M v 

1 − x v 1 −
 M a 

Pressure of air in Pa.
molar mass of dry air.
Compressibility factor
Universal constant of ideal gases
Temperature of air in K
molar fraction of water steam
molar mass of water

Molar mass of dry air, Ma
If it considers constant of air component
Ma = 0,028963 512440 kg· mol-1
If it can measure the concentration of CO2
Ma = [28,9635 + 12,011 (XCO2. - 0,0004)]* 10-3 kg·
mol-1

Compressibility factor, Z
[
]
(
p
p2
2
2
Z = 1 − a 0 + a1t + a 2 t + (b0 + b1t )x v + (c 0 + c1t )x v + 2 d + ex v2
T
T
Where:
p
Air pressure in Pa
T
Air temperature in K
t
Environmental temperature in oC
a0
1, 581 23 X 10-6 K Pa-1
a1
-2,9331 x 10-8 Pa-1
a2
1,1043 x 10-10 K-1 Pa-1
b0
5,707 x 10-6 K Pa-1
b1
-2,051 X 10-8 Pa-1
C0
1.9898 x 10-4 K Pa-1
C1
- 2,376 x 10-6 Pa-1
d
1,83 x 10-11K2Pa-2
e
-0,765 x 10-8 K2Pa-2
)

Universal Constant of ideal gases, R
R = 8.314510 ± 8,4 x lO-6 J . mol-1 . K-1
p
Molar
fraction of water steam, Xv
In function of relative humidity, h
sv
p sv (t )
x v = hf ( p, t )
p
p sv
In function of temperature of dew point, tr
p sv (t r )
X v = f ( p, t r )
p
Where:
Relative humidity
h
p sv
Pressure of saturated steam
f ( p, tr ) Fugacity factor
Enhancement factor f f(p,tr)
f = α + βp + γt 2
Where:
α
1,000 62
β
3.14 x 10-8 Pa-1
γ
5,6 x 10-7 K-2
p
Air pressure in Pa
T
Air temperature in OC or dew point
temperature (tr) in OC
Pressure of saturated steam, psv
D

p sv = 1Pax exp  AT 2 + BT + C + 
T

Where:
A 1,237 884 7 x 10-5 K-2
B -1,912 131 6 x 10-2 K-1
C 33,937 110 47
D -6,343 164 5 x 103 K
T Air temperature in K or
temperature (Tr) in K
dew point
UNCERTAINTY OF AIR DENSITY
SOURCES OF UNCERTAINTY
Atmospheric temperature
up = u 2p1 + u 2p 2 + u 2p 3
• Calibration of barometric
u p1 =
• Resolution of barometric
u p2 =
• Variation of
atmospheric pressure
during calibration
up3 =
UB
k
dB
12
p+ − p−
24
Environmental conditions
u t = u t21 + u t22 + u t23
• Calibration of thermometer
• Resolution of instrument
•
Variation of temperature
during calibration
Ut
u t1 =
k
ut 2 =
ut 3 =
dt
12
t+ −t−
24
Relative humidity of air
u h = u h21 + u h22 + u h23
• Calibration of hygrometer
Uh
u h1 =
k
• Resolution of hygrometer
uh2 =
• Variation of the air relative
humidity during calibration
u h3 =
dh
12
h+ − h−
24
Constant R of ideal gases
uR = 84x 10-7 J mol-1 K-1
Equation adjustment for the determination of
air density
u ec = (1x10 −4 )(0,9495) = 9,50 x10 −5 kgm −3
Sensitivity Coefficient
Pressure
 ∂ρa  ∂ρa ∂Z   ∂ρa ∂Z ∂X v ∂f   ∂ρa ∂Z ∂X v   ∂ρa ∂X v ∂f   ∂ρa ∂Xv 

 + 
 + 
 + 
 + 
cp = 
+ 

 ∂p  ∂Z ∂p   ∂Z ∂X v ∂f ∂p   ∂Z ∂X v ∂p   ∂X v ∂f ∂p   ∂Xv ∂ p 
Temperature
 ∂ρa ∂Z ∂T   ∂ρa ∂Z   ∂ρa ∂Z ∂Xv ∂f   ∂ρa ∂Z ∂Xv ∂Psv ∂T 
⋅ 
⋅
⋅
⋅
⋅  + 
⋅
⋅
⋅  + 
⋅ ⋅ +

 ∂Z ∂T ∂t   ∂Z ∂t   ∂Z ∂Xv ∂f ∂t   ∂Z ∂Xv ∂Psv ∂T ∂t 
ct = 





∂
∂
ρ
ρ
ρ
∂
∂
∂
∂
∂
∂
∂
T
X
f
X
P
T


a
a
v
a
v
sv
+ 

⋅ 
⋅
⋅
⋅  + 
⋅
⋅  + 
  ∂T ∂t   ∂Xv ∂f ∂t   ∂Xv ∂Psv ∂T ∂t 

Relative humidity.
 ∂ρa ∂Z ∂Xv   ∂ρa ∂Xv 
Ch = 
⋅
⋅
⋅
+

v
v
∂
Z
∂
X
∂
h
∂
X
∂
h

 

Where:
∂T
=1
∂t
∂Psv  
D 
D
= exp AT 2 + BT + C +  2 AT + B − 2
∂T
T 
T
 
∂f
= β = 3,14 x10 −8 Pa −1
∂p
∂f
= 2γt
∂t
∂X v
fPsv
=
∂h
p
∂x v hPsv
=
∂f
p



∂xv − hfPsv
=
∂p
p2
∂X v hf
=
p
∂Psv
[
]
(
)
[
]
(
)
2p
∂Z −1
2
2
a0 + a1t + a2t + (b0 + b1t )X v + (C0 + C1t ) X v + 2 d + exv2
=
∂p T
T
2
2
p
∂Z p
= 2 a0 + a1t + a2t 2 +(b0 +b1t)Xv +(Co +C1t)Xv2 − 3 d +ex2v
∂T T
T
∂Z − p
=
a1 + 2a 2 t + b1 X v + c1X v2
T
∂t
(
)
2 p 2 eX v
∂Z
− p
(b 0 + b1t + 2 c o x v + 2 c1tx v ) +
=
∂X v
T
T2
∂ρ
Mv  
Ma 

=
1 − Xv1 −
 
∂p ZRT 
 Ma  
∂ρ − pM a
= 2
∂Z
Z RT

 M v 
−
x
1


v 1 −
M
a 



∂ρ − pM a
=
∂T
ZRT 2

 M v 


X
1
−

v 1 −
 M a 

− pM a  M v
∂ρ
1 −
=
ZRT  M a
∂X v
∂ρ − pM a
=
∂R
ZR 2T




 M v 

− Xv1 −
 M a 

Constant R
 ∂ρ a 
CR = 

 ∂R 
CR = −0,114203618 kgm 3 j −1molK
C ec = 1
Equation
Uncertainty evalation
up =
∑ [c ⋅ u(x )]
i
2
i
t
Degrees of freedom
vef =
uγ4
n
4
xt
u
∑t v
t
=
u ρ4
u 4p
ut4 u h4 u R4 uec4
+ + +
+
v p vt vh vR vec
Expanded uncertainty
U = k ⋅up
:
Consider that the atmospheric pressure, temperature and
relative humidity are correlated, its uncertainty:
up =
∑ (c ⋅ u(x ))
i
i
i
2
N −1
+ 2∑
c c u (x )u (x )r (x , x )
∑
( )
N
i =1 j i +1
i
j
i
j
i
j
Where :
( )(
()( )
)
( )(
()( )
)
n
1
∑ t k − t pk − p
n(n − 1) k =1
r (t , p ) =
st s p
n
1
t k − t hk − h
∑
n(n − 1) k =1
r (t , h ) =
st sh
(
)(
( )( )
n
1
p k − p hk − h
∑
n(n − 1) k =1
r ( p, h ) =
s psh
)
Reference

Estimaciòn de la incertidumbre en la determinaciòn de la
densidad del aire, Luis Omar Becerra Santiago y marìa Elena
Guardado gonzàlez, CENAM , Abril 2003.
Equation for the determination of the density of Moir Air, R.S.
Davis, Metrologia 1992,29,67-70.
Equation for the dertimination of the density of moist air, P.
giacomo.Metrologia 18,33-40 1982.
Three methods of determining the density of moist air during
mass comparisons, A. Picard y –h. Fang. Metrologia 2002, 39,
31-40.
Discrepancies in air density determination between the
thermodynamic formula and a gravimetric method: evidence
for a new value of the mole fraction of argon in air, A Picard,
H. Fang, Metrologia 41, 396-400
Thanks for the
attention

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