Download Heat Stored by Greenhouse Gases

Heat Stored by Greenhouse Gases
By Nasif Nahle
Biology Cabinet: Institute of Scientific Research and Education on Biology, 614-C S.
Business IH 35, New Braunfels, Texas, 78130 USA; email: [email protected]
Published on ©27 April 2007 by Biology Cabinet. Last Revision by the author on 12
June 2008.
INDEX:
1.
2.
3.
4.
5.
6.
7.
8.
Introduction, General Concepts and Formulas
Heat absorbed by atmospheric CO2 and change of temperature.
Examples of Heat Transfer by Conduction-Convection Taken from Nature.
The Sun and its effects on the Earth’s Temperature.
Methane
Heat Absorbed by the Atmospheric CO2 from the Surface by Radiation
Arrhenius Formula
Real Change of Temperature Caused by the Heat Stored by the Atmospheric
CO2
9. Water Vapor and Atmospheric Temperature.
10. Further Reading
The changes made from the paper by Stephen E. Schwartz are marked in red.
To quote this article copy and paste the next TWO lines. Please, fill in the spaces of
day, month and year:
Nahle, Nasif. Heat Stored by Greenhouse Gases. 2007. Biology Cabinet. Read from
Biology Cabinet’s site: http://www.biocab.org/Heat_Stored.html
1. INTRODUCTION
When investigating the propagation of energy, we must take into account the science
of thermodynamics which allows us to predict the trajectory of the process; and the
science of heat transfer to know the modes by which energy is propagated from one
system to other systems. We say that heat is not the same as temperature because
heat is energy in transit. Heat can exist in rotational, vibrational and translational
motions of the particles of a system, whereas temperature is the measurement of
the average of the kinetic energy of the particles of a substance. The average of the
molecular kinetic energy depends on the translational motion of the particles of a
system.
The energy absorbed or stored by a substance causes an increase in the kinetic
energy of the particles that form that substance. This kinetic energy or motion
causes the particles to emit heat, which is transferred to other regions of that
substance or towards other systems with a lower energy density.
To understand heat transfer we have to keep in mind that heat is not a substance,
but energy that flows from one system toward other systems with lower density of
energy.
GENERAL FORMULAS AND LAWS:
1st Law of Thermodynamics: Energy can be changed from one form to another, but
it cannot be created or destroyed.
The mathematical expression of the 1st. law is as follows:
ΔU = ΔQ – ΔW
Where ΔU is the increase of the internal energy of a thermodynamic system, ΔQ is
the amount of heat applied to the thermodynamic system, and ΔW is the change of
work done by that thermodynamic system.
The formula means that the change in the internal energy of a system is equal to the
heat transferred to that system minus the work done by that system in its
environment.
2nd Law of Thermodynamics: In all transformations from one form of energy into
another form of energy, a quantity of energy is always dispersed towards other
states, generally in the form of heat.
The mathematical expression of the 2nd law is as follows:
ΔS/Δt ≥ 0
Where ΔS is the increase of the entropy, and Δt is time.
The formula denotes that the change in the entropy in a thermodynamic system is
always higher or equal to zero, and that time is the fundamental dimension in which
the system is doing work.
The formula permits us to deduce other conceptualizations of the 2nd law which
mean the same thing, for example:
1. No system can transform energy into useful forms of energy with an efficiency of
100 percent.
2. Energy cannot spontaneously rearrange from low density states to high density
states.
3. Heat is never spontaneously transferred from cold systems to hot systems.
4. The entropy of any thermodynamic system is constantly increasing over time.
GENERAL FORMULAS TO CALCULATE THE VARIATION OF TEMPERATURE:
Convection:
Δq/A = h (σ) (T14-T24)
Where Δq is heat variation, A is the area in square meters, h is the convective heat
transfer coefficient of a given substance, σ is the Stephan-Boltzmann constant
(5.6697 x 10-8 W/m2 K4), and T14-T24 (in Kelvin) is the difference between the higher
absolute temperature to the fourth power and the lower absolute temperature to the
forth power.
Change of Temperature:
ΔT = q /m (Cp)
Where ΔT is the tropospheric temperature variability, q is the amount of heat
absorbed by a given substance, m is the current volumetric mass of that substance,
and Cp is the specific heat of that substance at P = 1 atm and T = 300 K.
Conversion from ppmv to mg/m3:
W mg/m3 = ppmv (12.187) (MW) / 273.15 + °C
Where W is the density of the substance expressed in milligrams per cubic meter,
ppmv is the concentration of the substance expressed in parts per million by volume,
12.187 is a constant of proportionality, MW is the molar weight of the substance, and
273.15 + °C is the temperature expressed in Kelvin.
If temperature changes with time, as it does, the atmosphere's temperature:
q stored = m (Cp) (ΔT/Δt) (Pitts, Donald and Sissom, Leighton. Heat Transfer. 1998.
Pg. 1. Potter & Somerton. 1993. Page 58. (For quasiequilibrium process with Cp rel.
constant)).
Where mass (m) is the product of volume and density (Pitts & Sissom. 1998. Page 1),
Cp is the Specific heat of the substance, ΔT is the change of temperature (final
temperature - initial temperature), and Δt is the change of time. The result is the
relation between the heat absorbed by the substance and the heat removed from the
substance. (Pitts & Sissom. 1998. Page 1) (Wilson. 1994)
2. ALGORITHM FOR CO2:
KNOWN DATA:
The weight of 1000 L of air is 1.23 Kg, or 1.23 N (Manrique. 2002. Oxford. Page 290).
The density of the dry air at T = 0 °C and P = 101.325 kPa is 1.292 kg/m3.
The density of the mixed air at T = 25 °C and P = 101.325 kPa is 1.18 g/L, or 1.18
Kg/m3. (Manrique. 2002).
At ambient T = 25 °C and P = 1 atm, dry air has a density of 1.168 kg/m3 (Pitts &
Sissom. 1998. Page 344)
At present, 1 cubic meter of air contains 0.000690 Kg of CO2 (690 mg).
Δ [CO2] in the last 200 years = 101 ppmv = 0.000164 Kg / m3
Notice that for denoting concentration we enclose the symbol of the substance
between square brackets [...]
Example from nature: Data taken from the meteorological station in Monterrey,
Mexico: On June 22, 2007 at 18.05 UT, at the coordinates 25º 48´ North latitude
and 100º 19' West longitude, and an altitude of 513 meters ASL, the air temperature
at 1.5 m above ground level was 299.65 K (26.5 °C), whereas the ground
temperature was 300.15 K (27 °C), which is the load of heat transferred from the
ground to the mass of CO2 when 1 cubic meter of air contains 0.00069 kg.
CO2 Thermal Conductivity Coefficient of CO2 (k) = 0.016572 W/m K (Manrique. 2002.
Oxford)
A = 1 m^2
T soil = 300.15 K
T air at altitude 1.5 m = 299.65 K
ΔT = 0.5 K
Δq = -k (ΔT/d) or Δq = -k [(T1-T2)/d]
Δq = -0.016572 W/m K (0.5 K / 1.5 m) = - 0.005524 W (rate of heat transfer or
heat transferred from soil to carbon dioxide at 1.5 m above ground).
0.005524 W = 0.005524 J/s (http://www.techexpo.com/techdata/conversn.html)
Conversion of 0.005524 J/s to change of temperature:
The heat transfer occurred through one second, thus the amount of energy
transferred is:
q = 0.005524 J/s (1 s) = 0.005524 J
ΔT = q / m (Cp)
Known values:
q = 0.005524 J
m = 0.00069 Kg
Cp of CO2 = 871 J/Kg*°C (Pitts & Sissom. 1994. Shaum's) (Engels.1998) (Manrique.
2002. Oxford)
Introducing magnitudes:
ΔT = 0.005524 J / 0.00069 Kg [871 J/Kg*°C] = 0.00919 °C (rounding the cipher,
0.01 °C).
Heat Stored by 381 ppmv (0.00069 Kg) of CO2:
q = m (Cp) (ΔT) (Potter & Somerton. 1993. Page 58) (For a no-equilibrium process
with Cp rel. constant)
Introducing magnitudes:
q = 0.00069 Kg (871 J/Kg K) (0.01 K) = 0.0060099 J; rounding the cipher, 0.00601
J.
0.00601 J, which will cause a change of temperature of:
ΔT = q /m (Cp)
ΔT = 0.00601 J /0.00069 Kg [871 J/Kg K] = 0.00601 J / 0.60099 J K = 0.01 K; or
0.01 °C; then the calculation is correct.
Let’s apply the formula related to the heat transfer by radiation from hot sources of
heat:
Q = e σ A (Tr4 – Tc4)
Where Q is the net radiated power, e is the emissivity of the system, A is the
radiating area, Tr is the temperature of the radiator, and Tc is the temperature of the
surroundings.
The known values are:
e of CO2 at 299.65 K and 1 atm = 0.0196 (e is a dimensionless value)
σ = 5.6697 x 10-8 W/m2 K4
A = 1 m2
Tr or T of soil = 300.15 K [(300.15 K)4 = 8116212154.05 K4]
Tc or T of air = 299.65 K [(299.65 K)4 = 8062266098.565 K4]
Introducing magnitudes:
q = 0.0196 (5.6697 x 10-8 W/m2 K4) (1 m2) (8116212154.05 K4 – 8062266098.565
K4) = 0.0196 (5.6697 x 10-8 W/m2 K4) (1 m2) (53946055.485 K4) = 0.0599 W
0.06 W = 0.06 J/s
0.06 J/s is the rate of heat transferred from the soil to the atmospheric CO2.
0.06 (J/s) (1 s) = 0.06 J
Equivalence in change of temperature (ΔT):
ΔT = q/m (Cp)
Introducing magnitudes:
ΔT = 0.06 (J)/0.00069 Kg (871 J/Kg °C) = 0.06 (J)/0.60099 (J/°C) = 0.099 °C
(rounding the cipher, 0.1 °C)
The radiant heat stored (the heat being stored and removed from any system by
radiation) by 0.00069 Kg/m3 of CO2 is equivalent to 0.1 °C. Then an increase of 381
ppmv of atmospheric CO2 causes an increase in the tropospheric temperature.
However, the heat stored by CO2 is not equivalent to 0.1 °C. The reason being CO2 is
a poor absorber-emitter of heat and so it cannot store heat for long periods of time.
Empirically, we obtain a change in the tropospheric temperature for CO2 of 0.01 °C.
I'll describe the mathematical procedure in more detail below.
3. EXAMPLES FROM NATURE:
Earth receives 697.04 W/m2 of infrared radiation from 1367 W/m2 of the incoming
energy (light, ultraviolet, radio, etc.) from the Sun. (Maoz. 2007. Page 36). 14% of
incoming heat to Earth is absorbed by air.
Data from the meteorological station in Monterrey, Mexico located at 25º 48´ North
latitude and 100º 19' West longitude and an altitude of 513 meters ASL: On 31
March 2007 at 18:15 UT the soil absorbed approximately 453 W/m^2 of IR radiation
causing a ground temperature of 318.15 K (45°C). The air temperature was 300.15
K (27 °C), what was the tropospheric ΔT due to the absorptivity-emissivity of air?
For the answer, first we need to know the load of heat transferred from the soil to
the mixed air. Principally, we need to obtain the Grashoff Number and the Heat
Transfer Coefficient for those particular conditions:
Grashoff Number
When we are calculating the load of heat transferred from the surface to the air we
need to know the flux of the air toward the warm surface and toward the upper
levels. The rate of flux is known as the Grashoff number (Gr), and it describes a ratio
involving buoyancy and viscosity: buoyancy/viscosity. As a fluid adjacent to a
warmed surface starts to increase in temperature, the density of that fluid decreases.
The buoyancy causes the less dense fluid to lift up, so the adjacent colder fluid is
conveyed into contact with the warmer surface.
Gr L = g β (Ts – T ∞) D3 / v2
Where,
g is the gravitational constant (9.8 m/s2)
β is the volumetric expansion coefficient (1/T)
T1-T2 is the difference of temperature between two adjacent systems expressed in
Kelvin (18 K).
D3 is the distance between two systems to the third power (1 m)
v2 is the kinetic viscosity (2.076 x 10-5 m2 / s) to the second power.
Introducing magnitudes:
Gr L = (9.8 m/s2) (3.332 x 10-3 K-1) (18 K) (1 m)3 / (2.076 X 10-5)^2 m4 /s2 =
= 5.877648e-1 m4/s2 / 4.309776-10 m4 /s2 = 1.36 x 109
Heat Transfer Coefficient
The Heat Transfer Coefficient (Ћ) is the rate of heat transferred from a warmer
system to a colder system. It relates to the Grashoff number, the Prandtl number
and the thermal conductivity of the fluid. The Prandtl number is dimensionless and
refers to the ratio of momentum diffusivity (v, or dynamic viscosity) and the thermal
diffusivity (a). The heat transfer coefficient is determined by the next formula:
k
Ћ = ---------- (C) [(Gr) (Pr)]a
D^3
Where,
k is the thermal conductivity (for air, k = 0.03003 W/m K)
D or L is the distance between the two systems
C is a factor of correction for irregular surfaces facing up (soil)
Gr is the Grashoff Number (obtained in the previous calculus Gr = 1.36 x 10^5)
Pr is the Prandtl Number (0.697 for air)
a is the constant of proportionality for laminar natural systems (1/3 for surfaces
facing up).
Introducing magnitudes:
0.03003 W/m K
Ћ = ------------------------- (0.14) [(1.36 x 109) (0.697)]1/3 = 4.13 W/m2 K
1 m3
The heat transfer from soil to mixed air is:
q = Ћ A (Ts – T∞)
Where,
q is the heat absorbed by the colder system
Ћ is the convective heat transfer coefficient (obtained in the previous procedure =
4.13 W/m2 K)
A is the implied Area (1 square meter)
Ts - T∞ is the difference of temperature between the heated system and the colder
system.
Introducing magnitudes:
q = 4.13 W/m2 K (1 m)2 (18 K) = 74.4 W (rate of heat transfer or heat flow rate)
74.4 W = 74.4 J/s (http://www.iprocessmart.com/techsmart/conversions.htm)
E = 74.4 (J/s) (1 s) = 74.4 J = 17.782 cal-th
If m of mixed air = 1.18 Kg/m3 and the Cp of mixed air at 300.15 K = 1005.7 J/kg K
(240.37 cal/Kg K), then:
ΔT = q/m (Cp) = 17.8 cal / (1.18 Kg) (240.37 cal/Kg °C) = 17.8 cal/ 283.64 cal °C=
0.063 °C
0.063 °C was the ΔT caused by thermal transfer by conduction-convection from the
ground to the air mixture. Let's see what happened in 1998:
The heat absorbed by dry air from incoming Solar radiation is 697.04 W/m2 X 0.14
(absorptivity of dry air at T = 300.15 K, and P = 1 atm) = 18.7 W/m2 = 4.47 cal/s m
K.
Considering the whole mixture of air, the Δq by Solar Irradiance absorbed-emitted
by mixed air would be only 0.734 W/m2 K (0.175 cal/s m °C). From this quantity,
the CO2 can store 0.012 W/m2 K (0.003 cal/s m °C) for only one second, which is
equivalent to 0.01 °C/s m.
The change in the tropospheric temperature that occurred in 1998 averaged 0.52 °C
throughout the year (UAH). The discrepancy, regarding the change caused by carbon
dioxide was -0.51 °C. The mathematical expression from the experimental data is as
follows:
Known Data on 14 April 1998:
Mass of CO2 in the atmosphere = 0.000614 kg (obtained by multiplying density by
volume, that is, 0.000614 kg/m3 x 1 m3).
Cp = 871 J/kg K
ΔT = 0.62 K
Δt = 60 s
Indicated Formula:
q Stored = m (Cp) (ΔT) / Δt
Introducing magnitudes:
q Stored = 0.000614 Kg (871 J/kg K) (0.62 K) / 60 s
q Stored = 0.534794 J/K (0.0103 K/s) = 0.00553 J/s (Joules per second or energy
per unit time)
E = 0.000553 J/s (1 s) = 0.000553 J = 0.001321 cal-th
Equivalence in ΔT = q / m (Cp) = 0.001321 cal/ 0.000614 kg (208.17 cal/Kg °C) =
0.01 °C
Since each kilogram of CO2 received 0.001321 cal, the temperature of each Kg of
CO2, and therefore the entire volume of CO2, increased by only 0.01 °C.
Note: If we take the last report from Mauna Loa for this algorithm, the mass of CO2
would be 0.00069 Kg. The change of temperature would be 0.0062 °C. The
difference between the ΔT produced by 0.000614 Kg and the ΔT produced by
0.00069 Kg of CO2 is negligible (0.0062 - 0.00553 = 0.00067).
To cause a variation in the tropospheric temperature of 0.52 °C (average global
temperature anomaly in 1998; UAH) required 1627.6 ppmv of CO2, a density of
atmospheric CO2 which has never been recorded or documented anywhere in the last
420,000 years. (Petit et al. 1999)
The total change in the tropospheric temperature of 0.75 °C was given for the
duration of one minute of one year (1998) (UAH); however, CO2 increased the
tropospheric temperature by only 0.01 °C. We know now that 1934 was the warmest
year of the last century. Where did the other 0.74 °C come from? Answer: it came
from the Sun.
April 6, 2007, AT 19:01 UT, data taken from the meteorological station in Monterrey,
MX, located at 25º 48´ North latitude and 100º 19' West longitude, and an altitude
of 513 m ASL:
Ts = 316.95 K
T∞ = 305.45 K
Density of dry air (d) = 1.168 kg/m3
Dry Air volumetric expansion Coefficient (β) = 3.16 x 10-3/K
Kinetic Viscosity (v) = 1.741 x 105 m2/s
Dry air thermal conductivity (k) = 0.02753 W/m K
Corrective factor (C) = 0.14
Constant of proportionality (a) = 1/3
Grashoff Number:
Gr L = g β (Ts – T∞) D3 / v2
Gr L = 9.8 m/s2 (3.16 x 10-3 K-1) (316.95 K - 305.45 K) (1 m)
m2/s)2
3
/ (1.741 x 10-5
Gr L = 3.0968 x 10-2 m/s2 K-1 (11.5 K) (1 m3) / (3.031081 x 10-10 m4/s2)
Gr L = 1.175 x 109
Conductive Heat Transfer Coefficient:
k
Ћ = ------------- (C) [(Gr) (Pr)]
D3
a
Ћ = [0.02753 W/m K / 1 m3] (0.14) [(1.175 x 109) (0.7043)] 1/3
Ћ = 0.0038542 (938.854) = 3.6185 W/m2 K
The heat transfer from soil to mixed air by conduction-convection is:
q = Ћ A (Ts – T∞)
q = 3.6185 W/m2 K (1 m)
2
(11.5 K) = 41.61 W (rate of heat transfer)
41.61 W = 41.61 J/s (http://www.iprocessmart.com/techsmart/conversions.htm)
E = 41.61 J/s (1 s) = 41.61 J = 9.94 cal
ΔT = q / m (Cp)
ΔT = 9.94 cal/1.168 kg (240.37 cal/Kg °C) = 9.94 cal/280.75216 cal K= 0.0354 K =
0.0354 °C.
4. CHANGE OF THE TROPOSPHERIC TEMPERATURE BY SOLAR IRRADIANCE (This
theme is better developed at Solar Irradiance is Increasing. It was corrected in 30
June 2008 due to minor grammar errors)
The total incoming solar irradiance to the terrestrial surface is 697.04 W/m2. From
this amount of infrared radiation, the surface absorbs about 348.52 W/m2. The
atmosphere absorbs 317 W/m2. Considering the mass of air and its thermal capacity,
the Earth’s temperature should vary by 30 °C. The fluctuation of the solar irradiance
in the last 300 years has been 1.25 W/m2. 1.25 W/m2 causes a change of the
Earth's temperature of 0.56 °C, which is the maximum averaged change in
tropospheric temperature achieved during the 1990s (the average of change of
temperature in 1998 is 0.51 °C). (Hurrell & Trenberth. 1999)
Planet Earth would not be warming if the Sun's energy output (Solar Irradiance) was
not increasing. Favorably, our Sun is emitting more radiation now than it was 200
years ago, and so we should have no fear of a natural cycle that has occurred many
times over in the lifetime of our Solar System.
Heat always moves from places of higher density of heat to places of lower density of
heat, thus states the Second Law of Thermodynamics (Van Ness. 1969. Page 54). In
daylight (P. S. obviously under, Sunlight), air is always colder than soil (P. S.
obviously, the surface of soil); consequently, heat is transferred from the soil to the
air, not vice versa. By the same physical law, the heat emitted by the Sun -a source
of heat- is transferred to the Earth, which is a colder system.
The capacity of carbon dioxide to absorb-emit heat is much more limited than that of
oceans and soil; thus, carbon dioxide cannot have been the cause of the warming of
the Earth in 1998.
A fact well known to all scientists is that the absorptivity-emissivity thermal property
of carbon dioxide diminishes as its density increases and as the temperature
increases. This happens because the infrared radiation absorption margin is very
narrow (wavelengths from 12-18 micrometers) and so the opacity of carbon dioxide
to infrared radiation increases with altitude. As the column of CO2 gains height, its
opacity to infrared radiation increases.
The dispersion of emitted heat increases when the density of carbon dioxide
increases because there are more microstates toward which energy can diffuse. As a
result, the momentum of the carbon dioxide molecules decays each time heat is
transformed into molecular kinetic energy, and emitted heat disperses in greater
amounts towards deep space through the upper layers of the atmosphere. This
process -determined by the second law of thermodynamics -could explain the
observed paradoxical phenomenon of the coldness of the higher tropospheric layers
in contrast with the tropospheric layer above the Earths surface, which is always
warmer than the upper layers.
When the concentration of atmospheric carbon dioxide increases, the strong
absorption lines become saturated. Thereafter its absorptivity increases
logarithmically not linearly or exponentially; consequently, carbon dioxide convective
heat transfer capacity decreases considerably.
5. ALGORITHM FOR METHANE (CH4)
Molar Mass of Methane (CH4) = 16.0425 g/mol
Current mass of CH4 in air = 1.740 ppmv:
W in mg/m3 = [ppmv] (12.187) (MW) / (273.15 + °C) = 1.740 (12.87) (16.0425
g/mol) / 300 °C = 1.198 g/ m3 = 0.0012 Kg/ m3
Specific Heat of Methane
T (K)
kJ/Kg K
275
2.191
300
2.226
325
2.293
350
2.365
375
2.442
Concentration of CH4 = 1.74 ppmv
Converting concentration to Density = [ppmv] (12.187) (MW) / (273.15 + °C) =
1.740 ppmv (12.87) (16.0425 g/mol) / 300 °C = 1.198 g/m3 = 0.0012 Kg/m3
Concentration of CH4 = 1.74 ppmv
Mass of CH4 = 0.0012 Kg/m3 (1 m3) = 0.0012 kg
Cp CH4= 2 226 J/kg K
Δq = 0.000535 W/m2 = 0.00013 cal-th
ΔT = Δq / m (Cp)
ΔT = 0.00013 cal-th /0.0012 Kg (533.3 cal/Kg*°C) = 0.00013 cal / 0.64 cal*°C =
0.0002 °C
Consequently, Methane is not an important heat forcing gas at its current
concentration in the atmosphere.
6. THE CASE ON 14 APRIL 1998 (RADIATIVE "FORCING") (Notice the standard for
CO2 of 350 ppmv was determined empirically by Friederike Wagner et al. The
standard of 280 ppmv is misleading).
Known data:
δ CO2 in 1998 [(ppmv) ∞] converted to density = 0.00049 Kg/m3
δ CO2 standard [(ppmv) s] converted to density = 0.00045 Kg/m3 (280 ppmv is a
flawed standard. The lower standard determined by Friederike Wagner et al is 350
ppmv during the late Holocene).
T of air = 318.15 K
Notice that for denoting concentration we enclose the symbol of the substance or
units between square brackets [...]
Formula to be applied exactly as it is applied by the IPCC:
Δ T = [α] ln [(CO2) ∞ / (CO2) s] / 4 (σ) T3.
Introducing magnitudes:
ΔT = (5.35 W/m2) ln ([367 ppmv] ∞/ [280 ppmv] s)/4 (5.6697 x 10-8 W/m^2 K4)
(318.15 K) 3
ΔT = 5.35 W/m2 (0.126)/4 [5.6697 x 10-8 W/m2 K4] (318.15 K)
(W/m2) K =
= 0.19 K; rounding the cipher, 0.2 K, or 0.2 °C.
3
= 1.45 (W/m2)/7.3
However, the change of temperature on April 1998 was 0.786 °C (UAH), so carbon
dioxide did not cause the anomaly.
Let's now apply the REAL STANDARDS for the Northern Hemisphere on April 14,
1998:
Standard temperature for the Northern Hemisphere = 288.15 K (Potter & Somerton.
1993)
Global standard density of atmospheric CO2 = 350 ppmv (The US Department of
Labor Occupational Safety & Health Administration (OSHA) and the American Society
of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) determined
empirically the standard average for outdoor CO2 from 350 to 600 ppmv. The lower
standard (350 ppmv) determined by Friederike Wagner et al is the one we are
considering here).
Density of CO2 (in some places only) in 1998 = 367 ppmv.
Applying Arrhenius' formula we obtain (In red, Schwartz, Stephen E. 2007
adjustments):
ΔT = α (ln [CO2] ∞ / [CO2] s) / 4 (σ) T3.
ΔT = (5.29 W/m2) ln (367 ppmv/350 ppmv) / 4 (5.6697 x 10-8 W/m2 K) (288.15 K)
=
3
= 0.28 W/m2/5.43 W/m2 K = 0.046 K; rounding the cipher, 0.05 K,
ΔT = 0.05 K is compatible with the value of the heat stored obtained by applying the
algorithm used in some paragraphs above:
On April 14, 1998, the temperature changed by 0.08 K in one second. The heat
stored by 0.00056 Kg (367 ppmv) of atmospheric CO2 is:
q = m (Cp) (ΔT / Δt) (Potter & Somerton. 1993. Page 58) (For a no-equilibrium
process with Cp rel. constant)
Introducing magnitudes:
q = 0.00056 Kg (871 J/Kg K) (0.08 K/s) = 0.48776 J/K (0.08 K/s) = 0.039 J/s
E = p (t) = 0.039 J/s (1 s) = 0.039 J, or 0.0093 cal, which will cause a change of
temperature of:
ΔT = q /m (Cp)
ΔT = 0.0093 cal/0.00069 Kg (208.17 cal/Kg K) = 0.0093 cal/0.144 cal K = 0.065 K;
or 0.065 °C; then the calculation using real standards is correct.
This comparison between the algorithm to obtain the stored heat and the algorithm
to obtain the radiative "forcing" demonstrates two important features:
1. The algorithm q = m (Cp) (ΔT/Δt) denotes the three modes of heat transfer:
radiation, convection and conduction.
2. When we introduce real standards and apply the proper algorithms, the
temperature increase caused by CO2 is no more than 0.1 K.
7. CO2 SCIENCE: THE CASE ON JUNE 22, 2007 (RADIATIVE "FORCING"):
If we want to know what ΔT is caused by the q absorbed by the increase of CO2 since
1985, we need to apply the next formula:
q stored = m (Cp) (ΔT/Δt)
Where the mass m is the product of volume and density (Pitts & Sissom. 2000), Cp
is the Specific heat of the substance, ΔT is the change of temperature (final
temperature - initial temperature), and Δt is the change of time. The result is the
relation between the heat absorbed by the substance and the heat removed from the
substance (Pitts & Sissom. 2000) (Wilson. 1994).
Introducing magnitudes:
q stored = 0.000155 Kg (871 J/Kg K) (0.5 K/s) = 0.0675025 J/s
E = 0.0675025 J/s (1 s) = 0.0675025 J = 0.016 cal
To know the change of temperature caused by 0.0003334 cal, we need to use the
next formula:
ΔT = Δq / m (Cp)
Introducing magnitudes:
ΔT = 0.016 cal / 0.000155 Kg (208.1 cal/Kg K) = 0.00027 cal / 0.0322555 cal K =
0.00385 K; rounding the cipher, 0.004 K, or 0.004 °C.
Since each kilogram of CO2 received 0.016 cal, the temperature of each Kg of CO2,
and therefore the entire volume of CO2, increased by only 0.004 °C.
A common error among some authors is to calculate the anomaly taking into account
the whole mass of atmospheric CO2, when for any calculation we must take into
account only the increase of the mass of atmospheric CO2. The error consists of
taking the bulk mass of CO2as if it were entirely the product of human activity, when
in reality the increase in human CO2 contribution is only 34.29 ppmv out of a total of
381 ppmv (IPCC). This practice is misleading because the anomaly is caused not by
the total mass of CO2, but by an excess of CO2 from an arbitrarily fixed "standard"
density. There is however no such thing as a "standard" density of atmospheric CO2.
What is the heat load transferred to 101 ppmv (0.000157 Kg/m^3) of atmospheric
CO2 (101 ppmv is the total increase in the mass of carbon dioxide since 1985), when
the change of temperature on 2 August 2007 was only 1.23 K through one minute
(ΔT/Δt)?:
q stored = m (Cp) (ΔT/Δt)
q stored = 0.000157 Kg (871 J/Kg * K) (1.23 K/60 s) = 0.0028 J/s
E = q (t) = 0.0028 J/s (1 s) = 0.0028 J = 0.00067 cal-th
To know the change of temperature caused by 0.00067 cal:
ΔT = Δq / m (Cp)
ΔT = 0.00067 cal/0.000157 Kg (208.1 cal/Kg K) = 0.00067 cal/0.0326717 cal K =
0.02 K; or a change of temperature of 0.02 °C.
Since each kilogram of CO2 received 0.00067 cal, the temperature of each Kg of CO2,
and therefore the entire volume of CO2, increased by only 0.02 °C. Notice that the
change of temperature on 2 August 2007 was twice the change of temperature on 6
April 2007. From this we deduce that there is a link between Kelvin and energy (J)
established by the Stephan-Boltzmann's Law.
Does this mean that air temperature would increase by 0.02 °C per second until it
reached scorching temperatures? No, it does not, as almost all of the absorbed heat
is emitted in the very next second. Thus the temperature anomaly caused by CO2
cannot go up if the heat source does not increase the amount of energy transferred
to CO2.
Now let us study the extreme case on July 8, 2007:
The real radiative equilibrium temperature of Earth is 300.15 K which is caused by
the oceans, not the "greenhouse" effect. The change of temperature caused by the
heat transferred from the ground to the total mass of atmospheric CO2 by radiation
(radiative "forcing") was:
Formula to be applied:
ΔT = [α] ln [(CO2) ∞ / (CO2) s] / 4 (σ) T3.
Where ΔT is the change of temperature, α is the assumed coefficient of heat transfer
of CO2 by radiation (5.35 W/m2), CO2 ∞ is the current density of carbon dioxide
expressed in ppmv, CO2 s is the assumed "standard" density of carbon dioxide
expressed in ppmv, σ is the Stephan-Boltzmann constant (5.6697 x 10-8 W/m2 K4)
and T3 is the temperature to the third power expressed in Kelvin.
Introducing magnitudes:
ΔT = 5.29 W/m2 [ln (381 ppmv ∞/280 ppmv s)] /4 (5.6697 x 10-8 W/m2 K4) (300.15
K) 3. (In red, Schwartz, Stephen E. 2007 adjustments)
ΔT = 5.29 W/m2 (0.308) / (4 [5.6697 x 10-8 W/m2 K4] (300.15 K)
6.13 W/m2 K
= 0.27 K ((In red, Schwartz, Stephen E. 2007 adjustments).
3
= 1.63 W/m2/
0.27 K/s is only 1.24% of the temperature difference between the ground and the air,
which was 21.8 K. We can see that carbon dioxide is not able to cause the
temperature anomalies that have been observed on Earth.
From the most recent observations of the tropospheric temperatures and their
relationship with the density of carbon dioxide, it is possible that the “radiative
forcing coefficient” of carbon dioxide is neither 5.35 W/m2 nor 5.29 W/m2, but an
amount of between 1.78 W/m2 and 2.68 W/m2. This would be congruent with the
observations of nature made by many scientists up to date. (Monckton. 2007)
For example, in the first scenario the ΔT caused by 381 ppmv of carbon dioxide is
0.1 K:
ΔT = (1.78 W/m2) ln ([381 ppmv] ∞ / [280 ppmv] s) /4 (5.6697 x 10-8 W/m2 K4)
(300.15 K) 3
ΔT = (1.78 W/m2) 0.308 / 6.13 = 0.55 (W/m2)/6.13 W/m2 K = 0.089 K (0.1 K,
rounding the cipher).
In the second option for α = 2.68 W/m2, the change of temperature caused by 381
ppmv of CO2 is 0.1 K:
ΔT = (2.68 W/m2) ln (381 ppmv /280 ppmv) /4 (5.6697 x 10-8 W/m2 K4) (300.15 K)
3
ΔT = 2.68 W/m2 (0.308) / 6.13 W/m2 K = 0.83 (W/m2)/ 6.13 W/m2 K = 0.13 K
The latter is congruent with observations (Decadal trend = 0.12 K from UAH data),
however the values have been forced to obtain a preconceived result, the procedure
therefore is flawed. The real value for alpha is 0.423 W/m2 (Read Total Emittancy of
CO2), so the real change of temperature caused by CO2 is 0.01 °C.
8. THE REAL “FORCING” OF CARBON DIOXIDE
We have seen that Arrhenius’ Formula to know the change of temperature by carbon
dioxide is not functional, given the high uncertainties in the values of heat flux (α),
the “standard” concentration of carbon dioxide and the “standard” temperature (T3).
Formula to be applied:
q = e (σ) (A) [(Ts)4 – (Ta)4]
Where q is the heat transferred by radiation from one system to another, e is the
emissivity of the surface that absorbs energy, σ is the Stephan-Boltzmann constant,
A is the area of interchange of energy, Ta is the temperature of the absorbent
surface and Ts is the temperature of the emitter.
Known variables and constants:
Data taken from the meteorological station in Monterrey, Mexico: On June 22, 2007
at 18.05 UT, at the coordinates 25º 48´ North latitude and 100º 19' West longitude,
and an altitude of 513 meters ASL, the air temperature at 1.5 m above ground level
was 299.65 K (26.5 °C), whereas the ground temperature was 300.15 K (27 °C),
which is the load of heat transferred from the ground to the mass of CO2 when 1
cubic meter of air contains 0.00069 kg.
e (at 300.15 K and a partial pressure of 0.00034 atm-m) = 0.001 (it has no units
because it refers to an index).
σ = 5.6697 x 10-8 W/m2 K4
A = 1 m2
Ta = 299.65 K [(299.65 K)4 = 8062266098.565 K4]
Ts = 300.15 K [(300.15 K)4 = 8116212154.05 K4]
Ts4 – Ta4 = 53946055.485 K4
Introducing magnitudes:
q = 0.001 (5.6697 x 10-8 W/m2 K4) (1 m2) (53946055.485 K4) = 0.0031 W
If the transference of energy occurred each second, then the equivalent energy is:
q = 0.0031 W*s
0.0031 W*s = 0.0031 J
What is the change of temperature caused by the heat transfer of 0.0031 W*s?
Formula to be applied:
ΔT = q/m (Cp)
Where q is the heat transferred from a warm system to a colder system (for this
case, soil is the warm system and air is the cold system), m is the mass of the
interferer system (carbon dioxide) and Cp is the Specific Heat of the interferer
system (carbon dioxide) at 300.15 K and constant pressure of 1 atm.
Known variables and constants:
q = 0.0031 J
m = 0.00062 Kg
Cp = 842 J/Kg K
Introducing quantities:
ΔT = 0.0031 J /0.00062 Kg (842 J/Kg K) = 0.006 K
Six thousandths of one degree is a negligible change of temperature.
We can apply the formula to extreme cases, for example, the case on April 6, 2007,
when the temperature of the soil was 316.95 K and the temperature of the air was
305.45 K:
Ts4 – Ta4 = 1386835138.99 K4
Introducing magnitudes to the formula:
q = 0.001 (5.6697 x 10-8 W/m2 K4) (1 m2) (1386835138.99 K4) = 0.0786 W
In terms of energy, 0.0786 W*s = 0.0786 J
Now let us apply the formula to convert heat to change of temperature:
ΔT = 0.0786 J /0.00062 Kg (842 J/Kg K) = 0.0786 J / 0.522 J K = 0.15 K
The change of temperature caused by 0.0786 Joules of energy absorbed by 0.00062
Kg of CO2 in the atmosphere on April 6, 2007 was 0.15 °C through one second.
Considering that the difference between the temperature of the soil and the
temperature of the air was 11.5 °C, the amount of 0.15 °C is negligible (just 1.3% of
the total).
We would be mistaken if we were to think that the change of temperature was
caused by CO2 when, in reality, it was the Sun that heated up the soil. Carbon
dioxide only interfered with the energy emitted by the soil and absorbed a small
amount of that radiation (0.0786 Joules), but carbon dioxide did not cause any
warming. Please never forget two important points: the first is that carbon dioxide is
not a source of heat, and the second is that the main source of warming for the
Earth is the Sun.
9. WATER VAPOR:
Known data on June 22, 2007:
The concentration of atmospheric Steam = 35387 ppmv (3.15% of atmospheric
water vapor) = 0.026 Kg/m3
∂ x v = 0.026 Kg/m3 (1 m3) = 0.026 Kg (Pitts and Sissom. 1994).
MW of H2O vapor = 18.0151 u
q stored = m (Cp) (ΔT/Δt) = 0.026 Kg (2059.5 J/Kg K) (0.5 K/60 s) = 0.45 J/s
It is evident that water vapor is a much better absorber-emitter of heat than carbon
dioxide. Under the same conditions, water vapor transfers 160 times more heat than
carbon dioxide.
Nasif S. Nahle
February 05, 2007
10. FURTHER READING
Bakken, G. S., Gates, D. M., Strunk, Thomas H. and Kleiber, Max. Linearized Heat
Transfer Relations in Biology. Science. Vol. 183; pp. 976-978. 8 March 1974.
Boyer, Rodney F. Conceptos de Bioquímica. 2000. International Thompson Editores,
S. A. de C. V. México, D. F.
Haworth, M., Hesselbo, S. P., McElwain, J. C., Robinson, S. A., Brunt, J. W. MidCretaceous pCO2 based on stomata of the extinct conifer Pseudofrenelopsis
(Cheirolepidiaceae). Geology; September 2005; v. 33; no. 9; p. 749-752.
Manrique, José Ángel V. Transferencia de Calor. 2002. Oxford University Press.
England.
Maoz, Dan. Astrophysics. 2007. Princeton University Press. Princeton, New Jersey.
McGrew, Jay L., Bamford, Frank L and Thomas R. Rehm. Marangoni Flow: An
Additional Mechanism in Boiling Heat Transfer. Science. Vol. 153. No. 3740; pp.
1106 - 1107. 2 September 1966.
Petit, J.R., J. Jouzel, D. Raynaud, N.I. Barkov, J.-M. Barnola, I. Basile, M. Benders, J.
Chappellaz, M. Davis, G. Delayque, M. Delmotte, V.M. Kotlyakov, M. Legrand, V.Y.
Lipenkov, C. Lorius, L. Pépin, C. Ritz, E. Saltzman, and M. Stievenard. Climate and
Atmospheric History of the Past 420,000 Years from the Vostok Ice Core, Antarctica.
Nature, Vol. 399, June 3, 1999 pp.429-43.
Pitts, Donald and Sissom, Leighton. Heat Transfer. 1998. McGraw-Hill.
Potter, Merle C. and Somerton, Craig W. Thermodynamics for Engineers. Mc GrawHill. 1993.
Schwartz, Stephen E. 2007. Heat Capacity, Time Constant, and Sensitivity of Earth's
Climate System. Journal of Geophysical Research. [Revised 2007-07-16]
Van Ness, H. C. Understanding Thermodynamics. 1969. McGraw-Hill, New York.
Wagner, Friederike, Bohncke, Sjoerd J. P., Dilcher, David L., Kürschner, Wolfram M.,
Geel, Bas van, Visscher, Henk. Century-Scale Shifts in Early Holocene Atmospheric
CO2 Concentration. Science; 18 June 1999: Vol. 284. No. 5422, pp. 1971 - 1973
Wagner, F., Aaby, B., and Visscher, H. Rapid atmospheric CO2 changes associated
with the 8,200-years-B.P. cooling event. Proceedings of the National Academy of
Sciences. September 17, 2002; vol. 99; no. 19; pp. 12011-12014.
Wilson, Jerry D. College Physics-2nd Edition; Prentice Hall Inc. 1994.
http://www.uah.edu/News/newsread.php?newsID=210 (Last reading on 25 August
2007)
http://www.atmos.uah.edu/data/msu/t2lt/tltglhmam_5.2 (Last reading on 25 August
2007)
http://www.cgd.ucar.edu/cas/papers/bams99/ (Last reading on 25 August 2007)
http://scienceandpublicpolicy.org/monckton_papers/greenhouse_warming_what_gre
enhouse_warming_.html
(Last reading on 25 August 2007)
http://www.ipcc.ch/SPM2feb07.pdf (Last reading on 25 August 2007)
http://www.gsfc.nasa.gov/topstory/20011212methane.html (Last reading
August 2007)
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