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Clifford E Carnicom
September 17 2000


A preliminary model has been developed to estimate the length of time that is
required for a contrail to dissipate. It is assumed within this
discussion that the contrail is composed of water vapor (per
historical definition). The model developed agrees extremely well
with the historical behavior and observation of contrails. The model
is not intended to encompass all variables that may be in effect, but
it does model reasonably well the expected behavior of water at
flight altitudes. Any errors will be corrected if and as they are
brought to my attention. It will be noted that this model is not a
function of relative humidity, as no basis from thermodynamics
has yet been established for it’s inclusion. Any model based upon
the premise of “mixing” as the primary mechanism for dissipation
requires quantification to receive consideration. Cloud formation and the
introduction of aerosol particles to assist in their formation is an
entirely different discussion which is to be examined separately.
The conclusions that result from the study of this model are several:

1. Contrails composed of water vapor routinely dissipate, as the physics and
chemistry of this model will demonstrate. As a separate and distinct
set of events, clouds may form if temperature, relative humidity, and
aerosol conditions are favorable to their development. If “contrails”
by appearance transform into “clouds”, it can be concluded that the
material of composition is not water vapor.

2.The conditions under consideration show that the ice crystals
within a contrail can warm to the melting point and subsequently melt
with the heat provided by solar radiation.

3. As demonstrated both by historical observation and this model, the
time expected for contrail dissipation is relatively short, e.g., 2
minutes or less. This assumes the contrail is composed essentially
of water vapor, per the classic definition (condensed trail).

4. The rate of contrail dissipation is highly dependent upon the
the size of the ice crystal particles and the amount of solar radiation.
Dependence upon relative humidity is not evident. ‘Cloud’
formation from aircraft, should it occur, is dependent primarily upon
the temperature, the relative humidity, and the type and size of
aerosol particles(nuclei) that are introduced.

The basic form of the contrail dissipation model, based upon the
chemistry, mathematics and physics of thermodynamics is as follows:

time for dissipation = (mass of water crystal * (Q + heat of fusion))
/ power

where Q is the amount of heat required to increase the temperature of
a substance (ice).


t(sec) = (m (kg) * Ht(kj/kg)) / P(watts)

where t is the time required for contrail
dissipation(transformation), in seconds, m is the mass of the ice
crystal in kilograms, Ht is the heat of transformation of ice in
kilojoules per kilogram, and P is the power applied to the system in

Calculating the internal energy, or enthalpy, of water vapor often
involves several phase changes, as water varies between solid, liquid
and vapor under varying conditions of temperature and pressure. In
the case of a contrail composed of water vapor, the heat of
transformation will consist of two phases. The first is the amount
of heat required to raise the temperature of the ice crystal at a
sub-zero temperature to 0 deg. C., which will be designated as Q in
the present case. The second segment of heat required will be that
which melts the ice crystal to a liquid form. The primary processes
involved in contrail formation therefore appear to involve:

1. The emission of water vapor from the aircraft.
2. The freezing of the water vapor at sub-zero temperatures into ice
3. The warming of the ice crystals to the melting point through solar
4. The melting of the ice crystal with solar radiation to where the
water vapor once again no longer is visible. This returns the water
to the state from which it was emitted from the engine.

Let us now quantify the components of this model with elements that
are typical or representative of the conditions of contrail


Assume that we have a cubed particle size (nucleated ice crystal) of
dimension d on a side, measured in microns(designate as u). Given
also that the density of ice is .917gm/cm3, the mass of the particle

mass=(d(u) * (1E-6m/u))^3 * (1E6cm3/m3) * (.917gm/cm3) * (1E-3kg/cm3)


mass = (d^3 * 9.17E-16 cm3 gm kg m3) / ( m3 cm3 gm)

Q + Heat of Fusion:

Q is equal to the amount of heat required to increase the temperature
of the ice crystal from the ambient temperature to 0 deg. C. The
specific heat of ice is given as 4.21 kJ/(kg C) at 0 deg. C. The
specific heat varies only slightly with respect to temperature and
pressure, and this value will therefore be used. J refers to joules
of energy.

The heat of fusion of ice is 335kJ/kg. It requires this amount of
energy to melt ice.

Therefore, the amount of heat required to transform the ice crystal

dQ + heat of fusion = 4.21 kJ/(kg C) * dT + 355kJ/kg

where dQ is the amount of heat entering the ice crystal, the heat of
fusion is the amount of heat required to melt the ice crystal, and dT
is the temperature change from the ambient air to 0 deg. in Celsius.

The model now becomes:

t(sec) = (d^3 * (9.17E-16)cm^3 gm kg m^3 * ((((4.21kJ/kg)*dT)/(kg C))
+ 355kj/kg)) / P * (m^3 cm^3 gm)

Power (P):

The energy of solar radiation is given in terms of watts/ square
meter. Representative values measured range approximately from 200 to
700 watts/m^3. To arrive at the power applied to the ice crystal, we
will take the surface area of the crystal exposed perpendicularly to
the sunlight, and apply the solar radiation to it. The solar
radiation will be applied on a continuous basis to the surface area
until melting is complete.

Power absorbed = d^2 * (watts/m^2) * (1E-6m/u)^2

and since 1 watt = 1 joule/sec

Power absorbed = d^2 * (J/(m^2 s) * (1E-12) m^2/u^2

The model now becomes:

t(sec) = (d(u)^3 * (9.17E-16) cm3 gm kg m^3 * ((4.21kJ/kg * dT kJ/kg
C) + (335kj/kg))) / (d(u)^2 * (J/(m^2 s) * (1E-12) m^2 / u^2)


t(sec) = ((d(u) * (9.17E-13) * (4.21dT + 335) J cm^3 gm kg m^3 s
m^2) / (Watts * 1E-12 J m^2 m^3 cm^3 gm kg)

or t(sec) = (d(u) * (9.17E-13) * (4.21dT + 335)) sec / (Watts *

or t(sec) = (d(u) * .917 * (-4.21T + 335)) / Watts/m2

where d is measured in microns, T is the air temperature where the
contrail forms, measured in Celsius, and solar radiation is in watts
per square meter.

Representative cases and the application of this model will now be
considered. Research indicates that the expected size of particles
emitted from aircraft ranges between 30 and 200 microns (Goethe MB –
Ground Based Passive Remote Sensing of Ice Clouds with Scattered
Solar Radiation in the Near Infrared – Max Planck Inst Meteorol).
The temperature of the air at flight altitudes commonly approaches
-50 deg. C. Solar radiation commonly ranges between 400 and 700
watts per square meter.

In the tables presented, d is the dimension of the ice crystal along
one side of the cube, T is the temperature of the ambient air where
the contrail forms (.e.g, 35000ft. MSL), and P is the solar radiation
in Watts/sq. m. t is the length of time that it requires for the
contrail, or ice crystal to dissipate (i.e., transform from ice to
water vapor).

d(microns) T(deg. C.) P(watts/sq. m) t(sec)

1 -50 600 1
10 -50 600 8
30 -50 600 25
50 -50 600 42
100 -50 600 83

1 -40 400 1
10 -40 400 12
30 -40 400 35
50 -40 400 58
100 -40 400 115

1 -30 700 1
10 -30 700 6
30 -30 700 18
50 -30 700 33
100 -30 700 60

This model covers the expected size range of any particles expected
to be emitted by aircraft; most airborne particles range between
0-100 microns. It is of interest that the particle sizes considered
in this model are generally considered to be too large to serve as
cloud condensation nuclei; the average expected size of cloud
condensation nuclei is extremely small, and on the order of .1 to .2
microns. A 10 micron particle is considered extremely large with
respect to cloud condensation nuclei. This size distinction, when
coupled with the results of the model above, further indicate the
need to consider cloud formation as a separate and distinct physical
process from that of contrail dissipation. That analysis would
necessarily consider the significant role that aerosol particles,
deliberately or otherwise introduced, would have on the cloud
nucleation and formation process.

As can be seen, the results of this model agree extremely well with
the observed properties of contrails over their historical existence.
This work is based upon the physical processes, chemistry and
mathematics of thermodynamics with respect to water and the various
phase states. Consideration has also been given to the phenomenon of
sublimation, and it has been found to be not applicable due to the
extremely low atmospheric pressure requirements for sublimation to
occur(P<.006atm). The greatest variation within this model is seen
to relate to particle size. It is seen that the contrails composed
of the smaller particles dissipate within 30 seconds or less, and
that the contrails composed of even relatively large particles are
expected to dissipate within a couple of minutes at most.

If the dissipation of an observed contrail does not conform to the
model above, and the corresponding physics and chemistry and math of
same, then the logical conclusion that can be drawn is that the
material of emission is not likely to be water vapor. As mentioned earlier, the
physics of cloud formation are an entirely separate process, and are
highly dependent upon temperature, relative humidity, aerosol type
and the size of aerosol particles that are introduced. Any
alterations in the formation of cloud processes as they have been
repeatedly observed and recorded must necessarily consider the impact
of these aerosols, identified and unidentified, within the analysis.
Prior attention given to microscopic hydrated salts remains a
priority in this research.

Clifford E Carnicom
September 16 2000
Authored at Lake Heron, NM