Re: How do I estimate the temperature at a given distance from a gas flame?

I'm not certain what you mean.  Do you want to estimate the temperature of
the flame, from a distance?  Or do you want to estimate the temperature of
the air at some distance from the flame?  I'll try to provide an answer for
both questions.

To estimate the temperature of the flame itself:
The most direct method would require a spectrometer (e.g., a diffraction
grating and protracter). The presence or absence of particular emission
lines in the flame will give some idea as to the temperature in the flame.
  You can find a listing of atomic and molecular emission lines in a CRC
handbook or on the NIST website (at
physics.nist.gov/PhysRefData/contents-atomic.html). The relative intensity
of two related lines (an Oxygen-I and Oxygen-II pair, for instance) can
give a better temperature estimate, but is a more involved set of
measurements to make.

To estimate the temperature of the air near the flame:
This is one of those deceptively simple heat flow problems that sadistic
thermodynamics professors love.  Probably the simplest approach to the
problem is to neglect gravity (so the problem is spherically symmetric),
neglect radiative heat transfer (to simplify the equations -- this is
actually a good assumption in air) and assume steady-state.  Then, the
temperature T is a function of the distance r from the flame, where T(r)
obeys a stripped-down heat equation in 3D polar coordinates:
    (1/r^2) D/Dr( K r^2 DT/Dr ) = 0
where D/Dr is the partial derivative with r and K is the thermal
conductivity of air (in meters^2/second).  The boundary conditions would be
that T(r=0) is the temperature of the flame and T at large r is simply room
temperature (25 C).  You can integrate twice on r and apply these boundary
conditions to solve for T(r).

If you want to get a better estimate for the temperature near the flame,
then you pretty much have to include both gravity and convection.  Gravity
will force you to use cylindrical coordinates, but the problem will
fortunately be only 2D.  The heated air will tend to rise and expand
adiabatically, so you're going to have two equations:  the heat equation
for temperature T; and, another equation for the local air pressure in
terms of T, formed from the ideal gas equation and the relation for
adiabatic expansion of an ideal gas.  Including convection means placing an
upper limit on the temperature gradient, since convection will be active
whenever the temperature gradient exceeds some critical value.  You can
probably estimate the critical temperature gradient from thinking about
your second equation (very high gradient -> more free energy available than
in simple adiabatic equilibrium -> small-scale adiabatic mixing of the gas
will flatten the gradient).  You'll need to be careful about your boundary
conditions, but you should be able to arrive at a solution for T in a
finite amount of time.

I hope that this answers your question.