Hello all,

In the field of heat transfer in the mode of radiation there is an integral that can be used find the fraction of the total emission from a blackbody at a certain wave length.

The equation is: (please forgive the notation, my LaTex knowledge is minimal.)

Integrand = (E(lambda of a blackbody) / sigma*T^5 ))

F(0 to lambda) = Integrand d(lambda *T)

The limits of integration are

Lower = 0
Upper = lambda*T

Additional information:

lambda = wavelength (microns)
T = temperature of the blackbody (K)
sigma = Stefan-Boltzman constant = 5.670e-8 (W/m^2-K^4)

E is a function of wavelength and Temperature

Here is the euqation for E:

E(lambda of blackbody, T) = C1 / lambda^5*(exp(C2/lambda*T) - 1)

C1 = 2*PI*h*c^2 = 3.742e8 (W*microns^4 / m^2)
C2 = (h*c/k) = 1.439e4 (microns*K)

From Siegel's book "Thermal Radiation Heat Transfer" the integrand is only a function of the wavelength temperature product.

So basically the integral is set equal to a function f(lambda*T).

A table of the results are given. For example,

f(lambda*T) = 2200 (microns*K) the value of F(0 to lambda) = 0.100888

My question is this;

I am assuming that this integral was solved numerically, but how was that performed? I tried to used qsimp and qtrap but didn't obtain the correct results.

Any help would be greatly appreciated.

See attachment for a much "easier" read of the equations (crudely) described above.

EDIT: I did a quick search of Google books and the newest edition is there with as a limited preview. The solution methodology used to evaluate the function is descibed on pages 21 - 22.

Here is a link to page 21.

http://books.google.com/books?id=O389yQ0-fecC&lpg=PP1&pg=PA21#v=onepage&q=&f=false

END EDIT:

Thanks
Matt

The integral you need to compute simplifies to

http://i50.tinypic.com/23rai5c.gif

where http://i46.tinypic.com/sdo0w9.gif ; there is an infinite series given in the reference you gave which you can use for computations (or from which you can derive a nice Pade approximant); but if you insist on integration, you might want to look into the double-exponential quadrature routines used in NR.

Jan M.

Thanks for your input.

... there is an infinite series given in the reference you gave which you can use for computations ...

Yes, I have been using the series solution. It converges in a short number of iterations.

Thanks
Matt