Basic radiation problem
Basic radiation problem
(OP)
Hi all,
I am wondering if someone has the analytical solution (or point me to a reference book) of the following thermal problem:
I have a beam of cross section (Ac), conductivity (lambda), length (L), exposed area to space (As) and finally with an emissivity of 0.8. One end of the beam is fixed to 300K for instance and the rest of the beam is free to radiate to space (0K). I would like to know the temperature distribution along the bar.
Any comments greatly welcome!
Best regards,
Franck
I am wondering if someone has the analytical solution (or point me to a reference book) of the following thermal problem:
I have a beam of cross section (Ac), conductivity (lambda), length (L), exposed area to space (As) and finally with an emissivity of 0.8. One end of the beam is fixed to 300K for instance and the rest of the beam is free to radiate to space (0K). I would like to know the temperature distribution along the bar.
Any comments greatly welcome!
Best regards,
Franck





RE: Basic radiation problem
of course, this does depend upon you having taken a heat transfer course . . . are you a student?
from a previous posting, there is a heat transfer textbook at:
http://web.mit.edu/lienhard/www/ahtt.html
And this site is very good.
http://www.onesmartclick.com/engineering/heat-transfer.html
good luck!
-pmover
RE: Basic radiation problem
I am just checking if someone would have this info handy before doing the calculation myself.
Thanks for your response.
Best regards,
Franck
RE: Basic radiation problem
-k.dt/dx=S*0.8*(Tk^4-273^4) at x=L. For an exam question I'd simply say that the solution is trivial.
corus
RE: Basic radiation problem
RE: Basic radiation problem
RE: Basic radiation problem
Fortunately this is for a space application so no convection. I am ended up solving this simple problem with MathCad since the solution is not really trivial.
Best regards,
Franck
RE: Basic radiation problem
RE: Basic radiation problem
This is what I came up with:
assumptions-one dimensional steady state flow
finite length and uniform cross section beam
T(0)=300,T'(0)=300,T'(L)=0
T"(X)= (s*As*e/k*Ax)*(T^4-T#^4)
T(X)=300+300X-1/2*X^2*T#^4+
1/30*(s*As*e/k*Ax)*X^6
s and e are the radiation properties; X-the variable beam length; Ax-cross sectional area of beam;As-surface area of beam; T(x)-is the temperature gradient along beam; T'(0)- was estimatedand probably incorrect;T#-ambient temperature. I did not have a chance to plot this equation so try it out and tell me of the outcome.
RE: Basic radiation problem
RE: Basic radiation problem
Second order, 4th degree DE: T''(X)+M^2T^4+M^2T#^4
whereby M^2= s*Ps*e/kAx
One of the mistakes in my first reply was w/ As instead
of Ps(being the surface perimeter of the slender beam).
The solution that I gave you was based on Laplace Transform and one of the problem was to determine the I.C.
for T'(0). So far I can not get a solution to this non-linear D.E. but I think that you may try this approximation method:T(Xn)=T(Xn-1)-2*(T(Xn-1)/N+T#^4)^(1/4) on a spread sheet.
Note: N=s*e*Ps*Dx^2/k*Ax; Dx is the incremental delta x.
By the way is Ts=300 in Celsius or Farhenheit?
and T#=??? -if assumed at absolute zero above D.E. simplifies to two terms.
RE: Basic radiation problem
that is a tough problem period and a spread sheet calc only works for limited cases. try it is is a real learning experiece...
RE: Basic radiation problem
T(X)=T(0)+T'(0)*X + (2^-1)*M^2*X^6
where T(0) is the beam base temperature
T'(0)=dT/dX @X=0 and is a negative value
X = length measurement from beam base
M=s*Ps*e/k*Ax
also, the term involving the ambient temperature
was neglected being too low in value.
I ran a set of values other than that suggested by Franck on the above formula thru Excel and the shape of the curve was almost linear but still had a lazy S appearance.