Fourier heat trans equation problem 1

[PEinOHIO]

i'm checking some calculations, i can't figure out how they got the following equation from fouriers law:

q = (4)(pi)(k)(delta T) / (ln(t1/t2))

for a long wire
---------------

fouriers law is q = (k)(A)(delta T) / L

are they using characteristic length instead of wire length?
i'm not sure about the natural log of time either

 

[ione]

Not the right forum

http://www.eng-tips.com/threadminder.cfm?pid=391

Anyway...

First of all who are "they"?

Integrating Fourier’s law for an hollow cylinder of internal radius t1 and external radius t2 and considering a unitary length

Q = 2*PI*k*deltaT/ln(t2/t1)

 

[PEinOHIO]

'they' are the people's work i am checking, employees.

the equation they have is correct, it is an equation used for a probe method of measuring thermal conductivity. the 4 pi has to do with boundary conditions. thanks

 

[ione]

Sorry for having miuntepreted...

Carslaw, H. S. and Jaeger, J. C, Conduction of Heat in Solids, Oxford Press, 2nd ed., 1964, pp. 58-60, 344-345

 

[hacksaw]

seems that the usual equation has a 2pi rather than a 4pi factor for example:

http://www.engineersedge.com/heat_transfer/conduction_cylidrical_coor.htm

or

http://en.wikibooks.org/wiki/Heat_Transfer/Conduction

 

[ione]

hacksaw,

Google "measurement of apparent thermal conductivity probe method" and you'll see the equation reported in the OP is correct, as confirmed by PEinOHIO in his/her second post

 

[ione]

PEinOHIO,

To make you surer I've not invented the reference above, you can take a look at equation 1.1 of the attached paper (page 3)

 

[corus]

Looking at the paper ione posted, I think the simplest explanation is that the heat flux related to 2.Pi..etc relates to a heat transfer boundary condition on a cylindrical surface, whereas the 4.Pi..etc. relates to the solution of the transient heat transfer equation for relatively small r, which the thermal probe within the medium is assumed to be.

Tata

 

[hacksaw]

amazing forum always something to learn

 

[PEinOHIO]

yep, that is the same reference i found. i too thought the 4pi should have been 2pi, but i learned something haha. thanks for the help everyone!

happy friday

 

[Cockroach]

1) Wire length as measured,
2) Natural Log, not base 10.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada