Length of a rod cooled by free convection
Length of a rod cooled by free convection
(OP)
Let's say that a small a small cylindrical rod with no insulation is welded to a large insulated vessel with temperature x. What is the solution for finding the length of the rod based on a maximum allowed temperature y at the free end of the cylinder? I have decided not to consider radiation of heat in this case. The rod is cooled by free convection.
Thomas A





RE: Length of a rod cooled by free convection
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RE: Length of a rod cooled by free convection
RE: Length of a rod cooled by free convection
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RE: Length of a rod cooled by free convection
I am aware that I will not get the correct answer by ignoring radiation. Since I am trying to solve this in a general manner, very little is known on the surroundings, exept that it is cooler. Let say just say that the heat radiated from the "rod" will be a safety margin.
To IRstuff
OK I admit that I tried to simplify this case by calling it rod. But since my knowledge on Thermodynamics is currently limited the idea whas that I first need to be able to solve this case by using a massive rod. It is actually a piece of instrument tubing (O.D. is 10mm, I.D. is 7mm) connected to a pressure gauge. The temperature of the vessel is 250deg C and the temperature at the pressure gauge should not exceed 70deg C.
Thomas A
RE: Length of a rod cooled by free convection
Temperature goes down along the bar with the law (it doesn't make much difference whether it's solid or a pipe):
T(x)=Twe-nx
with n=√(hP/kA) and:
h=heat transfer coefficient
P=perimeter exposed to air (outer perimeter only)
k=conductivity of material
A=cross section area of bar
prex
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RE: Length of a rod cooled by free convection
RE: Length of a rod cooled by free convection
I believe your solution is for a fin of infinite length.
Solution for fin of finite length should be
(T(x)-Tambient)=(Tw-Tambient-C)e^nx + Ce^-nx
with n=?(hP/kA
Where C is determined from conditions at x=L
Regards
RE: Length of a rod cooled by free convection
prex
http://www.xcalcs.com : Online tools for structural design
http://www.megamag.it : Magnetic brakes for fun rides
http://www.levitans.com : Air bearing pads
RE: Length of a rod cooled by free convection
Regards
RE: Length of a rod cooled by free convection
prex
http://www.xcalcs.com : Online tools for structural design
http://www.megamag.it : Magnetic brakes for fun rides
http://www.levitans.com : Air bearing pads
RE: Length of a rod cooled by free convection
Refering to your solution, T(x)=Twe-nx, I believe
T(x) should be T(x)-Tambient and
Tw should be Tw- Tambient.
Further, using dt/dx=0 as a boundary condtion, the closed form solution will involve cosh(nx) and will probably be safer to use.
Regards
RE: Length of a rod cooled by free convection
Tw
dt
dx
To Sailoday28
Ref your answer 29 oct 07 13:32 you write n=?(hP/kA)should the question mark be a sqrt?
Thomas A
RE: Length of a rod cooled by free convection
Tw refers to temperature at location where x=0
Not understanding this terminology indicates you are weak or rusty in conduction heat transfer.
With no heat transfer at length x=L and letting
THETA = T - Tambient
THETAO = Tw -Tambient
THETAL = TL - Tambieet
n=sqrt(hP/kA)
THETAL/THETAO=1/COSH(nL) WHERE cosh is the hyperbolic cosine function.
Please work with an engineer having a heat transfer background.
Regards