I was reading about the heat equation, and Wikipedia solves it for a 1D rod of length L: http://en.wikipedia.org/wiki/Heat_equat ... ier_series
The solution there is an infinite series involving sines and exponentials. I was wondering about the limiting temperature of the entire rod as time t → ∞. It says on that page that the entire body reaches a uniform temperature. Since their solution's series general term has an exp(-kt) term (k > 0), it seems as though, looking at their solution, that the limiting temperature as t → ∞ should be 0. Is this true? Or does the series converge to something else when t → ∞ (it doesn't seem obvious to me just looking at the series). Should the limiting temperature actually be the average value of the initial temperature function f (1/L * integral of f from x = 0 to L)?
cont. due to character limit
Heat diffusion
Re: Heat diffusion
cont.
And what would be the steady state temperature if the object were a 2D plate instead of a 1D rod? Would it be the average value of the initial temperature function for that? (1/Area * double integral over the region of the function)?
And what would be the steady state temperature if the object were a 2D plate instead of a 1D rod? Would it be the average value of the initial temperature function for that? (1/Area * double integral over the region of the function)?
Re: Heat diffusion
A lot depends on the boundary conditions.
In the example of your link, the temperature u is fixed at zero at both ends of the rod. So, heat can be conducted out of the rod at either end. Hence, after a long time, the temperature of the whole rod will go to zero. (In practice u would probably be a relative temperature, because heat conduction is not a constant close to absolute zero of temperature.)
Continuing with the one dimensional problem: if the boundary conditions were zero heat conduction (an insulator at both ends), then in that case the state at t -> infinity would have the temperature homogenous and equal to the average value of the initial (or any other) distribution.
For the 2D problem with u=0 at the edge, again you end with u = 0 everywhere
For the disc with insulator at the edge, the final state will have the heat distributed uniformly and so uniform temperature.
Joe
In the example of your link, the temperature u is fixed at zero at both ends of the rod. So, heat can be conducted out of the rod at either end. Hence, after a long time, the temperature of the whole rod will go to zero. (In practice u would probably be a relative temperature, because heat conduction is not a constant close to absolute zero of temperature.)
Continuing with the one dimensional problem: if the boundary conditions were zero heat conduction (an insulator at both ends), then in that case the state at t -> infinity would have the temperature homogenous and equal to the average value of the initial (or any other) distribution.
For the 2D problem with u=0 at the edge, again you end with u = 0 everywhere
For the disc with insulator at the edge, the final state will have the heat distributed uniformly and so uniform temperature.
Joe
Re: Heat diffusion
Thanks, Joe.
So for keeping the rod or disk insulated at the ends/edges, does this mean that the derivative of the temperature with respect to position (∂u/∂x in the 1D case) at the edges is 0 for all t? Because at the bottom of this page ( equations 25 and 26 at http://mathworld.wolfram.com/HeatConduc ... ation.html), Wolfram gives solutions assuming boundary conditions with ∂u/∂x = 0 at the ends of the rods, so I'm wondering if this is what it means to be insulated.
So for keeping the rod or disk insulated at the ends/edges, does this mean that the derivative of the temperature with respect to position (∂u/∂x in the 1D case) at the edges is 0 for all t? Because at the bottom of this page ( equations 25 and 26 at http://mathworld.wolfram.com/HeatConduc ... ation.html), Wolfram gives solutions assuming boundary conditions with ∂u/∂x = 0 at the ends of the rods, so I'm wondering if this is what it means to be insulated.
Re: Heat diffusion
Correct. Directly from the diffusion equation, if there is no heat flow in a particular direction (here a consequence of having the insulator), the temperature gradient in that direction must be zero.So for keeping the rod or disk insulated at the ends/edges, does this mean that the derivative of the temperature with respect to position (∂u/∂x in the 1D case) at the edges is 0 for all t?
The other case you quoted (u = 0 at boundary) corresponds to a heat reservoir at u = 0. e.g. a large, well mixed volume of fluid at the reference temperature.
Joe
Re: Heat diffusion
Incidentally, we always referred to such problems as Carslaw and Jaeger problems. If you are interested, I recommend "Conduction of heat in solids" by Carslaw and Jaeger (from NSW and Tas respectively). A very good book, and more generally useful, because the diffusion equation arises not only in heat conduction but in many other areas as well.
Re: Heat diffusion
Thanks for mentioning that book Joe, it seems very interesting. And yes, this type of equation appears in many places, even in areas like quantitative finance.joe wrote:Incidentally, we always referred to such problems as Carslaw and Jaeger problems. If you are interested, I recommend "Conduction of heat in solids" by Carslaw and Jaeger (from NSW and Tas respectively). A very good book, and more generally useful, because the diffusion equation arises not only in heat conduction but in many other areas as well.