| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
10 Oct 2005 10:15:03 PM |
| Object: |
Heat equation, confusing boundary conditions? |
I'm attempting to solve the steady-state heat equation on
a2-dimensional region (i.e., Laplace's equation, del^2T=0). I'm having
a bit of trouble figuring out how to properly form the problem with the
boundary conditions present. Any help you could give, I am attempting
to get an analytical solution, even if it is an approximation.
Background:
Domain: semi-infinite strip - 0<x<w , 0<y<Infinity
isotropic medium, conductivity, k
Boundaries:
dT/dx=0 (insulated) at x=0 & x=w
The 3rd boundary is where I have the problem:
at y=0: 0<x<w (non-inclusive) k*dT/dx+h*T=h*T0 (convection, with fluid
temperature at T=T0)
AND, T(0,0)=T1, T(w,0)=T2
Those 2 fixed temperatures at the corners seem to be giving me
headaches. I've tried superposition, but that doesn't seem to work.
Someone told me he doesn't think you can use superposition for
different types of boundary condition (type 1 and type 3 here) on the
same boundary. Also, solving the convective portion reduces the problem
to a 1D problem, seemingly making superposition wacky as it wouldn't
account for the imposed boundary with variation in the other dimension.
I've tried using integral methods to represent the fixed temperatures
at point sources and convective loads, where I intended to modify heat
load to result in the appropriate temperature, but in both cases this
resulted in infinite corner temperatures, killing the attempt. I've
looked at conformal mapping, but haven't found a domain where I can
figure out how to apply the BC.
Any help you give would be greatly appreciated. I'll be happy to
answer any other questions you may have.
In addition: what I'm really trying to find is the temperature in the
center of the convected face {(x,y)=(w/2,0)} as a function of the two
end temperatures, the convective conditions, width, and the material
conductivity. It seems like it would be a one-dimensional problem, but
the fact that the capacity of the material in the region is going to
pull heat in the y-direction forces a 2-D solution. If you could offer
any advice on this, it would also be appreciated.
Thanks in advance,
Nick J.
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| User: "Robert Israel" |
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| Title: Re: Heat equation, confusing boundary conditions? |
11 Oct 2005 03:18:53 PM |
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wrote:
I'm attempting to solve the steady-state heat equation on
a2-dimensional region (i.e., Laplace's equation, del^2T=0). I'm having
a bit of trouble figuring out how to properly form the problem with the
boundary conditions present. Any help you could give, I am attempting
to get an analytical solution, even if it is an approximation.
Background:
Domain: semi-infinite strip - 0<x<w , 0<y<Infinity
isotropic medium, conductivity, k
Boundaries:
dT/dx=0 (insulated) at x=0 & x=w
The 3rd boundary is where I have the problem:
at y=0: 0<x<w (non-inclusive) k*dT/dx+h*T=h*T0 (convection, with fluid
temperature at T=T0)
AND, T(0,0)=T1, T(w,0)=T2
Those 2 fixed temperatures at the corners seem to be giving me
headaches. I've tried superposition, but that doesn't seem to work.
Someone told me he doesn't think you can use superposition for
different types of boundary condition (type 1 and type 3 here) on the
same boundary. Also, solving the convective portion reduces the problem
to a 1D problem, seemingly making superposition wacky as it wouldn't
account for the imposed boundary with variation in the other dimension.
I've tried using integral methods to represent the fixed temperatures
at point sources and convective loads, where I intended to modify heat
load to result in the appropriate temperature, but in both cases this
resulted in infinite corner temperatures, killing the attempt. I've
looked at conformal mapping, but haven't found a domain where I can
figure out how to apply the BC.
Any help you give would be greatly appreciated. I'll be happy to
answer any other questions you may have.
In addition: what I'm really trying to find is the temperature in the
center of the convected face {(x,y)=(w/2,0)} as a function of the two
end temperatures, the convective conditions, width, and the material
conductivity. It seems like it would be a one-dimensional problem, but
the fact that the capacity of the material in the region is going to
pull heat in the y-direction forces a 2-D solution. If you could offer
any advice on this, it would also be appreciated.
Thanks in advance,
Nick J.
and later (in another version of this thread):
by approximation, i meant possibly a variational, or galerkin, method
approximation to the solution. Also, on the boundary I meant dt/dy,
not dt/dx. Finally, regarding the single point temperatures, could the
BC not be stated at f(x)=dirac_delta(x)+dirac_delta(x-w)?
A Dirac delta in the boundary condition would perhaps cause T to go to
infinity at a certain point of the boundary (the precise limiting
behaviour
might be complicated), but certainly would not give it a certain finite
limit.
For k*dT/dy+h*T=h*T0 on y=0, 0 < x < w, and dT/dx = 0 on x = 0 and x =
w,
there is a family of solutions T(x,y) = a + b y where k b + h a = h T0.
Thus you can specify T(0,0) = a arbitrarily, but you'll always have
T(w,0) = T(0,0).
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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| User: "" |
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| Title: Re: Heat equation, confusing boundary conditions? |
10 Oct 2005 10:29:42 PM |
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In article <1129000503.850083.261750@z14g2000cwz.googlegroups.com>, writes:
I'm attempting to solve the steady-state heat equation on
a2-dimensional region (i.e., Laplace's equation, del^2T=0). I'm having
a bit of trouble figuring out how to properly form the problem with the
boundary conditions present. Any help you could give, I am attempting
to get an analytical solution, even if it is an approximation.
Background:
Domain: semi-infinite strip - 0<x<w , 0<y<Infinity
isotropic medium, conductivity, k
Boundaries:
dT/dx=0 (insulated) at x=0 & x=w
The 3rd boundary is where I have the problem:
at y=0: 0<x<w (non-inclusive) k*dT/dx+h*T=h*T0 (convection, with fluid
temperature at T=T0)
AND, T(0,0)=T1, T(w,0)=T2
Those 2 fixed temperatures at the corners seem to be giving me
headaches. I've tried superposition, but that doesn't seem to work.
Someone told me he doesn't think you can use superposition for
different types of boundary condition (type 1 and type 3 here) on the
same boundary. Also, solving the convective portion reduces the problem
to a 1D problem, seemingly making superposition wacky as it wouldn't
account for the imposed boundary with variation in the other dimension.
I've tried using integral methods to represent the fixed temperatures
at point sources and convective loads, where I intended to modify heat
load to result in the appropriate temperature, but in both cases this
resulted in infinite corner temperatures, killing the attempt. I've
looked at conformal mapping, but haven't found a domain where I can
figure out how to apply the BC.
Any help you give would be greatly appreciated. I'll be happy to
answer any other questions you may have.
In addition: what I'm really trying to find is the temperature in the
center of the convected face {(x,y)=(w/2,0)} as a function of the two
end temperatures, the convective conditions, width, and the material
conductivity. It seems like it would be a one-dimensional problem, but
the fact that the capacity of the material in the region is going to
pull heat in the y-direction forces a 2-D solution. If you could offer
any advice on this, it would also be appreciated.
Thanks in advance,
Nick J.
Something appears to be missing here. Absent any heat sources and
with constant coolant temperature along the one non-insulated boundary
you can't have any other time independent distribution than uniform
one. A single point at different temperature on the boundary contributes
nothing in the integration. So, where is the source?
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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| User: "" |
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| Title: Re: Heat equation, confusing boundary conditions? |
11 Oct 2005 12:22:33 AM |
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So, where is the source?
outside the domain? :)
isn't any prescribed temperature on a boundary assumed to be the result
of an exterior heat source, i.e., an infinitely strong heat source with
time constant of zero and a feedback loop maintaining a fixed
temperature at a point?
The temperatures at the corners aren't just initial conditions, they
are maintained in the steady'state.
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