Science > Physics > Temp gradient inside a sphere with heat source at the surface
| Topic: |
Science > Physics |
| User: |
"Florian" |
| Date: |
22 Nov 2007 05:03:20 PM |
| Object: |
Temp gradient inside a sphere with heat source at the surface |
Enough, I crosspost to sci.physics in order to expose your fraud.
May be a physicist will confirm my claim.
Here it is:
I claim that a temperature gradient will form close to the surface of an
homogenous sphere even if the heat source is located at or just beneath
the surface, because there is no escape way for the heat but the
surface.
Indeed, the heat that diffuse toward the center of the sphere accumulate
until it reaches an equilibrium. As a consequence, a sharp gradient of
temperature forms close to the surface (higher T at the center; lower T
at the surface).
I use the model of a wire with heat production at each extremity.
Let's say the wire is 12,8 unit long (1 unit= 1000 km for earth's
diameter), and heat is constantly produced at each extremity of the wire
with the production decreasing exponentially going toward the
center of the wire:
If u(x,t) is the temperature function, then the heat equation is:
∂u/∂t=∂2u/∂x2+exp(-x^2)+exp(-(x-12.8)^2)
At t=0 the temperature of the wire is set to zero (u(0,x)=0).
I fixed the boundary conditions to maintain the temperature at 0 C at
each extremity (u(t,0)=0 and u(t,12.8)=0) to translate the fact that
heat escapes through the extremities, i. e., for a sphere the surface is
constantly cooled down.
Then the wire is heated and its temperature rises until it reaches an
equilibrium.
Here is the numerical solution for that equation solved with
Mathematica:
http://nachon.free.fr/Phy/HeatModel.png
At the equilibrium, the temperature is hotter in the middle of the wire
and a gradient of temperature formed close to the extremities, as
expected.
The result would be equivalent for a sphere with the temperature being
hotter in the center of the sphere and a steep gradient of temperature
close the surface.
Could anyone confirm this, because some morons in talk.origins claim
that is *****.
They claim that in order to form a gradient with the higher temp at the
center of the sphere, the heat source has to be in the center of the
sphere. They claim it would not work if the source of heat was all over
the surface.
Thanx.
--
Florian
"Toute vérité passe par trois phases. D'abord, elle est ridiculisée;
ensuite, elle rencontre une vive opposition avant d'être acceptée comme
une totale évidence" - Arthur Schopenhauer
.
|
|
| User: "CWatters" |
|
| Title: Re: Temp gradient inside a sphere with heat source at the surface |
23 Nov 2007 11:41:23 AM |
|
|
"Florian" <first_name@last_name.net> wrote in message
news:1i806dc.d7zpzq1nrj5a8N%first_name@last_name.net...
Indeed, the heat that diffuse toward the center of the sphere accumulate
until it reaches an equilibrium. As a consequence, a sharp gradient of
temperature forms close to the surface (higher T at the center; lower T
at the surface).
Actually that's wrong. The center cannot be hotter than the surface. If it
was then energy would flow outwards not inwards. The center would start to
cool relative to the surface.
.
|
|
|
|
| User: "Ye Old One" |
|
| Title: Re: Temp gradient inside a sphere with heat source at the surface |
23 Nov 2007 04:21:15 AM |
|
|
On Fri, 23 Nov 2007 00:03:20 +0100, first_name@last_name.net (Florian)
enriched this group when s/he wrote:
Enough, I crosspost to sci.physics in order to expose your fraud.
May be a physicist will confirm my claim.
Here it is:
I claim that a temperature gradient will form close to the surface of an
homogenous sphere even if the heat source is located at or just beneath
the surface, because there is no escape way for the heat but the
surface.
I see you change your claim from the original.
Message-ID: <1i7ss3x.k1jxi81wan5f2N%first_name@last_name.net>
[quote]
....there is a gradient of temperature even when
the heat is only produced at the surface of a sphere, because there is
no escape way but the surface!
So the heat that diffuse toward the center of the sphere accumulate...
[end quote]
Message-ID: <1i7twh0.1ty41lp1nqfig7N%first_name@last_name.net>
[quote]
Even if the heat is produced at the surface of the sphere, the
higher temperature is still at the center, you moron!
check the heat equation below.
[end quote]
Message-ID: <1i7twh0.1ty41lp1nqfig7N%first_name@last_name.net>
[quote]
Even if the heat is produced at the surface of the sphere, the
higher temperature is still at the center, you moron!
check the heat equation below.
[end quote]
Indeed, the heat that diffuse toward the center of the sphere accumulate
until it reaches an equilibrium.
Correct, and at equilibrium there is no convection dummy.
As a consequence, a sharp gradient of
temperature forms close to the surface (higher T at the center; lower T
at the surface).
I use the model of a wire with heat production at each extremity.
And claimed that when heated at both ends the highest temperature
would be in the middle.
Let's say the wire is 12,8 unit long (1 unit= 1000 km for earth's
diameter), and heat is constantly produced at each extremity of the wire
with the production decreasing exponentially going toward the
center of the wire:
If u(x,t) is the temperature function, then the heat equation is:
?u/?t=?2u/?x2+exp(-x^2)+exp(-(x-12.8)^2)
At t=0 the temperature of the wire is set to zero (u(0,x)=0).
I fixed the boundary conditions to maintain the temperature at 0 C at
each extremity (u(t,0)=0 and u(t,12.8)=0) to translate the fact that
heat escapes through the extremities, i. e., for a sphere the surface is
constantly cooled down.
Then the wire is heated and its temperature rises until it reaches an
equilibrium.
That was NOT what you claimed dummy.
Here is the numerical solution for that equation solved with
Mathematica:
http://nachon.free.fr/Phy/HeatModel.png
At the equilibrium, the temperature is hotter in the middle of the wire
and a gradient of temperature formed close to the extremities, as
expected.
If there is equilibrium then no part is hotter - look up the meaning
of equilibrium dimwit.
The result would be equivalent for a sphere with the temperature being
hotter in the center of the sphere and a steep gradient of temperature
close the surface.
Could anyone confirm this, because some morons in talk.origins claim
that is *****.
It is.
They claim that in order to form a gradient with the higher temp at the
center of the sphere, the heat source has to be in the center of the
sphere. They claim it would not work if the source of heat was all over
the surface.
Correct.
Thanx.
--
Bob.
.
|
|
|
|
| User: "tadchem" |
|
| Title: Re: Temp gradient inside a sphere with heat source at the surface |
22 Nov 2007 07:24:46 PM |
|
|
On Nov 22, 6:03 pm, first_name@last_name.net (Florian) wrote:
<snip>
I use the model of a wire with heat production at each extremity.
Let's say the wire is 12,8 unit long (1 unit= 1000 km for earth's
diameter), and heat is constantly produced at each extremity of the wire
with the production decreasing exponentially going toward the
center of the wire:
Your wire is linear. The earth is not. This will drastically alter
any geometric considerations such as the shape of temperature fronts
moving from the surface inward (which maintain their spherical shape,
but steadily decrease in cross-sectional area).
If u(x,t) is the temperature function, then the heat equation is:
$B"_(Bu/$B"_(Bt=$B"_(B2u/$B"_(Bx2+exp(-x^2)+exp(-(x-12.8)^2)
This equation has the same shape as a rectal sphincter. I can guess
where it came from. Newton's Law of Cooling is a first order linear
differential equation. There is nothing Gaussian about it.
At t=0 the temperature of the wire is set to zero (u(0,x)=0).
I fixed the boundary conditions to maintain the temperature at 0 C at
each extremity (u(t,0)=0 and u(t,12.8)=0) to translate the fact that
heat escapes through the extremities, i. e., for a sphere the surface is
constantly cooled down.
If it is constantly being heated, how can it constantly be cooling
down at the same time? Are you using bipolar thermometers?
Then the wire is heated and its temperature rises until it reaches an
equilibrium.
Here is the numerical solution for that equation solved with
Mathematica:
Since your equation and your geometry are bogus, the "numerical
solution" is as contrived and meaningless as a claim for 20 m of sea
level rise in <20 years.
GIGO - Garbage In, Garbage Out.
BTW, in the *real* world, the deeper one goes into the earth, the
warmer it gets.
http://hypertextbook.com/facts/2003/YefimCavalier.shtml
"Many problems arise when digging so deep into the Earth. The most
obvious is the heat. For example, at 5 km the temperature reaches 70
degrees Celsius and therefore massive cooling equipment is needed to
allow workers to survive at such depths."
That's a thermal gradient of 14 degrees per km with the heat flux
OUTWARD. If you feel you absolutely *MUST* extrapolate numbers from a
limited trend, at least you should extrapolate real data rather than
'back-of-a-napkin" scribbled equations.
Tom Davidson
Richmond, VA
.
|
|
|
|
| User: "tadchem" |
|
| Title: Re: Temp gradient inside a sphere with heat source at the surface |
22 Nov 2007 07:07:57 PM |
|
|
On Nov 22, 6:03 pm, first_name@last_name.net (Florian) wrote:
Enough, I crosspost to sci.physics in order to expose your fraud.
May be a physicist will confirm my claim.
Here it is:
I claim that a temperature gradient will form close to the surface of an
homogenous sphere even if the heat source is located at or just beneath
the surface, because there is no escape way for the heat but the
surface.
Congratulations! You have just discovered what my granddaughter
figured out when she was 18 months old - that fire is hot.
"Fire (i.e. a heat source) is hot (causes a thermal gradient with heat
flowing away from the source)."
But then, since she was already a brilliant physicist she was able to
instantly generalize this, in a manner compatible with Newton's Law of
Cooling, to any material body without qualification as to homogeneity
of composition or geometry."
You are the latest nominee for the 2007 Captain Obvious award for
pointing out trivialities that even animals are capable of
recognizing.
Tom Davidson
Richmond, VA
.
|
|
|
|
|
Related Articles |
|
|
