Temp change through a wall 2

[josephn16]

Help please, I am a ChemE, and I know I should remember this, but i guess it's just been too long.

I have a known temperature on one side of a wall of known thickness and thermal conductivity. How do I calculate the temp on the outside of the wall?

 

[corus]

You can't. To know the temperature on the outside wall you need to know the heat transfer from that surface via either the heat flux or heat transfer coefficient. If the heat transfer includes radiation then the problem is non-linear, similarly if it involves natural convection. If the heat transfer coefficient is constant then the temperature variation is linear in cartesian co-ordinates and you solve T=ax+b, where a and b are unkown. At the outside surface To=a.xo+b and -ka=h(To-Ta), and on the inner surface Ti=a.xi+b. Solve for To, a and b.

 

[josephn16]

That's what I thought, thanks.

 

[25362]

Fourier's law is the basis for heat conduction:

Q = -kA(dT/dx)

Generally speaking the temperature change from one side to the other of a wall of thickness x and surface A (perpendicular to heat flow), with a thermal conductivity k, depends on these and on the heat flow Q:

T₂-T₁=(Q/A)(x/k)

x/k is known as thermal resistance.
Please note that T drops in the direction of heat flow.

For cylinders as in thick-walled pipes or vessels with inner radius r₁, outer radius r₂, and length L, with heat Q flowing outwards,

T₁-T₂= Q*ln(r₂/r₁)/(2*pi*L*k)

One always has to keep in mind the consistency of the units used.

 

[MortenA]

if its a relatively thinn steel pipe un-insulated then assuming that the outside wall temp is equal to the inside medium temperature is not totally off the mark.

Usually in heat transfer (for pipes) you consider:

1) Inside heat transfer from medium to solid
2) conduction through solid
3) Transfer from outside wall to outside medium

Usually either 1) or 3) will be much larger than 2) and also 1)>>3) or 3)>>1)

If you have a liquid flow in a pipe and air on the outside you will find that 1)>>3) so that the bulk of "resistance" will be from outside wall and to outside medium-

Therefore the outside wall temperature will be quite close to the inside medium temperature.
Best regards

Morten

 

[josephn16]

Thanks for all the answers, but I have no idea what Q or the heat transfer coefficient are. This was a question from a customer, wanting to possibly replace one of our materials with another, so I have no way of finding out heat flux. I thought I remembered something using a lot of assumptions (infinite plate, etc.) that could give me a ballpark number. I could probably use the temperature of the material now and get pretty close, since I know the properties of both materials.

 

[IRstuff]

Seems to me that the simple solution is to ask the customer what the temperature is on the other side of the wall.

TTFN

 

[corus]

If you know what the temperatures are now and the inside surface temperature is fixed then from 25362's theory the temperature difference is inversely proportional to the conductivity, assuming the same flux on the outside. So doubling the conductivity gives you half the temperature difference you have now.

 

[jproj]

josephn16:

Could you give us an idea of the temperatures and materials you are using (even wall thickness would help). If you have steam running through a thin steel pipe, the outside temperature is going to be less than a degree different from the steam temperature.... even if you're in the artic.

If you want to get a close outside wall temp. estimate, you need to know the external wall conditions (e.g. air temperature, wind speed). For a conservative number (highest wall temp) you can use the average high temperature of the area with no wind (natural convection).

Let us know some more specific info and we can try to help you out.

Regards,

jproj

 

[josephn16]

I am using fused silica 3 inches thick. Temp on the inside is 1200F. Thermal conductivity is 0.6 W/m K.

 

[corus]

Using mm and degrees C the outside temperature is
T=75*(20-648)/(75+0.0006/h)+648 where h is the heat transfer coefficient to an ambient temperature of 20C. h is usually about 10 W/m^2 C so your outside temperature is about 300 C.
I would guess that your silica is surrounded by a steel shell but the conductivity of steel is relatively high not to affect the sums.

 

[hacksaw]

allowing for the heat transfer from the outer surface i come up with 500-700 oC, depending on the shell diameter.

3" of refractory in such a furnace does not offer much resistance to heat flow and is one reason why it is rarely used alone.

good luck with your design...

 

[25362]

Assume the outside temperature is 25^oC, wind and radiation effects result in an external htc=15W/(m²*K); then by equating the heat flow from the external surface with that flowing across the 75 mm of fused silica, one can estimate the outside wall temperature t as follows:

Q=(T-t)(A)(0.6/0.075)=(t-25)(15)(A)

(649-t)(0.6/0.075)=(t-25)(15)

This results in: t=242^oC.

As one can see the resulting external wall temperature also depends on the outside htc, corus took 10, I used 15 W/(m²*K). There are graphs that help in estimating the outside htc w/o many iterations.