In an article from Wikipedia about heat transfer coefficients in the section Combining heat transfer coefficients there is an equation which describes the rate of heat transfer between the bulk of the fluid inside the pipe and the pipe external surface:

$$q=\left( {1\over{{1 \over h}+{t \over k}}} \right) \cdot A \cdot \Delta T$$

where: $q$ - heat transfer rate (W), $h$ - heat transfer coefficient (W/(m2·K)), $t$ - wall thickness (m), $k$ - wall thermal conductivity (W/m·K), $A$ - area (m2), $\Delta T$ - difference in temperature.

Can you explain it or give any reference?

  • 1
    That looks like the naive formula for heat transfer trough multiple layers of thermal insulators. It's basically analog to the formula for parallel resistance in an electronic circuit. Technically it only applies to infinite flat plates, and I am not sure that it's actually correct for pipes, where the different layers have different diameters and therefor different area, but I have never looked into that case. Unlike with lumped resistive elements, these simplified heat transfer equations are not quite as reliable, because there is no equivalent for wires with heat (lest we use heat pipes) – CuriousOne Dec 15 '14 at 15:29
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    When you have circular symmetry, the temperature gradient ends up being of the form $1/r$ where $r$ is the distance from the center of the cylinder. Integrating to get temperature, you get something like $T\propto \log{r/r_0}$ where $r_0$ is the inner diameter of the pipe. When the wall is thin compared to the radius, then you can expand: $\log{1 + dr/r_0}\approx dr/r_0$. So temperature is linear with distance for thin shells... which is the approximation used here. – Floris Dec 15 '14 at 16:13
up vote 1 down vote accepted

A formula such as this (being a analog to parallel electrical resistors, as pointed out in the comments) can get a little more complicated when different areas are involved. However, I think that the author of your formula restricted the discussion to one single heat transfer area in order to avoid dealing with that.

The heat transfer rate originally starts out stated in terms of temperature difference. In this case we're trying to work out the complications from dealing with a 3rd intermediary temperature. As such, both of the following statements should be true for your problem:

$$ q = h A \left( T_2 - T_1 \right) \\ q = \frac{k}{t} A \left( T_3 - T_2 \right) $$

This is if $T_2$ is the temperature on the inner surface of the pipe wall. Also, the definition of the total temperature drop is $\Delta T = T_3-T_1$

You should be able to solve these equations for $T_2$, then plug that back into one of those equations and get the final result you were asking about.

In reality it's the coefficients in front of $A$ that get a little complicated. You can see that they used the heat transfer coefficient for the pipe material times the thickness. If you really wanted to be accurate, you could do some integrals in cylindrical coordinates to get a format which is valid for pipes with a large thickness relative to its radius. But clearly that makes it more difficult to illustrate the basic principle.

You would also see more complexity if an external heat transfer coefficient was referenced to the external surface of the pipe, since now each equation you write will have a different area involved. This is really just a more simple version of the integral I was talking about for the pipe material. Clearly you can't have $A$ factored out of your final form, but an acceptable alternative is to group area along with the heat transfer coefficients. In this sense, we're entertaining a more general format of $q=H_n (T_{n+1}-T_n)$. You could write a number of those equations and justify the method for their combining to yourself. This would be my preferred perspective. The notational weirdness of your pipe example is sure to confuse people.

So my conclusion is that it's best to zoom out and look at the problem in more macroscopic terms. You have a heat transfer rate (energy/time) that flows over several resistances to heat transfer (H, units come from this definition), and the equations for this mechanism cause a series of temperatures along the path of the heat movement to arrange in a certain way.

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