[Physics] Why do people use the outer area of the inner pipe for overall heat transfer coefficients in heat exchangers

thermodynamics

I was reading the below document on heat exchangers and noticed that when people define their overall heat transfer coefficient, they use the outer diameter of the inner pipe.

http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node131.html

An illustration is shown below (taken from the above link)

enter image description here

The equation in question is shown below:

enter image description here

So here's what I know:

  • D refers to the outer diameter of the inner pipe (r2*2)
  • L refers to the length of your pipe
  • $h_o$ is the overall heat transfer coeffcient
  • $deltaT_{lm}$ refers to the log-mean temperature difference
  • We need the log mean temperature difference because we are dealing with cylindrical coordinates (similar to normal temperature difference for Cartesian coordinates). Derivation is in the article
  • We are defining an overall heat transfer coefficient as we see fit and people just happen to use the outer surface area of the inner pipe for the area when defining the overall heat transfer coefficient

Here's what I don't know:

  • Why would people choose to use the outer area of the inner pipe instead of the inner area of the inner pipe? Heat travels from hot to cold (talking about net travel here. I am aware that random molecular collisions are what transfer heat excluding radiation), so wouldn't it make sense to use the inner pipe inner area for ho since the inner area is the first spatial limiting factor for heat transfer? This is where the hot fluid is on the inside and the cold is on the outside.

Best Answer

This is defining the overall heat transfer rate, from $T_A$ to $T_B$ (which will basically define the performance of the heat exchanger).

At $T_B$, the entire diameter and length of the pipe with be the area for heat exchange.

The important part you are not accounting for is $h$, the heat transfer coefficient.

Earlier in the page you linked, a value for $h_0$ is given as:

$$\displaystyle \frac{1}{h_0} = \frac{r_2}{r_1h_1}+\frac{r_2}{k}\ln\left(\frac{r_2}{r_1}\right)+\frac{1}{h_2}$$ (and it was actually in mathjax on that site, I thankfully could just copy it).

The important part to see here is the term $$\frac {r_2}{r_1 h_1}$$ which actually accounts for the fact that the internal convection $h_1$ is not acting on the same area as the overall coefficient. $h_2$ is acting on the outer wall, so it does not have the scaling factor.

The derivation of this final equation accounts for the diameter quite well.

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