# Why do people use the outer area of the inner pipe for overall heat transfer coefficients in heat exchangers?

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)

The equation in question is shown below:

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.

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.

• You're right that the equation is sound because of the way they defined their overall heat transfer coefficient. This is more of a question as to why they define ho like they did instead of using the inner pipe's inner diameter for the derivation of ho. Why the motivation to use the outer area? – MechE Jul 31 '17 at 22:42
• @MechE Does it make a difference? You have $T_A$ and $T_B$ and need the transfer between them. If you do it the other way, you get the exact same thing but with $\frac {r_1}{r_2 h_2}$ and a regular $\frac 1 {h_1}$. – JMac Jul 31 '17 at 22:43
• No, I was just wondering there was some reasoning behind this standard. I guess people just chose that as the standard and stuck with it? Is that always the standard? Seems like you'd run into communication problems with people if it wasn't standardized as one person's h wouldn't match up with another's. – MechE Jul 31 '17 at 22:46
• Isn't that here the inner and outer diameter of the pipe are taken as the same? – Alchimista Jul 31 '17 at 22:54
• @MechE It's really no different. You just make sure you keep track of which D you choose as the overall, and the pick the right ratio to give the right coefficient. The trick is to make sure your format makes sense, you can't blindly apply any equation. This isn't necessarily any "standard", they had to pick one or the other arbitrarily to solve the equation, or come up with some messy middle grround. – JMac Jul 31 '17 at 23:32