Finding the convection of a node in a hot pipe

[theBEAST]

Homework Statement

I just completed a lab in class where we heated a copper pipe and measured the temperatures at 25 different locations. Each of these locations is called a node. The goal is to find the convection losses by using the First Law (q_c1 + q_c2 + q_c3 + q_c4 + q_rad + q_conv = dE_CV/dt).

Here is the temperature data for the 25 nodes:

Also, the ambient temperature is 22 °C.

Specifically I circled the nodes I would like to analyze - the node at 58.327 °C. So I assumed the system is in steady state and got:

q_c1 + q_c2 + q_c3 + q_c4 + q_rad + q_conv = 0

Conduction heat transfer: q = kA(T2-T1)/t
Convection heat transfer: q = hA(Tw-T∞)
Radiation heat transfer: q = σε(T2⁴-T1⁴)

Looking at the data, we see that the heat transfer by conduction is out of the middle node to the four surrounding nodes. Also, there is radiation out of that node as well. In other words both conduction and radiation is transferring out of the node.

By the First Law, this would imply that the heat transfer by convection is into the node. However, this does not make sense since the ambient temperature is lower than the node temperature so we should expect the heat to transfer out of the node.

Does anyone know what I am doing wrong?

 

[haruspex]

I presume the arrangement into a 5x5 grid corresponds the physical geometry. But it's a pipe, so how are they arranged on the pipe?
Where was the heat applied? Is the ambient temperature only outside the pipe or inside too? If inside too then this can't be steady state, so why should the rates add to zero?

 

[theBEAST]

I presume the arrangement into a 5x5 grid corresponds the physical geometry. But it's a pipe, so how are they arranged on the pipe?
Where was the heat applied? Is the ambient temperature only outside the pipe or inside too? If inside too then this can't be steady state, so why should the rates add to zero?

Yup, it corresponds to the physical geometry. One end of the pipe is attached to a heater, the rest of it is suspended in mid air. The pipe is in a wind tunnel and it's side walls are exposed to an air flow (2m/s at ambient temperature). The top five numbers of the grid correspond to the temperature closest to where the pipe is attached to the heater and that is why the numbers are the largest (see the image below).

The temperature inside the pipe is not known but I do know the pipe is thin walled and that the opposite end from the heater is closed off.

 

[haruspex]

To be honest, there are many anomalies in the readings. Were the measurements concurrent, therefore using different gauges (calibrated to read the same?), or is one gauge being moved around? If the latter, was a complete set of readings taken, then repeated a couple of times?
Is there any chance the pipe is rather inhomogeneous? Don't forget that a point reading does not give the temperature for the whole rectangle around it, so there will be heat flow between what appear to be diagonally adjacent cells.

 

[Chestermiller]

Hi Beast,

This problem looks a lot like a cooling fin. Here are some comments:

1. The temperature variations in the circumferential direction look much less than the axial variations. You can average the circumferential values to get the temperature at each axial cross section.

2. For the small temperature difference between the pipe and the air, the radiative heat flux is going to be negligible compared to the convective heat flux.

3. Are the thermocouples spaced uniformly in the axial direction? From the data, it doesn't look like they are. If they are uniformly spaced, then why isn't the temperature in the last few rows dropping off more rapidly with axial distance? The pipe temperature should be dropping with axial location toward the temperature of the air (22C).

4. I recommend that you look up correlations for convective heat transfer in flow over a cylinder. You should be able to get a pretty good estimate of the convective heat transfer coefficient. This should enable you to model the conductive heat flow down the pipe in combination with the convective heat loss to the air. Even if you don't get the convective heat transfer coefficient from the literature, you can still leave it as an adjustable parameter in the model, and back out the value from the data.

Chet