Why does the author assume the temperature to remain constant, in this case, along the axial direction?

Question

This is in regards to steady heat conduction taking place in a pipe.

Within the pipe a hot fluid flows and heat transfer first takes place via convection to the pipe and then via conduction within the pipe and then finally via convection from the pipe to the environment. As can be seen in the diagram, the author has taken the temperature at the inner surface of the pipe as constant $T_1$ as well as the temperature of the fluid is taken constant at $T_\infty1$.

However, I think when the fluid flows inside the pipe the surface temperature would be varying, it wont be constant at $T_1$. Similarly the average temperature of the fluid at every section will also not remain constant at $T_\infty1$(it will vary along the axial direction).

Is this only a sort of approximation? Is $T_1$ the average of the surface temperatures along axial direction and $T_\infty1$ the average of average fluid temperatures at cross sections along the pipe?

Excerpt from the book - https://drive.google.com/file/d/1WaBp4I-Nbx902G4stCCdRb01ZrDb_lLo/view?usp=sharing

Answer 1

For a steady state solution, (after a loooong time), because the temperature of the air is considered constant the heat lost $\dot{q}$ to the environment will be equal to the heat lost from the fluid $\dot{q}$. However the later (i.e. the heat lost from the fluid can be expressed in terms of the temperature difference in and out i.e.:

$$\dot{q} = \dot{m}c_p \Delta T $$ where:

• $c_p $ the heat capacity of the fluid
• $\Delta T $: the temperature difference in and out
• $\dot{m}$: the mass rate (which indirectly indicates the velocity of the fluid also)

For a steady state solution that temperature difference will be spread along the axis that the fluid moves. In most engineering applications though in the radial direction you can have a temperature difference in the order of 50 degrees C or greater in a few mm, while in the axial direction it can travel several meters in order for the temperature to fall 1 or two degrees (and it can be approximated in most cases as linear altohough its not).

So the temperature gradient is at least an order of a magnitude greater (my guess is at least a couple), so it is easy to neglect that change in temperature to simplify the solution.

• Radial direction $\frac{50 ^oC}{5[mm]}= 5000 \frac{^o C}{m} $
• Axial direction $\frac{1 ^oC}{1 [m]}= 1 \frac{^o C}{m} $

Answer 2

Your assumption is correct in all real world situations. the author has idealized the case for learning purposes.

in real world the heat of the pipe is transmitted to the air by a mix of conduction and conventioin itially. creating an expanding onion like stratifyed contour of warm air around the pipe.

This hot air will surround the pipe and will rise like a flame and then move the heat by convection.

This 'flame' is hotter rhan the ambient air and is more intense at the beginning of the pipe and will gradually become smaller as the liquid or water in the pipe gets cooler, even when the steady regime is stablished.

Cooling systems try to perform in this phase so as to use smaller pumps and less liquid mass to move around.