Consider the cylindrical tube shown in the figure below. Hot fluid enters the tube at a constant $($plug$)$ temperature and velocity. The fluid is hot enough that radiation is important. The outside of the tube is exposed to the ambient, which is at a much lower temperature. The objective is to find the temperature distribution $($both axial and radial$)$ within the tube, and the net heat loss from the tube.

a$)$

Clearly write down all governing equations and boundary conditions that you will employ to meet your objective. If you are solving more than one governing equation, you must also describe clearly in what sequence you will solve the governing equations, and how the coupling between the various equations will be manifested. Do not describe numerical methods. Just state what equations will be solved and in what sequence.

b$)$

How will you obtain the net heat loss from the tube?

To facilitate your analysis, make the following assumptions:

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The fluid is radiatively non$-$participating

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The tube wall is very thin

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The velocity distribution within the tube is already known $($i$.$e$.,$ there is no need to describe the equations needed to determine the velocity distribution$)$

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Angular symmetry prevails

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All thermo$-$physical properties and geometric dimensions are known

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The density of the fluid is constant