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I have a gas (assuming air) at $T$ = 500 K that enters a cylindrical pipe. The outlet target temperature is 330 K.

There will be heat transfer via: Forced convection from the gas to the inside of the pipe, conduction through the pipe thickness, convection to the ambient air environment ($T_\infty=300$ K) and radiation from the pipe surface.

I don't think I can assume a constant surface temperature to get a relatively accurate answer. The way I was thinking of solving this was to assume an inner and outer radius, split the pipe into segments, first segment from 500 K to 470 K, compute all the relative non-dimensional parameters and solve for heat loss per unit length in that particular section. Then I would advance to the next segment, from 470 K to 440 K, solve for the non-dimensional parameters and find $q$ per unit length for that segment.

I am not sure, however, how to determine the total length of the pipe in the end, given the heat loss per unit length in each section, since the heat loss is not really linear.

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Here's what I would do to solve this problem:

  • Set up a differential equation that governs the temperature in the pipe. Judging from what you've said, it seems reasonable to assume that for any cross-section of the pipe, the temperature will be assumed homogeneous, so your problem has one spatial dimension.
  • Unless you need to model the temperature of the pipe surface separately (which requires another separate -- probably differential -- equation), I'd just treat both the conduction and convection terms using a source term governing heat loss, like $h(T - T_{\infty})$, where $h$ is a heat transfer coefficient. You can model thermal radiation using a Stefan-Boltzmann-law-like relationship (that is, $\sigma A (T^{4} - T_{\infty}^{4})$, where $A$ will be the cross sectional surface area). However, if your inlet temperature is 500 K, and your ambient temperature is 300 K, radiative heat loss is probably negligible over the length and time scales you're considering, compared to conductive and convective heat losses (not to mention fluid advection within the pipe).
  • Your differential equation will be in terms of one spatial dimension and have one initial condition (ignore the boundary condition at the end of the pipe for now). So you have an initial value problem. Solve it using an ODE solver, and pick an arbitrary end of the domain -- you want it to be longer than the end of the pipe. A good ODE solver will take care of all of the discretization for you, so you won't need to worry about splitting the pipe into segments.
  • Plot the solution you got from the previous step. The temperature in the pipe should decrease monotonically going along the pipe away from its inlet. The temperature should never go below 300 K; wherever you find the temperature is at 330 K, that should be the length of your pipe. If the temperature does not fall below 330 K, then solve the differential equations over a larger domain (that is, increase the right end point of your integration interval) until there is a point where the temperature is below 330 K.
  • If you really need the exact length of the pipe, use an ODE solver with event location, and set the event function such that it triggers the ODE solver to stop when the temperature reaches 330 K.
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