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Let's say I am flowing a fluid through a pipe that has a constant surface temperature. Is it possible to determine the heat flux on the inner surface without knowing the pipe's length when the only two temperatures I know are the pipe surface temperature and the inlet flow temperature?

I am able to assume the flow is thermally and hydrodynamically developed. A fully developed pipe with constant surface temperature has a Nusselt number of 3.6568 which allows me to then determine the convection coefficient. I was going to calculate the heat flux using $$q''=\bar h\Delta T_{lm}$$

The log mean temperature difference would need to use the outlet temperature found by $$\frac{T_{s}-T(L)}{T_{s}-T_{in}}=exp(\frac{-\pi DL}{\dot m c_{p}}\bar h)$$

The problem lies in that I don't have the pipe's length so I can't calculate the outlet temperature or the LMTD. The only other method I could think of is using $$q''=h(T_{s}-T_{in})$$ but that will greatly underestimate the actual heat flux.

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You need the length because the fluid temperature is different at all points in the tube and you do not have these temperatures. you can however get the expression of temperature as a logarithmic function of length, which is essentially pipe outlet temperature for varying lengths of pipe. (your final equation will be a function of length) –  gregsan Mar 11 '14 at 6:42
@gregsan Are you sure there is no other way to find heat flux? This problem was on my professor's past exam and I am using it for practice. He must have made an error –  Greg Harrington Mar 11 '14 at 7:00

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