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Question: a long tube with diameter d and length l is...

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A long tube with diameter D and length L is subject to a sinusoidal heat flux given by q"=q0"*sin(Pi*x/L) where x is the distance from the pipe inlet. Note, if the flow is fully turbulent and fully developed (hydraulically) then it is fully developed thermally and h(x) is constant:

a.) Arrive at an expression for the total rate of heat-transfer into the fluid flowing through the tube

b.) Arrive at an expression for the outlet mean-flow temperature and sketch Tm(x) vs. x Is the relationship linear???? (Hint: Begin with the differential energy-balance dTm(x)/dx = (q"(x)P)/(m_dot*CP) where P is the pipe perimeter and integrate. You will have one integration constant: Solve for this constant by evaluating your temperature equation at the pipe inlet where the mean-flow temperature is Tmin).

c.) Arrive at an expression for the pipe surface temperature Ts(x) in terms of the local heat-flux, local mean-flow temperature and the heat-transfer coefficient. Sketch Ts(x) and Tm(x) vs. x on the same plot!  

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