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

###### Question details

A long tube with diameter **D** and length
**L** is subject to a sinusoidal heat flux given by
**q"=q _{0}"*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 **T _{m}(x)** vs.

**x**Is the relationship linear???? (Hint: Begin with the differential energy-balance

**dT**where

_{m}(x)/dx = (q"(x)P)/(m_dot*CP)**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

**T**).

_{min}c.) Arrive at an expression for the pipe surface temperature
**T _{s}(x)** in terms of the local heat-flux,
local mean-flow temperature and the heat-transfer coefficient.

**Sketch T**

_{s}(x) and T_{m}(x) vs. x on the same plot!