Why is heat transfer better from within circular tubes versus ex. square tubes? It is claimed that heat exchange, through the tube-suface, is best from within circular tubes versus either (not sure which is correct):


*

*from outside of the tube itself

*other shapes like for example a square tube


What is the reasoning behind this claim?
Let's say we are talking about a tube and shell heat exchanger, maybe that narrows it down a bit.
 A: The incoming waves will terminate against each other in the corners of the square pipe, meaning there is a deficit of heat in the faces of the square. 
If that's true, we should add more corners, and waste less material. 
A square has 4 corners, with not a lot of heat on the center of the faces. 
An octagon has 8 corners, with $ 1 \over 2 $ of the wastage. 
16 corners is $ 1 \over 4 $ of the wastage. 
32 is $ 1 \over 8 $
64 is $ 1 \over 16 $
4000 is $ 1 \over 1000 $
So as you can see the limit of 'waste' approaches 0 as the number of corners increases to infinity. 
So if you have infinite corners, a circle, you get really close to 0 wasted heat transfer area. Therefore circular tubes are just more efficient.
EDIT: If you're talking about fluid, a circle is still better than any cornered shape for transfer.
For example, a circle of radius r = 0.5 will fit into a square of side s = 1 perfectly. 
Although the square has more 'area', the maximum rate of flow of a liquid would be in the center radiating outward and slowing down.
If it radiates outward and decreases speed until contacting the walls, you want a shape that has all walls in contact (circle), otherwise your flow terminates at contact with the walls (square) and the corners are left out, or incredibly inefficient.
So if you are budgeted copper for a pipe of cross-sectional area A, make sure that it is not wasted what-so-ever. So for A = 1 for a square, you have $ 1 + 1 + 1 + 1 = 4 $. For a circle you have $ 2 \pi \sqrt{1 \over \pi} = 3.454 $. 
Less material, same area. Cost effective. 
A: the shape and area of the cross section of the pipe can change flow property along the stream. The fluid velocity in a pipe changes from zero at the surface because of the no-slip condition to a maximum at the pipe center.
In fluid flow, it is convenient to work with an average velocity which remains constant in incompressible flow.
Laminar flow in a round pipe prescribes that there are a bunch of circular layers (lamina) of liquid, each having a velocity determined only by their radial distance from the center of the tube. the center is moving fastest while the liquid touching the walls of the tube is stationary (due to the no-slip condition).But infact most of the liquid which has lower/near viscosity of water will flow at turbulant regime.
Temperature of liquid reduces/increases as it moves forward and the bends and joints creates pressure drop as well as turbulent flow which increases heat transfer.
The rectangular cross section will have unstable flow as the formation of eddies near to walls.and the effective hydraulic diameter(For heat transfer) will be lesser for rectangular section.

Another minor reason is Circular pipes can withstand large pressure differences between the inside and the outside without undergoing any significant distortion, but noncircular pipes cannot.
