Spherical Shells & Tube Walls

This section is about heat transfer through geometries other than a plane wall: a tube wall and a spherical shell were the two geometries covered in lectures. This requires using coordinates other than the usual x-y, but we are still using thermal resistance as the key modelling tool.

There are some Practise Exercises to help test yourself at the bottom of this page! :)

Tube Wall

Imagine a pipe, with conduction occurring through the walls such that heat travels into or out of the pipe radially. Real life examples might include:

Tubes inside a heat exchanger (this would be implemented as part of a bank of tubes)
An insulated pipe making up part of the central heating system inside the wall of a house
A sewer carrying a flow of warm-ish sewage under the cold ground

Bear in mind that boundary conditions apply as always and that there may be further heat transfer mechanisms taking place beyond the boundaries either side.

The geometry of a pipe means that applying a cylindrical coordinate system is ideal. But - bear in mind that we are working on the assumption that the pipe and thermofluidic conditions within it are axisymmetric.

Derivation - thermal resistance to heat conducting through a tube wall

(hopefully the extra detail compared to the lecture slides alone is helpful!)

We begin with the heat equation written in terms of r, rather than x:

This equation looks awful, but we can make some assumptions...

The RHS is zero at steady-state
The last term on LHS is zero if there is zero heat generation
The middle two terms on LHS are both zero if we assume axisymmetry and constant conditions along the pipe's length
Also, we assume constant thermal conductivity, k, which makes solving this differential equation easier later on

Taking these assumptions into account, the equation simplifies to:

Clearly, the multiplier 1/r on the LHS will now also cancel.

We can now think about solving this differential equation simply by integrating twice in succession. Integrating once:

Where c is a constant of integration. Integrating a second time after separating variables T and r:

Again, d is a constant of integration.

We can now solve for constants c and d using the boundary conditions Ts1 and Ts2 shown in the diagram above using simultaneous equations and a little algebra

If numerical values for Ts1, Ts2, r1 and r2 are known, these can be substituted at this point. Rearranging the second simultaneous equation to solve for d:

Hence we can write temperature as a function of r for a conducting tube wall:

Substituting into the usual equations for heat rate, q, and conduction resistance, R:

Spherical Shell

In this case, imagine an spherical (or approximately spherical) container with contents that differ in temperature from the outside. This could be a useful approximation for a chemical reactor, or a spherical storage tank not in thermal equilibrium. In cross-section:

For this geometry, there was no derivation in the lectures, so there isn't shown one here either because that's of limited revision usefulness. However, a summary of key results from a derivation like the one shown for the tube wall is shown here:

Practise Exercises! You can do it... :D

For each of the real-life examples of where you might find conduction through a tube wall, state the key type of heat transfer present at boundaries inside and outside the tube. For the sewer, assume the pipe is fully filled (in reality sewers are only ever partially filled).
1. Draw cross-sectional diagrams for each tube in Question 1 in the r-θ plane to show how temperature varies with r.
2. Shawn the Sheep can be approximated as a naked sheep inside a spherical shell made of wool, with an inner radius of 0.3m and an outer radius of 0.5m. If his skin is at 37˚C and the outside air temperature is 20˚C, calculate the following:
  1. The rate of heat flow from Shawn's torso into the outside world
  2. The thermal resistance of Shawn's wool

(The thermal conductivity of sheep wool can be looked up on Engineering Toolbox!)

Solutions

Heat exchanger tube: convecting fluid on both sides
Heating pipe: conduction through the solid additional insulation / wall on the outside, convecting fluid on the inside
Sewer: convecting sewage on the inside (possibly also conduction due to fouling, ewww), conduction through the ground / soil on the outside

1. q = 6.3W
2. R = 2.7W/K
3. by directly substituting into the spherical shellequations for heat rate and thermal resistance respectively