Theory

Heat transfer theory seeks to predict the energy transfer that takes place between material bodies as a result of temperature difference. This energy transfer is defined as heat. The three modes by which heat can be transferred from one place to another are conduction, convection and radiation. It is well known that a hot plate of metal will cool faster when placed in front of a fan than when placed in still air. With the fan, we say that the heat is convected away, and we call the process convection heat transfer. Convection involves the transfer of heat by motion and mixing of a fluid. Forced convection happens when the fluid is kept in motion by an external means, such as a turbine or a fan. Some examples of forced convection are stirring a mixture of ice and water, blowing on the surface of coffee in a cup, orienting a car radiator to face airflow, etc. Convection is called natural convection when motion and mixing of fluid is caused by density variation resulting from temperature differences within the fluid. The density of fluid near the hot surface is less than that of the colder fluid away from the heated surface, and gravity creates a buoyant force which lifts the heated fluid upward.

In the case of conduction through a solid of area A and thickness L, heat flow is given by

$$\frac{Q}{t}=\frac{kA\Delta T}{L}.............(1)$$

Where $\Delta T$ is the temperature difference across the thickness L, and k is the thermal conductivity of the object.

In the case of convection, the heat flow is proportional only to the surface area A of the object,

$$\frac{Q}{t}=hA\Delta T.............(2)$$

Where h is the convective heat transfer coefficient (units $Wm^{-2}K^{-1}$) which depends on the shape and orientation of the object. $\Delta T$ is the temperature difference between the surface of the object and the surrounding fluid.

Convection is an enhanced form of conduction, since the movement of the fluid helps carry heat transferred by conduction, so one would expect some relation between h and k. If the temperature of the cylinder is not much above that of the surrounding air, the moving fluid can be approximated as a stationary layer having some characteristic thickness L. Comparing equations (1) and (2), one immediately has the relation h = k/L. In fact, as the temperature of the cylinder increases, fluid motion increases and becomes turbulent, whereupon the fluid becomes more efficient at carrying heat, and h can turn out to be $10^{2}$ - $10^{4}$ times k/L.The proportionality between h and k/L is called the Nusselt number N,

$$N=\frac{h}{k/L}=\frac{hL}{k}..............(3)$$

Where k is thermal conductivity of air and L is the characteristic length. Note that N is a dimensionless quantity.

In our case, which does involve turbulent flow, we are interested in temperature variation along the length of a metal cylinder, so we will take the characteristic length L to be the length of the cylinder.

Applications:

Natural convection heat transfer is extensively used in the following areas of engineering:

1. Cooling of commercial high voltage electrical power transformers.
2. Heating of houses by electrical baseboard heaters.
3. Heat loss from steam pipe lines in power plants and heat gain in refrigerant pipe lines in air conditioning applications.
4. Cooling of reactor cores in nuclear power plants, though often the coolant is driven by pumps, resulting in more efficient heat transfer by forced convection.
5. Cooling of electronic devices (chips, transistors) by finned heat sinks, though a fan is often present to augment the natural convection with forced convection.