Common Thread: Fourier’s Law in Solidification

There are many applications of heat conduction in our everyday lives. Heat conduction is the process where heat, or internal energy is transferred within a body or from one body to another due to the collisions of particles. These particles may include molecules, atoms, and electrons. One common example of heat conduction in our everyday lives is cooking. An iron skillet is placed on top of the stovetop. Heat will always flow from a region of higher concentrated energy to a region of lower concentrated energy. When the stovetop is turned on, there is a temperature difference between the iron skillet and the stovetop. However, this temperature difference decays over time and as a result, the thermal equilibrium is achieved. The heat transfer is in the direction of decreasing temperature, which in this case is from the stovetop to the bottom of the iron skillet.

According to Fourier’s Law, if the material is isotropic, then the heat flow is proportional to the temperature gradient. The equation below shows that the heat flux density is equal to the product of the thermal conductivity and the negative temperature gradient.

q is the heat flux density expressed in J/(m2*s), k is thermal conductivity constant expressed in J/(m*s*K), and ?T is the temperature gradient expressed in K/m. The negative sign in the equation indicates that the heat flows down the temperature gradient. In other words, the heat flows from a higher temperature to a lower temperature. This equation can be derived into its one-dimensional form. The equation below is Fourier’s Law in the x-direction:

An element of a certain volume is located at Point P in the figure shown below. The edges of the element are parallel to the coordinate axes (x, y, and z). The lengths are 2dx, 2dy, and 2dz. At this time, any internal heat production is neglected.

The heat flux density is parallel to the x-axis at Point P and is denoted as qx. The heat flux is entering the element via face A which is perpendicular to the x-axis and its equation is given below:

The heat flux leaving the element via the same face A is the same as the equation above, however, the negative sign is changed to positive. As a result, the sum of the two heat flux entering and leaving is the total heat flux. The equation for the total heat flux in the x-direction is shown below:

This same application can be done for the y and z-axes. By doing so, the total heat flux through the element can be determined. The equation below depicts the total heat flux. It can be rewritten as another equation where ? is the density and cP is the specific heat at constant pressure.

The transfer of heat is important because there are certain processes that can only operate at high temperatures. The heat flow is important to attain a uniform temperature in a designated area such as a furnace chamber. During a chemical reaction, the temperature of the reactants and the products must be raised to the required values. However, there is always loss of heat to the surroundings and as a result, the chemical process will not fully occur. It is the job of the engineer to minimize the heat loss. The basics of heat transfer must be understood in order to efficiently calculate the heat flow and design the optimal flow path in the process. Heat transfer can occur via three ways, conduction, convection, and radiation.