Common Thread: Fourier’s Law in Solidification
Fourier’s Law is known as the law of heat conduction. Conduction refers to the transfer of heat (or internal energy) through direct contact which takes place in all 3 phases: solid, liquid, and gas. It is also a rate equation which determines the conduction heat flux from the temperature distribution in a medium. Heat will always flow from a region of higher concentrated energy to a region of lower concentrated energy and the heat transfer is in the direction of decreasing temperate. For example, a beaker of water at room temperature is placed on a hotplate. The heat transfer is in the direction from the surface of the hotplate to the bottom of the beaker. There is a temperature difference between the hotplate and the bottom of the beaker. However, this temperature difference decays over time and as a result, the thermal equilibrium is achieved.
According to Fourier’s Law, the time it takes for the heat to transfer through a material is proportional to the temperature gradient. The equation below shows that the density flux density is equal to the product of the thermal conductivity and the negative temperature gradient.
where 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 below equation 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: