If Tj-Ti is the difference between the temperatures at two points i, j inside the casting and s be the distance between the two points then the thermal gradient Gij would be given by Gij=(Tj-Ti )/s. The casting geometry affects the gradient to a large extent. The direction which is normal to the front of solidification, the gradients are generally higher, whereas if one moves from the centre of casting from the wall of mold, they decrease gradually.
As a result high gradients characterize the thin castings and points closer to mould wall. In contrast, low gradients characterize the thick castings' middle regions. The thermal gradient between two neighboring regions is s enhanced if the difference in the thickness of the section between them is high. In order to compensate for the contraction in volume during solidification, the motion of the feed metal is along the direction of the thermal gradients. Shrinkage porosity is caused by poor gradients at a spot which is isolated and hot.
Alongside the path there is a certain critical value that has to be exceeded by the thermal gradient for feeding to happen. The values suggested are 0.5K/mm for the castings of steel and 2K/mm for the castings of aluminum (both are in castings of sand). The shape of the castings affects the critical value. For instance, higher gradients are required by the sections that are circular than the sections that are rectangular and flat.
There is also a dependence on the requirement of quality. Higher gradients are required by those critical castings in which even the micro-porosity should be absent.
The kind of feeding at a location is influenced by the gradients and the temperature alongside the path of feed. Liquid's movement causes mass feeding when the gradient and the temperature are both high (proximate to the feeder).
In comparison inter-dendritic feeding happens if the temperature is high but the gradient is low (at long and thick sections which are near the centre).
High gradient and low temperature (at the end sections that are thin) causes the solid feeding to take place. If in the aforementioned three zones, the feeding is improper, than it leads to surface sink, micro porosity, and macro porosity.
The feed path lying between the hot-spot and the feeder must be clear. For the flow of liquid metal to the hot spot from feeder to happen the presence of enough thermal gradients is necessary. It is assumed that alongside the thermal gradient which is maximum, lays the feed path. The heat conduction law of Fourier can be used to determine the gradient. The law is as follows:
q = - K A ?T / ?s
G = (-1 / K) w
- In a given direction and a given point within the casting the thermal gradient is given by G = ?T / ?s and the flux of heat is given by w = q / A. In a direction that is tangential to the isotherm at a point the flux of heat and the gradient are both zero, whereas they are maximum in the direction perpendicular to it.
- In all the directions that originate from a point within the casting, the thermal gradient which is maximum has its direction and magnitude proportional to the resultant of vectors of thermal flux.