Most diffusion problems can be solved using a finite difference method of numerical analysis and Fick's first law of diffusion. Consider the solute profile shown in figure 1 and the change in composition at C1 produced by a flux into ( Jin) and a flux out of (Jout) a region of thickness h. The flux equations would look like the following:
Jin= [qin/delta(t)] = - D [(C1 - Co)/h]
Jout= [qout/delta(t)] = - D [(C2 - C1)/h]
where delta(t) is a finite time interval, D is the diffusivity of the solute, q represents the amount of solute per unit area entering or leaving the volume of interest and Ci is the solute per unit volume. The net amount of solute collected in time delta(t) will be
C1new - C1 = [(qin - qout)/h] = [(D x delta(t))/h^2 x (Co + C2 - 2C1)
This equation can be greatly simplified by setting [D x delta(t)/h^2] equal to 1/2 to give
C1new = (Co + C2/2
The only constraint is that the time interval and distance between composition measurements are fixed. This requirement is called the stability criterion for the finite difference method.
With a little effort, the spreadsheet can be turned into a powerful tool where carbon potential, base carbon content and diffusivity can be entered as variables to test new carburization cycles. Carbon leveling operations can also be modeled by lowering the surface carbon potential after the case depth is obtained. In the problem shown here, the time to attain a uniform carbon concentration, less than 0.9 wt.%, can be determined by changing the surface carbon concentration to 0.9 in column 93 and then inspecting the subsequent columns to determine the number of intervals required to level the carbon. Many spreadsheets are equipped with graphics packages and the effect of changing the variables can be immediately observed in these graphs at fixed number of iterations or times.