Using the decarburization expression derived in Problem 1 , plot the concentration profi...

Using the decarburization expression derived in Problem 1 , plot the concentration profile of carbon within 1 mm of the carbon-free surface after 1 hour in a vacuum at 1,000°C. Take the initial carbon content of the steel to be 0.3 wt %.

Carburization was described in Example 1 . The decarburization of a steel can also be described by using the error function. Starting with Equation 1 and taking cs = 0, derive an expression to describe the concentration profile of carbon as it diffuses out of a steel with initial concentration, c0. (This situation can be produced by placing the steel in a vacuum at elevated temperature.)

Example 1

Steel surfaces can be hardened by carburization, as discussed relative to Figure 1. During one such treatment at 1,000°C, there is a drop in carbon concentration from 5 to 4 at % carbon between 1 and 2 mm from the surface of the steel. Estimate the flux of carbon atoms into the steel in this near-surface region. (The density of γ-Fe at 1,000°C is 7.63 g/cm3.)

SOLUTION

First, we approximate

To obtain an absolute value for carbon-atom concentration, we must first know the concentration of iron atoms. From the given data and Appendix 1,

Therefore,

From Table 1,

Using Equation 1 gives us

Diffusivity Data for a Number of Metallic Systems a

a See Equation 2

Problem 1

Equation 1

Figure 1

Solution to Fick’s second law (Equation 3) for the case of a semi-infinite solid, constant surface concentration of the diffusing species cs, initial bulk concentration c0 , and a constant diffusion coefficient, D.

Equation 2

D = D0e−Q/RT