A(low carbon steel) camshaft needs to be case-hardened by the process of ‘carburization’ of the region near the surface. in order to do this, the component is embedded in carbon at a high temperature. this gives a constant surface concentration cs of carbon of 4.6 x 1029 atoms m-3 . the carbon is allowed to diffuse into the steel for a certain time t (120 minutes) at a temperature of 975°c. the parameters that describe the diffusion coefficient (d) of carbon in fe are the following: do = 2.7 x 10-7 m 2 sec-1 and q = 75 kj mol1 . assume there is no carbon already in the component.

a) first, using the appropriate equation in chapter 3 (link to chapter 3 attached in the end), calculate the diffusion coefficient of carbon in fe at 975°c. the ideal gas constant r is: 8.314 jmol-1 k-1 and °k = °c +273. make sure that you properly convert any units.

b) next, calculate the average depth of penetration d (the average distance an average atom moves in the diffusion time t of 120 minutes), see the equation in chapter 3 (link to chapter 3 attached in the end) of the text.

c) finally, calculate the detailed concentration depth profile c(x) of the diffusion of carbon into fe after the diffusion time of 120 minutes. to calculate c(x), use the relevant solution to fick’s second law (also called the diffusion equation) for the case of diffusion from a constant surface concentration cs . this solution is (see chapter 3 of the text): c(x, t) = cs [1-erf (x/(2(dt)1/2))] where erf is known as the error function, x is the distance into the material from the surface and t is the diffusion time. note that the time t is fixed in this case. it’s best to present this depth profile by calculating the ratio c(x)/cs and plotting this ratio as a function of distance x. unless you are a bit lucky and your calculator happens to have the erf function built in, it’s probably best to use matlab (which has the function erf immediately available) for this calculation. if this is not convenient, then go online and find a power series expansion for erf and program that into your calculator. for a suitable range for the depth x, you should choose 0 < x < 4d or thereabouts.