Question

An FCC iron-carbon alloy initially containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere wherein the surface carbon concentration is maintained at 1.0 wt%. If after 51 h the concentration of carbon is 0.35 wt% at a position 3.9 mm below the surface, determine the temperature at which the treatment was carried out. You will need to use data in the two tables below to solve this problem.

Answer 1

Step-by-step explanation:

`\text{From the information given:}`

`C_o = 0.20 \ wt\% \\ \\ C_s = 1 \ wt\% \\ \\ t = 51 \ h \\ \\ x = 3.9 \ mm \\ \\ C_x = 0.35 \ wt\%`

`\text{Using Fick's 2{nd} \ law \ of \ diffusion;} \\ \\ (C_x- C_o)/(C_s-C_o)= 1 - erf ( (x)/(2√(Dt)))`

`(0.35-0.20)/(1-0.20)= 1 - erf ( (x)/(2√(Dt)))`

`0.1875 = 1 - erf ( (x)/(2√(DT))) \\ \\ erf ( (x)/(2√(DT))) = 1 - 0.1875 \\ \\ erf ( (x)/(2√(DT))) = 0.8125`

`\text{To find the value of Z by Obtaining Data from Tabulation of Error Function}`
`\text{Table Values:}`

Z erf(z)

0.90 → 0.7970

0.95 → 0.8209

? → 0.8225

∴

`(z-0.90)/(0.95-0.90)= (0.8125-0.7970)/(0.8209-0.7970)`

`(z-0.90)/(0.05)= (0.0155)/(0.0239)`

`z = 0.9324`

`\text{To determine the diffusion coefficient;}`

`erf (0.9324) = 0.8125 = erf ((x)/(2√(Dt))) \\ \\`

`(x)/(2 √(Dt))= 0.9324 \\ \\ (3.9 * 10^(-3))/(2 * √(D* 51 * 3600)) = 0.92324 \\ \\ √(D) = 4.88 * 10^(-6) \\ \\ D = \sqrt{4.88 * 10^(-6)} \\ \\ D = 2.38 * 10^(-11) \ m^2 /s`