Question

Determine the carburizing time necessary to achieve a carbon concentration of 0.48 wt% at a position 3.2 mm into an iron-carbon alloy that initially contains 0.20 wt% C. The surface concentration is to be maintained at 1.0 wt% C, and the treatment is to be conducted at 1070°C. Assume that D₀ = 2.2 × 10^-5 m²/s and Q_d = 173 kJ/mol. The following table may be useful.

om Equation 6.5 calculate the Gaussian error function:

erf(x2Dt) =

Answer 1

Given : Carbon concentration at surface Final Carbon concentration at distance x Initial Carbon concentration Distance Temperture (Cs) (Cx) (Co) 1 wt%C 0.48 wt%C 0.2 wt%C 3.2 mm 1070 C 0.0032 m 1343.15 K (T) D. 2.2 * 10 m/s 173 kJ/mol 173000 J/mol - 173000 First we need to find Diffusion coefficient at given temperature QD D Do exp 2.2 * 10-5 exp RT 8.314-1343.15 4.114 * 10-12 m-/s The governing equation obtained by applying boundary conditions to Ficks Second Law: (CO) Cs-Co (1) 1 - erf ( 2 VDP where erf D is Error function is Diffusivity at given temperature is Carburizing time From equation (1), erf ( 2 ore 1- (CO Cs-Co 0.48-0.2 1-( 1-0.2 0.65 We need to find value z for which error function erf(z) gives value of 0.65

The value is not directly available. For erf(z) = 0.64938, z= 0.66 For erf(z) = 0.65663, z= 0.67 Using Linear Interpolation, z for which erf(z) = 0.5 is calculated as : z = 0.66 + (0.67 -0.66) (0.65663-64938) (0.65 -0.64938) 0.66086 Thus, erf(0.66086)= 0.65 Thus, erf ( z Your erf(0.66086) Cancelling the error functions, 0.66086 2 VDt х Thus, Dt х 2* 0.66086 х Dt - 0.86086) (2+0.86086)2 t D 1 4.114. 10-12 0.0032 2-0.66086 78267 seconds 78267 hrs 60 - 60 21.74 hrs Carburizing time required 21.74 hrs