Formulas you may need to solve the questions: C (t, x ) = - A x 2 = exp Vt 4Dt C(t, x) = 2 Co erf 2VDE C(t, x) = A + B . erf ,pt x C(t, x) - C1 = Lerf x C2 - C1 2 2VDE d dz erf (z) = exp (-z2) erf(x - 100) = +1; erf (0) = 0 2 erf(z) 0 0 0.025 0.0282 2 0.05 0.0564 0.10 0. 1 125 erfc(z) 0.15 0. 1680 0.20 0.2227 erf(z) 0.25 0.2763 0.30 0.3286 (2/VIT )exp(-z?) 0.35 0.3794 0.40 0.4284 0.45 0.4755 0.50 0.5205 0.55 0.5633 0.60 0.6039 0.65 0.6420 0.70 0.6778 0.75 0.7112 -2 0 2 0.80 0.7421 Any Real Variable z D = 172 = = [tot12 (R2) = (n)12 (RZ) (n)12 Izra2 rza2 6T 6T 6T 6 R2 = X2 + Y2 + 72 = 3X2 (X2) 2T D = (X2) + ( Y2 ) 4T D = f = [tot Cigar = fwivCfa2 f = = 1 + (cos0i,i+1) NIN 1 - (cosei,i+1) ~ 1 -1. To reduce the carbon content Co in a steel sheet sample with a thickness of 10 cm, the steel sample is heat treated in an oxygen and hydrogen rich environment at T = 200 C, Dcarbon = 3 x 10-6 cm sec for the decarbonization process. a. Define the initial and boundary conditions of this diffusion problem. b. Find the equation to predict the carbon concentration profile in the steel sample, detail your assumptions to solve the question. c. Derive the expression for the flux of the carbon out of the sample. d. After the steel sheet sample has been decarbonized for 5 minutes, approximately how deep from the surface of the sample will the carbon content be reduced to a third of the initial carbon concentration. e. What is the diffusion mechanism in this process, interstitial diffusion mechanism or vacancy mediated substitutional diffusion mechanism