Problem: Determine the carburizing time necessary to achieve a carbon concentrat

Question

Problem: Determine the carburizing time necessary to achieve a carbon concentration listed below at a position listed below into an iron-carbon alloy that initially contains 0.11 wt% C. The surface concentration is to be maintained at 1.2 wt% C, and the treatment is to be conducted at the temperature listed below. Assume that Do = 6.2x10-5 m2/s and Qd = 166 kJ/mol. Utilize the Tabulation Error Function Value Table from your Examination Booklet.

--Given Values--

Position (mm) = 4.2
Temperature in Celsius = 1082
Final Concentration (wt%C) = 0.53

Part A:

Calculate the Gaussian Error function, erf(z), from Ficke's Second Law Formula (Round your answer to the ten thousandths place, 0.0000)

        Your Answer =

Part B:

Find the value for x/(2*sqrt(Dt)) = z utilizing the Error Function Table in your Examination Booklet. (Round your answer up to the tenthousandths place , 0.0000)

        Your Answer =

Part C:

Calculate D, the diffusion coefficient, in m2/s at the specified temperature.

        Your Answer =

Part D:

What is the carburizing time in hours?

        Your Answer =

A Tabulation of Diffusion Data Diffa Host Metal Species (m /s) Aci vation Energy Interstitial Diffusion Ch a Fe (BCC) 1.1 x 106 87.4 500 1.4 x 10 9000 1.4 x 10 2.3 x 10-5 900 5.9x 10 148 1100 5.3 x 10 Fe (BCC) 5.0 x 1017 500 3.1 x 10 4.0 x 10 16 y Fe (FCC) 500 Self-Diffusion )a 2.8 x 104 a-Fe (BCC 251 Fes y-Fe (FCC)a 5.0 x 10 1.1 x 10 1100 7.7 x 10 u 2.5 x 10-5 500 7.5 x 1019 Cud 200 2.3 x 104 500 Al 144 4.2 x 10 1.5 x 10 9.6x 10 13 Mg Mg 136 500 1.5x 10 94 500 6.6 x 10 Zn 1.8 x 10 1.2 x 10 Mo Mo 461 500 1.9 x 10 500 285 Ni Ni 1.0 x 10 Interdiffusion (Vacancy) 500 Cu 2.4 x 10 189 4.0 x 10 8.7 x 10-13 2.1 x 1 124 500 Al 6.5 x 10 136 500 4.2 x 10 1.2 x 1 4 Mg 130 500 1.9 x 10 2.7 x 10 500 Ni 256 1.3 x 10 5.4 x 10 20 Cu 1.9 x 10 230 Tabulation of Error Function Values erf E 0.55 0.5633 1.3 0.9340 0.025 0.0282 0.60 0.6039 1.4 0.9523 0.05 0.0564 0.65 0.6420 1.5 0.9661 0.10 0.1125 0.70 0.6778 1.6 0.9763 0.15 0.1680 0.75 0.7112 1.7 0.9838 0.20 0.2227 0.80 0.7421 1.8 0.9891 0.25 0.2763 0.85 0.7707 1.9 0.9928 0.3286 0.30 0.90 0.7970 2.0 0.9953 0.35 0.3794 0.95 0.8209 2.2 0.9981 0.4284 0.9993 0.40 1.0 0.8427 2.4 1.1 0.8802 2.6 0.9998 0.45 0.4755 0.50 0.5205 1.2 0.9103 2.8 0.9999

Explanation / Answer

Part A:

Fick's second law,

(Cx - C0) / (Cs - C0) = 1 - erf [x / 2 (Dt)1/2]

where Cx = Concentration at distance x from surface at time t = 0.53%

C0 = Initial concentration = 0.11%

Cs = Surface concentration = 1.2%

erf = Gaussian error function

x = Distance from surface = 4.2 mm = 0.0042 m

D = Diffusivity co-efficient

t = time

(Cx - C0) / (Cs - C0) = 1 - erf [x / 2 (Dt)1/2]

(0.53 - 0.11) / (1.2 - 0.11) = 1 - erf [x / 2 (Dt)1/2]

0.3853 = 1 - erf [x / 2 (Dt)1/2]

erf [x / 2 (Dt)1/2] = 1 - 0.4 = 0.6

erf [z] = 0.6 where z = x / 2 (Dt)1/2

(erf [z] - erf [z1]) / (erf [z2] - erf [z1]) = (z - z1) / (z2 - z1)

(0.6 - 0.5205) / (0.5633- 0.5205) = (z - 0.50) / (0.55- 0.50)

1.8575 = (z - 0.50) / 0.05

z - 0.50 = 1.8575 x 0.05

z = 0.0929 + 0.50 = 0.1429

z = 0.1429

Part B:

x / 2 (Dt)1/2 = 0.1429

Part C:

Calculation of D, the diffusion coefficient:

D = D0 e (-Qd / RT)

where D0 = Diffusion constant = 6.2x10-5 m2/s

Qd = Activation energy =  166 kJ/mol = 166000 J/mol

R = Gas constant = 8.314 J/K mol

T = Absolute temperature = 1082°C = 1355 K

D = D0 e (-Qd / RT) = 6.2x10-5 e (-166000/ 8.314 x 1355) = 2.47x10-11 m2/s

Part D:

t = [x / (2x0.1429)]2 / D = [0.0042 / (2x0.1429)]2 / 2.47x10-11 = 8743336 s = 2428.7 hours

Carburizing time in hours = 2428.7 hours

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