This is a material science question. Can anyone help me with this? Thank you! A

Question

This is a material science question. Can anyone help me with this? Thank you!

A carbon steel plate with a thermal diffusivity of (a = 0.1 cm/s) with a thickness of L = 3[cmis initially at (T; = 25 °C). At t= 0, its temperature at the bottom surface is raised to 100 °C (T) and kept at this level. At t= 5[s], what is the temperature at the top surface? Judging from your answer, can the plate be considered semi-infinite for this combination of (T) and (t)? Explain. T(x, t) - T, Ti - T 27. = erf (Zal) [1] erf" B2 Gaussian Error Function erfw 0,00 0.00000 0.02 0.02256 0.04 0.04511 0.06 0.06762 0.08 0.09008 0.10 0.11246 0.12 0.13476 0.14 0.15695 0.16 0.17901 0.18 0.20094 0.20 0.22270 0.22 0.24430 0.26570 0.26 0.28690 0.28 0.30788 0 30 0.32863 0.32 0.34913 0.34 0.36936 0.36 0.38 0.40 0.44 0.48 0.52 0.56 0.60 0.64 0.68 072 0.76 0.80 0.84 0.88 0.92 0.96 1.00 0.38933 0.40901 042839 0.46622 0.50275 0.53790 0.57162 0.60386 0.63459 0.66378 0.69143 0.71754 0.74210 0.76514 0.78669 0.80677 0.82542 0.84270 1.04 1.08 L12 1.16 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.20 2.40 2.60 2.80 3,00 0.85865 0.87333 0.88679 0.89910 091031 0.93401 0.95228 0.96611 0.97635 0.98379 0.98909 0.99279 0.99532 0.99814 0.99931 0.99976 0.99992 099998 0.24 'The Gaussian error function is defined as efw-4S de The complementary error function is defined as erfc w-|-erf w

Explanation / Answer

the given equation can be written as

(TS-T(x,t)/(Ts-Ti)= erf(x/2sqrt(alpha*t) (1)

given TS=100, Ti=25 deg,c, x= thickness = 3cm and alpha= 0,1cm2/s

from Eq.1m erf(x/2Sqrt(alpha*t)= erf(3/(2*sqrt(0.1*5)=2.12

from the data given, based on extrapolation, erf(x/2*sqrt(alpha*t)= 0.9965 =(100-T(3,5)/(100-25)

hence T(3,5)= 100-75*0.9965= 25.26, since even after 5 second at 3cm depth, there is no significant change in temperature it is considered as semi-infinite.

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