Answer:
The depth of the damage is given as [tex]x = 1.357*10^{-3}m[/tex]
Explanation:
From Fick's second law
[tex]\frac{C_{x}-C_{0}}{C_{s}-C_{0}} = 1 -erf [\frac{x}{2\sqrt{Dt} } ] ...(1)[/tex]
Where
[tex]C_x[/tex] is the temperature when the skin is damage
[tex]C_0[/tex] is the initial temperature of the skin
[tex]Cs[/tex] is the temperature of the oven
erf is a mathematical function known as error function
D is the thermal diffusivity
x is the depth of the damage
From the question we are given that
[tex]C_x[/tex] = 62°C
[tex]C_0[/tex] = 33°C
D = [tex]2.5 * 10^{-7} m^2/s.[/tex]
Substituting values into equation one
[tex]\frac{62-33}{200-33} = 1 -erf [\frac{x}{2\sqrt{2.5*10^{-7}} *2} ][/tex]
[tex]erf[\frac{x}{1.414*10^{-3}} ] = 0.826\\[/tex]
Let assume that [tex]z =\frac{x}{1.414*10^{-3}} ... (2)[/tex]
Hence
[tex]erf(z) = 0.826[/tex]
Looking at the table of error function shown on the first uploaded image
We can use interpolation to obtain the value of z in this manner
[tex]\frac{z-0.95}{0.8260-0.8209} = 1[/tex]
z = 0.96
Substituting the value of z into equation 2
[tex]0.96 = \frac{x}{1.414*10^{-3}}[/tex]
[tex]x = 1.357*10^{-3}m[/tex]
