The temperature T of the ground at depth X is given by T Ts

The temperature, T, of the ground at depth X is given by T - T_s/T_i - T_s = erf(c/2squarert at) Where, T_s is surface temperature, T_i is initial temperature (=20 degree C),t is time and erf is \"error function\". How deep should a major water pipe be buried to keep the water from freezing if the surface is at -15 degree C for 60 days? Assume alpha = 1.38 times 10^-7 m^2/s.

Solution

Program :

Ti=20;                                  %Initial Temperature
Ts=-15;                                 %Surface Temperature
a=1.38*(10^(-7));                           %Constant Alpha
t = 60*24*3600;                             %Time in seconds
T = 0;                                    %Final Temperature
y = (T-Ts)/(Ti-Ts);                       
z=erfinv(y);
x = 2*sqrt(a*t)*z                       %Depth

Output :

x =0.6770