From the experimental studies, values for Crustal rocks are density ρ=2650 kg/m^3, thermal conductivity K_c= 2.51 W /m K and Specific heat Cp =500 J/Kg. K. Approximate the penetration depth (The daily temperature variation = 7.27 ×10^{-5} /s).
Solution:
Density
\rho = 2650 kg/m^3
Thermal
Conductivity K_c=2.5 W/m K
Specific
Heat C_p=700 J/Kg. K
To
Calculate the Thermal diffusivity K_d is
K_d=
\frac{K_c}{\rho c_p}
K_d=
\frac{2.5}{2650 \times 700}
K_d=
1.35 \times 10^{-6} \frac{m^2}{s}
Decay
depth (or) Penetration depth (d) is
d
=\sqrt{\frac{2 K_d}{\omega}}
d
=\sqrt{\frac{2\times 1.35\times 10^{-6}}{7.27\times 10^{-5}}}
d=0.192
m
d=
19.2 cm
Extra
information:
Heat
flux (q) is defined as the amount of heat passing through a unit area in a unit
time.
q= \frac{J}{s m^2}
q=\frac{W}{m^2}
Derive the Work units,
Work
=Force \times Displacement
Work
= mass \times acceleration \times Displacement
Work
= mass \times \frac{Velocity}{time} \times Displacement
Work
= kg \times \frac{m}{s^2} \times m
Work
= kg \times \frac{m^2}{s^2}
kg \times \frac{m^2}{s^2} = Joules
Watts is defined as
Watts= \frac{Joules}{second}
Finally derive the units of Thermal conductivity K_c is
K_c=\frac{q}{\frac{dT}{dZ}}
q= Heat flux
\frac{dT}{dZ}=geothermal gradient
\frac{dT}{dZ}
units= \frac{K}{m}
K_c
units=\frac{Joules}{m. K}
Reference : Fundamental of Geophysics, William Lowrie.
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