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Expected GATE-2022 Geophysics Solution (6)

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 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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