49) In a cratonic region, radioactive heat generation decreases exponentially with depth. Assuming characteristic depth as 10km and surface heat generation as $3\mu Wm^{-3}$ and neglecting mantle heat flow , the heat production per unit volume for a 30 km thick layer will be_______________$\mu Wm^{-3}$ .[round off to 2 decimal places].
(Thanks to Chandrasekhar and Rajkumar, AKNU)
Solution:
Given that Depth(D)= 10 km
Layer thickness(Z) = 30 km
we know that
$A(z)=A_{0}e^{\frac{-Z}{D}}$
by integrating both sides
of the above equation with repspect to the thickness layer from 0 to 30 km then
we get
$\int_{0}^{30}A(z) z
=A_{0}\int_{0}^{30}e^{\frac{-Z}{D}}dz. $
$\int_{0}^{30}A(z) z
=A_{0}(\frac{e^{\frac{-Z}{D}}}{-1/D})$
$[A(z) z]_0^{30}
=-A_{0}D[e^\frac{-Z}{D}]_{0}^{30} $
$A(z) [30-0] = -A_{0}D [e^\frac{-30}{10}-e^\frac{0}{10}]$
$30 A(z) = -A_{0}D[e^{-3}-1]$
$30A(z) =A_{0}D[1-e^{-3}]$
$30A(z) =A_{0}D[0.95]$
$A(z) =\frac{A_{0}D}{30}[0.95]$
$A(z)=\frac{3\times10}{30}\times0.95$
$A(z) =0.95 \mu Wm^{-3}$
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