Use Laplace Transform to solve the following problems related with semi infinite solid applications. Use tables to find the inverse Laplace Transform.

$4\mathrm{.}7$ On a hot day the wood surface gets heated to $50C$ to a considerable depth. Sudden sharp showers cool the surface to $20C$ and maintain the surface at this temperature level. Determine the temperature at $2$ cm depth after $40$ minutes. The material properties are:

density $=2115k\frac{g}{m}3.$ Specific heat $=920\frac{J}{k}gK.$ Conductivity $=0\mathrm{.}062\frac{W}{m}K.$

Also calculate the heat flow from the surface upto the time, instantaneous heat flow at the surface and $2$ cm depth.

$4\mathrm{.}8$ Derive an expression for the heat flux per unit area at depth x and time $$ when

a semi$-$infinite solid is suddenly exposed to an instantaneous energy pulse at the surface of strength $\frac{Q_0}{A}.$

$4\mathrm{.}9$ The temperature of road surface at $9$ am was $5C$ to a good depth. From $9$ am to $12$ noon the surface is exposed to solar radiation of $650\frac{W}{m}2$ of which $0\mathrm{.}4$ times is absorbed$.$ Determine the surface temperature and the temperature at a depth of $4$ cm $.$

Take $=8\mathrm{.}26{10}^{-7}\frac{m^2}{s}$ and $k=0\mathrm{.}552\frac{W}{m}K.$

$4\mathrm{.}10$ A semi$-$infinite solid of stainless steel $(18\%Cr,8\%Ni)$ is initially at a uniform temperature of $0C.$ The surface is pulsed with a laser with $10M\frac{J}{m}2$ instantaneous energy. Calculate the temperature at the surface and depth of $1$ cm after a time of $3$ s $.$Use Laplace Transform to solve the following problems related with semi infinite solid applications. Use tables to find the inverse Laplace Transform.

$4\mathrm{.}7$ On a hot day the wood surface gets heated to $50C$ to a considerable depth. Sudden sharp showers cool the surface to $20C$ and maintain the surface at this temperature level. Determine the temperature at $2$ cm depth after $40$ minutes. The material properties are:

density $=2115k\frac{g}{m}3.$ Specific heat $=920\frac{J}{k}gK.$ Conductivity $=0\mathrm{.}062\frac{W}{m}K.$

Also calculate the heat flow from the surface upto the time, instantaneous heat flow at the surface and $2$ cm depth.

$4\mathrm{.}8$ Derive an expression for the heat flux per unit area at depth x and time $$ when

a semi$-$infinite solid is suddenly exposed to an instantaneous energy pulse at the surface of strength $\frac{Q_0}{A}.$

$4\mathrm{.}9$ The temperature of road surface at $9$ am was $5C$ to a good depth. From $9$ am to $12$ noon the surface is exposed to solar radiation of $650\frac{W}{m}2$ of which $0\mathrm{.}4$ times is absorbed$.$ Determine the surface temperature and the temperature at a depth of $4$ cm $.$

Take $=8\mathrm{.}26{10}^{-7}\frac{m^2}{s}$ and $k=0\mathrm{.}552\frac{W}{m}K.$

$4\mathrm{.}10$ A semi$-$infinite solid of stainless steel $(18\%Cr,8\%Ni)$ is initially at a uniform temperature of $0C.$ The surface is pulsed with a laser with $10M\frac{J}{m}2$ instantaneous energy. Calculate the temperature at the surface and depth of $1$ cm after a time of $3$ s $.$