Question

Question: To find: The rate of heat flow across a sphere \(S\) of radius \(a\).

Short answer

The rate of heat flow across a sphere \(S\) of radius \(a\) is \(4\pi Kc\).

Step-by-step solution

Concept of Surface Integral and Formula Used.

A surface integral is a surface integration generalisation of multiple integrals. It can be thought of as the line integral's double integral equivalent. It is possible to integrate a scalar field or a vector field over a surface.

Formula used:

By taking the center of the ball to be at the center of the radius, then the \(u(x,y,z)\) is,

\(u(x,y,z) = \frac{c}{{\sqrt {{x^2} + {y^2} + {z^2}} }}\)

Calculate the Surface Integral.

Find\(\nabla u\).

\(\begin{array}{l}\nabla u = \left( {{\rm{i}}\frac{\partial }{{\partial x}} + {\rm{j}}\frac{\partial }{{\partial y}} + {\rm{k}}\frac{\partial }{{\partial z}}} \right)\left( {\frac{C}{{\sqrt {{x^2} + {y^2} + {z^2}} }}} \right) = {\rm{i}}\frac{\partial }{{\partial x}}\left( {\frac{C}{{\sqrt {{x^2} + {y^2} + {z^2}} }}} \right) + {\rm{j}}\frac{\partial }{{\partial y}}\left( {\frac{C}{{\sqrt {{x^2} + {y^2} + {z^2}} }}} \right) + {\rm{k}}\frac{\partial }{{\partial z}}\left( {\frac{C}{{\sqrt {{x^2} + {y^2} + {z^2}} }}} \right)\\ = - {\rm{i}}\frac{{Cx}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{\frac{3}{2}}}}} - {\rm{j}}\frac{{Cy}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{\frac{3}{2}}}}} - {\rm{k}}\frac{{Cz}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{\frac{3}{2}}}}}\\ = - \frac{C}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{\frac{3}{2}}}}}(x{\rm{i}} + y{\rm{j}} + z{\rm{k}})\end{array}\)

Find\({\rm{F}}\).

As the heat flow is towards the center, modify the equation (2) as follows,

\({\rm{F}} = - K\nabla u\)

\(\begin{array}{l}{\rm{ Substitute }} - \frac{C}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{\frac{3}{2}}}}}(x{\rm{i}} + y{\rm{j}} + z{\rm{k}}){\rm{ for }}\nabla u,\\\begin{array}{*{20}{c}}{{\rm{F}} = - K\left( { - \frac{C}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{\frac{3}{2}}}}}(x{\rm{i}} + y{\rm{j}} + z{\rm{k}})} \right)}\\{ = \frac{{KC}}{{{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{\frac{3}{2}}}}}(x{\rm{i}} + y{\rm{j}} + z{\rm{k}})}\end{array}\end{array}\)

As the outward unit normal to the sphere\({x^2} + {y^2} + {z^2} = {a^2}\)at the point\((x,y,z)\).

\({\rm{n}} = \frac{1}{a}(x{\rm{i}} + y{\rm{j}} + z{\rm{k}})\)

Find \({\rm{F}} \cdot {\rm{n}}\).

Substitute \(\frac{{KC}}{{{a^2}}}\) for \({\rm{F}} \cdot {\rm{n}}\) in equation (1),

Thus, the rate of heat flow across a sphere \(S\) of radius \(a\) is \(4\pi Kc\).