Consider a conduction problem similar to the one given in Figure 5.5

, except the boundary condition of the right side (at \(x=L\) ) is no longer held at a fixed temperature and is instead in contact with a fluid at temperature \(T_{\infty}\). The boundary condition on the right side is given by a convection boundary condition, i.e.:

\[ \vec{q}^{\prime \prime} \cdot \vec{n}=h\left(T-T_{\infty}\right) \]

where \(h\) is a parameter known as the heat transfer coefficient. Obtain an expression for the non-dimensional steady state temperature in terms of a non-dimensional parameter known as the Biot number \((B i)\), given by: \(B i=\frac{h L}{k}\).