5. V/A for Simple Geometries

Plane Wall: For a plane wall of thickness \(2L\), width \(W\), and height \(H\) for one dimensional heat flow across the plane \(WH\), the heat transfer area on one side is \(WH\); and since heat is flowing from both the front and rear faces to the ambient, the total heat transfer area is \(2WH\). Volume of the plane wall is \((2L)WH\). Therefore, \(V/A=L\).

Cylinder: For heat flow through cylindrical surface (with insulated flat ends) \[\frac{V}{A} = \frac{\pi r_o^2 H}{2\pi r_o H} = \frac{r_o}{2}\]

Sphere: \[\frac{V}{A} = \frac{(4/3)\pi r_o^3}{4\pi r_o^2} = \frac{r_o}{3}\]