For same volume the ratio of surface areas of sphere : cylinder : cube: is 4.84 : 5.54 : 6

Explanation:

The surface area to volume ratio is an important parameter in the design of risers used in casting. The ratio is important because it affects the efficiency of heat transfer during the solidification of the metal in the casting process. The lower the ratio, the better the efficiency of heat transfer.

The ratio of surface area to volume for a unit volume of riser is minimum in case of a spherical riser. This can be explained as follows:

1. Definition of surface area to volume ratio: The surface area to volume ratio is the ratio of the surface area of an object to its volume. It is expressed as A/V, where A is the surface area and V is the volume of the object.

2. The formula for surface area and volume of a sphere: For a sphere of radius r, the surface area and volume are given by the formulas:

Surface area = 4πr2
Volume = (4/3)πr3

3. Calculation of surface area to volume ratio for a sphere: Substituting the values of surface area and volume in the formula for surface area to volume ratio, we get:

A/V = (4πr2)/[(4/3)πr3] = 3/r

4. Comparison of surface area to volume ratio for different shapes of risers: For a cylindrical riser, the surface area to volume ratio is given by 2/r, for a hemispherical riser it is given by 2.5/r, and for a cuboidal riser it is given by 6/l, where l is the length of the riser.

5. Conclusion: From the above calculations, it can be seen that the surface area to volume ratio for a unit volume of riser is minimum in case of a spherical riser, as the ratio is inversely proportional to the radius of the sphere. Therefore, a spherical riser is the most efficient in terms of heat transfer during the solidification of the metal in the casting process.