This question was previously asked in

UPPSC AE Mechanical 2013 Official Paper II

Option 1 : equal to the rate of heat transfer by convection and is maximum

**Explanation:**

**Concept of the critical thickness of insulation:**

- In a plane wall thicker the insulation, lower the heat transfer rate. This is due to the fact the outer surface have always the
**same area**. **But in cylindrical and spherical coordinates,**the addition of insulation also increases the outer surface area which decreases the convection resistance at the outer surface.- At the same time
**conduction resistance increases due to increase in insulation thickness.**So, to optimize the balance of conductive and convective resistance, a critical radius is decided.**At this critical radius, the equivalent conduction and convection resistance will be minimum**and heat transfer will be maximum. - Moreover, in some cases, a decrease in the convection resistance due to the increase in surface area can be more significant than an increase in conduction resistance due to thicker insulation.

As a result, the total resistance may actually decrease resulting in increased heat flow.

Thickness up to which heat flow increases and beyond which heat flow decreases is termed as **critical thickness**. For cylinders and spheres, it is the **critical radius (r _{c }).**

That can be derived from the **critical radius of insulation** depends on the thermal conductivity of the insulation** k** and the external convection heat transfer coefficient **h**.

**For cylinder**

**\({{\bf{r}}_{\bf{c}}} = \;\frac{k}{h}\)**

**For sphere**

**\({{\bf{r}}_{\bf{c}}} = \;\frac{{2k}}{h}\)**