Thermal insulation is the reduction of heat transfer (the transfer of thermal energy between objects of differing temperature) between objects in thermal contact or in range of radiative influence. Thermal insulation can be achieved with specially engineered methods or processes, as well as with suitable object shapes and materials.
Heat flow is an inevitable consequence of contact between objects of differing temperature. Thermal insulation provides a region of insulation in which thermal conduction is reduced or thermal radiation is reflected rather than absorbed by the lower-temperature body.
The insulating capability of a material is measured with thermal conductivity (k). Low thermal conductivity is equivalent to high insulating capability (R-value). In thermal engineering, other important properties of insulating materials are product density (ρ) and specific heat capacity (c).
Low thermal conductivity (k) materials reduce heat fluxes. The smaller the k value, the larger the corresponding thermal resistance (R) value. Thermal conductivity is measured in watts-per-meter per kelvin (W·m−1·K−1), represented as k. As the thickness of insulating material increases, the thermal resistance—or R-value—also increases.
For a cylinder, the convective thermal resistance is inversely proportional to the surface area and therefore the radius of the cylinder, while the thermal resistance of a cylindrical shell (the insulation layer) depends on the ratio between outside and inside radius, not on the radius itself. Suppose for example that we double the outside radius of a cylinder by applying insulation. We have added a fixed amount of conductive resistance (equal to ln(2)/(2πkL)) but at the same time we have halved the value of the convective resistance. Because convective resistance tends to infinity when the radius approaches zero, at small enough radiuses the decrease in convective resistance will be larger than the added conductive resistance, resulting in lower total resistance.
This implies that a critical radius exists at which the heat transfer is maximum. Above this critical radius, added insulation decreases the heat transfer. For insulated cylinders, the critical radius is given by the equation 
This equation shows that the critical radius depends only on the heat transfer coefficient and the thermal conductivity of the insulation. If the radius of the uninsulated cylinder is larger than the critical radius for insulation, the addition of any amount of insulation will decrease the heat transfer.