A bakelite coating is to be used with a 10 -mm-diameter...

A bakelite coating is to be used with a 10 -mm-diameter conducting rod, whose surface is maintained at $200^{\circ} \mathrm{C}$ by passage of an electrical current. The rod is in a fluid at $25^{\circ} \mathrm{C}$, and the convection coefficient is $140 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. What is the critical radius associated with the coating? What is the heat transfer rate per unit length for the bare rod and for the rod with a coating of bakelite that corresponds to the critical radius? How much bakelite should be added to reduce the heat transfer associated with the bare rod by $25 \%$ ?

Key Concepts

Critical Radius of Insulation

This concept refers to the specific outer radius of an insulation layer on a cylindrical object at which the total thermal resistance to heat transfer is maximized. Below this radius, adding insulation can actually increase the overall heat loss because the conduction resistance becomes too low relative to the increase in the convective heat transfer area. The critical radius is typically computed by setting the derivative of the total thermal resistance with respect to the outer radius equal to zero, resulting in a simple relationship between the insulation’s thermal conductivity and the convective heat transfer coefficient.

Thermal Resistance in Cylindrical Systems

In cylindrical geometries, the heat transfer via conduction is characterized by a logarithmic dependence on the ratio of the outer to inner radii. The conduction thermal resistance is given by the formula R_cond = ln(r2/r1)/(2?kL), where r1 is the radius of the cylinder, r2 is the outer radius of the insulation, k is the thermal conductivity of the insulating material, and L is the length of the cylinder. This formulation is essential for understanding how insulation thickness impacts the overall resistance to steady?state heat transfer.

Conduction Heat Transfer in Cylindrical Coordinates

Conduction in cylindrical coordinates differs from the planar case due to the radial dependence of the cross-sectional area through which heat flows. The logarithmic form of the conduction resistance reflects how heat spreads over an increasing area as one moves radially outward, making the resistance depend on the ratio of radii and the material's thermal conductivity. This approach is critical when modeling heat transfer in rods, pipes, or any cylindrical structure.

Convection Heat Transfer

Convection involves the transfer of heat from a solid surface to a fluid moving past the surface. The convective heat transfer rate is governed by the convection coefficient, h, and the external surface area through which heat is lost. For cylindrical objects, the convective thermal resistance is calculated using the surface area of the outer diameter of the insulation. This concept is crucial when coupling conduction through insulation with the subsequent heat loss to the surrounding environment.

Overall Heat Transfer Rate in Composite Systems

The analysis of a system with insulation and surrounding convection typically involves the construction of a thermal resistance network in series, where the total resistance is the sum of the conduction resistance through the insulation and the convective resistance at the outer surface. This network setting provides a means to calculate the overall heat transfer rate per unit length, and it forms the basis for evaluating modifications, such as adding a layer of insulation to achieve a desired reduction in heat loss.

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