Question

Can the thermal resistance concept be used for a solid cylinder or sphere in steady operation? Explain.

Short answer

Answer: Yes, the concept of thermal resistance can be utilized for solid cylinders and spheres in steady operation. The governing equations for both geometries can be expressed in terms of thermal resistance, which is useful for analyzing heat transfer in such systems.

Step-by-step solution

Understand the concept of thermal resistance

Thermal resistance is a parameter that describes the resistance to heat flow across a material or between two surfaces in contact. It is the reciprocal of thermal conductivity, which is a property of a substance that indicates its ability to transfer heat. Mathematically, thermal resistance (R) can be expressed as: R = \frac{L}{kA} Where L is the thickness of the material, k is the thermal conductivity, and A is the surface area for heat transfer.

Consider steady-state conditions

Steady-state conditions refer to situations where the temperature distribution within a system does not change with time. For a solid cylinder or sphere, thermal energy flows radially (outwards or inwards) under steady-state conditions, and the temperature distribution remains constant over time.

Apply thermal resistance concept to solid cylinders

For a solid cylinder, the radial heat flow can be described using the thermal resistance concept. The governing equation for cylindrical coordinates is: q = \frac{2 \pi kL (\Delta T)}{ln(r2/r1)} Where q represents the heat flow (in watts), ∆T is the temperature difference between the inner and outer surfaces, r2 > r1, and L is the length of the cylinder. The thermal resistance (R_cylinder) can be written as: R_cylinder = \frac{ln(r2/r1)}{2 \pi kL} Thus, the thermal resistance concept can be utilized for a solid cylinder in steady operation since the governing equation can be expressed in terms of the resistance.

Apply the thermal resistance concept to solid spheres

Similarly, the radial heat flow in a solid sphere can be described using the thermal resistance concept. The governing equation for spherical coordinates is: q = \frac{4 \pi kR (\Delta T)}{(r2^3-r1^3)/(r2-r1)} Where R is the average sphere radius, r2 > r1, ∆T is the temperature difference between the inner and outer surfaces, and q represents the heat flow (in watts). The thermal resistance (R_sphere) can be written as: R_sphere = \frac{(r2^3-r1^3)/(r2-r1)}{4 \pi kR} This equation shows that the concept of thermal resistance can also be applied to spheres in steady-state operation.

Conclusion

In conclusion, the thermal resistance concept is applicable to solid cylinders and spheres in steady-state operation. The governing equations for both geometries can be expressed in terms of thermal resistance, signifying that this approach is useful for analyzing heat transfer in such systems.