A 0.1-W small cylindrical resistor mounted on a lower part of a vertical circuit board is \(0.3\) in long and has a diameter of \(0.2 \mathrm{in}\). The view of the resistor is largely blocked by another circuit board facing it, and the heat transfer through the connecting wires is negligible. The air is free to flow through the large parallel flow passages between the boards as a result of natural convection currents. If the air temperature at the vicinity of the resistor is \(120^{\circ} \mathrm{F}\), determine the approximate surface temperature of the resistor. Evaluate air properties at a film temperature of \(170^{\circ} \mathrm{F}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption? Answer: \(211^{\circ} \mathrm{F}\)

The Grashof number is a dimensionless quantity in fluid dynamics that is a measure of the ratio of buoyancy to viscous forces in natural convection. It is utilized to predict the pattern of fluid flow where buoyancy significantly influences movement, such as the flow of air caused by temperature differences.

The formula for calculating the Grashof number is: Gr = \(\frac{g \cdot \beta \cdot (T_s - T_\infty) \cdot L^3}{u^2}\), where \(g\) is the acceleration due to gravity, \(\beta\) is the coefficient of thermal expansion of the fluid, \(T_s\) is the surface temperature, \(T_\infty\) is the temperature of the fluid far from the surface, \(L\) is the characteristic length, and \(u\) is the kinematic viscosity.

In a problem involving a cylindrical resistor, the Grashof number helps indicate if the natural convection is strong enough to be influential in the heat transfer process. To use this number effectively, one must pre-define conditions and properties such as the film temperature and establish the initial temperature difference between the heated surface and the surrounding fluid.

The Nusselt number represents the enhancement of heat transfer through a fluid layer as a result of convection relative to heat transfer through pure conduction. In essence, it is the ratio of the total heat transfer rate to the conductive heat transfer rate under the same temperature gradient.

The correlating formula for the Nusselt number around a cylinder in natural convection is: NuNucylinder)}=0.53 Ra\(^{0.25}\), which indicates that the Nusselt number depends on the fourth root of the Rayleigh number (\(Ra\)), a product of the Grashof and Prandtl numbers. This correlation provides insight into how effectively the cylinder, in this case a resistor, is able to dissipate heat into the surrounding air by natural convection. The Nusselt number is directly utilized to calculate the convective heat transfer coefficient, \(h\), which is integral for determining the surface temperature of the resistor.

Thermal conductivity, denoted by \(k\), is a material property that measures a substance's ability to conduct heat. In the context of natural convection, thermal conductivity is critical for understanding and evaluating heat transfer between the surface of an object, such as our cylindrical resistor, and the surrounding fluid, the air.

The relationship between thermal conductivity and heat transfer is directly incorporated into the determination of the Nusselt number: \(h = \frac{Nu \cdot k}{L}\). Here, the Nusselt number describes the convective heat transfer relative to the diameter \(L\) of the cylinder, and \(k\) is the thermal conductivity of air. In practical problem-solving, knowing the thermal conductivity of the air at a specific temperature helps to compute the rate at which heat is transferred from the resistor's surface to the air.

Convection currents refer to the movement of fluid that occurs when warmer, and therefore less dense, portions of a fluid rise, while cooler, denser portions sink. This process is a key mechanism in natural convection heat transfer, as it facilitates the distribution of thermal energy in a fluid.

When heating a cylindrical resistor on a circuit board, the resistor's surface temperature increases, causing the adjacent air layers to warm up and rise due to the reduced density. As warmer air moves away from the resistor, it is replaced by cooler air, forming convection currents. The efficient creation of these currents is crucial for dissipating heat from the resistor and preventing overheating, which could damage the electronic components.

The strength and pattern of these currents are directly related to the Grashof number, which quantifies the impact of buoyancy forces compared to viscous forces in the fluid, and the resulting convection contributes fundamentally to the Nusselt number's characterisation of heat transfer efficacy.