An aluminum plate of \(25 \mathrm{~mm}\) thick \((k=235 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is attached on a copper plate with thickness of \(10 \mathrm{~mm}\). The copper plate is heated electrically to dissipate a uniform heat flux of \(5300 \mathrm{~W} / \mathrm{m}^{2}\). The upper surface of the aluminum plate is exposed to convection heat transfer in a condition such that the convection heat transfer coefficient is \(67 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and the surrounding room temperature is \(20^{\circ} \mathrm{C}\). Other surfaces of the two attached plates are insulated such that heat only dissipates through the upper surface of the aluminum plate. If the surface of the copper plate that is attached to the aluminum plate has a temperature of \(100^{\circ} \mathrm{C}\), determine the thermal contact conductance of the aluminum/copper interface.

Heat transfer is a fundamental concept in thermal physics and engineering, describing the movement of heat energy from one place to another. Heat can be transferred in three primary ways: conduction, convection, and radiation. In the context of the aluminum and copper plates exercise, the focus is on conduction and convection.

Conduction is the transfer of heat through a solid material, which occurs when molecules or atoms at a higher temperature energize neighboring particles. Thermal conductivity, denoted as 'k' in the exercise for aluminum (k=235 W/m·K), directly affects the rate of energy transfer, hence influencing how quickly or slowly heat moves through the material.

Convection, on the other hand, is the heat transfer that occurs within fluids (liquids and gases) due to the movement of the fluid itself. This process is influenced by the convection heat transfer coefficient 'h', which in this scenario is given as 67 W/m²·K. For the aluminum plate exposed at the top, the air around it acts as the fluid that either takes away or delivers heat to the plate through convective processes. Understanding these two mechanisms of heat transfer is critical in deriving an accurate solution for the aluminum/copper interface's thermal contact conductance.

In a scenario where two different metals are in contact, such as aluminum and copper plates in the exercise, heat conduction at their interface becomes a key area of focus. The heat must first conduct through one metal and then transfer to the other, each with their own characteristic thermal conductivity values.

The formula for calculating conduction heat transfer is given by Fourier's law of conduction, as shown in the solution steps: $$Q_{cond} = \frac{k \cdot A \cdot (T_{Al\_Copper} - T_{Al\_Surface})}{L}$$
On the surface of the aluminumplate exposed to air, convection plays a role. The convection heat transfer can be calculated using the formula: $$Q_{conv} = h \cdot A \cdot (T_{Al\_Surface} - T_{\infty})$$where 'h' is the heat transfer coefficient and 'T_{\infty}' is the surrounding temperature.

Exercise Improvement Advice

It's essential to provide clarity on the relationship between conduction and convection in the heat dissipation process. Stating that the heat generated by the copper plate must equal the heat lost by the aluminum plate through conduction and convection could enhance comprehension. Additionally, depicting the path of heat flow with diagrams could visually reinforce the understanding of these concepts.

Thermal conductivity, denoted as 'k' in physical equations, is a measure of a material's ability to conduct heat. The unit Watt per meter-Kelvin (W/m·K) indicates how much heat in Watts can be transferred through a meter-thick sample of the material when there is a temperature difference of one Kelvin across the sample. In our exercise, the aluminum plate has a thermal conductivity of 235 W/m·K, meaning it is quite good at conducting heat.

Factors that impact thermal conductivity include the material's composition, temperature, and phase (solid, liquid, or gas). Metals, such as aluminum and copper, typically have higher thermal conductivity because their delocalized electrons can transfer kinetic energy quickly between atoms.

The thermal conductivity comes into play when determining the temperature drop across a material and influences how efficiently heat can be transferred from the interface. As part of the solution process, it has been used to calculate the rate of heat conduction through the aluminum plate. This property, along with the thickness of the material, directly impacts the thermal contact conductance, which is essential for engineering applications where heat dissipation is critical, such as in computer processors or mechanical systems.