A 5 -mm-thick stainless steel strip \((k=21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=\) \(8000 \mathrm{~kg} / \mathrm{m}^{3}\), and \(\left.c_{p}=570 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) is being heat treated as it moves through a furnace at a speed of \(1 \mathrm{~cm} / \mathrm{s}\). The air temperature in the furnace is maintained at \(900^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the furnace length is \(3 \mathrm{~m}\) and the stainless steel strip enters it at \(20^{\circ} \mathrm{C}\), determine the surface temperature gradient of the strip at mid-length of the furnace. Hint: Use the lumped system analysis to calculate the plate surface temperature. Make sure to verify the application of this method to this problem.

The lumped system analysis is a simplification technique used in heat transfer when we assume that the temperature within a solid object is uniform. This means that the temperature difference within the object is negligible compared to the temperature difference between the object and its surroundings. To justify this assumption, we calculate the Biot number (Bi), which is a dimensionless quantity given by \(Bi = h_c \cdot L_c / k\), where \(h_c\) is the convection heat transfer coefficient, \(L_c\) is the characteristic length (typically, the object’s thickness), and \(k\) is the thermal conductivity.

In the given problem, we found that the Bi number is less than 0.1, making the lumped system analysis applicable. It allowed us to use an exponential equation to calculate the surface temperature of the stainless steel strip. This greatly simplifies the calculations and provides a reasonable approximation of temperature distribution, assuming that heat transfer by conduction within the object occurs much faster than heat transfer by convection from the surface to the environment.

Understanding the validity and application of the lumped system analysis is crucial for students tackling heat transfer problems as it can simplify complex situations into more manageable calculations. Knowing when it can be applied, and the associated implications, forms the foundation for further studies in thermal systems and engineering designs.

Conduction is the process of heat transfer through a material without any movement of the material itself. This mode of heat transfer is governed by Fourier's law, which states that the rate of heat transfer through a material is proportional to the negative gradient of temperatures and the area through which the heat is being transferred, mathematically expressed as \( Q = -k A \frac{dT}{dx} \), where \( Q \) is the rate of heat transfer, \( k \) is the thermal conductivity, \( A \) is the cross-sectional area, and \(- \frac{dT}{dx}\) is the temperature gradient.

Thermal conductivity, \( k \), is a property of the material and measures its ability to conduct heat. Materials with high \( k \) values are good conductors and readily transfer heat, whereas materials with low \( k \) values are insulators and resist heat flow. Understanding conduction is essential for solving various engineering problems related to heat transfer in solids—ranging from designing effective thermal insulation to ensuring appropriate heat dissipation in electronic components.

Convection is the heat transfer mode that involves the movement of fluid (which can be a liquid or gas) and the ensuing energy exchange between the fluid and a surface. In the exercise, we consider the forced convection process, which occurs when a fluid's motion is induced by external means, such as a pump or, in this case, the movement of the stainless steel strip through the furnace.

The convective heat transfer coefficient, \( h_c \), is a measure of the convective heat transfer between the fluid and the surface. Newton's law of cooling describes this exchange with the equation \( Q = h_c A_s (T_s - T_{\infty}) \) where \( Q \) is the heat transfer rate, \( A_s \) is the surface area, \( T_s \) is the surface temperature, and \( T_{\infty} \) is the fluid temperature far from the surface.

When dealing with convection problems, it is important to know the rate at which heat is being transferred, as it affects the temperature distribution along the surface of the material, as well as the overall system's thermal performance. By mastering this concept, students can tackle a wide array of practical engineering challenges, such as cooling of electronic equipment, design of heat exchangers, and heating and ventilation system optimization.