A 5-m-long strip of sheet metal is being transported on a conveyor at a velocity of \(5 \mathrm{~m} / \mathrm{s}\), while the coating on the upper surface is being cured by infrared lamps. The coating on the upper surface of the metal strip has an absorptivity of \(0.6\) and an emissivity of \(0.7\), while the surrounding ambient air temperature is \(25^{\circ} \mathrm{C}\). Radiation heat transfer occurs only on the upper surface, while convection heat transfer occurs on both upper and lower surfaces of the sheet metal. If the infrared lamps supply a constant heat flux of \(5000 \mathrm{~W} / \mathrm{m}^{2}\), determine the surface temperature of the sheet metal. Evaluate the properties of air at \(80^{\circ} \mathrm{C}\).

Understanding radiation heat transfer is key to solving problems involving thermal energy emission from surfaces. This type of heat transfer occurs without the need for a medium and can happen through the vacuum of space. In our exercise, radiation heat transfer plays a crucial role as the upper surface of the metal strip emits energy after being heated by infrared lamps.

Radiation heat transfer is primarily dependent on the temperature difference between a surface and its surroundings, the surface's emissivity (a measure of how well a surface emits thermal radiation), and the Stefan-Boltzmann constant. The formula to calculate the radiative heat transfer rate from a surface is expressed by the Stefan-Boltzmann law, which will be explored further in another section. Recognizing these factors allows us to accurately assess how much heat is lost due to radiation when attempting to determine the surface temperature of the sheet metal in our problem.

Convection heat transfer is another method by which heat is moved from one place to another. This type of transfer is characterized by the movement of molecules within fluids (such as air or water), which carry thermal energy as they move. In the case of our sheet metal, convection occurs on both the upper and lower surfaces, where the ambient air takes away some of the heat.

The rate of convection heat transfer can be described using the formula: $$Q_{conv} = hA(T_{s} - T_{\text{\tiny\textcircled{r}}})$$where h is the heat transfer coefficient, A is the surface area, T_{s} is the surface temperature of the metal, and T_{\text{\tiny\textcircled{r}}} (typically denoted as T_{\text{\tiny\textinfinity}}) is the fluid temperature far from the surface (ambient temperature). This relationship tells us that the heat transfer rate is proportional to the surface area, the temperature difference, and the convective heat transfer coefficient, which depends on the properties of the fluid and the flow conditions.

The Stefan-Boltzmann law is a cornerstone in thermal physics, relating the radiative energy emitted from a black body to the fourth power of its absolute temperature. In its mathematical form, it states:$$Q_{rad} = \text{\textepsilon} \text{\textsigma} A (T_{s}^{4} - T_{\text{\tiny\textcircled{r}}}^{4})$$Here, \text{\textepsilon} represents the emissivity of the surface, \text{\textsigma} the Stefan-Boltzmann constant (5.67 \times 10^{-8} \text{W/m}^{2}\text{K}^{4}), A the area, T_{s} the absolute surface temperature (in Kelvins), and T_{\text{\tiny\textcircled{r}}} the absolute temperature of the surroundings.

Our exercise requires us to apply this law to determine how much heat is being radiated away from the metal's surface. The emissivity value given in the problem indicates that the surface does not emit radiation as efficiently as a perfect black body (which would have an emissivity of 1). This law is critical for realistic heat transfer calculations in engineering and environmental sciences.

The energy balance equation is a statement of the conservation of energy principle and is used to solve many heat transfer problems, including the one about the sheet metal being cured by infrared lamps. It essentially says that the energy entering a system must equal the energy leaving it plus any change in the energy stored within the system.

In steady-state conditions, where there is no change in stored energy, the energy balance equation simplifies to equate incoming and outgoing energy rates. For our exercise, the equation is:$$Q_{in} = Q_{conv,t} + Q_{rad} + Q_{conv,b}$$This indicates that the heat provided by the infrared lamps to the metal (Q_{in}) is dissipated through convection at the top (Q_{conv,t}) and bottom (Q_{conv,b}) surfaces, and through radiation from the top surface (Q_{rad}). To find the surface temperature (T_{s}), we must rearrange and solve this equation, accounting for all the given values and material properties.