A hot liquid \(\left(c_{p}=950 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) flows at a flow rate of \(0.005 \mathrm{~kg} / \mathrm{s}\) inside a tube with a diameter of \(25 \mathrm{~mm}\). At the tube exit, the liquid temperature decreases by \(8^{\circ} \mathrm{C}\) from its temperature at the inlet. The hot liquid causes the tube surface temperature to be \(120^{\circ} \mathrm{C}\). To prevent thermal burn hazards, the tube is enclosed with a concentric outer cover of \(5 \mathrm{~cm}\) in diameter. Determine whether the outer cover temperature is below \(45^{\circ} \mathrm{C}\) to prevent thermal burns in contact with human skin. Evaluate the properties of air in the concentric enclosure at \(80^{\circ} \mathrm{C}\) and 1 atm pressure. Is this a good assumption?

We are given the flow rate of the hot liquid \((0.005 \mathrm{~kg/s})\) and the decrease in temperature at the tube exit (\((8\mathrm{^{\circ}C})\)). Thus, we can calculate the heat loss rate, \(Q\), using the following equation: $$ Q = \dot{m}c_{p}\Delta T $$ where \(\dot{m}\) is the mass flow rate, \(c_{p}\) is the specific heat capacity of the liquid, and \(\Delta T\) is the temperature difference between the inlet and exit. $$ Q = (0.005 \mathrm{~kg/s})(950 \mathrm{~J/kg\cdot K})(8 \mathrm{^{\circ}C}) = 38 \mathrm{W} $$

For heat transfer through the concentric tube, we have the heat transfer equation for a cylindrical enclosure, which also includes conduction through the walls and convection: $$ Q = U A (T_{i} - T_{o}) $$ where \(U\) is the overall heat transfer coefficient, \(A\) is the surface area of the inner tube, \(T_{i}\) is the temperature of the tube's inner surface, and \(T_{o}\) is the temperature of the tube's outer surface. The inner surface area, \(A\), can be calculated as: $$ A = \pi D_{i} L $$ where \(D_{i}\) is the inner diameter of the tube and \(L\) is the length of the tube. In this exercise, we don't have information about the material and thickness of the tube or the convection coefficient of air inside. So, we can't determine the value of \(U\). However, we can certainly deduce the value of \(A\) and simplify the equation as follows: $$ T_{o} = T_{i} - \frac{Q}{UA} $$

Due to insufficient data, we're unable to determine the outer temperature precisely. To make sure the outer cover temperature is below \(45^{\circ} \mathrm{C}\), we need more information about the tube (thickness, material) and convection coefficient of air inside the concentric tube. However, we can deduce certain insights regarding the assumptions. The liquid temperature decreases by \(8^{\circ} \mathrm{C}\) while flowing, meaning its temperature at inlet is at least \(128^{\circ} \mathrm{C}\). If we assume that the temperature of the outer surface will be somewhere between the inlet liquid temperature and the safe temperature, we must trust that the insulation of the tube and presence of concentric cover will ensure that the safety temperature is maintained.

The air in the concentric enclosure can be assumed to be the main heat transfer medium between the hot liquid tube and the outer tube. At \(80^{\circ} \mathrm{C}\) and 1 atm pressure, the air properties can be evaluated and found in standard property tables. By comparing the values of necessary properties at the given condition to our problem, we can validate our assumptions. Without further information about the tube and convection properties, we cannot conclusively determine whether the outer cover temperature will remain below the \(45^{\circ} \mathrm{C}\) threshold for safety. More data about the tube and airflow properties are necessary to achieve a definite solution.