Long aluminum wires of diameter \(3 \mathrm{~mm}(\rho=\) \(2702 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=0.896 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=236 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and \(\alpha=\) \(9.75 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) ) are extruded at a temperature of \(350^{\circ} \mathrm{C}\) and exposed to atmospheric air at \(30^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Determine how long it will take for the wire temperature to drop to \(50^{\circ} \mathrm{C}\). (b) If the wire is extruded at a velocity of \(10 \mathrm{~m} / \mathrm{min}\), determine how far the wire travels after extrusion by the time its temperature drops to \(50^{\circ} \mathrm{C}\). What change in the cooling process would you propose to shorten this distance? (c) Assuming the aluminum wire leaves the extrusion room at \(50^{\circ} \mathrm{C}\), determine the rate of heat transfer from the wire to the extrusion room.

First, check the lumped capacitance method's validity by calculating the Biot Number (Bi). The Biot Number compares the conduction resistance inside the body and the convection resistance at its surface. The lumped capacitance method is valid if \({Bi}<0.1\). \[ Bi = \frac{hL_c}{k} \] where: \(h\) is the heat transfer coefficient \((35 \ \mathrm{W/m^2 \cdot K})\), \(L_c\) is the characteristic length, which is equal to the radius (r) of the aluminum wire \(L_c = \frac{D}{2} = \frac{0.003}{2} \ \mathrm{m}\), and \(k\) is the thermal conductivity of the aluminum \((236 \ \mathrm{W/m \cdot K})\). Calculate the Biot Number: \[ Bi = \frac{35 \cdot (0.003/2)}{236} \]

Now, we use the lumped capacitance method to calculate the time it takes for the wire's temperature to drop from 350℃ to 50℃. The temperature decreases exponential according to the formula: \[ \frac{T - T_\infty}{T_i - T_\infty} = exp(-\frac{hA}{\rho V c_p}t) \] where: \(T\) is the final temperature of the wire \((50^{\circ}C)\), \(T_\infty\) is the atmospheric air temperature \((30^{\circ}C)\), \(T_i\) is the initial temperature of the wire \((350^{\circ}C)\), \(A\) is the wire's surface area \(A = \pi D L\), with L being the length of the wire, \(V\) is the wire's volume \(V = \pi (\frac{D}{2})^2 L\), \(\rho\) is the aluminum's density \((2702 \ \mathrm{kg/m^3})\), \(c_p\) is the specific heat of the aluminum \((0.896 \ \mathrm{kJ/kg \cdot K})\), and t is the time we need to determine. The surface area to volume ratio of the wire is: \[ A/V = \frac{\pi DL}{\pi (\frac{D}{2})^2 L} = \frac{4}{D} \] Rearrange the formula to isolate t: \[ t = -\frac{\rho V c_p}{hA} \ ln (\frac{T - T_\infty}{T_i - T_\infty}) = - \frac{\rho c_p}{h(A/V)} \ ln (\frac{T - T_\infty}{T_i - T_\infty}) \] Now, substitute the given values and perform the calculation to obtain t.

To calculate the distance traveled by the wire, we will use its cooling time and extrusion velocity: \[ d = v \cdot t \] where: \(d\) is the distance traveled, \(t\) is the time it takes for the wire's temperature to drop to 50℃ (calculated in step 2), and \(v\) is the extrusion velocity given as \(10 \mathrm{~m/min}\). Calculate the distance:

The rate of heat transfer (Q) from the wire to the extrusion room can be calculated using the following formula: \[ Q = hA(T-T_\infty) \] For this calculation, we will first find the length (L) of the aluminum wire when its temperature reaches 50℃. Since we know the distance traveled (d) and the diameter (D), we can calculate L using the formula: \[ L = \frac{d}{\pi D} \] Now, we will find the surface area (A) of the wire using the formula: \[ A = \pi DL \] Finally, substitute the given values and perform the calculations to obtain the heat transfer rate.