A 40-cm-long, 800-W electric resistance heating element with diameter \(0.5 \mathrm{~cm}\) and surface temperature \(120^{\circ} \mathrm{C}\) is immersed in \(75 \mathrm{~kg}\) of water initially at \(20^{\circ} \mathrm{C}\). Determine how long it will take for this heater to raise the water temperature to \(80^{\circ} \mathrm{C}\). Also, determine the convection heat transfer coefficients at the beginning and at the end of the heating process.

Answer: To find the convection heat transfer coefficients at the beginning and the end of the heating process, use the formula: \(h =\frac{P_\text{heater}}{(T_\text{heater} - T_\text{water}) \cdot A}\) For the beginning of the heating process, plug in the initial surface temperature of the heating element (120°C), the initial temperature of the water (20°C), and the surface area calculated in Step 3: \(h_\text{start} = \frac{0.8 \mathrm{\ kW}}{(120^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C}) \cdot A}\) For the end of the heating process, use the final temperature of the water (80°C): \(h_\text{end} =\frac{0.8 \mathrm{\ kW}}{(120^{\circ} \mathrm{C} - 80^{\circ} \mathrm{C}) \cdot A}\) Calculate h_start and h_end to find the convection heat transfer coefficients at the beginning and the end of the heating process, respectively.