Question

To defrost ice accumulated on the outer surface of an automobile windshield, warm air is blown over the inner surface of the windshield. Consider an automobile windshield $\left(k_{w}=1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)\( with an overall height of \)0.5 \mathrm{~m}$ and thickness of \(5 \mathrm{~mm}\). The outside air (1 atm) ambient temperature is \(-20^{\circ} \mathrm{C}\), and the average airflow velocity over the outer windshield surface is \(80 \mathrm{~km} / \mathrm{h}\), while the ambient temperature inside the automobile is \(25^{\circ} \mathrm{C}\). Determine the value of the convection heat transfer coefficient for the warm air blowing over the inner surface of the windshield that is needed to cause the accumulated ice to begin melting. Assume the windshield surface can be treated as a flat plate surface.

Short answer

#Step 2: Heat transfer required to melt the ice# #tag_title# Determine the heat transfer required to melt the ice #tag_content# To determine the heat transfer necessary to cause the ice to melt, we use the equation: \(q_{melt} = m\cdot L_f\) Where: \(q_{melt}\) - heat transfer required to melt the ice (W) \(m\) - mass of the ice (kg) \(L_f\) - latent heat of fusion for ice (J/kg) To find the mass of the ice, we can use the equation: \(m = \rho V\) Where: \(\rho\) - density of ice (kg/m³) \(V\) - volume of the ice layer (m³) The volume of the ice layer can be calculated as the product of the windshield area and the thickness of the ice layer. #Step 3: Calculate the required convection heat transfer coefficient# #tag_title# Calculate the required convection heat transfer coefficient #tag_content# To find the required convection heat transfer coefficient for the air inside the car to keep the ice melted, we can use the equation: \(h = \frac{q_{melt}}{A(T_{in} - T_{ice})}\) Where: \(h\) - convection heat transfer coefficient (W/m²*K) \(T_{ice}\) - temperature of ice (°C) The ice temperature is assumed to be at its melting point, 0°C. The heat transfer required to melt the ice must be equal to the heat transfer through the windshield. #Step 4: Calculate the actual value# #tag_title# Calculate the actual value #tag_content# Now that we have the necessary equations, we can calculate the actual value of the convection heat transfer coefficient with the provided information. 1. Calculate the heat transfer through the windshield using the conductive heat transfer equation. 2. Determine the heat transfer required to melt the ice using mass and latent heat of fusion. 3. Calculate the required convection heat transfer coefficient with the determined heat transfers from the previous steps. #Conclusion# By following these steps, students should be able to calculate the required convection heat transfer coefficient for the air inside the car to keep the ice melted on a windshield on a cold winter day.

Step-by-step solution

Determine the heat transfer through the windshield

To obtain the heat transfer from the inside of the car to the outer surface of the windshield, we can use the conductive heat transfer equation, which is as follows: \(q=k_{w} A \frac{T_{in}-T_{out}}{L}\) Where: \(q\) - heat transfer (W) \(k_{w}\) - Conductivity of windshield material (W/m*K) \(A\) - Area of the windshield (m²) \(T_{in}\) - Temperature inside the car (°C) \(T_{out}\) - Temperature outside the car (°C) \(L\) - Thickness of the windshield (m) Since the overall height of the windshield is given, we can assume it to be a square to simplify the problem. Thus, the area of the windshield can be calculated as A = height\(^{2}\).