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 ( \(1 \mathrm{~atm}\) ) 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, necessary to cause the accumulated ice to begin melting. Assume the windshield surface can be treated as a flat plate surface.

Short answer

In this problem, we are given the properties of an automobile windshield, the temperature and velocity of the outside air, and the temperature of the inside air. We are asked to find the convection heat transfer coefficient of warm air blowing over the inner surface of the windshield to cause the accumulated ice to start melting. We determined the heat transfer through the windshield using the given properties and temperature differences, finding a value of 63 W. We then equated this to the convection heat transfer on the inner surface and found that the inner surface temperature needed for the ice to start melting is approximately 20°C. Finally, we solved for the convection heat transfer coefficient, which was found to be 63 W/m²⋅K. Answer: The convection heat transfer coefficient required for the warm air blowing over the inner surface of the windshield to cause the accumulated ice to begin melting is h = 63 W/m²⋅K.

Step-by-step solution

Determine Heat Transfer through Windshield

Let's first calculate the heat transfer through the windshield using the formula: \(Q = k_w\cdot A\cdot \frac{\Delta T}{d}\) We're given: - Thermal conductivity, \(k_w = 1.4\,\mathrm{W/m\cdot K}\) - Thickness, \(d = 5\,\mathrm{mm} = 0.005\,\mathrm{m}\) - Overall Height, \(H = 0.5\,\mathrm{m}\) We can assume the width of the windshield to be \(1\,\mathrm{m}\) (although not required, it will cancel out later), so the area \(A = 0.5 * 1 = 0.5\,\mathrm{m}^2\). The temperature difference across the windshield is given by the inner and outer surface temperature difference: \(\Delta T = T_{\text{air}} - T_\infty = 25^{\circ}\mathrm{C} - (-20^{\circ}\mathrm{C}) = 45^{\circ}\mathrm{C}\). Now we can calculate the heat transfer through the windshield: \(Q = 1.4\cdot 0.5 \cdot \frac{45}{0.005} = 63 \,\mathrm{W}\)

Applying Convection Heat Transfer Equation

We have to find the convection heat transfer coefficient, \(h\), for the warm air blowing over the inner surface of the windshield. We will use the equation for convection heat transfer on the inner surface: \(Q = h\cdot A\cdot (T_{\text{air}} - T_{\text{inner}})\) We know that the heat transfer through the windshield must be equal to the heat needed by the inner surface and the outer surface to be in equilibrium: \(Q_\text{cond} = Q_\text{conv}\) Therefore, \(63\,\mathrm{W} = h\cdot 0.5\cdot (25^{\circ} \mathrm{C} - T_{\text{inner}})\).

Solve for the Convection Heat Transfer Coefficient

To solve for \(h\), we need the inner surface temperature, \(T_\text{inner}\). Since we want the ice to start melting, the outer surface temperature should be equal to the melting point of ice, \(0^{\circ}\mathrm{C}\). Using the formula for the temperature difference across the windshield (like in step 1), we can find the inner surface temperature: \(T_{\text{inner}} = T_\text{air} - \frac{Q\cdot d}{k_w\cdot A} = 25 - \frac{63\cdot 0.005}{1.4\cdot 0.5} \approx 20^{\circ}\mathrm{C}\) Now, we can solve for \(h\) using the convection heat transfer equation: \(h = \frac{63}{0.5\cdot (25 - 20)} = 63\) So, the convection heat transfer coefficient required for the warm air blowing over the inner surface of the windshield to cause the accumulated ice to begin melting is \(h = 63\,\mathrm{W/m^2\cdot K}\).

Key concepts

Thermal Conductivity

Thermal conductivity, denoted as k or k_{w} in specific cases, is a measure of a material's ability to conduct heat. It plays a crucial role in understanding how quickly heat can pass through materials, such as the windshield in our exercise. Here, we're looking at an automobile windshield with a thermal conductivity of 1.4 W/m·K, which is a moderate value indicating that the windshield is neither a perfect insulator nor a superb conductor of heat.

When warm air is blown over the inner surface of the windshield, heat transfers from the warm side to the cold side by conduction through the glass. This process can be calculated using a simple formula that relates thermal conductivity to the rate of heat transfer (Q), the area through which heat is conducted (A), the temperature difference (ΔT), and the thickness of the material (d). The formula used is: Q = k_w·A·ΔT/d. In practical terms, the thermal conductivity helps us to determine how effective a material, such as the windshield glass, is at transferring heat from the warm interior of a car to the icy exterior.

Heat Transfer through Windshield

Heat transfer through the windshield is an example of conduction, which occurs when there is a temperature difference across a material. In our exercise, the windshield has accumulated ice on the outside, and we aim to defrost it by transferring heat through the windshield's glass.

The effectiveness of this process relies on several factors: the temperature inside the car (25°C), the external temperature (-20°C), and the windshield's characteristics such as thickness and thermal conductivity. Using the formula Q = k_w·A·ΔT/d, we calculated the amount of heat transfer necessary to start melting the ice. The calculated heat transfer rate (Q) tells us how much heat energy per second must pass through the windshield to achieve the desired effect.

Importance in Vehicle Design

Understanding heat transfer through windshields is critical in vehicle design for passenger comfort and safety. Engineers ensure that the glass used in vehicles provides sufficient insulation during cold weather but also enables quick defrosting to maintain visibility. This example beautifully illustrates the balance manufacturers must strike between thermal efficiency and functional design.

Convection Heat Transfer Equation

The convection heat transfer equation is fundamental when calculating the heat exchange between a surface and the fluid moving over it. This equation is represented as Q = h·A·(T_surface - T_fluid), where Q is the heat transfer rate, h is the convection heat transfer coefficient, A is the surface area, and (T_surface - T_fluid) is the temperature difference between the surface and the fluid.

In the context of our windshield defrosting scenario, we use this equation to determine the h value that would make the warm air inside the car melt the ice on the outside. By equating the heat transferred through the windshield by conduction (Q_cond) to that which must be removed by convection (Q_conv), we can solve for the heat transfer coefficient. This coefficient ultimately dictates the effectiveness of the heating system to defrost the windshield — a principle that is not only applicable to car cabins but extends to various engineering and environmental applications where temperature regulation is key.