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 with thickness of 5 mm and thermal conductivity of 1.4 W/m·K. The outside ambient temperature is 210°C and the convection heat transfer coefficient is 200 W/m2 ·K, while the ambient temperature inside the automobile is 25°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

To 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, we first need to establish the heat transfer through the windshield.

For steady-state one-dimensional heat transfer through a plane wall, the heat transfer rate (\(q\)) is given by Fourier's law:

\[ q = \frac{k}{d} \cdot A \cdot (T_{hot} - T_{cold}) \]

where: - \(k\) is the thermal conductivity of the windshield material (1.4 W/m·K), - \(d\) is the thickness of the windshield (5 mm or 0.005 m), - \(A\) is the surface area (this will cancel out from both sides of the equation, so we don’t need its value), - \(T_{hot}\) is the temperature of the warm air inside the automobile (25°C), and - \(T_{cold}\) is the outside ambient temperature (-210°C, assuming you meant -210°C since 210°C does not make sense in this context).

However, considering the ice is at 0°C when it is at the point of starting to melt, the outside ambient temperature does not directly affect the calculation for the ice-melt scenario.

For the windshield's outer surface (where ice is building up), we use the convection heat transfer rate given by Newton's law of cooling:

\[ q = h_{out} \cdot A \cdot (T_{surface} - T_{ambient\_out}) \]

where \(h_{out}\) is the convection heat transfer coefficient of the outside air blowing over the outer surface of the windshield (200 W/m²·K).

To cause the ice to melt, heat transfer in must equal heat transfer out. In other words, the heat flow rate through the windshield must match the convective heat loss at the outer surface.

Setting these equations equal to each other and solving for the inner heat transfer coefficient \(h_{in}\):

\[ \frac{k}{d} \cdot (T_{hot} - T_{melt}) = h_{in} \cdot (T_{hot} - T_{surface}) \]

As \(T_{melt}\) (temperature where ice begins to melt) is 0°C, and assuming \(T_{surface}\) is also 0°C, because ice is assumed to be at the point of melting, the equation simplifies to:

\[ \frac{1.4}{0.005} \cdot (25 - 0) = h_{in} \cdot (25 - 0) \]

This simplifies further to:

\[ h_{in} = \frac{1.4}{0.005} \]

Calculating \(h_{in}\):

\[ h_{in} = 280 \text{ W/m²·K} \]

Therefore, the convection heat transfer coefficient for the warm air blowing over the inner surface of the windshield necessary to cause the ice to begin melting would be 280 W/m²·K.

Extra: In heat transfer problems like this one, the process of conduction and convection is at play. Conduction is the transfer of heat through a solid material (the windshield in this case) without any movement of the material itself. It is described by Fourier's law, which relates the heat flux (heat transfer per unit area per unit time) to the thermal conductivity, the thickness of the material, and the temperature difference across the material.

Convection, on the other hand, is the heat transfer due to the movement of a fluid (here, "fluid" refers to both gases and liquids) over a surface. Convective heat transfer can be forced (by a fan or a pump, for example) or natural (caused by buoyancy effects due to temperature differences in the fluid). Newton's law of cooling is used to describe the rate of convective heat transfer.

Thermal conductivity (k) is a material property that indicates the ability of a material to conduct heat. Materials with higher thermal conductivity can transfer heat more quickly. The convection heat transfer coefficient (h) quantifies the convective heat transfer ability of the fluid and is influenced by the properties of the fluid, the flow velocity, and other factors.

In the context of defrosting a windshield, the overall goal is to transfer enough heat to the ice to bring it to its melting point and to provide additional heat to cause the phase change from solid to liquid, without significantly increasing the temperature of the inner cabin air, which should remain comfortable for the passengers.