Question

Consider a window glass consisting of two 4-mmthick glass sheets pressed tightly against each other. Compare the heat transfer rate through this window with that of one consisting of a single 8-mm-thick glass sheet under identical conditions.

Short answer

Answer: To determine the heat transfer rate, we compared the thermal resistance of both configurations using the Fourier's law of heat conduction. Under identical conditions, if the thermal resistance of the single 8mm thick glass sheet (R1) is lower than that of the two 4mm thick glass sheets pressed together (R2), then the single 8mm thick glass sheet has a higher heat transfer rate. Conversely, if R1 is greater than R2, the two 4mm thick glass sheets pressed together have a higher heat transfer rate.

Step-by-step solution

Identify the relevant formula for heat transfer rate

We can use Fourier's law of heat conduction for calculating the heat transfer rate (Q) through a solid. The formula is given by: Q = k * A * (T1 - T2) / d where: Q is the heat transfer rate (W), k is the thermal conductivity of the material (W/mK), A is the cross-sectional area through which heat is transferred (m^2), T1 and T2 are the temperatures on either side of the glass (K), d is the thickness of the glass (m).

Calculate thermal resistance for each glass configuration

To compare the heat transfer rates through both types of window glasses, we have to determine the equivalent thermal resistance for each configuration. A lower thermal resistance means that the material is more efficient in conducting heat, therefore, we can compare the thermal resistance of both window glass configurations to determine which one has a higher heat transfer rate. For the single 8mm thick glass sheet, the thermal resistance (R1) can be calculated as: R1 = d1 / (k * A) For the two 4mm thick glass sheets pressed together, we can treat them as a single 8mm thick glass sheet because they are in thermal series, and calculate the thermal resistance (R2) as follows: R2 = d2 / (k * A) where d1 = 0.008m (8mm), and d2 = 0.004m (4mm).

Compare the thermal resistance of both configurations

Now that we have the thermal resistance for both configurations, we can compare them and determine which glass has a higher heat transfer rate. If R1 < R2, it means that the single 8mm thick glass sheet has a lower thermal resistance and therefore a higher heat transfer rate. Conversely, if R1 > R2, it means that the two 4mm thick glass sheets pressed together have a higher heat transfer rate.

Comparison under identical conditions

We were asked to compare the heat transfer rate under identical conditions. Since the thermal conductivity (k) and the cross-sectional area (A) remain constant for both configurations, and the temperatures (T1 and T2) are also identical as specified, then the only differing factor would be the thickness (d). Thus, by comparing the thermal resistance of both configurations, we can determine the heat transfer rate.

Conclusion

To compare the heat transfer rate through two different window glass configurations, we used Fourier's law of heat conduction formula to calculate heat transfer rates and thermal resistance. By comparing the thermal resistance of the two configurations under identical conditions, we can determine which one has a higher heat transfer rate.

Key concepts

Fourier's law of heat conduction

Fourier's law of heat conduction is a fundamental principle that describes the way heat energy transfers through materials. The law states that the heat transfer rate (\( Q \)) through a solid is directly proportional to the temperature difference across the material (\( T_1 - T_2 \)), the area (\( A \)) through which heat is transferred, and the thermal conductivity of the material (\( k \)), and inversely proportional to the thickness (\( d \)) of the material.Fourier's Law equation:\[\begin{equation} Q = \frac{k \times A \times (T_1 - T_2)}{d} \end{equation}\]By using this formula, students can calculate how much heat energy will pass through a given material with known properties. In the exercise provided, this law helps to understand the difference in heat transfer rates between a single thick glass and two thinner glasses pressed together. To enhance comprehension, it's helpful to illustrate this concept with common experiences, such as feeling the warmth of a hot cup through different thicknesses of gloves—thinner gloves allow more heat to transfer, just as thinner glass sheets may transfer more heat than a single thick one.

Thermal resistance

Thermal resistance is a concept used to quantify how resistant a material is to heat flow. The higher the thermal resistance, the lower the heat transfer rate. It's analogous to electrical resistance, which measures how much a material opposes the flow of electric current. In heat transfer, thermal resistance (\( R \)) is defined as the ratio of the temperature difference across the material to the heat transfer rate.Thermal Resistance equation:\[\begin{equation} R = \frac{d}{k \times A} \end{equation}\]In solving the provided exercise, understanding thermal resistance is crucial. It allows students to compare the efficiency of different glass configurations in preventing heat loss. If we consider two glass sheets pressed together versus a single thicker one, we can calculate and compare their respective thermal resistances to see which configuration better impedes heat flow under the same conditions. Visualizations such as comparing layers of clothing can help students grasp how adding or removing layers affect warmth, similar to how the glass layers affect heat transfer.

Thermal conductivity

Thermal conductivity (\( k \)) is a material-specific property that indicates how well a substance can conduct heat. Materials with high thermal conductivity, such as metals, are good conductors of heat, whereas materials with low thermal conductivity, such as glass or plastic, are considered insulators.Thermal conductivity plays a pivotal role in Fourier's law of heat conduction, as seen in the equation mentioned earlier. In the context of the textbook exercise, thermal conductivity is the constant that determines how easily heat can pass through the glass—regardless of whether the glass is a single thick sheet or two thinner sheets pressed together. By understanding thermal conductivity, students can reason why certain materials are chosen for specific applications, like why glass is used for window panes instead of metal. It's beneficial to provide real-world examples, such as comparing a metal spoon that quickly becomes hot in a cup of tea to a wooden spoon that remains cool, to illustrate the concept of thermal conductivity.