Question

The inner and outer glasses of a \(2-\mathrm{m} \times 2\) -m double-pane window are at \(18^{\circ} \mathrm{C}\) and \(6^{\circ} \mathrm{C}\), respectively. If the glasses are very nearly isothermal and the rate of heat transfer through the window is \(110 \mathrm{W}\), determine the rates of entropy transfer through both sides of the window and the rate of entropy generation within the window, in \(\mathrm{W} / \mathrm{K}\).

Short answer

Question: Calculate the rates of entropy transfer through the inner and outer glasses, as well as the rate of entropy generation within a double-pane window with given temperatures and heat transfer rate. Given: - Inner glass temperature: \(T_{in} = 18^{\circ} \mathrm{C}\) - Outer glass temperature: \(T_{out} = 6^{\circ} \mathrm{C}\) - Rate of heat transfer through the window: \(\dot{Q} = 110 \mathrm{W}\) Answer: - Rate of entropy transfer through inner glass: \(\dot{S}_{in} = 0.378 \frac{\mathrm{W}}{\mathrm{K}}\) - Rate of entropy transfer through outer glass: \(\dot{S}_{out} = 0.394 \frac{\mathrm{W}}{\mathrm{K}}\) - Rate of entropy generation within the window: \(\dot{S}_{gen} = 0.016 \frac{\mathrm{W}}{\mathrm{K}}\)

Step-by-step solution

Identify the given values

We are given the temperatures of the inner and outer glasses, \(T_{in} = 18^{\circ} \mathrm{C}\) and \(T_{out} = 6^{\circ} \mathrm{C}\), as well as the rate of heat transfer through the window, \(\dot{Q} = 110 \mathrm{W}\). We will need these values when calculating the rates of entropy transfer and generation.

Convert temperatures to Kelvin

Before proceeding with the calculations, we need to convert the given temperatures to Kelvin. \(T_{in} = 18^{\circ} \mathrm{C} + 273.15 = 291.15 \mathrm{K}\) and \(T_{out} = 6^{\circ} \mathrm{C} + 273.15 = 279.15 \mathrm{K}\).

Calculate the rate of entropy transfer through the inner glass

The equation for the rate of entropy transfer is \(\dot{S} = \frac{\dot{Q}}{T}\). We will first calculate the rate of entropy transfer through the inner glass, \(\dot{S}_{in}\): $$\dot{S}_{in} = \frac{\dot{Q}}{T_{in}} = \frac{110 \mathrm{W}}{291.15 \mathrm{K}} = 0.378 \frac{\mathrm{W}}{\mathrm{K}}$$

Calculate the rate of entropy transfer through the outer glass

Similarly, we can calculate the rate of entropy transfer through the outer glass, \(\dot{S}_{out}\): $$\dot{S}_{out} = \frac{\dot{Q}}{T_{out}} = \frac{110 \mathrm{W}}{279.15 \mathrm{K}} = 0.394 \frac{\mathrm{W}}{\mathrm{K}}$$

Calculate the rate of entropy generation within the window

The rate of entropy generation is the difference between the rates of entropy transfer through the inner and outer glasses, \(\dot{S}_{gen}=\dot{S}_{out}-\dot{S}_{in}\): $$\dot{S}_{gen} = 0.394 \frac{\mathrm{W}}{\mathrm{K}} - 0.378 \frac{\mathrm{W}}{\mathrm{K}} = 0.016 \frac{\mathrm{W}}{\mathrm{K}}$$ Now we have the rates of entropy transfer through both sides of the window and the rate of entropy generation within the window: - Rate of entropy transfer through inner glass: \(\dot{S}_{in} = 0.378 \frac{\mathrm{W}}{\mathrm{K}}\) - Rate of entropy transfer through outer glass: \(\dot{S}_{out} = 0.394 \frac{\mathrm{W}}{\mathrm{K}}\) - Rate of entropy generation within the window: \(\dot{S}_{gen} = 0.016 \frac{\mathrm{W}}{\mathrm{K}}\)

Key concepts

Thermodynamic Properties

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In particular, it describes how thermal energy is converted to and from other forms of energy and how it affects matter.

Thermodynamic properties are characteristics that define the state of a system, and when we speak of such properties in the context of energy and heat transfer, two key concepts often come into play: temperature and entropy. Temperature is a measure of the thermal energy within a system, indicating how hot or cold an entity is. Entropy, on the other hand, is a measure of the disorder in a system and is a central concept in the second law of thermodynamics, which highlights that the total entropy of an isolated system can never decrease over time.

Understanding these properties is pivotal when analyzing energy transfer in systems such as the double-pane window from our exercise, which involves both heat transfer and entropy changes.

Heat Transfer

Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as conduction, convection, and radiation.

In the exercise we're considering, the heat transfer through the double-pane window is predominantly by conduction, where heat passes through solids from a region of high temperature to a region of low temperature. The rate of heat transfer, denoted by \(\dot{Q}\), is an important quantity that tells us how much thermal energy is being transferred per unit time.

Importance of Heat Transfer in Everyday Applications

This principle finds applications in everyday life from the design of buildings to keep them warm during winters to the functioning of refrigerators, and much more.
In our scenario, knowing the rate of heat transfer allows us to assess how effectively the window insulates the room from external temperatures.

Entropy Generation

Entropy generation is indicative of the irreversibility within a process and it represents the energy that is not available to do work. During heat transfer, entropy is not only transferred but can be generated, especially when there is a temperature gradient.

The laws of thermodynamics dictate that in an isolated system, the entropy will remain constant if the process is reversible and will increase if the process is irreversible. In real-world applications, all processes increase the total entropy of the system, which is a measure of the energy dispersion in a process.

Practical Implications of Entropy

This has real-world implications, such as in the efficiency of engines or the effectiveness of insulation in buildings. In the case of the double-pane window, the generation of entropy signifies inefficiencies in the system that might lead to energy loss. The total entropy generation is significant as it quantifies the magnitude of energy dispersion due to temperature differences.

Temperature Conversion

Temperature conversion is a basic but essential thermodynamic task that involves changing the temperature reading from one scale to another. The Celsius and Kelvin scales are commonly used in thermodynamic calculations.

The Kelvin scale is an absolute temperature scale where 0 K (zero Kelvin) is the point at which particles within a substance have minimum thermal motion, commonly known as absolute zero. To convert from Celsius to Kelvin, which is necessary for accurate thermodynamic calculations, one must add 273.15 to the Celsius temperature.

Why Use Kelvin for Thermodynamic Calculations?

The use of Kelvin is crucial because it accurately relates thermal energy to other thermodynamic properties, such as entropy. For instance, the rate of entropy transfer through the window hinges on the absolute temperatures (in Kelvin) rather than temperatures in Celsius, as observed in our exercise, because Kelvin ensures a proportionate relationship between the heat transfer and resulting entropy change.