Question

The inner and outer surfaces of a \(2-m \times 2-m\) win dow glass in winter are \(10^{\circ} \mathrm{C}\) and \(3^{\circ} \mathrm{C}\), respectively. If the rate of heat loss through the window is \(3.2 \mathrm{kJ} / \mathrm{s}\), determine the amount of heat loss, in \(\mathrm{kJ},\) through the glass over a period of 5 h. Also, determine the rate of entropy generation during this process within the glass.

Short answer

Answer: The total amount of heat loss through the window glass over a period of 5 hours is 57,600 kJ, and the rate of entropy generation within the glass is 11.44 J/(K*s).

Step-by-step solution

Calculate the total heat loss

To find the total heat loss over 5 hours, we can multiply the rate of heat loss by the total time in seconds. Our given rate of heat loss is 3.2 kJ/s, and there are 5 hours * 60 min/hour * 60 s/min = 18000 s in 5 hours. Total heat loss (Q) = (rate of heat loss) * (time) Q = 3.2 kJ/s * 18000 s

Calculate the total heat loss

Now, we can calculate the heat loss over the period of 5 hours. Q = 3.2 kJ/s * 18000 s = 57600 kJ The total amount of heat loss through the glass over a period of 5 hours is 57,600 kJ.

Calculate the entropy generation rate

Next, we'll determine the rate of entropy generation within the glass. First, we need to find the average temperature (T_avg) of the glass: T_avg = (T_inner + T_outer) / 2 = (10 + 3) / 2 = 6.5°C = 279.65 K Now, we need to convert the rate of heat loss to a rate of heat transfer (q): q = 3.2 kJ/s * 1000 J/kJ = 3200 W The entropy generation rate (S_gen) can be calculated using the heat transfer rate (q) and the average temperature (T_avg): S_gen = q / T_avg

Calculate the rate of entropy generation

Finally, we can calculate the rate of entropy generation within the glass. S_gen = 3200 W / 279.65 K = 11.44 J/(K*s) The rate of entropy generation within the glass is 11.44 J/(K*s).

Key concepts

Entropy Generation Rate

The concept of entropy is a fundamental element in the second law of thermodynamics, often associated with a measure of disorder or randomness.

The entropy generation rate is particularly significant when analyzing irreversible processes, such as heat loss. In thermodynamics, when heat is transferred through a medium—like the glass in our exercise—there is always some degree of irreversibility, which in turn generates entropy.

To calculate the entropy generation rate within the glass, we initially find the average temperature of the glass, then we use this value to divide the heat transfer rate expressed in watts (joules per second). The increase of entropy signifies that energy has become more spread out and less available to do work, which has implications for energy efficiency and system performance.

Heat Transfer

Heat transfer is the process of thermal energy moving from a warmer space to a cooler one. It is the way heat moves through materials and across boundaries due to a temperature difference.

In the context of the textbook example, heat transfer is happening between the warmer interior (10°C) and the colder exterior (3°C) of the glass. The heat loss rate, measured in kilojoules per second (or watts), represents the amount of heat energy transferring through the glass per unit of time.

Understanding heat transfer is essential for calculating energy efficiencies, designing insulation in buildings, and even for weather forecasting. The methodology applied to ascertain total heat loss involves multiplying this rate by the total time during which the process occurs.

Thermal Conductivity

Thermal conductivity is a material-specific property that quantifies a substance's ability to conduct heat. Materials with high thermal conductivity, such as metals, are good conductors of heat, while those with low thermal conductivity, like glass or insulation materials, are considered thermal insulators.

However, the textbook exercise does not directly address thermal conductivity, knowing the thermal conductivity value would allow us to understand why a certain rate of heat loss occurs through the window glass. Thermal conductivity often plays a key role in determining the thickness and type of materials used for thermal insulation in construction and various industrial applications.

Temperature Gradient

A temperature gradient is the rate of temperature change relative to distance within a substance. It is the driving force behind heat transfer by conduction, where heat flows from the warmer part toward the cooler part of a material.

The larger the temperature gradient, the more rapid the heat transfer. In the given exercise, the temperature gradient is present across the thickness of the window glass, between the inner surface temperature of 10°C and the outer surface temperature of 3°C. Essentially, the temperature gradient tells us how quickly the temperature changes as we move from the interior to the exterior side of the glass and directly influences the rate of energy transfer across the medium.