Problem 28 - College Physics

Question

(. A large factory furnace maintained at \(175^{\circ} \mathrm{C}\) at its outer surface is wrapped in an insulating blanket of thermal conduc- tivity 0.055 \(\mathrm{W} /(\mathrm{m} \cdot \mathrm{K})\) which is thick enough that the outer surface of the insulation is at \(42^{\circ} \mathrm{C}\) while heat escapes from the furnace at a steady rate of 125 W for each square meter of surface area. By how much does each square meter of the furnace change the entropy of the factory every second?

Step-by-Step Solution

Answer

Each square meter increases the entropy by 0.1176 W/K per second.

Step 1: Identify the Given Values

We are given the inner temperature \( T_{ ext{hot}} = 175^{\circ}C + 273.15 = 448.15 \, K \), the outer temperature \( T_{ ext{cold}} = 42^{\circ}C + 273.15 = 315.15 \, K \), the heat conduction \( q = 125 \, W/m^2 \), and the thermal conductivity \( k = 0.055 \, W/(m \cdot K) \).

Step 2: Understanding Entropy Change Principle

The change in entropy \( \Delta S \) is calculated using the formula for entropy change due to heat transfer: \( \Delta S = \frac{q}{T} \) where \( T \) is the absolute temperature.

Step 3: Calculating the Entropy Change for the Cold Side

Calculate the change in entropy at the cold side: \( \Delta S_{ ext{cold}} = \frac{125 \, W}{315.15 \, K} \) Plugging in the values, \( \Delta S_{ ext{cold}} \approx 0.3965 \, \text{W/K} \).

Step 4: Calculating the Entropy Change for the Hot Side

For the hot side of the system, the entropy change is negative because heat is conducted away: \( \Delta S_{ ext{hot}} = -\frac{125 \, W}{448.15 \, K} \). Plugging in the values, \( \Delta S_{ ext{hot}} \approx -0.2789 \, \text{W/K} \).

Step 5: Total Entropy Change in the System

The total change in entropy of the system is the sum of changes from the hot and cold sides: \( \Delta S = \Delta S_{ ext{cold}} + \Delta S_{ ext{hot}} \). Therefore, \( \Delta S \approx 0.3965 \, \text{W/K} - 0.2789 \, \text{W/K} = 0.1176 \, \text{W/K} \).

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics focusing on the relationships between heat, work, and energy. It plays a crucial role in understanding how heat and energy transfer within systems, like that in the scenario of the factory furnace. Thermodynamic principles help us understand processes like combustion, refrigeration, and heat engines. Several key concepts arise from thermodynamics, such as:

• Energy Conservation: Energy remains constant in an isolated system, albeit changing forms between kinetic energy, gravitational potential energy, etc.
• Entropy: A measure of disorder or randomness in a system. Entropy tends to increase in real-world processes, aligning with the second law of thermodynamics.
• Thermal Equilibrium: A state where there are no net thermal energy transfers between objects in contact due to their temperatures being equal.

In the factory furnace, thermodynamics aids us in quantifying how entropy changes as heat moves through different temperature gradients.

Heat Conduction

Heat conduction happens when thermal energy passes through materials, typically from warmer areas to cooler ones. This transport takes place due to collisions between molecules, transferring kinetic energy. Understanding heat conduction is vital in many applications, including the design of furnaces and insulation materials. The rate of heat conducting through a material relies on:

• Thermal Conductivity: A material's ability to conduct heat. In our furnace scenario, with a specified thermal conductivity, it affects how quickly heat moves through the insulating blanket.
• Temperature Gradient: The difference in temperature across the material. A larger gradient usually results in more heat transfer.

In our case, the furnace's outer surface is hotter than the insulated outer layer, driving heat flow. Calculating this helps us determine the efficiency of insulation and predict energy loses.

Insulation

Insulation is crucial in managing energy efficiency by reducing the rate of heat transfer. It plays a significant part in retaining desired temperatures within spaces, whether keeping heat inside a space, like the factory furnace, or preventing external heat from entering a cold space. A good insulation material:

• Has low thermal conductivity, making it harder for heat to pass through quickly.
• Maintains stability under varied temperature conditions, ensuring prolonged effectiveness.

For the furnace, the insulating blanket minimizes heat loss, keeping energy consumption and costs lower. Without effective insulation, much more energy would be required to maintain the desired temperature of the furnace, impacting efficiency and economic costs.

Temperature Conversion

Converting temperatures between different scales is an important skill, especially in scientific applications that use different temperature units. The most common scales are Celsius, Fahrenheit, and Kelvin. Here's how to make these conversions:

• From Celsius to Kelvin: Add 273.15, since 0°C corresponds to 273.15 K, the absolute zero in the Kelvin scale.
• From Celsius to Fahrenheit: Use the formula, Fahrenheit = (Celsius × 9/5) + 32.

In the exercise, we converted from Celsius to Kelvin, necessary for thermodynamic calculations as Kelvin is an absolute scale and directly relates to energy in physics. This conversion allows us to measure variables like entropy correctly, as seen in determining the changes in the factory furnace's system.