Problem 92 - Pearson Physics

Question

Suppose an OTEC system operates with surface water at \(22^{\circ} \mathrm{C}\) and deep water at \(4.0^{\circ} \mathrm{C}\). What is the maximum efficiency this system could have? A. \(6.1 \%\) C. \(9.4 \%\) B. \(8.2 \%\) D. \(18 \%\)

Step-by-Step Solution

Answer

The maximum efficiency is \(6.1\%\). The correct answer is A.

Step 1: Understand the Carnot Efficiency Formula

The Carnot efficiency represents the maximum efficiency any heat engine can achieve as determined by the second law of thermodynamics. It is expressed as \( \eta = 1 - \frac{T_c}{T_h} \), where \( T_h \) is the temperature of the hot reservoir and \( T_c \) is the temperature of the cold reservoir. Both temperatures need to be in Kelvin.

Step 2: Convert Temperatures to Kelvin

Convert the given temperatures from Celsius to Kelvin. The surface water temperature is \(22^{\circ}C\), so in Kelvin it would be \( T_h = 22 + 273.15 = 295.15\, K \). The deep water temperature is \(4^{\circ}C\), so in Kelvin it would be \( T_c = 4 + 273.15 = 277.15\, K \).

Step 3: Calculate the Carnot Efficiency

Substitute the Kelvin temperatures into the Carnot efficiency formula: \( \eta = 1 - \frac{277.15}{295.15} \). This calculation results in \( \eta = 1 - 0.9389 = 0.0611 \), or in percentage terms, \( 6.1\% \).

Step 4: Choose the Correct Multiple Choice Answer

Compare the calculated efficiency of \(6.1\%\) with the provided options. The correct answer is \( A.\ 6.1\% \).

Key Concepts

Understanding Heat Engines

A heat engine is a fascinating device mainly used to convert heat energy into mechanical work. It operates between two thermal reservoirs: a hot reservoir and a cold reservoir. The heat engine absorbs heat from the hot reservoir, does some work (like moving a piston), and then releases some heat to the cold reservoir.

The efficiency of a heat engine is a critical factor, indicating how well it transforms heat into work. No engine can be 100% efficient due to losses such as friction and sound. The Carnot engine, a theoretical construct, provides a benchmark for the maximum efficiency a heat engine can achieve. The idea is based on the principles of thermodynamics and acts as an ideal standard to which all real-world engines are compared. When exploring topics like Ocean Thermal Energy Conversion (OTEC), understanding how the heat engine operates is essential, as it involves extracting energy based on temperature differences in ocean water.

Fundamentals of Thermodynamics

Thermodynamics is the branch of physics concerned with heat and temperature and their relation to energy and work. It encompasses a few key laws that describe how energy moves and changes form. This serves as the foundation for understanding how systems like heat engines function.

The second law of thermodynamics is pivotal for heat engines. It states that heat will naturally flow from a higher temperature body to a lower temperature body. This principle dictates the direction of energy transfer and ensures that no process can be perfectly efficient. In the context of a heat engine, this translates into the creation of work by transferring heat from a hot reservoir to a cold one, underpinning the necessity to consider Carnot efficiency when evaluating systems such as OTEC.

Importance of Temperature Conversion

Temperature conversion is a basic but crucial operation, especially when dealing with equations related to thermodynamics. Scientists and engineers often use the Kelvin scale in their calculations because it starts at absolute zero, making it ideal for measuring thermodynamic temperature.

• Celsius to Kelvin: To convert Celsius to Kelvin, simply add 273.15. For example, to convert the surface water temperature of 22°C, you calculate: \(T_h = 22 + 273.15 = 295.15 \text{ K}\).
• It’s important to ensure that both temperatures in the Carnot efficiency formula are in Kelvin to obtain accurate results.

Temperatures must be converted properly to avoid errors in efficiency calculations, as seen when predicting the performance of systems like those used in OTEC.

Exploring Ocean Thermal Energy Conversion (OTEC)

Ocean Thermal Energy Conversion (OTEC) is an innovative process that leverages the temperature gradient between warm surface water and cold deep-sea water to generate electricity. It utilizes the principles of heat engines to convert this temperature difference into mechanical energy and then into electricity.

The idea is environmentally appealing, as it offers a renewable and consistent source of energy, particularly in tropical regions where the temperature difference is sufficient. OTEC systems rely on the fundamental thermodynamic principle that dictates energy conversion efficiency, as calculated through the Carnot efficiency formula. By maximizing these efficiency parameters, OTEC demonstrates real-world applications of theoretical thermodynamic principles, harnessing the ocean's energy effectively when designed and operated optimally.