Question

Ocean temperature energy conversion (OTEC) power plants generate power by utilizing the naturally occurring decrease with depth of the temperature of ocean water. Near Florida, the ocean surface temperature is \(27^{\circ} \mathrm{C}\), while at a depth of \(700 \mathrm{~m}\) the temperature is \(7^{\circ} \mathrm{C}\). (a) Determine the maximum thermal efficiency for any power cycle operating between these temperatures. (b) The thermal efficiency of existing OTEC plants is approximately 2 percent. Compare this with the result of part (a) and comment.

Short answer

The maximum thermal efficiency is 6.66%. Current OTEC plants have an efficiency of 2%, which is about 30% of the maximum efficiency.

Step-by-step solution

Understand the Problem

Determine the maximum thermal efficiency for a power cycle based on the given temperature differences and compare it to the efficiency of existing OTEC plants.

Identify Given Temperatures

The surface temperature of the ocean is given as \( T_h = 27^{\text{°C}} \). The temperature at a depth of 700 m is \( T_c = 7^{\text{°C}} \).

Convert Temperatures to Kelvin

Convert the temperatures from Celsius to Kelvin:\[ T_h = 27 + 273.15 = 300.15 \text{ K} \]\[ T_c = 7 + 273.15 = 280.15 \text{ K} \]

Apply Carnot Efficiency Formula

The maximum thermal efficiency for any power cycle operating between two temperatures can be calculated using the Carnot efficiency formula:\[ \eta_{\text{max}} = 1 - \frac{T_c}{T_h} \]

Calculate Maximum Thermal Efficiency

Substitute the converted temperatures into the Carnot efficiency formula:\[ \eta_{\text{max}} = 1 - \frac{280.15}{300.15} \approx 0.0666 \text{ or } 6.66\text{%} \]

Compare with OTEC Efficiency

The given efficiency for existing OTEC plants is approximately 2%. Compare this to the calculated maximum efficiency:\[ 2\text{%} \text{ (OTEC Efficiency)} < 6.66\text{%} \text{ (Maximum Efficiency)} \]This shows that the current OTEC plants are operating at about \(\frac{2}{6.66} \approx 0.3 \) or 30% of the maximum possible efficiency.

Key concepts

OTEC efficiency

Ocean Thermal Energy Conversion (OTEC) is a process that converts the thermal energy stored in the ocean into electrical power. The efficiency of OTEC systems is fundamental for understanding how much electrical energy can be generated compared to the thermal energy available in the ocean. Efficiency is defined as the ratio of useful work output to the total energy input. In the context of OTEC, the term specifically refers to how effectively the system can convert the temperature difference between warm surface water and cold deep water into electrical energy. Currently, the thermal efficiency of existing OTEC plants is about 2 percent. This means that only 2 percent of the thermal energy available is being converted into electrical power. Given the significant temperature gradient between the warm surface ocean water and the colder deep ocean water, this relatively low efficiency suggests there is substantial room for improvement in OTEC technology.

Carnot efficiency

To understand the potential upper limit to the efficiency of an OTEC system, we use the concept of Carnot efficiency, which represents the maximum possible efficiency of any heat engine operating between two temperatures. This theoretical maximum is derived from the Carnot cycle, which is an idealized thermodynamic cycle proposed by French physicist Sadi Carnot. The formula for Carnot efficiency is given by:\[ \text{Carnot efficiency}, \, \theta_{\text{max}} = 1 - \frac{T_c}{T_h} \]where \(T_h\) is the high temperature (in Kelvin) and \(T_c\) is the low temperature (in Kelvin). To calculate the maximum thermal efficiency for the OTEC system, we convert the given temperatures from Celsius to Kelvin. For example, in the given problem, the surface temperature of the ocean near Florida is 27°C and the temperature at a depth of 700 meters is 7°C. Converting these to Kelvin, we have:\[T_h = 27 + 273.15 = 300.15 \, K \]\[T_c = 7 + 273.15 = 280.15 \, K \]Substituting these values into the Carnot efficiency formula, we get:\[ \theta_{\text{max}} = 1 - \frac{280.15}{300.15} \ \theta_{\text{max}} \ \theta_{\text{max}} = 0.0666 \ \] or 6.66%. This means that even in an ideal scenario, the maximum efficiency of the OTEC system would be about 6.66%. Current OTEC plants operating at around 2% efficiency are therefore operating at roughly 30% of their theoretical maximum efficiency.

temperature conversion

One crucial aspect of calculating and understanding efficiencies, particularly thermal efficiencies, involves converting temperatures from one unit to another. Temperature in thermodynamic equations is generally represented in Kelvin, which is the absolute temperature scale. This is important because thermodynamic efficiency calculations, such as the Carnot efficiency, require the use of the absolute temperature scale to avoid negative values and ensure proper calculation. For instance, to convert a temperature from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. For example:

• The surface temperature of 27°C converts to Kelvin as: 27 + 273.15 = 300.15 K.
• The temperature at 700 meters depth of 7°C converts to Kelvin as: 7 + 273.15 = 280.15 K.

Proper temperature conversion ensures accurate and meaningful efficiency calculations. Understanding and practicing these conversions is critical for students studying thermodynamics and energy systems.