Question

It can be shown that the thermal efficiency of a combined gas-steam power plant \(\eta_{\mathrm{cc}}\) can be expressed in terms of the thermal efficiencies of the gas- and the steamturbine cycles as $$\eta_{\mathrm{cc}}=\eta_{g}+\eta_{s}-\eta_{8} \eta_{\mathrm{s}}$$ Prove that the value of \(\eta_{\mathrm{cc}}\) is greater than either of \(\eta_{g}\) or \(\eta_{s}\) That is, the combined cycle is more efficient than either of the gas-turbine or steam-turbine cycles alone.

Short answer

Short Answer: The thermal efficiency of a combined gas-steam power plant, represented by η_cc, is greater than either of the individual cycle efficiencies (η_g and η_s) as demonstrated through the derived inequalities η_s(1 - η_g) > 0 and η_g(1 - η_s) > 0. This shows that the combined cycle is more efficient than either the gas-turbine cycle or steam-turbine cycle alone.

Step-by-step solution

Write down the inequalities

We want to show that the combined cycle efficiency is greater than the individual cycle efficiencies. We need to show that both of the following inequalities are true: 1. \(\eta_{\mathrm{cc}} > \eta_{g}\) 2. \(\eta_{\mathrm{cc}} > \eta_{s}\)

Rewrite the inequalities in terms of the given equation

We want to manipulate the given equation, which is: $$\eta_{\mathrm{cc}} = \eta_{g} + \eta_{s} - \eta_{g} \eta_{s}$$ For the first inequality: 1. Subtract \(\eta_{g}\) from both sides: $$\eta_{\mathrm{cc}} - \eta_{g} > 0$$ For the second inequality: 2. Subtract \(\eta_{s}\) from both sides: $$\eta_{\mathrm{cc}} - \eta_{s} > 0$$

Substitute the given equation for \(\eta_{\mathrm{cc}}\) back into both inequalities and simplify

For the first inequality, substitute \(\eta_{\mathrm{cc}}\) back in using the given equation and simplify: $$ \begin{aligned} (\eta_{g} + \eta_{s} - \eta_{g} \eta_{s}) - \eta_{g} &> 0 \\ \eta_{s} - \eta_{g} \eta_{s} &> 0 \end{aligned} $$ Factor out \(\eta_{s}\): $$ \eta_{s}(1 - \eta_{g}) > 0 $$ For the second inequality, substitute \(\eta_{\mathrm{cc}}\) back in using the given equation and simplify: $$ \begin{aligned} (\eta_{g} + \eta_{s} - \eta_{g} \eta_{s}) - \eta_{s} &> 0 \\ \eta_{g} - \eta_{g} \eta_{s} &> 0 \end{aligned} $$ Factor out \(\eta_{g}\): $$\eta_{g}(1 - \eta_{s}) > 0 $$

Analyze the inequalities

Now we have: $$ \eta_{s}(1 - \eta_{g}) > 0 $$ and $$ \eta_{g}(1 - \eta_{s}) > 0 $$ We know that the efficiencies of the gas-turbine cycle and steam-turbine cycle are both non-negative and less than or equal to 1 (\(0 \leq \eta_{g}, \eta_{s} \leq 1\)). Therefore, \((1-\eta_{g})\) and \((1-\eta_{s})\) are also non-negative and less than or equal to 1. Since \(\eta_{s}\) and \((1 - \eta_{g})\) are non-negative, their product is also non-negative: $$ \eta_{s}(1 - \eta_{g}) \geq 0 $$ Similarly, since \(\eta_{g}\) and \((1 - \eta_{s})\) are non-negative, their product is also non-negative: $$ \eta_{g}(1 - \eta_{s}) \geq 0 $$ Since we are given that the cycle efficiencies are less than 1, this implies that $$ \eta_{s}(1 - \eta_{g}) > 0 $$ and $$ \eta_{g}(1 - \eta_{s}) > 0 $$ This means that: $$\eta_{\mathrm{cc}} > \eta_{g}$$ and $$\eta_{\mathrm{cc}} > \eta_{s}$$ Hence, we have proved that the thermal efficiency of the combined gas-steam power plant, \(\eta_{\mathrm{cc}}\), is greater than either of the individual cycle efficiencies, \(\eta_{g}\) and \(\eta_{s}\). This demonstrates that the combined cycle is more efficient than either of the individual cycles (gas-turbine or steam-turbine) alone.

Key concepts

Thermal Efficiency

Thermal efficiency is a measure of the performance of a power plant. It is defined as the ratio of the work output to the heat input, often expressed as a percentage. In simpler terms, it indicates how well a power plant converts the heat from fuel into useful work. The higher the thermal efficiency, the better the plant is at converting heat into work while minimizing energy losses.

When we consider a combined cycle power plant, its thermal efficiency is enhanced by capturing the waste heat from one cycle to power another. In the equation provided, \(\eta_{\mathrm{cc}} = \eta_{g} + \eta_{s} - \eta_{g} \eta_{s}\), you can see that the combined cycle efficiency \(\eta_{\mathrm{cc}}\) improves due to the interplay between the gas turbine and the steam turbine efficiencies (\(\eta_{g}\) and \(\eta_{s}\)).

Understanding this concept is crucial for realizing why combined cycle power plants are more efficient than single cycle plants. They leverage the remaining thermal energy from one cycle to boost the other, thus improving overall system efficiency.

Gas-Turbine Cycle

The gas-turbine cycle, also known as the Brayton cycle, is fundamental to modern power generation and aviation. This cycle involves compressing air, mixing it with fuel to produce a high-temperature, high-pressure gas, and then expanding this gas through a turbine to produce work.

In terms of thermal efficiency, gas turbines favor high temperatures. However, inefficiencies arise due to heat loss to the surroundings and the limit to the maximum temperature that can be achieved without damaging the turbine.

Importance in Combined Cycles

Despite the limitations in temperature and efficiency when used separately, gas turbines are essential for combined cycles. The waste heat from a gas-turbine cycle can be utilized to run a steam-turbine cycle, significantly increasing the efficiency of the overall system. Moreover, the quick start-up and variability of the gas-turbine cycle provide flexibility in power generation, which is incredibly valuable for meeting fluctuating energy demands.

Steam-Turbine Cycle

The steam-turbine cycle, also known as the Rankine cycle, is a classic power generation process and an integral part of thermal power plants. This cycle converts heat into mechanical work by boiling water to create steam and channeling that steam through turbines.

While the steam-turbine cycle can be efficient, it's limited by the maximum temperature of the steam and the quality of the heat source. Condensation and reheating stages are used within this cycle to optimize efficiency but some energy inevitably escapes as waste heat during the process.

Integration in Combined Cycles

The uniqueness of the steam-turbine cycle in a combined cycle setup lies in its ability to harness the residual heat from the gas turbine. This synergy allows the combined cycle plant to achieve efficiencies that neither cycle could attain individually. The steam-turbine cycle's incorporation into a combined cycle power plant typifies how integrating different thermal processes can lead to significant improvements in energy utilization and power plant performance.