Question

Steam at $32\mathrm{MPa},{520}^{\circ }\mathrm{C}$ enters the first stage of a supercritical reheat cycle including three turbine stages. Steam exiting the first-stage turbine at pressure $p$ is reheated at constant pressure to ${440}^{\circ }\mathrm{C}$, and steam exiting the second-stage turbine at $0.5\mathrm{MPa}$ is reheated at constant pressure to ${360}^{\circ }\mathrm{C}$. Each turbine stage and the pump has an isentropic efficiency of $85\mathrm{\%}$. The condenser pressure is $8\mathrm{kPa}$. (a) For $p=4\mathrm{MPa}$, determine the net work per unit mass of steam flowing, in $\mathrm{kJ}/\mathrm{kg}$, and the thermal efficiency. (b) Plot the quantities of part (a) versus $p$ ranging from $0.5$ to $10\mathrm{MPa}$.

Short answer

Using given data and steam tables, calculate enthalpies at each stage, determine isentropic efficiencies, and compute the net work and thermal efficiency for each reheat pressure.

Step-by-step solution

- Identify the given data

The given values are:$p_1=32\text{ MPa}$, $T_1={520}^{\circ }\text{C}$, $T_{reheat1}={440}^{\circ }\text{C}$, $T_{reheat2}={360}^{\circ }\text{C}$, $p_2=0.5\text{ MPa}$, $p_{cond}=8\text{kPa}$, ${\eta }_s=0.85$. The reheat pressures are given as constants and there are intermediate pressure values to determine.

- Determine the properties of steam at each state

Using steam tables or software, determine the enthalpy and entropy for each stage:$T_1$ at $p_1=32\text{ MPa}$,$T_{reheat1}$ at $p_{reheat1}=4\text{ MPa}$,$T_{reheat2}$ at $p_{reheat2}=0.5\text{ MPa}$,$T_{condenser}$ at $p_{cond}=8\text{kPa}$.

- Calculate the isentropic enthalpies at each expansion

For each turbine stage, calculate the isentropic enthalpy drop:For turbine 1: using $h_1$ and entropy $s_1=s_{2}s$, find the isentropic enthalpy $h_{2s}$ at $p=4\text{ MPa}$.Apply isentropic efficiency:$h_2=h_1-{\eta }_s(h_1-h_{2s})$. Repeat this for the subsequent stages.

- Calculate heat added during reheating

Calculate the heat added during each reheating process using: $q_{reheat1}=h_3-h_2$ and $q_{reheat2}=h_5-h_4$.

- Calculate net work output

The total work produced by the turbines is:$(W_t=(h_1-h_2)+(h_3-h_4)+(h_5-h_6))$.The net work output per unit mass of steam $(W_{net})$ is:$W_{net}=W_t-W_p$ where $W_p$ is the pump work.

- Calculate thermal efficiency

The thermal efficiency is given by:$\eta =\frac{W_{net}}{q_{in}}$. Calculate the total heat addition $q_{in}$, sum of initial heat addition and reheats.

- Vary p from 0.5 to 10 MPa and repeat

Plot the net work and thermal efficiency versus the varying pressure range.

Key concepts

isentropic efficiency

Isentropic efficiency is a measure of how close a real-world process approaches the ideal, isentropic process. In the context of a thermodynamic cycle, it's essential for assessing the performance of turbines and pumps.

Isentropic efficiency (${\eta }_s$) for turbines and pumps can be defined as: For turbines: $${\eta }_s=\frac{h_1-h_2}{h_1-h_{2s}}$$ For pumps: $${\eta }_s=\frac{h_{2s}-h_1}{h_2-h_1}$$where $h_1$ and $h_2$ are the actual enthalpies before and after the process and $h_{2s}$ is the enthalpy after isentropic expansion/compression.

This efficiency impacts the overall cycle efficiency and the total work output. A higher isentropic efficiency indicates a process that is closer to being perfectly reversible.

steam reheating

Steam reheating is a method used to increase the thermal efficiency of a steam cycle. When steam is expanded through a turbine, it cools down and its pressure drops. Rather than allowing the steam to continue expanding and losing more energy, it is extracted partway through the expansion process and reheated at a constant pressure.

This reheated steam can then return to a turbine for further expansion. Reheating steam:

• Increases the average temperature at which heat is added, enhancing thermal efficiency.
• Improves the quality of steam at the turbine exit, reducing moisture content.
• Raises the cycle's total output work, as seen in equations like $q_{reheat1}=h_3-h_2$ and $q_{reheat2}=h_5-h_4$.
  
  The process typically involves several reheats, like in the given problem where the steam is reheated twice.

thermal efficiency

Thermal efficiency is a key performance metric for thermodynamic cycles. It represents the ratio of net work output to the total heat input.

The thermal efficiency ($\eta$) is calculated using:
$$\eta =\frac{W_{net}}{q_{in}}$$Where:

• $\eta$ is the thermal efficiency.
• $W_{net}$ is the net work per unit mass of steam.
• $q_{in}$ is the total heat input.

In a supercritical reheat cycle, heat input occurs at different points: during the initial heating, and through reheating in intermediate stages.

Increasing the thermal efficiency means improving the cycle's ability to convert heat into useful work or power. Various strategies, such as increasing the reheat temperature or improving component efficiencies, can enhance thermal efficiency.

enthalpy calculation

Enthalpy is a fundamental property used to quantify energy changes within thermodynamic cycles.

In the context of the given problem, enthalpy calculations are essential for determining energy transformations at each stage of the process. Calculating the enthalpy at different points involves using steam tables or thermodynamic software, looking up the values for specific temperatures and pressures (like at state $h_1$ with $p_1=32\text{MPa}$ and $T_1={520}^{\circ }\text{C}$).

Key enthalpy relationships include: For isentropic processes: obtaining $h_{2s}$ with known $s_1$ and pressure $p_2$.
For real processes: calculating actual enthalpy using efficiencies, like $h_2=h_1-{\eta }_s(h_1-h_{2s})$.

These enthalpy values are crucial for further work and heat transfer calculations in each cycle stage.

thermodynamic cycle analysis

Thermodynamic cycle analysis involves comprehensively understanding the behavior and performance of the cycle stages.

For a supercritical reheat cycle: Analyzing involves calculating states, properties, and performance metrics through various stages: Initial heating, followed by subsequent expansions and reheats till the condenser.

Key steps include:

• Determining state properties (enthalpy, entropy) at various points using steam tables or specialized software.
• Computing work done by turbines and pumps considering efficiencies.
• Applying energy balances to calculate heat inputs and outputs.

The overall goal of cycle analysis is to compute the net-work output and thermal efficiency, as well as illustrating performance variations by plotting quantities like net work and thermal efficiency against intermediate pressures like in the provided exercise.

This analysis helps optimize operational variables to improve cycle efficiency and output, making it indispensable for engineers and thermodynamic experts.