Question

The net work output and the thermal efficiency for the Carnot and the simple ideal Rankine cycles with steam as the working fluid are to be calculated and compared. Steam enters the turbine in both cases at \(5 \mathrm{MPa}\) as a saturated vapor, and the condenser pressure is 50 kPa. In the Rankine cycle, the condenser exit state is saturated liquid and in the Carnot cycle, the boiler inlet state is saturated liquid. Draw the \(T\) -s diagrams for both cycles.

Short answer

Question: Compare the net work output and thermal efficiency of the Carnot and Rankine cycles when steam is the working fluid, given the steam entry pressure, condenser pressure, and the states of the cycle.

Step-by-step solution

1. Draw the T-s diagrams for the Carnot and Rankine cycles

First, we need to identify the four processes in each cycle and draw the T-s diagrams. The Carnot cycle consists of isoentropic expansion (process 1-2), isothermal heat rejection (process 2-3), isoentropic compression (process 3-4), and isothermal heat addition (process 4-1). In contrast, the Rankine cycle consists of isentropic expansion (process 1-2'), isobaric heat rejection (process 2'-3'), isentropic compression (process 3'-4'), and isobaric heat addition (process 4'-1).

2. Calculate the heat and work quantities for each cycle

For both the Carnot and Rankine cycles, we can use the properties of steam and the first law of thermodynamics to find the heat and work involved in each process: - Isoentropic expansion: \(W_{turbine} = h_1 - h_2\) (or \(h_{2'}\) for Rankine) - Isoenthalpic/ isobaric heat rejection: \(Q_{out} = h_2 - h_3\) (or \(h_{2'} - h_{3'}\) for Rankine) - Isoentropic compression: \(W_{pump} = h_4 - h_3\) (or \(h_{4'} - h_{3'}\) for Rankine) - Isoenthalpic/ isobaric heat addition: \(Q_{in} = h_1 - h_4\) (or \(h_{1} - h_{4'}\) for Rankine)

3. Calculate the net work output and the thermal efficiency for both cycles

Using the calculated heat and work quantities, we can find the net work output and thermal efficiency for both cycles. - Net work output: \(W_{net} = W_{turbine} - W_{pump}\) - Thermal efficiency: \(\eta = \frac{W_{net}}{Q_{in}}\)

4. Compare the results

After computing the net work output and thermal efficiency for both the Carnot and Rankine cycles, compare the results to see which cycle has better performance.

Key concepts

Thermal Efficiency Calculation

Thermal efficiency is a crucial measure of a heat engine's performance, reflecting how well it converts heat into work. It is calculated as the ratio of net work output to the heat input. Mathematically, it is represented as \[ \eta = \frac{W_{net}}{Q_{in}} \.\] This formula is derived from the first law of thermodynamics, which states energy cannot be created or destroyed, only transformed.

In practical terms, for steam cycles like the Carnot and Rankine cycles, \(W_{net}\) is the difference between the work produced during the expansion process in the turbine and the work consumed during compression in the pump. Heat input \(Q_{in}\) refers to the heat added to the system, typically in a boiler. A higher thermal efficiency indicates that more of the heat input is being converted to useful work, which is the goal for energy conservation and cost efficiency.

T-s Diagrams

Understanding the Temperature-Entropy (T-s) diagrams is fundamental to analyzing thermodynamic cycles. These diagrams graphically present the changes in temperature and entropy as a system undergoes various processes. For the Carnot and Rankine cycles, T-s diagrams exhibit crucial characteristics such as isentropic processes where entropy remains constant, and isothermal processes where temperature is unchanged.

In these diagrams, the vertical axis represents temperature and the horizontal axis represents entropy. The area under the curve of a process on a T-s diagram represents the heat transfer during that process. For example, in both Carnot and Rankine cycles, the expansion and compression processes are typically plotted as vertical lines because there is no heat transfer involved (isentropic). The heat addition and rejection processes appear as horizontal lines for the Carnot cycle (isothermal), whereas they follow the saturation curve for the Rankine cycle (isobaric).

First Law of Thermodynamics

The first law of thermodynamics, also known as the principle of the conservation of energy, underpins all thermal efficiency calculations. It states that the energy of an isolated system is constant. Energy can be transferred to or from the system in the form of work or heat, but never created or destroyed.

In the context of the Carnot and Rankine cycles, the first law can be used to calculate the work and heat transfer at each stage of the cycle. For instance, in the isentropic expansion stage for steam turbines, the work done by the turbine is the difference in enthalpy between the initial and final states. Similarly, for the compression stage in pumps, the work input can also be found through the difference in enthalpy. For both cycles, the first law ensures energy balance is maintained and is essential to finding the system's thermal efficiency.

Steam as Working Fluid

Steam is the working fluid of choice in many thermodynamic cycles due to its excellent energy-carrying capacity and the convenience of phase changes at manageable temperatures and pressures. When steam expands within a turbine, it performs work that can be converted into electricity or used for mechanical drives.

In both the Carnot and Rankine cycles, steam undergoes phase changes and transfers its latent heat during condensation and evaporation. Utilizing steam allows these cycles to exploit its thermodynamic properties, including high latent heat of vaporization, meaning a large amount of energy can be transferred with smaller mass flow rates compared to other fluids. This efficiency in carrying energy makes steam an optimal medium in many industrial and power generation applications.