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Water is the working fluid in a Rankine cycle. Steam exits the steam generator at \(1500 \mathrm{lb} / / \mathrm{in}^{2}\) and \(1100^{\circ} \mathrm{F}\). Due to heat transfer and frictional effects in the line connecting the steam generator and turbine, the pressure and temperature at the turbine inlet are reduced to \(1400 \mathrm{lbf} / \mathrm{in}^{2}\) and \(1000^{\circ} \mathrm{F}\), respectively. Both the turbine and pump have isentropic efficiencies of \(85 \%\). Pressure at the condenser inlet is 2 lbf/ in. \({ }^{2}\), but due to frictional effects the condensate exits the condenser at a pressure of \(1.5\) lbflin. \({ }^{2}\) and a temperature of \(110^{\circ} \mathrm{F}\). The condensate is pumped to 1600 lbf/in. \({ }^{2}\) before entering the steam generator. The net power output of the cycle is \(1 \times 10^{9}\) Btu/h. Cooling water experiences a temperature increase from \(60^{\circ} \mathrm{F}\) to \(76^{\circ} \mathrm{F}\), with negligible pressure drop, as it passes through the condenser. Determine for the cycle (a) the mass flow rate of steam, in lh/h. (b) the rate of heat transfer, in Btu/h, to the working fluid passing through the steam generator. (c) the thermal efficiency- (d) the mass flow rate of cooling water, in \(\mathrm{lb} / \mathrm{h}\).

Step by step solution

01

- Analyze Given Data

Extract the given data from the problem statement: initial and final pressures and temperatures for various components, isentropic efficiencies, the net power output, and the increase in cooling water temperature.

02

- Determine Work and Heat Transfer for Turbine and Pump

Use the given data and isentropic efficiencies to calculate the actual work output of the turbine and the actual work input for the pump.

03

- Use Energy Balance for Condenser

Use the first law of thermodynamics to determine the heat rejected in the condenser using the temperature change of the cooling water.

04

- Calculate Net Work and Efficiency

Use the net power output and the turbine and pump work to calculate the net work of the cycle. Then, calculate the thermal efficiency from the net work and the heat added in the steam generator.

05

- Find Mass Flow Rate of Steam

Using the net power output and the turbine actual work, calculate the mass flow rate of the steam in lb/h.

06

- Determine Heat Addition Rate

Calculate the rate of heat addition to the working fluid passing through the steam generator.

07

- Calculate Mass Flow Rate of Cooling Water

Using the heat rejected in the condenser and the temperature increase of the cooling water, calculate the mass flow rate of the cooling water.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

thermodynamic cycles

A thermodynamic cycle is a series of processes that involve heat and work transfer to convert energy from one form to another. In the Rankine cycle, water is used as the working fluid. The cycle consists of four main processes:

• Isentropic compression (pumping of water)
• Isobaric heat addition (boiling of water to generate steam)
• Isentropic expansion (steam expands through a turbine to do work)
• Isobaric heat rejection (condensation of steam back to water)

By completing these processes, the Rankine cycle converts heat into work, typically used for power generation in thermal power plants.

isentropic efficiency

Isentropic efficiency measures how close a real process is to an ideal isentropic (constant entropy) process. It indicates the performance of turbines, compressors, and pumps by comparing actual work to the work done in an ideal scenario.
For a turbine, the isentropic efficiency \( \eta_{T} \) is given by: \[ \eta_{T} = \frac{h_{1} - h_{2a}}{h_{1} - h_{2s}} \] where:

• \( h_{1} \) is the enthalpy at the turbine inlet
• \( h_{2a} \) is the actual enthalpy at the turbine outlet
• \( h_{2s} \) is the isentropic enthalpy at the turbine outlet

Pumps also have isentropic efficiency, calculated similarly, ensuring the amount of work supplied matches closer to ideal conditions.

mass flow rate calculation

The mass flow rate is the amount of mass passing through a point per unit of time. In the context of the Rankine cycle, the mass flow rate of steam (\dot{m_{s}} ) is crucial to understand how much steam is flowing through each component per hour. To calculate it, we use the net power output and the actual work done by the turbine:
\[ \dot{m_{s}} = \frac{ \text{Net Power Output} }{ \text{Turbine Work per unit mass} } \] Similarly, the mass flow rate of cooling water (\dot{m_c}) in the condenser can be found by using the heat rejected and the temperature rise:
\[ \dot{m_c} = \frac{ Q_{out} }{ c_{pw} ( T_{cw_{out}} - T_{cw_{in}} ) } \] where:

• \(c_{pw}\) is the specific heat capacity of water
• \(T_{cw_{out}}\) is the temperature of cooling water leaving the condenser
• \(T_{cw_{in}}\) is the temperature of cooling water entering the condenser

thermal efficiency

Thermal efficiency of a thermodynamic cycle measures how effectively the cycle converts heat input into useful work. For the Rankine cycle, it is expressed as:
\[ \eta_{th} = \frac{ W_{net} }{ Q_{in} } \] where:

• \(W_{net}\) is the net work output for the cycle
• \(Q_{in}\) is the heat input in the steam generator

Efficiency is typically presented as a percentage, so multiplying the result by 100 helps indicate the performance of the cycle.
Higher thermal efficiency implies that more of the input heat is converted to work, indicating a more effective power generation process. Factors such as turbine and pump efficiencies affect the overall thermal efficiency of the Rankine cycle.