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Chapter 8: Problem 93

Steam enters the turbine of a simple vapor power plant at 100 bar, \(520^{\circ} \mathrm{C}\) and expands adiabatically, exiting at \(0.08\) bar with a quality of \(90 \%\). Condensate leaves the condenser as saturated liquid at \(0.08\) bar. Liquid exits the pump at 100 bar, \(43^{\circ} \mathrm{C}\). The specific exergy of the fuel entering the combustor unit of the steam generator is estimated to be \(14,700 \mathrm{~kJ} / \mathrm{kg}\). No exergy is carried in by the combustion air. The exergy of the stack gases leaving the steam generator is estimated to be \(150 \mathrm{~kJ}\) per kg of fuel. The mass flow rate of the steam is \(3.92 \mathrm{~kg}\) per \(\mathrm{kg}\) of fuel. Cooling water enters the condenser at \(T_{0}=20^{\circ} \mathrm{C}, p_{0}=1 \mathrm{~atm}\) and exits at \(35^{\circ} \mathrm{C}, 1 \mathrm{~atm}\). Develop a full accounting of the exergy entering the plant with the fuel.

Short Answer

Expert verified
14,550 kJ/kg

Step by step solution

01

Identify the exergy of the fuel

The specific exergy of the fuel entering the combustor unit is provided: \[ e_{fuel} = 14,700 \text{ kJ/kg} \]
02

Calculate the exergy of the stack gases

The exergy of the stack gases leaving the steam generator is given as: \[ e_{stack\text{ }gases} = 150 \text{ kJ/kg} \]
03

Determine the fuel mass flow rate per kg of fuel

The mass flow rate of the steam per kg of fuel is provided: \[ \text{mass flow rate of steam} = 3.92 \text{ kg/kg of fuel} \]
04

Calculate the total exergy entering the plant with the fuel

The total exergy entering the plant with the fuel is calculated by subtracting the exergy of the stack gases from the exergy of the fuel: \[ \text{Exergy entering the plant} = e_{fuel} - e_{stack\text{ }gases} = 14,700 - 150 = 14,550 \text{ kJ/kg} \]

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Key Concepts

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

Exergy Calculation
Exergy represents the maximum useful work that can be extracted from a system as it reaches equilibrium with its surroundings. Understanding exergy is essential for assessing the efficiency and sustainability of energy systems.

In the context of the vapor power plant problem, we start by determining the specific exergy of the fuel. The given specific exergy value is 14,700 kJ/kg. This initial step sets the foundation for subsequent exergy calculations.

Next, we consider the exergy of the stack gases, which is 150 kJ/kg. Stack gases are the exhaust gases emitted from the combustion process. Though their exergy is lower, it still needs to be accounted for.

We then calculate the total exergy entering the plant with the fuel. This is done by subtracting the exergy of the stack gases from the specific exergy of the fuel:
\[\text{Exergy entering the plant} = e_{fuel} - e_{stack\text{ }gases} = 14,700 - 150 = 14,550 \text{ kJ/kg} \]

By understanding these values, we can better evaluate the potential work output and efficiency of the vapor power plant.
Adiabatic Expansion
Adiabatic expansion refers to a process in which a gas or vapor expands without exchanging heat with its surroundings. In our vapor power plant, the steam expands adiabatically in the turbine.

During adiabatic expansion, the internal energy of the steam decreases, leading to a drop in temperature and pressure. This results in the generation of mechanical work that drives the turbine.

We are given that the steam enters the turbine at 100 bar and 520掳C and exits at 0.08 bar with a quality of 90%. The term 'quality' refers to the ratio of the mass of vapor to the total mass of the mixture, indicating that 90% of the steam is vapor and 10% is liquid at the exit.

The adiabatic nature of the process ensures that no heat is lost to the environment, maximizing the conversion of internal energy into work. Understanding adiabatic expansion is crucial for optimizing the efficiency of turbines in vapor power plants.
Mass Flow Rate
Mass flow rate is an important parameter in analyzing the performance of vapor power plants. It refers to the amount of mass flowing through a given surface per unit time.

In this problem, the mass flow rate of the steam per kg of fuel is provided as 3.92 kg/kg of fuel. This value helps us understand the scale of the process and determine the total amount of steam generated relative to the fuel consumed.

Accurately calculating the mass flow rate allows us to ensure the proper functioning and efficiency of the power plant. It factors into various other calculations, such as determining the total exergy entering the plant and evaluating the energy conversion efficiency.

By understanding the concept of mass flow rate, we can better predict and control the operational parameters of vapor power plants, leading to more efficient and sustainable energy production.

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Most popular questions from this chapter

A power plant operates on a regenerative vapor power cycle with one open feedwater heater. Steam enters the first turbine stage at \(12 \mathrm{MPa}, 560^{\circ} \mathrm{C}\) and expands to \(1 \mathrm{MPa}\), where some of the steam is extracted and diverted to the open feedwater heater operating at \(1 \mathrm{MPa}\). The remaining steam expands through the second turbine stage to the condenser pressure of \(6 \mathrm{kPa}\). Saturated liquid exits the open feedwater heater at \(1 \mathrm{MPa}\). The net power output for the cycle is \(330 \mathrm{MW}\). For isentropic processes in the turbines and pumps, determine (a) the cycle thermal efficiency. (b) the mass flow rate into the first turbine stage, in \(\mathrm{kg} / \mathrm{s}\). (c) the rate of entropy production in the open feedwater heater, in \(k W / K\).

Water is the working fluid in an ideal regenerative Rankine cycle with one closed feedwater heater. Superheated vapor enters the turbine at \(16 \mathrm{MPa}, 560^{\circ} \mathrm{C}\), and the condenser pressure is \(8 \mathrm{kPa}\). The cycle has a closed feedwater heater using extracted steam at \(1 \mathrm{MPa}\). Condensate drains from the feedwater heater as saturated liquid at \(1 \mathrm{MPa}\) and is trapped into the condenser. The feedwater leaves the heater at \(16 \mathrm{MPa}\) and a temperature equal to the saturation temperature at \(1 \mathrm{MPa}\). The mass flow rate of steam entering the first-stage turbine is \(120 \mathrm{~kg} / \mathrm{s}\). Determine (a) the net power developed, in \(\mathrm{kW}\). (b) the rate of heat transfer to the steam passing through the boiler, in \(\mathrm{kW}\). (c) the thermal efficiency. (d) the mass flow rate of condenser cooling water, in \(\mathrm{kg} / \mathrm{s}\), if the cooling water undergoes a temperature increase of \(18^{\circ} \mathrm{C}\) with negligible pressure change in passing through the condenser.

Water is the working fluid in a regenerative Rankine cycle with one closed feedwater heater and one open feedwater heater. Steam enters the turbine at \(1400 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}\) and \(1000^{\circ} \mathrm{F}\) and expands to \(500 \mathrm{lbf}^{\prime} \mathrm{in}^{2}\), where some of the steam is extracted and diverted to the closed feedwater heater. Condensate exiting the closed feedwater heater as saturated liquid at \(500 \mathrm{lbf} /\) in. \({ }^{2}\) undergoes a throttling process to \(120 \mathrm{lbf} /\) in. \({ }^{2}\) as it passes through a trap into the open feedwater heater. The feedwater leaves the closed feedwater heater at \(1400 \mathrm{lbf} \mathrm{bin}^{2}\) and a temperature equal to the saturation temperature at \(500 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}\) The remaining steam expands through the second-stage turbine to \(120 \mathrm{lbf} / \mathrm{in}^{2}\). where some of the steam is extracted and diverted to the open feedwater heater operating at \(120 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}\) Saturated liquid exits the open feedwater heater at \(120 \mathrm{lbf} / \mathrm{in}^{2}{ }^{2}\) The remaining steam expands through the third-stage turbine to the condenser pressure of \(2 \mathrm{lbf} / \mathrm{in}^{2}\) The turbine stages and the pumps each operate adiabatically with isentropic efficiencies of \(85 \%\). Flow through the condenser, closed feedwater heater, open feedwater heater, and steam generator is at constant pressure. The net power output of the cycle is \(1 \times 10^{9} \mathrm{~B} t \mathrm{~h} / \mathrm{h}\). Determine for the cycle (a) the mass flow rate of steam entering the first stage of the turbine, in lb/h. (b) the rate of heat transfer, in Btu/h, to the working fluid passing through the steam generator. (c) the thermal efficiency.

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 \(\mathrm{lb} / \mathrm{h}\).

Superheated steam at \(8 \mathrm{MPa}\) and \(480^{\circ} \mathrm{C}\) leaves the steam generator of a vapor power plant. Heat transfer and frictional effects in the line connecting the steam generator and the turbine reduce the pressure and temperature at the turbine inlet to \(7.6 \mathrm{MPa}\) and \(440^{\circ} \mathrm{C}\), respectively. The pressure at the exit of the turbine is \(10 \mathrm{kPa}\), and the turbine operates adiabatically. Liquid leaves the condenser at \(8 \mathrm{kPa}, 36^{\circ} \mathrm{C}\). The pressure is increased to \(8.6\) MPa across the pump. The turbine and pump isentropic efficiencies are \(88 \%\). The mass flow rate of steam is \(79.53 \mathrm{~kg} / \mathrm{s}\). Determine (a) the net power output, in \(\mathrm{kW}\). (b) the thermal efficiency. (c) the rate of heat transfer from the line connecting the steam generator and the turbine, in \(\mathrm{kW}\). (d) the mass flow rate of condenser cooling water, in \(\mathrm{kg} / \mathrm{s}\), if the cooling water enters at \(15^{\circ} \mathrm{C}\) and exits at \(35^{\circ} \mathrm{C}\) with negligible pressure change.
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