m = 2 kg/s
p1 = 40 bar T1 = 400°C V1 = 10 m/s
p2 = 15 bar V2 = 665 m/s
1
1
2
2 Insulation
Control volume boundary
T1 = 400°C
p = 15 bar p = 40 bar
v T
Fig. E4.3
➊
c c c c EXAMPLE 4.3 c
4.7 Turbines 159 From Table A-4, h1 5 3213.6 kJ/kg. The velocities V1 and V2 are given. Inserting values and converting the units of the kinetic energy terms to kJ/kg results in
h253213.6 kJ/kg1 c(10)22(665)2 2 d am2
s2b ` 1 N
1 kg?m/s2` ` 1 kJ 103 N?m` 53213.62221.152992.5 kJ/kg
Finally, referring to Table A-4 at p2 5 15 bar with h2 5 2992.5 kJ/kg, the specific volume at the exit is 𝜐2 5 0.1627 m3/kg. The exit area is then
A25 (2 kg/s)(0.1627 m3/kg)
665 m/s 54.8931024 m2
➊ Although equilibrium property relations apply at the inlet and exit of the control volume, the intervening states of the steam are not necessarily equi- librium states. Accordingly, the expansion through the nozzle is represented on the T–𝜐 diagram as a dashed line.
➋ Care must be taken in converting the units for specific kinetic energy to kJ/kg.
Ability to…
❑ apply the steady-state energy rate balance to a control volume.
❑ apply the mass flow rate expression, Eq. 4.4b.
❑ develop an engineering model.
❑ retrieve property data for water.
✓Skills Developed
Evaluate the nozzle inlet area, in m2. Ans. 1.47 3 1022 m2.
4.7 Turbines
A turbine is a device in which power is developed as a result of a gas or liquid pass- ing through a set of blades attached to a shaft free to rotate. A schematic of an axial- flow steam or gas turbine is shown in Fig. 4.9. Such turbines are widely used for power generation in vapor power plants, gas turbine power plants, and aircraft engines (see Chaps. 8 and 9). In these applications, superheated steam or a gas enters the turbine and expands to a lower pressure as power is generated.
A hydraulic turbine coupled to a generator installed in a dam is shown in Fig. 4.10. As water flows from higher to lower elevation through the turbine, the turbine provides shaft power to the generator. The generator converts shaft power to electricity. This type of generation is called hydro- power. Today, hydropower is a leading renewable means for producing electricity, and it is one of the least expensive ways to do so. Electricity can also be produced from flowing water by using turbines to tap into currents in oceans and rivers.
Turbines are also key components of wind-turbine power plants that, like hydropower plants, are renewable means for generating electricity.
turbine
Stationary blades Rotating blades
Fig. 4.9 Schematic of an axial-flow steam or gas turbine.
➋
160 Chapter 4 Control Volume Analysis Using Energy
Reservoir
Dam
Generator Intake
Hydraulic turbine
Fig. 4.10 Hydraulic turbine installed in a dam.
ENERGY & ENVIRONMENT Industrial-scale wind turbines can stand as tall as a 30-story building and produce electricity at a rate meeting the needs of hundreds of typical U.S. homes. The three-bladed rotors of these wind turbines have a diameter nearly the length of a football field and can operate in winds up to 90 km per hour. They feature microprocessor control of all functions, ensuring each blade is pitched at the correct angle for current wind conditions. Wind farms consisting of dozens of such turbines increasingly dot the landscape over the globe.
In the United States, wind farms at favorable sites in several Great Plains states alone could supply much of the nation’s electricity needs—provided the electrical grid is upgraded and expanded (see Horizons on p. 372). Offshore wind farms along the U.S. coastline could also contribute sig- nificantly to meeting national needs. Experts say wind variability can be managed by producing maximum power when winds are strong and storing some, or all, of the power by various means, including pumped-hydro storage and compressed-air storage, for distribution when consumer demand is highest and electricity has its greatest economic value (see box in Sec. 4.8.3).
Wind power can produce electricity today at costs competitive with all alternative means and within a few years is expected to be among the least costly ways to do it. Wind energy plants take less time to build than conventional power plants and are modular, allowing additional units to be added as warranted. While generating electricity, wind-turbine plants produce no global warming gases or other emissions.
The industrial-scale wind turbines considered thus far are not the only ones available. Com- panies manufacture smaller, relatively inexpensive wind turbines that can generate electricity with wind speeds as low as 3 or 4 miles per hour. These low-wind turbines are suitable for small businesses, farms, groups of neighbors, or individual users.
4.7 Turbines 161
4.7.1 Steam and Gas Turbine Modeling Considerations
With a proper selection of the control volume enclosing a steam or gas turbine, the net kinetic energy of the matter flowing across the boundary is usually small enough to be neglected. The net potential energy of the flowing matter also is typically neg- ligible. Thus, the underlined terms of Eq. 4.20a (repeated below) drop out, leaving the power, enthalpy, and heat transfer terms, as shown by Eq. (a)
05Q#
cv2W#
cv1m# c(h12h2)1 (V122V22)
2 1g(z12z2)d 05Q#
cv2W#
cv1m#(h12h2) (a)
where m# is the mass flow rate. The only heat transfer between the turbine and surround- ings normally would be unavoidable (or stray) heat transfer, and this is often small enough relative to the power and enthalpy terms that it also can be neglected, giving simply
W#
cv5m#(h12h2) (b)
4.7.2 Application to a Steam Turbine
In this section, modeling considerations for turbines are illustrated by application to a case involving the practically important steam turbine. Objectives in this example include assessing the significance of the heat transfer and kinetic energy terms of the energy balance and illustrating the appropriate use of unit conversion factors.
Calculating Heat Transfer from a Steam Turbine
Steam enters a turbine operating at steady state with a mass flow rate of 4600 kg/h. The turbine develops a power output of 1000 kW. At the inlet, the pressure is 60 bar, the temperature is 4008C, and the velocity is 10 m/s. At the exit, the pressure is 0.1 bar, the quality is 0.9 (90%), and the velocity is 30 m/s. Calculate the rate of heat transfer between the turbine and surroundings, in kW.
SOLUTION
Known: A steam turbine operates at steady state. The mass flow rate, power output, and states of the steam at the inlet and exit are known.
Find: Calculate the rate of heat transfer.
Schematic and Given Data:
Engineering Model:
1. The control volume shown on the accompanying figure is at steady state.