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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Aerospace Series (PEP)
- Sprache: Englisch
Aerospace propulsion devices embody some of the most advanced technologies, ranging from materials, fluid control, and heat transfer and combustion. In order to maximize the performance, sophisticated testing and computer simulation tools are developed and used.
Aerospace Propulsion comprehensively covers the mechanics and thermal-fluid aspects of aerospace propulsion, starting from the fundamental principles, and covering applications to gas-turbine and space propulsion (rocket) systems. It presents modern analytical methods using MATLAB and other advanced software and includes essential elements of both gas-turbine and rocket propulsion systems. Gas turbine coverage includes thermodynamic analysis, turbine components, diffusers, compressors, turbines, nozzles, compressor-turbine matching, combustors and afterburners. Rocket coverage includes chemical rockets, electrical rockets, nuclear and solar sail.
Key features:
- Both gas-turbine and rocket propulsion covered in a single volume
- Presents modern analytical methods and examples
- Combines fundamentals and applications, including space applications
- Accompanied by a website containing MATLAB examples, problem sets and solutions
Aerospace Propulsion is a comprehensive textbook for senior undergraduate graduate and aerospace propulsion courses, and is also an excellent reference for researchers and practicing engineers working in this area.
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Veröffentlichungsjahr: 2013
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This edition first published 2014
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Library of Congress Cataloging-in-Publication Data
Lee, T.-W. (Tae-Woo)Aerospace propulsion / TW Lee.
1 online resource.
Includes bibliographical references and index.
Description based on print version record and CIP data provided by publisher; resource not viewed.
ISBN 978-1-118-53465-6 (Adobe PDF) – ISBN 978-1-118-53487-8 (ePub) – ISBN978-1-118-30798-4 (cloth) 1. Airplanes–Jet propulsion. 2. Rocketry. I. Title.
TL709
629.1′1–dc23
2013027339
A catalogue record for this book is available from the British Library.
ISBN: 978-1-118-30798-4
Series Preface
There are books in the Aerospace Series that deal with propulsion systems for aircraft. They generally treat the engine and its control system as an integral part of the aircraft – as an installed system. The interactions between the propulsion system and the aircraft systems are described.
The power plant of an airborne vehicle is critical to its performance and its safe operation, so it is vital for engineers working in this field to understand the fundamentals of the propulsion system. This book provides a different viewpoint to that of the systems books: it is very much an analytical view of the power plant itself, and it should be read as a complement to the other propulsion books. The author introduces the reader to the principles of thrust and the gas turbine engine before providing a comprehensive mathematical treatment of the major components of the propulsion mechanism and the complex aerodynamic and thermodynamic processes within various engine types – both air-breathing and rocket. This is to provide a basis for developing an understanding of propulsion systems and the modeling tools that can be used to provide a comprehensive and practical knowledge for use in research and industry.
MATLAB® models are provided to reinforce the explanations, and exercises are also set for the diligent student to pursue.
The book covers gas turbine (aeronautical) systems and rocket propulsion (astronautic) systems and is hence of interest to engineers working in the fields of aircraft, missiles and space vehicles. Some novel propulsion systems are also described, that may be pertinent to emerging fields of aerospace transportation systems, setting out to meet environmental objectives.
This is a book for those engineers who wish to understand the fundamental principles of aerospace propulsion systems.
Peter Belobaba, Jonathan Cooper and Allan Seabridge
Preface
Aerospace propulsion devices embody some of the most advanced technologies, ranging from materials, fluid control and heat transfer and combustion. In order to maximize performance, sophisticated testing and computer simulation tools are developed and used. In undergraduate or introductory graduate courses in aerospace propulsion, we only cover the basic elements of fluid mechanics, thermodynamics, heat transfer and combustion science, so that either in industry or in research labs the students/engineers can address some of the modern design and development aspects.
Compressor aerodynamics, for example, is a dynamic process involving rotating blades that see different flows at different radial and axial locations. Cascade and transonic flow behavior can make the analyses more complex and interesting. In turbine flows, the gas temperature is high, and thus various material and heat transfer issues become quite important. Owing to the rotating nature of turbine and compressor fluids, intricate flow control between the axis and the blade section needs to be used, while allowing for cooling flow passage from the compressor to the turbine blades. Combustor flow is even more complex, since liquid-phase fuel needs to be sprayed, atomized, evaporated and burned in a compact volume. High heat release and requirements for downstream dilution and cooling again make the flow design quite difficult and challenging. All of these processes – spray atomization, phase change, combustion, heat transfer (convection and radiation) and mixing – occur in turbulent flows, and no computational tools can accurately reproduce real flows without lengthy modeling and calibration. Any one of the issues mentioned above, such as spray atomization, turbulent flow or combustion, is an unsolved problem in science and engineering, and this is the reason for industry and research labs developing expensive testing and computational analysis methods. This aspect makes aerospace propulsion an important part of engineering curricula, as it provides an interdisciplinary and “tough” training ground for aerospace engineers.
As noted above, owing to the multiple engineering topics involved, we only go into basic elements of aerospace propulsion. After some of the basics are covered, we try to expose the students to projects involving computational fluid dynamic (CFD) software, since this is frequently used in industry and in research labs. There are commercial CFD packages that can be readily made available to the students, using educational licenses. With online documentation and examples, students can learn to operate these codes, individually or in group projects. In addition, the gas-turbine lab at ASU allows the students to use actual testing data for performance analyses. These elements cannot be included in this book without stretching the physical and mental limits, but they are essential components in an aerospace propulsion course, to link the underlying science and engineering to practical applications.
I have included discussions of both gas-turbine and rocket propulsion, for combined or separate aerospace propulsion courses. There are some good interrelations between aeronautical (gas-turbine) and astronautical (rocket) propulsion, based on the same knowledge set. In addition, many students opt to take both aeronautical and astronautical propulsion, unless a combined course is offered, since their final career choices are made many years downstream.
Thank you for reading up to this point, and potentially beyond.
1
Introduction to Propulsion Systems
Propulsion systems include some of the most advanced technologies. The high performance requirements, at low system weight, necessitate advanced thermal-fluid design, materials and system integration. The thrust, generated through a simple-looking principle of conservation of momentum (or Newton's second law), enables many human capabilities, such as high-speed civil transport (approximately 12 hours for trans-Pacific flights), affordable personal aircraft, advanced military aircrafts (e.g. F-22 Raptor, Sukhoi), Earth orbital operations (Space Shuttle) and numerous satellites, planetary probes and possible missions. The propulsion technology can also lead to potentially destructive uses, as in cruise missiles, intercontinental ballistic missiles and many other weapons propelled at high speeds.
A typical gas-engine shown in Figure 1.1 achieves the high exit momentum through a sequence of devices that include compressor, combustor, turbine and nozzle. The ambient air is ingested in gas-turbine engines. The compressor consists of a series of rotating blades, which aerodynamically is a set of airfoils using rotary motion to generate a pressure differential as the air traverses the blade elements. The air pressure is increased in the compressor, and sent into the combustor where the fuel is injected, mixed with the air, and burned. The air energy (enthalpy) increase is now used in the turbines to convert some of the thermal energy (enthalpy) into shaft power. This shaft power is used to power the compressor, by simply having a common axis between the turbine and the compressor in turbojet engines. However, in turbofan engines, the turbine power is used to run both the compressor and the fan. The fan adds enthalpy to the air stream in the fan section. The energy available at the end of the turbine section is converted to air kinetic energy in the nozzle. The high kinetic energy of the exhaust stream also has high momentum, which is useful in generating thrust. Ramjets are a much simpler form of turbojet engines, where “ram compression” of incoming stream at supersonic speeds is sufficient to elevate the pressure of the air. Fuel then needs to be injected into this high-pressure air stream and the resulting flame stabilized in the ramjet combustor, for sustained thrust.
Figure 1.1 A typical gas-turbine engine. Copyright United Technologies Corporation.
Advances in practically all aspect of engineering, including propulsion technology, can be found in the Lockheed Martin F-22 Raptor (Figure 1.2) that entered service in 2005. New materials such as advanced alloys and composite materials are used in the Raptor airframe, aerodynamic surfaces and engine components. The power plant in the F-22 consists of Pratt-Whitney afterburning turbofans (F119-PW-100) with a high efficiency, which provide supersonic cruise speeds with long range and unmatched agility with pitch-vectoring thrust nozzles. But these technological advances came with a high price tag. Many of the new technologies were researched and developed specifically as part of the F-22 project. If all the development costs are added in, the F-22 carries a price tag of over $300 million per aircraft. Table 1.1 shows some of the main specifications of the F-22, including some of the propulsion characteristics.
Figure 1.2 F-22 Raptor, with advanced embedded technologies, including the power plant (F119-PW-100). Courtesy of US Department of Defence.
Table 1.1 F-22 specifications.
Length62.1 ft (18.9 m)Wingspan44.5 ft (13.56 m)Height16.8 ft (5.08 m)Maximum take-off weight80 000 lb (36 288 kg)Power plantTwo Pratt-Whitney F119-PW-100 pitch-vectoring turbofans with afterburnersTotal thrust70 000 lbMaximum speedHigh altitude: Mach 2.42 or 1600 mph (2570 km/h)Low altitude: Mach 1.72 or 1140 mph (1826 km/h)Ceiling65 000 ft (20 000 m)Range2000 miles (5600 km)Rate of climbN/A (classified)Thrust-to-weight ratio1.26Maximum g-load−3/+9.5The Pratt-Whitney F119-PW-100 engine is another component in the F-22 that is arguably the most advanced in aircraft technology. Each of these engines generates more thrust without the afterburner than most conventional engines with full afterburner power on, and its supersonic thrust is also about twice that of the other engines in the class. Using two of these engines to develop a total thrust of 70 000 pounds, the F-22 can travel at supersonic speeds without the afterburners for fuel-efficient high-speed cruise to the target area. This level of thrust is more than the aircraft weight, and enables the F-22 to fly vertically upward much like a rocket. The F119 is also unique in fully integrating the vector thrust nozzle into the engine/airframe combination, for a 20-degree up/down redirection of thrust for high-g turn capabilities. The thrust vectoring is designed to enhance the turn rates by up to 50% in comparison to using control surfaces alone. The F119 engine achieves all these functional characteristics with 40% fewer parts than conventional engines to furnish exceptional reliability, and maintenance and repair access. In a design method called integrated product development, inputs from assembly line workers and air force mechanics were incorporated to streamline the entire sequence of engine production, maintenance and repairs. These design innovations are expected to reduce the support equipment, labor and spare parts in demand by approximately half. Similar to the mid-fuselage airframe, the turbine stage, consisting of the disk and blades, is constructed in a single integrated metal piece for high integrity at lower weight, better performance and thermal insulation for the turbine disk cavity. The fan and compressor blade designs went through extensive permutations and modifications using computational fluid dynamic (CFD) simulations, resulting in unprecedented efficiency in both sections. Hardware cut-and-try of different designs would have cost way too much time and money. High-strength and degradation-resistant Alloy C was used in key components such as the compressors, turbines and nozzles to allow the engine to run at higher temperatures, one of the important contributing factors to the increased thrust and durability of F119 engines. The combustor – the hottest component in the engine – uses oxidation-resistant, thermally insulating cobalt coatings. A digital electronic engine control device called FADEC (FADEC is generally meant to signify ‘Full Authority Digital Engine Control’ the level of redundancy is at the discretion of the engine manufacturer) not only fine tunes the engine operating parameters to deliver the highest performance at the maximum efficiency, but also establishes responsive and precise engine operating parameters with inputs from the pilot control of the throttle and the engine/flight sensors.
As is well known, the F-22 has unique stealth capabilities, in spite of its size. In addition to the external geometry and surfaces, the jet-engine exhaust is a critical component in minimizing infrared signatures that can be detected by forward-looking infrared (FLIR) or IR sensors in heat-seeking missiles. The exhaust of the F-22 is designed to absorb the heat by using ceramic components, rather than conduct heat to the outside surface. Also, the horizontal stabilizers are placed to shield the thermal emission as much as possible.
The F-35 Lightning II Joint Strike Fighter (JSF) Program represents the effort to provide a capable, multi-mission aircraft while containing the budget. The F-35's price tag is about half that of the F-22 Raptor. The argument for wide adoption of this scaled-back aircraft is that the F-22's capabilities are best directed against opponents with similar technological capabilities, and with the changed geo-political environment the United States forces are less likely to be involved in such encounters. A unique variant of the F-35 (Figure 1.3) is the marine STOVL version, F-35B, also planned for adoption by the British Royal Navy to replace the Sea Harrier. The short take-off is facilitated by a number of auxiliary nozzles to divert the thrust. In a normal engine, the jet exhaust is pushed out of the nozzle at the rear of the engine to provide only forward thrust. In engines with thrust reversers, the fan stream is redirected to the forward direction to generate negative thrust. The same concept can be used to redirect the thrust to other directions by using auxiliary nozzles. For the F-35B, there is a lift nozzle that takes the fan exhaust and directs it vertically downward. Also, the pitch nozzle at the main nozzle can be turned to add a vertical component to the thrust. For control of the aircraft during this tricky maneuver, there are four additional nozzles. Two roll nozzles control the roll angle by sending a small fraction of the main exhaust at off-horizontal angles, while two yaw nozzles generate thrust in the forward and backward offset angles.
Figure 1.3 F-35 Joint Strike Fighter. Courtesy of US Department of Defence.
Rockets, on the other hand, carry all the working fluid (both fuel and oxidizers) on board. The main reason for carrying both the fuel and oxidizer is so that rockets can operate in an air-less environment (e.g. underwater or in outer space), but this also means zero incoming momentum. In addition, some rocket devices can be quite simple in design. Solid-propellant rockets, for example, only require the propellant and a nozzle (Figure 1.4). The documented use of rocketry dates back to 900 AD in China, where “black powder” was used as crude flame throwers (“fire lance”), grenades, siege weapons and other devices that delivered shock effects against the Mongols in the 10th century. Black powder consists of readily available ingredients – charcoal, sulfur and saltpeter (potassium nitrate), and was probably discovered by accident and perfected through trial-and-error. Combustion of black powder goes roughly as
Figure 1.4 Solid-propellant rocket engine. Courtesy of US Department of Defence.
This technology was quickly adopted by the Mongols, and spread to Europe and other parts of the world. Rockets using liquid propellants are, in comparison, relatively new technologies, having been developed in the early 1900s. At the other extreme, modern liquid-propellant rockets contain some of the most advanced technologies (Figure 1.5), due to the high operating pressure and temperatures, in addition to the use of cryogenic propellants such as liquid oxygen and liquid hydrogen. The high operating pressure requires sophisticated pumping devices, while high temperature necessitates advanced combustion control and cooling technologies.
Figure 1.5 Liquid-propellant rocket engine (space shuttle main engine). Courtesy of NASA.
A large altitude change during a rocket flight requires modified designs for each of the stages. At launch, the ambient pressure is roughly equal to the sea level atmospheric pressure, while the pressure decreases with increasing altitude. This results in larger pressure thrust; however, at higher altitudes the nozzle exit pressure becomes greater than the ambient pressure and the nozzle operates in an under-expanded mode. This operation is less than optimum, and the gas expansion continues downstream, creating diamond-shaped shock waves. Upper stages are designed with this aspect in mind, where a larger expansion ratio in the nozzle is used. The first stage of a Delta II launch vehicle, for example, has a nozzle expansion ratio of 12. The propellant is liquid oxygen and RP-1 (a kerosene-based hydrocarbon), which is burned in the combustion chamber at a mixture ratio (O/F) of 2.25 and pressure of 4800 kPa. This combination results in a specific impulse of 255 s. The next stage, on the other hand, has a nozzle expansion ratio of 65. The propellant combination of nitrogen tetroxide and Aerozine 50 (hydrazine/unsymmetrical dimethyl hydrazine) is used at a mixture ratio of 1.90 and chamber pressure of 5700 kPa (830 psia), which provides a specific impulse of 320 s. The space shuttle main engine (SSME) has an even larger nozzle expansion ratio of 77.5.
Liquid oxygen and liquid hydrogen used in the SSME generates a high combustion chamber temperature, and also produces combustion product gases with a low molecular weight. These factors are optimum for producing large exit velocity and thus thrust. For this reason, a liquid hydrogen/oxygen combination is also used in the Atlas Centaur upper stage, the Ariane-4 third stage and the Ariane-5 core stage.
In addition to the boost, rockets are used for various orbit maneuvers, such as station-keeping and attitude adjustments. Various factors can contribute to deviations from the target orbit. Gravitational forces of the sun and moon, for example, can cause the orbital inclination to change by approximately one degree per year. The velocity increment that needs to be expended to compensate for this drift is roughly 50 m/s. Other smaller factors that lead to orbit deviations are the elliptical shape of the Earth's equator and the “solar wind” which is the radiation pressure due to the sun's radiation. Attitude adjustments are performed with a relatively large number of small thrusters, since all three degrees of freedom need to be accessed in addition to start/stop maneuvers. For example, the Ford Aerospace Intelsat V satellite had an array of four 0.44 N (0.1 lbf) thrusters for roll control, ten 2.0 N (0.45 lbf) thrusters for pitch and yaw control and station-keeping, and two 22.2 N (5.0 lbf) thrusters for repositioning and reorientation.
Since the thrust required for orbit maneuvers is small, simpler rocket boosters such as solid propellant or monopropellants can be used. For example, typical satellites in geosynchronous orbits launched during the 1980s were equipped with solid-propellant boosters for apogee maneuver and monopropellant hydrazine thrusters for station-keeping and attitude control. The solid propellant consisted of HTPB (fuel/binder) and ammonium perchlorate (oxidizer). Hydrazine is a monopropellant containing both fuel and oxidizer components in its chemical structure, and only requires a catalytic grid for decomposition. An interesting combination of electric and thermal thrust is the use of electrical heat for the hydrazine monopropellant, which increases the specific thrust.
For more recent satellites, electric or electromagnetic thrusters with high specific thrust are used for low propellant mass requirements and therefore longer mission durations. Arcjets, for example, use an electric arc to superheat propellants such as hydrazine, which nearly doubles the specific impulse to over 500 s with typical thrust levels of 0.20 N. Arcjet thrusters are used on Intelsat VIII and Lockheed Martin A2100 satellites, and Iridium satellites. Another type of electric propulsion system with even higher specific impulse (2000–4000 s) is the ion thruster (Figure 1.6), using xenon as propellant, which produces a typical thrust of less than 0.1 N. Xenon is an inert monatomic gas with a high atomic weight (131 kg/kmol). Xenon atoms are ionized by high-speed electrons, and then these positively charged ions are accelerated to a speed of some 34 000 m/s in an electric field of 750 V in thousands of ion beams. The momentum of these ion beams produces a thrust in the order of 10 mN.
Figure 1.6 Ion propulsion devices. Courtesy of NASA.
A combination of electric and magnetic fields can also be used in so-called Hall thrusters. Other exotic space propulsion devices include solar sails and nuclear propulsion, still at the experimental stage (Figure 1.7).
Figure 1.7 Some novel propulsion concepts. Courtesy of NASA.
1.1 Conservation of Momentum
We can see from the above examples that all propulsion devices generate some high-speed exhaust stream, through a variety of means. Thus, we can say that the objective of propulsion devices, in general, is to obtain excess momentum (higher exit momentum than incoming) by generating high-speed exhaust jets. A simple version (a more precise description is provided in Chapter 2) of the conservation equation of momentum (Newton's second law) can be used to illustrate how this process will work in producing positive thrust.
(1.1)
Or in mathematical form,
(1.2)
M=vehicle massU=vehicle velocityF=thrust forceIn a propulsion system, the momentum of some fluid with mass Δm will go from ΔmUin at the inlet to ΔmUout at the exit in time Δt. So we can approximate the left-hand side in (Eq. 1.2) as . Here, we can factor out the Δm since we are dealing with the same fluid mass. Moreover, the fluid mass divided by the transit time, Δm /Δt, is the mass flow rate. Thus, we can rewrite (Eq. 1.2) as follows.
(1.3)
Equation (1.3) shows that the higher the exiting momentum with respect to the incoming momentum, the higher the thrust will be, which is the objective of a propulsion device. We may also note that high exiting momentum can be achieved by high exit velocity, large mass flow rate, or both.
1.2 Conservation of Energy (the First Law of Thermodynamics) and Other Thermodynamic Relationships
In this book, we mostly focus on the thermal-fluid aspect of propulsion systems, starting from thermodynamics, fluid dynamics, heat transfer and combustion (chemical reaction). Let us set down some baseline thermodynamic relationships that we will be using in this book. The most important element of thermodynamics is the first law, or the conservation of energy, which simply states that the energy contained in the control volume (Ecv) changes at the rate determined by the heat input () minus the power output () and the net energy input consisting of the enthalpy and the kinetic energy).
(1.4)
For steady state, the left-hand side is set to zero, so that in compressors, turbines, combustors, nozzles and other propulsion components the heat, power and net energy flow rates are all balanced according to (Eq. 1.4). And this equation can also be used to define the stagnation enthalpy, which is the total energy of the fluid including the enthalpy and kinetic energy.
(1.5)
Equation (1.4) states that in the absence of heat transfer and power, the stagnation enthalpy will remain the same during a flow process.
For a closed system, there is no energy flux into or out of the volume, and the first law can be written in a differential form.
(1.6)
Equation (1.6) states that the internal energy in the system changes as a function of the heat input and work output. Changes in internal energy or enthalpy can be calculated using the specific heats of the fluid.
(1.7)
(1.8)
For ideal gases, the pressure (p), density (ρ) and temperature (T) are related by the ideal gas equation of state.
(1.9)
For ideal gases, the relationship between cp, cv, and the specific gas constant R follows from the definition of enthalpy.
(1.10)
(1.11)
Using ,
(1.12)
(1.13)
For isentropic processes, involving ideal gases and constant specific heats, we have the following relationships between the state variables.
(1.14)
(1.15)
Stagnation properties are defined as the condition reached when the flow decelerates to zero speed, isentropically. Using (Eq. 1.5), with constant cp,
(1.16)
We have used (Eq. 1.12) and for ideal gases. (Equation 1.15) shows that the stagnation temperature increases as the square of the Mach number, U /a. Using isentropic relationship, we also have
(1.17)
1.3 One-Dimensional Gas Dynamics
For propulsion systems operating in supersonic flows, some elements of gas dynamics are useful. In adiabatic flows, the stagnation enthalpy is conserved, so that the use of (Eq. 1.16) results in a relationship between the static temperatures between two points (1 and 2) in the flow.
(1.18)
Using ideal gas equation of state, and using the fact that for steady-state one-dimensional (area A = const) flows ρU = ρMa = ρM, we have
(1.19)
Combining Eqs. (1.18) and (1.19) and solving for the pressure ratio, we get
(1.20)
Conservation of momentum in one-dimensional flows can be written as
(1.21)
Using ρU2 = γpM2, we can rewrite (Eq. 1.21) as
(1.22)
Eliminating pressure from Eqs. (1.20) and (1.22) gives us the relationship between upstream and downstream Mach numbers across, for example, a normal shock.
(1.23)
Then, the ratio of other parameters can also be written as a function of the upstream Mach number.
(1.24)
(1.25)
Equations (Equations (1.23)–(1.25) are referred to as the normal shock relationships and are tabulated in the Appendix C.
For isentropic flows in ducts, the local Mach number is a function of the cross-sectional area.
(1.26)
A *=throat area (M * = 1)1.4 Heat Transfer
Due to the high temperatures in some components, heat transfer is an important element of propulsion science. There are three modes of heat transfer: conduction, convection and radiation. Each is a subject in itself, but here we briefly state the basic laws of these heat transfer modes. Conduction is due to the molecule-to-molecule transfer of thermal energy, and is described by Fourier's law of conduction.
(1.27)
k=thermal conductivity [W/(mK)]The heat is transferred “down” (from hot to cold) the temperature gradient – hence the negative sign. In solids, conduction occurs due to lattice vibration and movement of energy carriers such as electrons in conductors. The latter is the reason why most good electrical conductors are also good thermal conductors. The difference in transfer of heat with electrons is also the basis for devices to measure temperature, thermocouples.
Although heat is also transferred through conduction in fluids, larger amounts of heat can be moved through the motion of the fluid mass itself. Mass times the specific heat is the energy content of the fluid, and if this mass is moved through fluid motion then heat transfer results. This mode of heat transfer is called convection, and is approximated through Newton's law of cooling.
(1.28)
h=heat transfer coefficient [W/(m2 K)]The heat transfer coefficient, h, is expressed through correlations of Nusselt numbers (Nu).
(1.29)
L is the characteristic length of the object. Dimensional arguments show that the Nusselt number is a function of the Reynolds number (flow effects), Prandtl number (fluid properties) and flow geometry. For example, for turbulent flow over a flat plate, the average Nusselt number is given by
(1.30)
Radiation heat transfer occurs due to photon energy, and is determined by the Planck distribution multiplied by the spectral emissivity, λ.
(1.31)
λ=wavelength [μm]C1=C2=The Planck distribution in (Eq. 1.31) can be integrated over the wavelength range, to yield the Stafan–Boltzmann law.
(1.32)
Since surfaces typically both emit and receive radiation energy, and the emissivity and absorptivity are approximately (exactly at spectral, directional level) equal, the net radiation energy can be written as
(1.33)
1.5 Standard Atmospheric Air Properties
At sea level (zero altitude), the standard atmosphere air properties are as follows.
(1.34a)
(1.34b)
(1.34c)
(1.34d)
(1.34e)
As the altitude increases from sea level, the atmospheric pressure decreases according to hydrostatics, that is, the weight of the air above. The variations of pressure and other air properties are tabulated in Appendix A. A simplified model can also be used to approximate the air pressure.
(1.35a)
(1.35b)
; ; g = gravitational acceleration = 9.80665 m/s2
The temperature profile is somewhat more complex, as shown in Figure 1.8. In the troposphere (h = 1 to 11 000 m), the temperature decreases linearly as a function of altitude. Then the temperature becomes constant at TTP = 216.65 K in the tropopause. This is the typical region for the long-range cruise altitude, so that change in altitude does not result in any appreciable change in temperature. Absence of temperature gradients also means that there is little air movement in the tropopause. Above the tropopause, the temperature begins to increase, due to the absorption of ultraviolet components of solar radiation by the ozone present in the stratosphere. The temperature again becomes constant at an altitude of approximately 50 km, and this region is called the stratopause. Above the stratopause, the temperature decreases again with increasing altitude in the region termed the mesosphere. At high altitudes above 100 km, the temperature rises again due to rarified, but finite amount of oxygen which absorbs ultraviolet radiation. This region is called the thermosphere.
Figure 1.8 The temperature profile in the Earth's atmosphere. Courtesy of NASA.
1.6 Unit Conversion
In US industry, imperial units such as lbf, lbm and BTU are used in conjunction with the standard (SI) units. If some conversion relations are remembered, then accurate accounting of the units can be maintained. For example, 1 lbf (pound force) is the force that 1 lbm (pound mass) is subject to under the Earth's gravitational field, through F = mg.
(1.36)
This is used to calculate the force due to momentum flow rates, as in (Eq. 1.3). For example, for a mass flow rate of 100 lbm/s, and flow velocity difference of 250 ft/s (incident) and 500 ft/s (exit), then the thrust force is in the unit of lbm-ft/s2, which is not a unit of force in the US unit system.
(1.37)
The unit of force in the US unit system is lbf, which is obtained as follows.
(1.38)
Then, (Eq. 1.37) becomes
(1.39)
Conversion of energy (such as thermal energy) to kinetic energy is somewhat more interesting. A common unit of thermal energy in the US unit system is the BTU (British thermal unit).
(1.40)
The following example illustrates the unit conversion.
Other commonly used conversion factors are given in Table 1.2.
Table 1.2 Conversion factors.
ParameterMetricMetric/imperialAcceleration1 m/s2 = 100 cm/s21 m/s2 = 3.2808 ft/s21 ft/s2 = 0.3048 m/s2Density1 g/cm3 = 1 kg/liter = 1000 kg/m31 g/cm3 = 62.428 lbm/ft3 = 0.036 127 lbm/in31 lbm/in3 = 1728 lbm/ft31 kg/m3 = 0.062 428 lbm/ft3 = 1.9404 × 10−3 slug/ft3Energy1 kJ = 1000 J = 1000 N·m1 kJ/kg = 1000 m2/s21 kWh = 3600 kJ1 cal = 4.184 J1 Cal = 4.1868 kJ1 kJ = 0.94 782 BTU1 BTU = 1.055 056 kJ = 5.40 395 psia·ft3 = 778.169 lbf·ft1 BTU/lbm = 25 037 ft2/s2 = 2.326* kJ/kg1 kJ/kg = 0.430 BTU/lbm1 kWh = 3412.14 BTUForce1 N = 1 kg·m/s2 = 105 dyne1 kgf = 9.80 665 N1 N = 0.22 481 lbf1 lbf = 32.174 lbm·ft/s2 = 4.44 822 NLength1 m = 100 cm = 1000 mm = 106 µm1 km = 1000 m1 astronomical unit (au) = 1.496 × 1011 m1 m = 39.370 in = 3.2808 ft = 1.0926 yd1 ft = 12 in = 0.3048 m1 mile = 5280 ft = 1.6093 km1 in = 2.54 cm1 nautical mile = 1852.0 m1 mil = 0.001 in = 0.0 000 254 mMass1 kg = 1000 g1 metric ton = 1000 kg1 kg = 2.2 046 226 lbm1 lbm = 0.45 359 237 kg1 ounce = 28.3495 g1 slug = 32.174 lbm = 14.5939 kg1 short ton = 2000 lbm = 907.1847 kgPower1 W = 1 J/s1 kW = 1000 W = 1.341 hp1 hp = 745.7 W1 kW = 3412.14 BTU/h = 737.56 lbf ft/s1 hp = 550 lbf·ft/s = 0.7068 BTU/s =42.14 BTU/min = 2544.5 BTU/h = 0.74 570 kW1 BTU/h = 1.055 056 kJ/h1 ton of refrigeration = 200 BTU/minPressure1 Pa = 1N/m21 kPa = 103 Pa = 10−3 MPa1 atm = 101.325 kPa= 1.01 325 bars = 760 mmHg at 0 °C = 1.03 323 kgf/cm21 mmHg = 0.1333 kPa1 Pa = 1.4504 × 10−4 psia = 0.020 886 lbf/ft21 psia = 144 lbf/ft2 = 6.894 757 kPa1 atm = 14.696 psia = 2116.2 lbf/ft2 = 29.92 inHg at 30°F1 inHg = 3.387 kPaSpecific heat1 kJ/kg·°C = 1 kJ/kg·K = 1 J/g·°C1 BTU/lbm·°F = 4.1868 kJ/kg·°C1 BTU/lbmol·R = 4.1868 kJ/kmol·K1 kJ/kg·°C = 0.23 885 BTU/lbm·°F = 0.23 885 BTU/lbm·RSpecific volume1 m3/kg = 1000 L/kg = 1000 cm3/g1 m3/kg = 16.02 ft3/lbm1 ft3/lbm = 0.062 428 m3/kgTemperatureT (K) = T (°C) + 273.15ΔT (K) = ΔT (°C)T (R) = T (°F) + 459.67 = 1.8T (K)T (°F) = 1.8T (°C) + 32ΔT (°F) = ΔT (R) = 1.8ΔT (K)Velocity1 m/s = 3.60 km/h1 m/s = 3.2808 ft/s = 2.237mi/h1 km/h = 0.2278 m/s = 0.6214 mi/h = 0.9113 ft/s1 mi/h = 1.46 667 ft/s1 mi/h = 1.609 km/h1 knot = 1.15 155 mi/hViscosity, dynamic1 kg/m·s = 1 N·s/m2 = 1 Pa·s =10 poise1 centipoise = 10−2 poise = 0.001 Pa·s1 kg/m·s = 2419.1 lbf/ft·h = 0.020 886 lbf·s/ft2= 5.8016 × 10−6 lbf·h/ft2Viscosity, kinematic1 m2/s = 104 cm2/s1 stoke = 1 cm2/s =10−4 m2/s1 m2/s = 10.764 ft2/s = 3.875 × 104 ft2/h1 m2/s = 10.764 ft2/sVolume1 m3 = 1000 liter (L) =106 cm3 (cc)1 m3 = 6.1024 × 104 in3 = 35.315 ft3 = 264.17 gal (US)1 ft3 = 0.028 321 m3 = 28.32 L1 US gallon = 231 in3 = 3.7854 L = 4 qt = 8 pt1 fl ounce = 29.5735 cm3 = 0.0 295 735 L
