Appendix A Units and Dimensions It is essential to distinguish between units and dimensions. Broadly speaking, physical parameters have the separate properties of size and dimension. A unit is a, more or less arbitrarily defined, amount or quantity in terms of which a parameter is defined. A dimension represents the definition of an inherent property, independent of the system of units in which it is expressed. For example, the dimension mass expresses the amount of material of which a body is constructed, the distance between the wing tips (wingspan) of an aeroplane has the dimension length. The mass of a body can be expressed in kilograms or in pounds, the wingspan can be expressed in metres or in feet. Many systems of units exist, each of which with their own advantages and drawbacks. Throughout this book the internationally accepted dynamical system SI is used, except in a few places as specially noted. The Imperial set of units still plays an important roll in aviation, in particular in the United States. Fundamental dimensions and units Dimensions can be written in symbolic form by placing them between square brackets. There are four fundamental units in terms of which the dimensions of all other physical quantities may be expressed. Purely mechanical rela- tionships are derived in terms of mass [M], length [L], and time [T]; thermo- dynamical relationships contain the temperature [θ ] as well. A fundamental equation governing dynamical systems is derived from Newton’s second law of motion. This states that an external force F acting on a body is proportional to the product of its mass m and the acceleration a produced by the force: F = k F ma. The constant of proportionality k F is de- 511
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Appendix AUnits and Dimensions
It is essential to distinguish between units and dimensions. Broadly speaking,physical parameters have the separate properties of size and dimension. Aunit is a, more or less arbitrarily defined, amount or quantity in terms ofwhich a parameter is defined. A dimension represents the definition of aninherent property, independent of the system of units in which it is expressed.For example, the dimension mass expresses the amount of material of whicha body is constructed, the distance between the wing tips (wingspan) of anaeroplane has the dimension length. The mass of a body can be expressed inkilograms or in pounds, the wingspan can be expressed in metres or in feet.
Many systems of units exist, each of which with their own advantagesand drawbacks. Throughout this book the internationally accepted dynamicalsystem SI is used, except in a few places as specially noted. The Imperial setof units still plays an important roll in aviation, in particular in the UnitedStates.
Fundamental dimensions and units
Dimensions can be written in symbolic form by placing them between squarebrackets. There are four fundamental units in terms of which the dimensionsof all other physical quantities may be expressed. Purely mechanical rela-tionships are derived in terms of mass [M], length [L], and time [T]; thermo-dynamical relationships contain the temperature [θ] as well.
A fundamental equation governing dynamical systems is derived fromNewton’s second law of motion. This states that an external force F actingon a body is proportional to the product of its mass m and the acceleration a
produced by the force: F = kF ma. The constant of proportionality kF is de-
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512 A Units and Dimensions
Table A.1 Dimensions and SI units used for dynamical systems.
Quantity Dimension Unit name Symbol Explanation
length [L] metre m fundamental unitmass [M] kilogram kg fundamental unittime [T] second s fundamental unitarea [L2] – m2 length×lengthvolume [L3] – m3 area×lengthvelocity [LT−1] – m s−1 length/timeacceleration [LT−2] – m s−2 velocity/timemoment of inertia [ML2] – kg m2 mass×areadensity [ML−3] – kg m−3 mass/volumemass flow rate [MT−1] – kg s−1 mass/timeforce [MLT−2] Newton N, kg m s−2 mass×accelerationmoment [ML2T−2] – N m force×lengthpressure, stress [ML−1T−2] Pascal Pa, N m−2 force/areamomentum [MLT−1] – N s, kg m s−1 mass×velocitymomentum flow [MLT−2] – N, kg m s−2 mass×velocity/timework or energy [ML2T−2] Joule J, N m force×lengthpower [ML2T−3] Watt W, N m s−1 work or energy/timeangle 1 radian rad length/lengthangular velocity [T−1] – rad s−1 angle/timeangular acceleration [T−2] – rad s−2 angular velocity/timefrequency [T−1] Hertz Hz 1/time
termined by the definition of the units of force, mass and acceleration usedin the equation. In general, if the system of units is changed, so also is theconstant kF . It is useful, of course, to select the units so that the equationbecomes F = ma. In a consistent system of units, the force, mass, and timeare defined so that kF = 1. For this to be true, the unit of force has to be thatforce which, when acting upon a unit mass, produces a unit acceleration.
International System of Units
In most parts of the world the Système International d’Unités, commonlyabbreviated to SI units, is accepted for most branches of science and engi-neering. The SI system uses the following fundamental units:
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• Mass: the kilogram (symbol kg) is equivalent to the international stan-dard held in Sèvres near Paris.
• Length: the metre (symbol m), preserved in the past as a prototype, ispresently defined as the distance (m) travelled by light in a vacuum in299,792,458−1 seconds.
• Time: the second (symbol s) is the fundamental unit of time, defined interms of the natural periodicity of the radiation of a cesium-133 atom.
• Temperature: the unit Kelvin (symbol K) is identical in size with thedegree Celsius (symbol ◦C), but it denotes the absolute (or thermody-namical) temperature, measured from the absolute zero. The degree Cel-sius is one hundredth part of the temperature rise involved when purewater is heated from the triple point (273.15 K) to boiling temperatureat standard pressure. The temperature in degrees Celsius is thereforeT (C) = T (K) − 273.15.
Having defined the four fundamental dimensions and their units, all otherphysical quantities can be established, as in Table A.1. Velocity, for exam-ple, is defined as the distance travelled in unit time. It has the dimension[LT−1] and is measured in metres per second (m s−1, or m/s). The followingadditional remarks are made in relation to Table A.1:
• The SI system defines the Newton (symbol N) as the fundamental unitfor force, imparting an acceleration of 1 m s−2 to one kilogram of mass.From Newton’s equation, its dimension is derived as [MLT−2]. By con-trast to some other systems of units, the definition of a newton is com-pletely unrelated to the acceleration due to gravity. Clearly, the SI systemforms a consistent system.
• The fundamental unit of (gas) pressure or (material) stress is denotedpascal (symbol Pa). The bar is defined as 105 Pa, the millibar1 (mb)amounts to 102 Pa. A frequently used alternative unit of gas pressure isthe physical atmosphere (symbol atm), which is equal to the pressure un-der a 760 mm high column of mercury: 1.01325 × 105 Pa. The standardatmosphere is set at an air pressure of 1 atm at sea level. The techni-cal atmosphere (symbol at) is equal to the pressure under a 10 m highcolumn of water, g × 104 Pa. This requires a definition of the accelera-tion due to gravity, which is taken as the value at 45◦ northern latitude:g = 9.80665 m s−2.
• The (dimensionless) radian is defined as the angle subtended at the centreof a circle by an arc equal in length to the radius. One radian is equal to180/π = 57.296◦.
1 The preferred symbol is the hectopascal, hPa.
514 A Units and Dimensions
Fractions and multiples
Sometimes, the fundamental units defined above are inconveniently large orsmall for a particular case. In such cases, the quantity can be expressed interms of some fraction or multiple of the fundamental unit. A prefix attachedto a unit makes a new unit. The following prefixes may be used to indicatedecimal fractions or multiples of SI units.
Fraction Prefix Symbol Multiple Prefix Symbol
10−1 deci d 10 deca da10−2 centi c 102 hecto h10−3 milli m 103 kilo k10−6 micro µ 106 mega M
Imperial units
Until about 1968, the Imperial (or British Engineering) set of units was inuse in some parts of the world, the United Kingdom in particular. It uses thefundamental units foot (symbol ft) for length and pound (symbol lbm) formass, the unit for time is the second. The corresponding unit for force, thepoundal, produces an acceleration of 1 ft s−2 to 1 lbm. The Imperial Systemis therefore a consistent one. Since the poundal is considered as an unpracti-cally small force, it is often replaced by the pound force (symbol lbf), whichis defined as the weight of one pound mass. The pound force is therefore g
times as large as the poundal. However, used with 1 pound mass and 1 ft s−2,it does not constitute a consistent set of units. Therefore, the slug has beendefined as a mass equal to g times the pound mass, dictating that a standardvalue is used for the acceleration due to gravity (32.174 ft s−2). The Imperialsystem uses the Kelvin or the degree Celsius (“centigrade”) as the standardunit of temperature.
Although the SI system constitutes the generally accepted internationalstandard, many Imperial units are still in use, especially in the practice ofaircraft operation and in the US engineering world. For example, use is stillmade of the temperature scales Fahrenheit (F) and Rankine (R). The Rankineis an absolute temperature coupled to the Fahrenheit scale and is not to beconfused with the former Réaumur temperature unit. The conversion fromdegrees Fahrenheit to Kelvin is as follows: T (K) = 273.15 + 5/9{T (F) −32}. The system of units based on the foot, pound, second and rankine is
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Table A.2 Table for converting British FPSR units into SI units.
∗The unit of power in the (former) Technical System of Units is also known as the(metric) horsepower. It was derived from the kilogram as a fundamental unit forforce (kgf). Its value of 735.5 W is marginally smaller than the horsepower of theImperial system.
sometimes called the FPSR system. If their units are used in engineeringcomputations, it is recommendable to convert them into SI units with thehelp of Table A.2.
Appendix BPrinciples of Aerostatics
Ballooning originates from the early 18th century, and it is the oldest – andfor more than a century the only – form of aviation; see Sections 1.2 and 2.2.Despite recent competition from (more expensive) satellites, scientific andmeteorological balloons have preserved their place, while the popularity ofrecreational ballooning continues to grow. Because the physical principlesof ballooning form a clarifying illustration of the equation of state, someattention will be paid in this appendix to aerostatics.
Gross and net lift
From the equilibrium of a volume element of air in a static atmosphere, wederived in Section 2.6 that the pressure on the upper side of the element islower than on the lower side. This pressure difference is compensated by theweight of the air contained by the element and it is still present if the elementis replaced by an arbitrary body with the same geometry. The atmospherewill therefore exert a force on the body equal to the weight of the removedair. Using the aerostatic equation, we have thus given an explanation of thefamous law of Archimedes (287–212 BC). Applying this law to a balloonwith a volume Q, it says that the gross lift LG exerted on the balloon is equalto its volume multiplied by the specific weight of atmospheric air
LG = watQ = ρat g Q, (B.1)
with w and ρ denoting the specific weight and the density, respectively, ofthe atmosphere (index at). The weight of the internal lifting gas (index gas),forming the contents of the balloon, has to be subtracted from the gross liftto obtain the net lift LN ,
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518 B Principles of Aerostatics
LN = LB − Wgas = g Q(ρat − ρgas). (B.2)
The net lift is positive if ρgas < ρat. To comply with this condition the fol-lowing methods can be distinguished:
1. The balloon is filled with hot air. Because the air pressure in the balloonexceeds the ambient pressure only marginally, the difference betweenthe densities of the atmosphere and the hot air follows directly from theequation of state.
2. The balloon is filled with a gas which is “lighter than air”, in other words,the lifting gas – such as helium (He) – has a smaller molecular mass thanair. Hydrogen (H2) is the lightest gas but, in view of its high flammability,it is no longer used in manned balloons.
Hot-air balloons
A hot-air balloon has an inlet opening at the bottom so that the internal airpressure is equal to the ambient air pressure: pgas = pat. Lift results from thedifference in density between the hot internal air and the atmospheric air. Theinlet air is heated by means of a (LPG) gas burner flame below the opening.The gas is burnt intermittingly to control the average internal temperature.
The temperature difference between the hot air and the atmosphere �T =Tgas − Tat, is used to rewrite Equation (B.2) as follows:
LN = ρat g Q
(1 − ρgas
ρat
)= ρat g Q
(1 − Tat
Tgas
)= ρat g Q
�T
Tat + �T.
(B.3)By varying the gas burner heat added the value of �T is adjusted, making theballoon to ascend or descend. Equation (B.3) shows that the net lift largelydepends on the atmospheric air temperature. For example, we assume a bal-loon to be launched at an outside air temperature of 17◦C, and the inside airto be heated by �T = 80◦C. For an atmospheric density ρat = 1.25 kg/m3 itis found that LN = 2.65Q. At sea level the balloon will lift 2.65 N per cubicmetre of hot gas. However, if this balloon were to be launched on a hot daywith an ambient temperature of 37◦C and the same ambient pressure, we thenfind �T = 60◦C for the same hot air temperature, and ρat = 1.17 kg/m3 forthe ambient density, Equation (B.3) now indicates that the net lift per cubicmetre is merely 1.86 N or 30% less than for the earlier case. If the balloon’sempty weight is assumed to be the same in both cases, then the availableuseful load is reduced by the same 30%. Such a significant temperature de-
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pendence must be thoroughly taken into consideration when preparing for ahot-air ballooning flight.
Gas balloon
As for a hot-air balloon, the pressure of the lifting gas in a gas balloon isapproximately equal to the ambient pressure.1 By contrast, the lifting gastemperature is not much different from the outside air temperature, thoughit may be heated up appreciably by the sun, or cooled down when the bal-loon drifts below clouds. The equation of state dictates that the lifting gasdensity ρgas and the atmospheric air density ρat have a ratio similar to themolecular masses,
ρgas
ρat= Rat
Rgas= M̂gas
M̂at
. (B.4)
The net lift can be expressed according to Equation (B.2) either proportionalto the gas volume
LN = ρat g Q
(1 − ρgas
ρat
)= ρat g Q
(
1 − M̂gas
M̂at
)
(B.5)
or to the gas weight,
LN = ρgas g Q
(ρat
ρgas− 1
)= Wgas
(M̂at
M̂gas
− 1
)
. (B.6)
According to Equation (B.5), the net lift at sea level for a balloon filled withhelium gas (M̂ = 4) amounts to about 10 N per cubic metre. For an arbitrarygas volume, the lift is proportional to the ambient density and therefore de-creases at higher altitudes. Conversely, the net lift for a constant gas weightaccording to Equation (B.6) is also constant. By using the previous relation-ships, the altitude control of a gas balloon will be explained hereafter.
Open gas balloon
The gas in an open balloon is in contact with the surrounding atmospherevia a nozzle at its bottom, which is permanently open during flight. In level
1 Some gas balloons can accommodate a significant overpressure which allows them to attainan altitude up to 40 km without tearing. Their skin is manufactured from an extremely lightmaterial reinforced with high-strength fibres.
520 B Principles of Aerostatics
flight the net lift and the balloon’s weight Wb, including the useful load, are inequilibrium: LN = Wb. When a balloon ascends, ρat decreases and, becausethe (fully inflated) volume remains the same, gas escapes from the balloon sothat ρgas also decreases. According to Equations (B.5) and (B.6), the net liftdecreases so that LN < Wb. This counteracts the ascending motion and helpsthe balloon to maintain a steady rate of ascent. Conversely, in a descendingmotion, the gas weight is kept constant and the balloon is allowed to take onatmospheric air which does not contribute to the lift. For a constant amountof gas, the lift stays constant and – apart from the air drag on the balloon –the descending motion is not counteracted. An open gas balloon is thereforeindifferent to the rate of descent, which can only be reduced by off-loadingballast (sand). A fast descent – for example, while landing – can be executedby opening a gas valve at the top of the balloon.
Closed gas balloon
During its launch, a closed balloon will only be partially filled with gas, sothat the net lift is marginally greater than the weight: LN > Wb. Initially theballoon will ascend with constant acceleration, though the increasing speedwill magnify the air drag and cause the acceleration to reduce. After a while,the balloon will ascend at a steady rate. Due to the decreasing air pressure,the balloon will begin to expand until it becomes fully inflated. To preventthe balloon from tearing open, the gas valve is opened and the ascendingflight is continued as an open balloon, until the altitude limit is reached, asexplained below.
Ceiling of a gas balloon
Open balloons are used in ballooning sport at relatively low altitudes. Bycontrast, the purpose of closed balloons is to reach high altitudes, often pen-etrating the stratosphere. The ceiling of a closed balloon is reached whenthe net lift equals the balloon’s all-up weight. Expressed as the minimumatmospheric density achievable, this is determined by Equation (B.5)
ρat = Wb
g Qmax(1 − M̂gas/M̂at). (B.7)
For example, let us assume that we have a balloon with a volumeQmax = 576,000 m3 and a mass of Wb/g = 2,000 kg. Using M̂gas = 4 and
Flight Physics 521
M̂at = 28.96, we derive the density altitude at the ceiling from Equa-tion (B.7): ρat = 0.0040 kg/m3. According to the data for the standard at-mosphere (Section 2.6), the corresponding altitude is approximately 40 km.
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