Location (1)
Paris, France Perth, Australia Port-Moresby, PNG Pretoria, South Africa Quebec, Canada
g ( m / s ' ) (2) 9.80926 9.794 9.782 9.78615 9.80726
Location (1)
Quito, Ecuador Sapporo, Japan Reykjavik, Iceland Taipei, Taiwan Teheran, Iran
g(m/s2) (2) 9.7726 9.80476 9.82265 9.7895 9.7939
Location (1)
Thule, Greenland Tokyo,Japan Vancouver, Canada Ushuaia, Argentina
g (m/s^) (2) 9.82914 9.79787 9.80921 9.81465 Reference: Morelli (1971).
A1.1.2 Properties of water
Temperature
CC)
(1) 0 5 10 15 20 25 30 35 40
Density Pw (kg/m^) (2)
999.9 1000.0 999.7 999.1 998.2 997.1 995.7 994.1 992.2
Dynamic viscosity Mw(Pas)
(3)
1.792 X 10"^
1.519 X 10-3 1.308 X 10-3 1.140 X 10-3 1.005 X 10-3 0.894 X 10-3 0.801 X 10-3 0.723 X 10-3 0.656 X 10-3
Surface tension a-(N/m) (4) 0.0762 0.0754 0.0748 0.0741 0.0736 0.0726 0.0718 0.0710 0.0701
Vapour pressure i^v(Pa) (5) 0.6 X 103 0.9 X 103 1.2 X 103 1.7 X 103 2.5 X 103 3.2 X 103 4.3 X 103 5.7 X 103 7.5 X 103
Bulk modulus of elasticity {Ey,) (Pa) (6)
2.04 X 10^
2.06 X 10^
2.11 X 10^
2.14 X 10^
2.20 X 10^
2.22 X 10^
2.23 X 10^
2.24 X 10^
2.27 X 10^
Reference: Streeter and Wylie (1981).
A1.1.3 Gas properties Basic equations
The state equation of perfect gas is:
P = pRT (A1.3)
where P is the absolute pressure (in Pascal), p is the gas density (in kg/m^), T is the absolute temperature (in Kelvin) and R is the gas constant (in J/kgK) (see table below).
For a perfect gas, the specific heat at constant pressure Cp and the specific heat at constant volume Cy are related to the gas constant as:
^ y — 1 Cp = C,R
(A1.4a) (A1.4b) where y is the specific heat ratio (i.e. y = C^/Cy).
During an isentropic transformation of perfect gas, the following relationships hold:
P_
7p(l-7)/7
constant constant
(A1.5a) (A1.5b)
A1.1 Constants and fluid properties 121 Physical properties
Gas
(1)
Formula
(2)
Gas constant R (J/kgK) (3)
Specific heat Cp (J/kgK) (4)
Q (J/kgK) (5)
Specific heat ratio y
(6) Perfect gas
Mono-atomic gas Di-atomic gas Poly-atomic gas Real gas^
Air Helium Nitrogen Oxygen Water vapour
(e.g. He) (e.g. O2) (e.g. CH4)
He N2 O2 H2O
287 2077.4 297 260 462
(7/2)R 4R 1.004 5.233 1.038 0.917 1.863
(5/2)R 3R 0.716 3.153 0.741 0.657 1.403
7/5 4/3 1.40 1.67 1.40 1.40 1.33
Note: ^ at low pressures and at 299.83 K (Streeter and Wylie, 1981).
Compressibility and bulk modulus of elasticity
The compressibility of a fluid is a measure of change in volume and density when the fluid is subjected to a change of pressure. It is defined as:
'° pdp
The reciprocal function of the compressibility is called the bulk modulus of elasticity:
dp For a perfect gas, the bulk modulus of elasticity equals:
E^, = yP Adiabatic transformation for a perfect gas EY,= P Isothermal transformation for a perfect gas
(A1.6)
(A1.7a)
(A 1.7b) (A 1.7c) Celerity of sound
Introduction
The celerity of sound in a medium is:
C = ^
^ s o u n d -i ->^
(A1.8) where P is the pressure and p is the density. It may be rewritten in terms of the bulk modulus of elasticity E^:
a
sound (A1.9)Equation (A1.7) applies to both liquids and gases.
Sound celerity in gas
For an isentropic process and a perfect gas, equation (A1.9) yields:
^sound
= 4yRT (ALIO)
where y and R are the specific heat ratio and gas constant respectively (see above).
The dimensionless velocity of compressible fluid is called the Sarrau-Mach number:
Ma=- (Al.ll) V
^sound
Classical values
Celerity of sound in water at 20°C: 1485 m/s.
Celerity of sound in dry air and sea level at 20°C: 343 m/s.
A1.1.4 Atmospheric parameters Air pressure
The standard atmosphere or normal pressure at sea level equals:
Pstd = 1 atm = 360mm of Hg = 101 325 Pa (A1.12) where Hg is the chemical symbol of mercury. Unit conversion tables are provided in Appendix
A1.2. The atmospheric pressure varies with the elevation above the sea level (i.e. altitude). For dry air, the atmospheric pressure at the altitude z equals:
where Tis the absolute temperature in Kelvin and equation (A 1.13) is expressed in SI units.
Reference: Miller (1971).
Air temperature
In the troposphere (i.e. z < 10000 m), the air temperature decreases with altitude, on the aver- age, at a rate of 6.5 X lO^K/m (i.e. 6.5 K/km).
Table A 1.1 presents the distributions of average air temperatures (Miller, 1971) and corres- ponding atmospheric pressures with the altitude (equation (A1.13)).
Viscosity of air
Viscosity and density of air at 1.0 atm:
Temperature (K) /lair (Pa s) Pair (kg/^^^) (1) (2) (3) 300 18.4X10"^ 1.177 400 22.7 X 10-^ 0.883 500 26.7 X 10-^ 0.705 600 29.9 X 10-^ 0.588
A1.2 Unit conversions 123
Table Al.l Distributions of air temperature and air pressure as fiinctions of the altitude (for dry air and standard acceleration of gravity)
Altitude z(m) (1)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000
Mean air temperature (K) (2)
288.2 285.0 281.7 278.4 275.2 272.0 268.7 265.5 262.2 259.0 255.7 252.5 249.2 246.0 242.8 239.5 236.3 233.0 229.8 226.5 223.3
Atmospheric pressure (equation (Al.13)) (Pa) (3)
1.013 X 10^
9.546 X 10^
8.987 X lO'*
8.456 X 10^
7.949 X 10^
7.468 X 10^
7.011 X 10^
6.576 X 10^
6.164 X 10^
5.773 X 10^
5.402 X 10^
5.051 X 10^
4.718 X 10^
4.404 X 10^
4.106 X 10^
3.825 X 10^
3.560 X 10^
3.310 X 10^
3.075 X 10^
2.853 X 10^
2.644 X lO'*
Atmospheric pressure (equation (Al.l3)) (atm) (4)
1.000 0.942 0.887 0.834 0.785 0.737 0.692 0.649 0.608 0.570 0.533 0.498 0.466 0.435 0.405 0.378 0.351 0.327 0.303 0.282 0.261
The viscosity of air at standard atmosphere is commonly fitted by the Sutherland formula (Sutherland, 1883):
M^i, = 17.16 X 10"^ T ^"' 383.7
273.1 r+110.6 (A1.14)
A simpler correlation is:
^ N O . 7 6
J^:M(I1=\L\
(A1.15)
where ^igir is in Pas, and the temperature 7and reference temperature T^, are expressed in Kelvin.
* * * A1.2 UNIT CONVERSIONS
A1.2.1 Introduction
The systems of units derived from the metric system have gradually been replaced by a single system, called the Systeme International d'Unites (SI unit system, or International System of Units). The present lecture notes are presented in SI units.
Some countries continue to use British and American units, and conversion tables are provided in this appendix. Basic references in unit conversions include Degremont (1979) and ISO (1979).
Principles and rules
Unit symbols are written in small letters (i.e. m for metre, kg for kilogramme) but a capital is used for the first letter when the name of the unit derives fi-om a surname (e.g. Pa after Blaise Pascal, N after Isaac Newton).
Multiples and submultiples of SI units are formed by adding one prefix to the name of the unit:
e.g. km for kilometre, cm for centimetre, dam for decametre, jxm for micrometre (or micron).
Multiple/submultiple factor Prefix Symbol 1 X 10^
1 X 10^
1 X 10^
1 X 10^
1 X 10^
1 X 10- 1 X 10"
1 X 10 1 X 10"^
1 X 10-^
giga mega kilo hecto deca deci centi milli micro nano
G M k d da d c m M- n
The basic SI units are the metre, kilogramme, second. Ampere, Kelvin, mole and candela.
Supplementary units are the radian and the steradian. All other SI units derive fi'om the basic units.
A1.2.2 Units and conversion factors
Quantity (1) Length
Area Volume
Velocity Acceleration Mass Density Force
Moment of force Pressure
Unit (symbol) (2)
1 inch (in) 1 foot (ft) 1 yard (yd) Imil 1 mile
1 square inch (in^) 1 square foot (ft^) 1 litre (/) 1 cubic inch (in^) 1 cubic foot (ft^) 1 gallon UK (gal UK) 1 gallon US (gal US) 1 barrel US
1 foot per second (ft/s) 1 mile per hour (mph)
1 foot per second squared (ft/s^) 1 pound (lb or Ibm)
1 ton UK 1 ton US
1 pound per cubic foot (Ib/ft^) 1 kilogram-force (kgf) 1 pound force (Ibf) 1 foot pound force (ft Ibf) 1 Pascal (Pa)
1 standard atmosphere (atm)
I b a r 1 torr
1 conventional metre of water (mofHsO)
1 conventional metre of Mercury (m of Hg)
1 pound per square inch (PSI)
Conversion (3)
= 25.4 X 10"^m
= 0.3048 m
= 0.9144m
= 25.4 X 10"^m
= 1.609344 m
= 6.4516 X lO-^m^
= 0.092 903 06 m^
= 1.0 X 10"^m^
= 16.387064 X lO-^m^
= 28.3168 X lO-^m^
= 4.54609 X lO-^m^
= 3.78541 X lO-^m^
= 158.987 X 1 0 - % ^
= 0.3048 m/s
= 0.447 04 m/s
= 0.3048 m/s^
= 0.453 592 37 kg
= 1016.05 kg
= 907.185 kg
= 16.0185 kg/m^
= 9.806 65 N (exactly)
= 4.448 221 615 260 5 N
= 1.35582Nm
= IN/m^
= 101 325 Pa
= 760 mm of Mercury at normal pressure (i.e. m m o f Hg)
= lO^Pa
= 133.322 Pa
= 9.80665 X lO^Pa
= 1.333224 X lO^Pa
= 6.894 7572 X 10^ Pa
Comments (4) Exactly Exactly Exactly 1/1000 inch Exactly Exactly Exactly
Exactly. Previous symbol:
Exactly Exactly
For petroleum, etc.
Exactly Exactly Exactly Exactly
Exactly
Exactly
Exactly Exactly
L
(continued)
A1.3 Mathematics 125 Quantity
(1) Temperature
Dynamic viscosity
Kinematic viscosity Surface tension Work energy
Power
Unit (symbol) (2)
r(°c)
r (Fahrenheit) r(Rankine) IPas IPas INs/m^
1 Poise (P) 1 milliPoise (mP)
1 square foot per second (ft-^/s) lm2/s
Im^/s 1 dyne/cm 1 dyne/cm 1 Joule (J) 1 Joule (J) 1 Watt hour (Wh) 1 electronvolt (eV) lErg
1 foot pound force (ft Ibf) 1 Watt (W)
1 foot pound force per second (ft Ibf/s) 1 horsepower (hp)
Conversion (3)
= r(K)-273.16
= r(°C)(9/5) + 32
= (9/5)r(K)
= 0.006 720 Ibm/ft/s
= 10 Poises
= IPas
= 0.1 Pas
= l.OX 10^ Pas
= 0.092 903 Om^/s
= 10.7639 ft2/s
= 10^ Stokes
= 0.99987 X lO-^N/m
= 5.709 X 10"^ pound/inch
= I N m
= I W s
= 3.600 X lO^J
= 1.60219 X 10"^^J
= 10-^J
= 1.355 82 J
= IJ/s
= 1.355 82 W
= 745.700 W
Comments (4)
0°C is 0.01 K below the temperature of the triple point of water
Exactly Exactly Exactly Exactly
Exactly Exactly
A1.3 MATHEMATICS Summary
1. Introduction 2. Vector operations
3. Differential and differentiation
4. Trigonometric functions 5. Hyperbolic functions
6. Complex numbers 7. Polynomial equations
A1.3.1 Introduction References
Beyer, W.H. (1982). ''CRC Standard Mathematical Tables:' CRC Press Inc., Boca Raton, Florida, USA.
Kom, G.A. and Kom, T.M. (1961). ''Mathematical Handbook for Scientist and Engineers T McGraw-lliW, New York, USA.
Spiegel, M.R. (1968). "Mathematical Handbook of Formulas and Tables T McGraw-Hill Inc., New York, USA.
Notation
X, y, z Cartesian coordinates polar coordinates r,e,z
d_
dx d_ d_
3y' dz
partial differentiation with respect to the x-coordinate partial differential (Cartesian coordinate)
-\ -\
— — partial differential (polar coordinate) a r ' dd
_ partial differential with respect of the time t dt
_ absolute derivative
Sjj identity matrix element: Sn = 1 and S^y = 0 (for / different ofj) N\ iV-factorial: M = l X 2 X 3 X 4 X . . . X ( A ^ - l ) X i V
Constants
e constant such as ln(e) = 1: e = 2.718 281 828 459 045 235 360 287 IT 77 = 3.141 592 653 589 793 238 462 643
V2 V 2 = 1.414 213 562 373 095 048 8 V3 V3 = 1.732 050 807 568 877 293 5
A1.3.2 Vector operations Definitions
Considering a three-dimensional space, the coordinates of a point M or of a vector A can be expressed in a Cartesian system of coordinates or a cylindrical (or polar) system of coordinates as:
Point M^
Vector^
Cartesian system of coordinates
The relationship between the cartesian
Vector operations
Scalar product of two vectors
^ X 5 =
Cylindrical system of coordinates (Ar, AQ, A,)
coordinates and the polar coordinates of any p
r'=x'+y'
tan 0 = ^
X
-- \A\ X \B\ X cos{A,B) —> -^
2 2 2
where | ^ | = ^^^ + Ay + A^ . Two non-zero vectors are perpendicular to each other if and only if their scalar product is null.
Vector product
A AB=l(AyXB,-A,XBy) + "j(A,XB,-A,XB,) + l(A,XBy-AyXB,)
where / J and k are the unity vectors in the x-, y- and z-directions, respectively.A1.3 Mathematics 127