For The Use Of Students And Examination Candidates
Approved by the Interprovincial Power Engineering Curriculum Committee and the Provincial Chief Inspectors' Association's Committee for the standardization of Power Engineer's Examinations n Canada.
Duplication of this material for student in-class use or for examination purposes is permitted without written approval.
www.powerengineering.ca
Printed July 2003
Table of Contents
TOPIC PAGE
SI Multiples..........................................................................................1
Basic Units (distance, area, volume, mass, density)............................2
Mathematical Formulae.......................................................................5
Applied Mechanics.............................................................................10
Thermodynamics.................................................................................21
Fluid Mechanics..................................................................................28
Electricity............................................................................................30
Periodic Table.....................................................................................34
Names in the Metric System
VALUE
EXPONENT
SYMBOL
PREFIX
1 000 000 000 000
1012
T
tera
1 000 000 000
109
G
giga
1 000 000
106
M
mega
1 000
103
k
kilo
100
102
h
hecto
10
101
da
deca
0.1
10-1
d
deci
0.01
10-2
c
centi
0.001
10-3
m
milli
0.000 001
10-6
μ
micro
0.000 000 001
10-9
n
nano
0.000 000 000 001
10-12
p
pico
Conversion Chart for Metric Units
To
Milli-
To
Centi-
To
Deci-
To
Metre,
Gram,
Litre
To
Deca-
To
Hecto-
To
Kilo-
Kilo-
x 106
x 105
x 104
x 103
x 102
x 101
To Convert
Hecto-
x 105
x 104
x 103
x 102
x 101
x 10-1
Deca-
x 104
x 103
x 102
x 101
x 10-1
x 10-2
Metre,
Gram,
Litre
x 103
x 102
x 101
x 10-1
x 10-2
x 10-3
Deci-
x 102
x 101
x 10-1
x 10-2
x 10-3
x 10-4
Centi-
x 101
x 10-1
x 10-2
x 10-3
x 10-4
x 10-5
Milli-
x 10-1
x 10-2
x 10-3
x 10-4
x 10-5
x 10-6
Page 1
BASIC UNITS
SI IMPERIAL
DISTANCE
1 metre (1 m) = 10 decimetres (10 dm) 12 in. = 1 ft
= 100 centimetres (100 cm) 3 ft = 1 yd
= 1000 millimetres (1000 mm) 5280 ft = 1 mile
1760 yd = 1 mile
1 decametre (1 dam) = 10 m
1 hectometre (1 hm) = 100 m
1 kilometre (1 km) = 1000 m
Conversions:
1 in. = 25.4 mm
1 ft = 30.48 cm
1 mile = 1.61 km
1 yd = 0.914 m
1 m = 3.28 ft
Area
1 sq metre (1 m2) = 10 000 cm2 1 ft2 = 144 in.2
= 1 000 000 mm2 1 yd2 = 9 ft2
1 sq mile = 640 acre = 1 section
1 sq hectometre (1 hm2) = 10 000 m2
= 1 hectare (1 ha)
1 sq km (1 km2) = 1 000 000 m2
Conversions:
1 in.2 = 6.45 cm2 = 645 mm2
1 m2 = 10.8 ft2
1 acre = 0.405 ha
1 sq mile = 2.59 km2 Page 2
SI IMPERIAL
Volume
1 m3 = 1 000 000 cm3 1 ft3 = 1728 in.3
= 1 x 109 mm3 1 yd3 = 27 ft3
1 dm3 = 1 litre 1(liquid) U.S. gallon = 231 in.3
1 litre = 1000 cm3 = 4 (liquid) quarts
1 mL = 1 cm3 1 U.S. barrel (bbl) = 42 U.S. gal.
1 m3 = 1000 litres 1 imperial gallon = 1.2 U.S. gal.
Conversions:
1 in.3 = 16.4 cm3
1 m3 = 35.3 ft3
1 litre = 61 in.3
1 U.S.gal = 3.78 litres
1 U.S. bbl = 159 litres
1 litre/s = 15.9 U.S. gal/min
Mass and Weight
1 kilogram (1 kg) = 1000 grams 2000 lb = 1 ton (short)
1000 kg = 1 tonne 1 long ton = 2240 lb
Conversions:
1 kg (on Earth) results in a weight of 2.2 lb
Density
volumemass density mass= volumeweight densityweight =
⎟⎠⎞⎜⎝⎛=3mkg Vm ρ ⎟⎠⎞⎜⎝⎛=3ftlb Vw ρ
Conversions:
(onEarth)amassdensityof1kgm3resultsinaweightdensityof0.0623lbft3 Page 3
SI Imperial
RELATIVE DENSITY
In SI R.D. is a comparison of mass density In Imperial the corresponding quantity is
to a standard. For solids and liquids the specific gravity; for solids and liquids a
standard is fresh water. comparison of weight density to that of water.
Conversions:
In both systems the same numbers
hold for R.D. as for S.G. since
these are equivalent ratios.
RELATIVE DENSITY (SPECIFIC GRAVITY) OF VARIOUS SUBSTANCES
Water (fresh)...............1.00 Mica............................2.9
Water (sea average)....1.03 Nickel.........................8.6
Aluminum...................2.56 Oil (linseed)................0.94
Antimony....................6.70 Oil (olive)...................0.92
Bismuth.......................9.80 Oil (petroleum)...........0.76-0.86
Brass...........................8.40 Oil (turpentine)...........0.87
Brick...........................2.1 Paraffin.......................0.86
Calcium.......................1.58 Platinum....................21.5
Carbon (diamond).......3.4 Sand (dry)...................1.42
Carbon (graphite)........2.3 Silicon.........................2.6
Carbon (charcoal).......1.8 Silver.........................10.57
Chromium...................6.5 Slate............................2.1-2.8
Clay.............................1.9 Sodium........................0.97
Coal.............................1.36-1.4 Steel (mild).................7.87
Cobalt.........................8.6 Sulphur.......................2.07
Copper........................8.77 Tin...............................7.3
Cork............................0.24 Tungsten...................19.1
Glass (crown)..............2.5 Wood (ash).................0.75
Glass (flint).................3.5 Wood (beech).............0.7-0.8
Gold..........................19.3 Wood (ebony).............1.1-1.2
Iron (cast)....................7.21 Wood (elm).................0.66
Iron (wrought)............7.78 Wood (lignum-vitae)..1.3
Lead..........................11.4 Wood (oak).................0.7-1.0
Magnesium.................1.74 Wood (pine)................0.56
Manganese..................8.0 Wood (teak)................0.8
Mercury....................13.6 Zinc.............................7.0
Page 4
Greek Alphabet
Alpha α Iota ι Rho ρ
Beta β Kappa κ Sigma Σ, σ
Gamma γ Lambda λ Tau τ
Delta Δ Mu μ Upsilon υ
Epsilon ε Nu ν Phi Φ, φ
Zeta ζ Xi ξ Kai χ
Eta η Omicron Ο Psi ψ
Theta θ Pi π Omega Ω, ω
MATHEMATICAL FORMULAE
Algebra
1. Expansion Formulae
(x + y)2 = x2 + 2xy + y2
(x - y)2 = x2 - 2xy + y2
x2 - y2 = (x - y) (x + y)
(x + y)3 = x3 + 3x2y + 3xy2 + y3
x3 + y3 = (x + y) (x2 - xy + y2)
(x - y)3 = x3 - 3x2y + 3xy2 - y3
x3 - y3 = (x - y) (x2 + xy + y2)
2. Quadratic Equation
If ax2 + bx + c = 0,
Then x = 2aac4b b-2−± Page 5
Trigonometry
1. Basic Ratios
hy A Sin =, hx A cos=, xy A tan =
2. Pythagoras' Law
x2 + y2 = h2
3. Trigonometric Function Values
Sin is positive from 0° to 90° and positive from 90° to 180°
Cos is positive from 0° to 90° and negative from 90° to 180°
Tan is positive from 0° to 90° and negative from 90° to 180°
4. Solution of Triangles
a. Sine Law
CSin cBSin bA Sin a==
b. Cosine Law
c2 = a2 + b2 - 2 ab Cos C
a2 = b2 + c2 - 2 bc Cos A
b2 = a2 + c2 - 2 ac Cos B
Page 6
Geometry
1. Areas of Triangles
a. All Triangles
2heightlar perpendicu x base Area=
Area 2BSin ac 2CSin ab2ASin bc ===
and,
c) - (s b) - (s a)-(s s Area=
where, s is half the sum of the sides, or s = 2c b a++
b. Equilateral Triangles
Area = 0.433 x side2
2. Circumference of a Circle
C = πd
3. Area of a Circle
A = πr2 = 2r x ncecircumfere = 2d4π = 0.7854d2
4. Area of a Sector of a Circle
A = 2r x arc
A = 2r x π360θ° (θ = angle in degrees)
A = 2rθ2° (θ = angle in radians)
Page 7
5. Area of a Segment of a Circle
A = area of sector – area of triangle
Also approximate area = 0.608-hd h 342
6. Ellipse
A = Dd4π
Approx. circumference = ()2d D π+
7. Area of Trapezoid
A = h 2b a⎟⎠⎞⎜⎝⎛+
8. Area of Hexagon
A = 2.6s2 where s is the length of one side
9. Area of Octagon
A = 4.83s2 where s is the length of one side
10. Sphere
Total surface area A =4πr2
Surface area of segment As = πdh
Volume V = 3r π34
Volume of segment
Vs=πh23(3r–h)Vs=πh6(h2+3a2)wherea=radiusofsegmentbase
Page 8
11. Volume of a Cylinder
V = Ld4π2 where L is cylinder length
12. Pyramid
Volume
V = 31 base area x perpendicular height
Volume of frustum
VF = )Aa a (A 3h++ where h is the perpendicular height, A and a are areas as shown
13. Cone
Area of curved surface of cone:
A = 2DL π
Area of curved surface of frustum
AF = 2d)L (D π+
Volume of cone:
V=basearea×perpendicularheight3
Volume of frustum:
VF=perpendicularheight×π(R2+r2+Rr)3
Page 9
APPLIED MECHANICS
Scalar - a property described by a magnitude only
Vector - a property described by a magnitude and a direction
Velocity - vector property equal to displacementtime
The magnitude of velocity may be referred to as speed
In SI the basic unit is ms, in Imperial fts
Other common units are kmh,mih
Conversions: sft 3.28 sm 1=
hmi 0.621 hkm 1=
Speed of sound in dry air is 331 ms at 0°C and increases by about 0.61 ms for each °C rise
Speed of light in vacuum equals 3 x 108 ms
Acceleration - vector property equal to changeinvelocitytime
In SI the basic unit is 2sm, in Imperial 2sft
Conversion: 12sm = 3.28 2sft
Acceleration due to gravity, symbol "g", is 9.81 2sm or 32.2 2sft
Page 10
LINEAR VELOCITY AND ACCELERATION
u initial velocity
v final velocity
t elapsed time
s displacement
a acceleration
v=u+ats=v+u2ts=ut+12at2v2=u2+2as
Angular Velocity and Acceleration
θ angular displacement (radians)
ω angular velocity (radians/s); ω1 = initial, ω2 = final
α angular acceleration (radians/s2)
ω2 = ω1 + α t
θ = ω1 + ω2 x t
2
θ = ω1 t + ½ α t2
ω22 = ω12 + 2 α θ
linear displacement, s = r θ
linear velocity, v = r ω
linear, or tangential acceleration, aT = r α
Page 11
Tangential, Centripetal and Total Acceleration
Tangential acceleration aT is due to angular acceleration α
aT=rα
Centripetal (Centrifugal) acceleration ac is due to change in direction only
ac = v2/r = r ω2
Total acceleration, a, of a rotating point experiencing angular acceleration is the vector sum
of aT and ac
a = aT + ac
FORCE
Vector quantity, a push or pull which changes the shape and/or motion of an object
In SI the unit of force is the newton, N, defined as a kgms2
In Imperial the unit of force is the pound lb
Conversion: 9.81 N = 2.2 lb
Weight
The gravitational force of attraction between a mass, m, and the mass of the Earth
In SI weight can be calculated from
Weight = F = mg , where g = 9.81 m/s2
In Imperial, the mass of an object (rarely used), in slugs, can be calculated from the known weight in pounds
m=Weightgg=32.2fts2
Page 12
Newton's Second Law of Motion
An unbalanced force F will cause an object of mass m to accelerate a, according to:
F = ma (Imperial F = wg a, where w is weight)
Torque Equation
T = I α where T is the acceleration torque in Nm, I is the moment of inertia in kg m2 and α is the angular acceleration in radians/s2
Momentum
Vector quantity, symbol p,
p = mv (Imperial p = wg v, where w is weight)
in SI unit is kgms
Work
Scalar quantity, equal to the (vector) product of a force and the displacement of an object. In simple systems, where W is work, F force and s distance
W = Fs
In SI the unit of work is the joule, J, or kilojoule, kJ
1 J = 1 Nm
In Imperial the unit of work is the ft-lb
Energy
Energy is the ability to do work, the units are the same as for work; J, kJ, and ft-lb
Page 13
Kinetic Energy
Energy due to motion
Ek=12mv2
In Imperial this is usually expressed as Ek=w2gv2 where w is weight
Kinetic Energy of Rotation
22Rωmk21 E= where k is radius of gyration, ω is angular velocity in rad/s
or
2RIω21 E= where I = mk2 is the moment of inertia
CENTRIPETAL (CENTRIFUGAL) FORCE
rmv F2C= where r is the radius
or
FC = m ω2 r where ω is angular velocity in rad/s
Potential Energy
Energy due to position in a force field, such as gravity
Ep = m g h
In Imperial this is usually expressed Ep = w h where w is weight, and h is height above some specified datum
Page 14
Thermal Energy
In SI the common units of thermal energy are J, and kJ, (and kJ/kg for specific quantities)
In Imperial, the units of thermal energy are British Thermal Units (Btu)
Conversions: 1 Btu = 1055 J
1 Btu = 778 ft-lb
Electrical Energy
In SI the units of electrical energy are J, kJ and kilowatt hours kWh. In Imperial, the unit of electrical energy is the kWh
Conversions: 1 kWh = 3600 kJ
1 kWh = 3412 Btu = 2.66 x 106 ft-lb
Power
A scalar quantity, equal to the rate of doing work
In SI the unit is the Watt W (or kW)
1W=1Js
In Imperial, the units are:
Mechanical Power - ft–lbs, horsepower h.p.
Thermal Power - Btus
Electrical Power - W, kW, or h.p.
Conversions: 746 W = 1 h.p.
1 h.p. = 550 ft–lbs
1 kW = 0.948 Btus
Page 15
Pressure
A vector quantity, force per unit area
In SI the basic units of pressure are pascals Pa and kPa
1Pa=1Nm2
In Imperial, the basic unit is the pound per square inch, psi
Atmospheric Pressure
At sea level atmospheric pressure equals 101.3 kPa or 14.7 psi
Pressure Conversions
1 psi = 6.895 kPa
Pressure may be expressed in standard units, or in units of static fluid head, in both SI and Imperial systems
Common equivalencies are:
1 kPa = 0.294 in. mercury = 7.5 mm mercury
1 kPa = 4.02 in. water = 102 mm water
1 psi = 2.03 in. mercury = 51.7 mm mercury
1 psi = 27.7 in. water = 703 mm water
1 m H2O = 9.81 kPa
Other pressure unit conversions:
1 bar = 14.5 psi = 100 kPa
1 kg/cm2 = 98.1 kPa = 14.2 psi = 0.981 bar
1 atmosphere (atm) = 101.3 kPa = 14.7 psi
Page 16
Simple Harmonic Motion
Velocity of P = sm x- R ω22
Acceleration of P = ω2 x m/s2
The period or time of a complete oscillation = ωπ2 seconds
General formula for the period of S.H.M.
T = 2πonacceleratintdisplaceme
Simple Pendulum
T = 2π gL T = period or time in seconds for a double swing
L = length in metres
The Conical Pendulum
R/H = tan θ= Fc/W = ω2 R/g
Page 17
Lifting Machines
W = load lifted, F = force applied
M.A. = effortload = FW
V.R. (velocity ratio) = distance loaddistanceeffort
η = efficiency = V.R.M.A.
1. Lifting Blocks
V.R. = number of rope strands supporting the load block
2. Wheel & Differential Axle
Velocity ratio = 2)r -(r π2Rπ21
= 1r -r 2R2 R
Or, using diameters instead of radii,
Velocity ratio = )d - (d2D1
3. Inclined Plane
V.R. = heightlength
4. Screw Jack
V.R. = threadofpitch leverage of ncecircumfere Page 18
Indicated Power
I.P. = Pm A L N where I.P. is power in W, Pm is mean or "average" effective pressure in Pa, A is piston area in m2, L is length of stroke in m and N is number of power strokes per second
Brake Power
B.P. = Tω where B.P. is brake power in W, T is torque in Nm and ω is angular velocity in radian/second
STRESS, STRAIN and MODULUS OF ELASTICITY
Direct stress = APareaload=
Direct strain = L length originalextensionΔ=
Modulus of elasticity
E = Δ=Δ=APLL/P/Astraindirect stressdirect
Shear stress τ = shearunder areaforce
Shear strain = Lx
Modulus of rigidity
G = strainshear stressshear Page 19
General Torsion Equation (Shafts of circular cross-section)
TJ=τr=GθL
)d (d32π )r - (r 2π J32πdr2π J 4241424144−==== Shaft HollowFor 2. Shaft SolidFor 1.
T = torque or twisting moment in newton metres
J = polar second moment of area of cross-section about shaft axis.
τ = shear stress at outer fibres in pascals
r = radius of shaft in metres
G = modulus of rigidity in pascals
θ = angle of twist in radians
L = length of shaft in metres
d = diameter of shaft in metres
Relationship Between Bending Stress and External Bending Moment
MI=σy=ER
1. For Rectangle
M = external bending moment in newton metres
I = second moment of area in m4
σ = bending stress at outer fibres in pascals
y = distance from centroid to outer fibres in metres
E = modulus of elasticity in pascals
R = radius of currative in metres
I = 12BD3
2. For Solid Shaft
I=πD464
Page 20
THERMODYNAMICS
Temperature Scales
°)32F (95 C−°= °F = 32 C 59+°
°R = °F + 460 (R Rankine) K = °C + 273 (K Kelvin)
Sensible Heat Equation
Q = mcΔT
m is mass
c is specific heat
ΔT is temperature change
Latent Heat
Latent heat of fusion of ice = 335 kJ/kg
Latent heat of steam from and at 100°C = 2257 kJ/kg
1 tonne of refrigeration = 335 000 kJ/day
= 233 kJ/min
Gas Laws
1. Boyle’s Law
When gas temperature is constant
PV = constant or
P1V1 = P2V2
where P is absolute pressure and V is volume
2. Charles’ Law
When gas pressure is constant, constant TV=
or V1T1=V2T2 , where V is volume and T is absolute temperature
Page 21
3. Gay-Lussac's Law
When gas volume is constant, constantTP=
Or 2211TPTP= , where P is absolute pressure and T is absolute temperature
4. General Gas Law
P1V1T1=P2V2T2=constant
P V = m R T where P = absolute pressure (kPa)
V = volume (m3)
T = absolute temp (K)
m = mass (kg)
R = characteristic constant (kJ/kgK)
Also
PV = nRoT where P = absolute pressure (kPa)
V = volume (m3)
T = absolute temperature K
N = the number of kmoles of gas
Ro = the universal gas constant 8.314 kJ/kmol/K
SPECIFIC HEATS OF GASES
Specific Heat at Specific Heat at Ratio of Specific
Constant Pressure Constant Volume Heats
kJ/kgK kJ/kgK γ = cp / cv
GAS or or
kJ/kg oC kJ/kg oC
Air 1.005 0.718 1.40
Ammonia 2.060 1.561 1.32
Carbon Dioxide 0.825 0.630 1.31
Carbon Monoxide 1.051 0.751 1.40
Helium 5.234 3.153 1.66
Hydrogen 14.235 10.096 1.41
Hydrogen Sulphide 1.105 0.85 1.30
Methane 2.177 1.675 1.30
Nitrogen 1.043 0.745 1.40
Oxygen 0.913 0.652 1.40
Sulphur Dioxide 0.632 0.451 1.40
Page 22
Efficiency of Heat Engines
Carnot Cycle η=T1–T2T1 where T1 and T2 are absolute temperatures of heat source and sink
Air Standard Efficiencies
1. Spark Ignition Gas and Oil Engines (Constant Volume Cycle or Otto Cycle)
1) - (γvr1 - 1 η= where rv = compression ratio = volumeclearanceolumecylinder v
γ = volume)(constant heat specificpressure)(constant heat specific
2. Diesel Cycle
)1 - γ(R r)1(R - 1 η1 - γvγ−= where r = ratio of compression
R = ratio of cut-off volume to clearance volume
3. High Speed Diesel (Dual-Combustion) Cycle
[]1) - k(βγ 1) -(k r1 - kβ - 1 η1 - γvγ+=
where rv = volumeclearanceolumecylinder v
k = combustion Vconstant of beginningat pressue absoluten)(combustio heating Vconstant of endat pressue absolute
β = volumeclearancen)(combustio heating Pconstant of endat volume
4. Gas Turbines (Constant Pressure or Brayton Cycle)
⎟⎟⎠⎞⎜⎜⎝⎛−=γ1γpr1 - 1 η
Page 23
where rp = pressure ratio = pressure intake compressorpressure discharge compressor
Page 24
Heat Transfer by Conduction
Q=λAtΔTdwhereQ=heattransferredinjoulesλ=thermalconductivityorcoeficientofheattransferinJ×mm2×s×°CorWm×°CA=areainm2t=timeinsecondsΔT=temperaturedifferencebetweensurfacesin°Cd=thicknessoflayerinm
COEFFICIENTS OF THERMAL CONDUCTIVITY
Material
Coefficient of
Thermal Conductivity
W/m °C
Air
Aluminum
Brass
Brick
Concrete
Copper
Cork
Felt
Glass
Glass, fibre
Iron, cast
Plastic, cellular
Steel
Wood
Wallboard, paper
0.025
206
104
0.6
0.85
380
0.043
0.038
1.0
0.04
70
0.04
60
0.15
0.076
Page 25
Thermal Expansion of Solids
Increase in length = L α (T2 – T1 )
where L = original length
α = coefficient of linear expansion
(T2 – T1 ) = rise in temperature
Increase in volume = V β (T2 – T1 )
Where V = original volume
β = coefficient of volumetric expansion
(T2 – T1 ) = rise in temperature
coefficient of volumetric expansion = coefficient of linear expansion x 3
β = 3α
Page 26
Chemical Heating Value of a Fuel
Chemical Heating Value MJ per kg of fuel = 22O33.7 C + 144 H - + 9.3 S8()
C is the mass of carbon per kg of fuel
H2 is the mass of hydrogen per kg of fuel
O2 is the mass of oxygen per kg of fuel
S is the mass of sulphur per kg of fuel
Theoretical Air Required to Burn Fuel
Air (kg per kg of fuel) = 228O100C + 8 H - + S 38[()]23
Air Supplied from Analysis of Flue Gases
Air in kg per kg of fuel = N233(CO2+CO)×C
C is the percentage of carbon in fuel by mass
N2 is the percentage of nitrogen in flue gas by volume
CO2 is the percentage of carbon dioxide in flue gas by volume
CO is the percentage of carbon monoxide in flue gas by volume
Boiler Formulae
Equivalent evaporation = kJ/kg 2257)h - (h m21s
Factor of evaporation = kJ/kg 2257)h - (h 21
Boiler efficiency = fuel of valuecalorific x m)h - (h mf21s
where = mass flow rate of steam sm
h1 = enthalpy of steam produced in boiler
h2 = enthalpy of feedwater to boiler
= mass flow rate of fuelfm
Page 27
FLUID MECHANICS
Discharge from an Orifice
Let A = cross-sectional area of the orifice = (π/4)d2
and Ac = cross-sectional area of the jet at the vena conrtacta = ((π/4)2cd
then Ac = CcA
or Cc = 2ccddAA⎟⎠⎞⎜⎝⎛=
where Cc is the coefficient of contraction
At the vena contracta, the volumetric flow rate Q of the fluid is given by
Q=areaofthejetatthevenacontracta×actualvelocity=AcvorQ=CcACv2gh
The coefficients of contraction and velocity are combined to give the coefficient of discharge, Cd
i.e.Cd=CcCvandQ=CdA2gh
Typically, values for Cd vary between 0.6 and 0.65
Circular orifice: Q = 0.62 A 2gh
Where Q = flow (m3/s) A = area (m2) h = head (m)
Rectangular notch: Q = 0.62 (B x H) 232gh
Where B = breadth (m) H = head (m above sill)
Triangular Right Angled Notch: Q = 2.635 H5/2
Where H = head (m above sill)
Page 28
Bernoulli’s Theory
H = 2gvwP h 2++
H = total head (metres) w = force of gravity on 1 m3 of fluid (N)
h = height above datum level (metres) v = velocity of water (metres per second)
P = pressure (N/m2 or Pa)
Loss of Head in Pipes Due to Friction
Loss of head in metres = fLdv22g
L = length in metres v = velocity of flow in metres per second
d = diameter in metres f = constant value of 0.01 in large pipes to 0.02 in small pipes
Note: This equation is expressed in some textbooks as
Loss = 4fLdv22g where the f values range from 0.0025 to 0.005
Actual Pipe Dimensions
Page 29
ELECTRICITY
Ohm's Law
I = RE
or E = IR
where I = current (amperes)
E = electromotive force (volts)
R = resistance (ohms)
Conductor Resistivity
R = ρaL
where ρ = specific resistance (or resistivity) (ohm metres, Ω·m)
L = length (metres)
a = area of cross-section (square metres)
Temperature correction
Rt = Ro (1 + αt)
where Ro = resistance at 0ºC (Ω)
Rt = resistance at tºC (Ω)
α = temperature coefficient which has an average value for copper of 0.004 28 (Ω/ΩºC)
R2 = R1)αt (1)αt (112++
where R1 = resistance at t1 (Ω)
R2 = resistance at t2 (Ω)
α Values Ω/ΩºC
copper 0.00428
platinum 0.00385
nickel 0.00672
tungsten 0.0045
aluminum 0.0040
Page 30
Dynamo Formulae
Average e.m.f. generated in each conductor = 2ΦNpZ60c
where Z = total number of armature conductors
c = number of parallel paths through winding between positive and negative brushes where c = 2 (wave winding), c = 2p (lap winding)
Φ = useful flux per pole (webers), entering or leaving the armature
p = number of pairs of poles
N = speed (revolutions per minute)
Generator Terminal volts = EG – IaRa
Motor Terminal volts = EB + IaRa
where EG = generated e.m.f.
EB = generated back e.m.f.
Ia = armature current
Ra = armature resistance
Alternating Current
R.M.S. value of sine curve = 0.707 maximum value
Mean value of sine curve = 0.637 maximum value
Form factor of sinusoidal = 11.10.6370.707 Mean value valueR.M.S.==
Frequency of alternator = 60pN cycles per second
Where p = number of pairs of poles
N = rotational speed in r/min
Page 31
Slip of Induction Motor
100 x field of Speedrotor of speed - field of speed Slip
Inductive Reactance
Reactance of AC circuit (X) = 2πfL ohms
where L = inductance of circuit (henries)
Inductance of an iron cored solenoid = henries10x LμAT256.182
where T = turns on coil
μ = magnetic permeablility of core
A = area of core (square centimetres)
L = length (centimetres)
Capacitance Reactance
Capacitance reactance of AC circuit = πfC21ohms
where C = capacitance (farads)
Total reactance = ohmsfC 2π1- πfL2⎟⎠⎞⎜⎝⎛
Impedence (Z) = 22reactance)(e)(resistanc+
= ohms )fC π21 - fL (2π R22+
Current in AC Circuit
impedance voltsimpressed Current =
Page 32
Power Factor
p.f. = amperes x volts wattstrue
also p.f. = cos Φ, where Φ is the angle of lag or lead
Three Phase Alternators
Star connected
Line voltage = 3 x phase voltage
Line current = phase current
Delta connected
Line voltage = phase voltage
Line current = 3 x phase current
Three phase power
P = 3 EL IL cos Φ
EL = line voltage
IL = line current
cos Φ = power factor
Page 33
Page 34
ION NAMES AND FORMULAE
MONATOMIC POLYATOMIC
Ag+ silver ion BO33- borate ion
Al3+ aluminum ion C2H3O2- acetate ion
Au+ and Au2+ gold ion ClO- hypochlorite ion
Be2+ beryllium ion ClO2- chlorite ion
Ca2+ calcium ion ClO3- chlorate ion
Co2+ and Co3+ cobalt ion ClO4- perchlorate ion
Cr2+ and Cr3+ chromium ion CN- cyanide ion
Cu+ and Cu2+ copper ion CO32- carbonate ion
Fe2+ and Fe3+ iron ion C2O42- oxalate ion
K+ potassium ion CrO42- chromate ion
Li+ lithium ion Cr2O72- dichromate ion
Mg2+ magnesium ion HCO3- hydrogen carbonate or bicarbonate ion
Na+ sodium ion H3O+ hydronium ion
Zn2+ zinc ion HPO42- hydrogen phosphate ion
H2PO4- dihydrogen phosphate ion
HSO3- hydrogen sulphite or bisulphite ion
HSO4- hydrogen sulphate or bisulphate ion
MnO4- permanganate ion
N3- azide ion
NH4+ ammonium ion
NO2- nitrite ion
NO3- nitrate ion
O22- peroxide ion
OCN- cyanate ion
OH- hydroxide ion
PO33- phosphite ion
PO43- phosphate ion
SCN- thiocyanate ion
SO32- sulphite ion
SO42- sulphate ion
S2O32- thiosulphate ion
Page 35
This material is owned by Power Engineering Training Systems and may not be modified from its original form. Duplication of this material for student use in-class or for examination purposes is permitted without written approval.
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Printed in Canada
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www.powerengineering.ca
Printed July 2003
Table of Contents
TOPIC PAGE
SI Multiples..........................................................................................1
Basic Units (distance, area, volume, mass, density)............................2
Mathematical Formulae.......................................................................5
Applied Mechanics.............................................................................10
Thermodynamics.................................................................................21
Fluid Mechanics..................................................................................28
Electricity............................................................................................30
Periodic Table.....................................................................................34
Names in the Metric System
VALUE
EXPONENT
SYMBOL
PREFIX
1 000 000 000 000
1012
T
tera
1 000 000 000
109
G
giga
1 000 000
106
M
mega
1 000
103
k
kilo
100
102
h
hecto
10
101
da
deca
0.1
10-1
d
deci
0.01
10-2
c
centi
0.001
10-3
m
milli
0.000 001
10-6
μ
micro
0.000 000 001
10-9
n
nano
0.000 000 000 001
10-12
p
pico
Conversion Chart for Metric Units
To
Milli-
To
Centi-
To
Deci-
To
Metre,
Gram,
Litre
To
Deca-
To
Hecto-
To
Kilo-
Kilo-
x 106
x 105
x 104
x 103
x 102
x 101
To Convert
Hecto-
x 105
x 104
x 103
x 102
x 101
x 10-1
Deca-
x 104
x 103
x 102
x 101
x 10-1
x 10-2
Metre,
Gram,
Litre
x 103
x 102
x 101
x 10-1
x 10-2
x 10-3
Deci-
x 102
x 101
x 10-1
x 10-2
x 10-3
x 10-4
Centi-
x 101
x 10-1
x 10-2
x 10-3
x 10-4
x 10-5
Milli-
x 10-1
x 10-2
x 10-3
x 10-4
x 10-5
x 10-6
Page 1
BASIC UNITS
SI IMPERIAL
DISTANCE
1 metre (1 m) = 10 decimetres (10 dm) 12 in. = 1 ft
= 100 centimetres (100 cm) 3 ft = 1 yd
= 1000 millimetres (1000 mm) 5280 ft = 1 mile
1760 yd = 1 mile
1 decametre (1 dam) = 10 m
1 hectometre (1 hm) = 100 m
1 kilometre (1 km) = 1000 m
Conversions:
1 in. = 25.4 mm
1 ft = 30.48 cm
1 mile = 1.61 km
1 yd = 0.914 m
1 m = 3.28 ft
Area
1 sq metre (1 m2) = 10 000 cm2 1 ft2 = 144 in.2
= 1 000 000 mm2 1 yd2 = 9 ft2
1 sq mile = 640 acre = 1 section
1 sq hectometre (1 hm2) = 10 000 m2
= 1 hectare (1 ha)
1 sq km (1 km2) = 1 000 000 m2
Conversions:
1 in.2 = 6.45 cm2 = 645 mm2
1 m2 = 10.8 ft2
1 acre = 0.405 ha
1 sq mile = 2.59 km2 Page 2
SI IMPERIAL
Volume
1 m3 = 1 000 000 cm3 1 ft3 = 1728 in.3
= 1 x 109 mm3 1 yd3 = 27 ft3
1 dm3 = 1 litre 1(liquid) U.S. gallon = 231 in.3
1 litre = 1000 cm3 = 4 (liquid) quarts
1 mL = 1 cm3 1 U.S. barrel (bbl) = 42 U.S. gal.
1 m3 = 1000 litres 1 imperial gallon = 1.2 U.S. gal.
Conversions:
1 in.3 = 16.4 cm3
1 m3 = 35.3 ft3
1 litre = 61 in.3
1 U.S.gal = 3.78 litres
1 U.S. bbl = 159 litres
1 litre/s = 15.9 U.S. gal/min
Mass and Weight
1 kilogram (1 kg) = 1000 grams 2000 lb = 1 ton (short)
1000 kg = 1 tonne 1 long ton = 2240 lb
Conversions:
1 kg (on Earth) results in a weight of 2.2 lb
Density
volumemass density mass= volumeweight densityweight =
⎟⎠⎞⎜⎝⎛=3mkg Vm ρ ⎟⎠⎞⎜⎝⎛=3ftlb Vw ρ
Conversions:
(onEarth)amassdensityof1kgm3resultsinaweightdensityof0.0623lbft3 Page 3
SI Imperial
RELATIVE DENSITY
In SI R.D. is a comparison of mass density In Imperial the corresponding quantity is
to a standard. For solids and liquids the specific gravity; for solids and liquids a
standard is fresh water. comparison of weight density to that of water.
Conversions:
In both systems the same numbers
hold for R.D. as for S.G. since
these are equivalent ratios.
RELATIVE DENSITY (SPECIFIC GRAVITY) OF VARIOUS SUBSTANCES
Water (fresh)...............1.00 Mica............................2.9
Water (sea average)....1.03 Nickel.........................8.6
Aluminum...................2.56 Oil (linseed)................0.94
Antimony....................6.70 Oil (olive)...................0.92
Bismuth.......................9.80 Oil (petroleum)...........0.76-0.86
Brass...........................8.40 Oil (turpentine)...........0.87
Brick...........................2.1 Paraffin.......................0.86
Calcium.......................1.58 Platinum....................21.5
Carbon (diamond).......3.4 Sand (dry)...................1.42
Carbon (graphite)........2.3 Silicon.........................2.6
Carbon (charcoal).......1.8 Silver.........................10.57
Chromium...................6.5 Slate............................2.1-2.8
Clay.............................1.9 Sodium........................0.97
Coal.............................1.36-1.4 Steel (mild).................7.87
Cobalt.........................8.6 Sulphur.......................2.07
Copper........................8.77 Tin...............................7.3
Cork............................0.24 Tungsten...................19.1
Glass (crown)..............2.5 Wood (ash).................0.75
Glass (flint).................3.5 Wood (beech).............0.7-0.8
Gold..........................19.3 Wood (ebony).............1.1-1.2
Iron (cast)....................7.21 Wood (elm).................0.66
Iron (wrought)............7.78 Wood (lignum-vitae)..1.3
Lead..........................11.4 Wood (oak).................0.7-1.0
Magnesium.................1.74 Wood (pine)................0.56
Manganese..................8.0 Wood (teak)................0.8
Mercury....................13.6 Zinc.............................7.0
Page 4
Greek Alphabet
Alpha α Iota ι Rho ρ
Beta β Kappa κ Sigma Σ, σ
Gamma γ Lambda λ Tau τ
Delta Δ Mu μ Upsilon υ
Epsilon ε Nu ν Phi Φ, φ
Zeta ζ Xi ξ Kai χ
Eta η Omicron Ο Psi ψ
Theta θ Pi π Omega Ω, ω
MATHEMATICAL FORMULAE
Algebra
1. Expansion Formulae
(x + y)2 = x2 + 2xy + y2
(x - y)2 = x2 - 2xy + y2
x2 - y2 = (x - y) (x + y)
(x + y)3 = x3 + 3x2y + 3xy2 + y3
x3 + y3 = (x + y) (x2 - xy + y2)
(x - y)3 = x3 - 3x2y + 3xy2 - y3
x3 - y3 = (x - y) (x2 + xy + y2)
2. Quadratic Equation
If ax2 + bx + c = 0,
Then x = 2aac4b b-2−± Page 5
Trigonometry
1. Basic Ratios
hy A Sin =, hx A cos=, xy A tan =
2. Pythagoras' Law
x2 + y2 = h2
3. Trigonometric Function Values
Sin is positive from 0° to 90° and positive from 90° to 180°
Cos is positive from 0° to 90° and negative from 90° to 180°
Tan is positive from 0° to 90° and negative from 90° to 180°
4. Solution of Triangles
a. Sine Law
CSin cBSin bA Sin a==
b. Cosine Law
c2 = a2 + b2 - 2 ab Cos C
a2 = b2 + c2 - 2 bc Cos A
b2 = a2 + c2 - 2 ac Cos B
Page 6
Geometry
1. Areas of Triangles
a. All Triangles
2heightlar perpendicu x base Area=
Area 2BSin ac 2CSin ab2ASin bc ===
and,
c) - (s b) - (s a)-(s s Area=
where, s is half the sum of the sides, or s = 2c b a++
b. Equilateral Triangles
Area = 0.433 x side2
2. Circumference of a Circle
C = πd
3. Area of a Circle
A = πr2 = 2r x ncecircumfere = 2d4π = 0.7854d2
4. Area of a Sector of a Circle
A = 2r x arc
A = 2r x π360θ° (θ = angle in degrees)
A = 2rθ2° (θ = angle in radians)
Page 7
5. Area of a Segment of a Circle
A = area of sector – area of triangle
Also approximate area = 0.608-hd h 342
6. Ellipse
A = Dd4π
Approx. circumference = ()2d D π+
7. Area of Trapezoid
A = h 2b a⎟⎠⎞⎜⎝⎛+
8. Area of Hexagon
A = 2.6s2 where s is the length of one side
9. Area of Octagon
A = 4.83s2 where s is the length of one side
10. Sphere
Total surface area A =4πr2
Surface area of segment As = πdh
Volume V = 3r π34
Volume of segment
Vs=πh23(3r–h)Vs=πh6(h2+3a2)wherea=radiusofsegmentbase
Page 8
11. Volume of a Cylinder
V = Ld4π2 where L is cylinder length
12. Pyramid
Volume
V = 31 base area x perpendicular height
Volume of frustum
VF = )Aa a (A 3h++ where h is the perpendicular height, A and a are areas as shown
13. Cone
Area of curved surface of cone:
A = 2DL π
Area of curved surface of frustum
AF = 2d)L (D π+
Volume of cone:
V=basearea×perpendicularheight3
Volume of frustum:
VF=perpendicularheight×π(R2+r2+Rr)3
Page 9
APPLIED MECHANICS
Scalar - a property described by a magnitude only
Vector - a property described by a magnitude and a direction
Velocity - vector property equal to displacementtime
The magnitude of velocity may be referred to as speed
In SI the basic unit is ms, in Imperial fts
Other common units are kmh,mih
Conversions: sft 3.28 sm 1=
hmi 0.621 hkm 1=
Speed of sound in dry air is 331 ms at 0°C and increases by about 0.61 ms for each °C rise
Speed of light in vacuum equals 3 x 108 ms
Acceleration - vector property equal to changeinvelocitytime
In SI the basic unit is 2sm, in Imperial 2sft
Conversion: 12sm = 3.28 2sft
Acceleration due to gravity, symbol "g", is 9.81 2sm or 32.2 2sft
Page 10
LINEAR VELOCITY AND ACCELERATION
u initial velocity
v final velocity
t elapsed time
s displacement
a acceleration
v=u+ats=v+u2ts=ut+12at2v2=u2+2as
Angular Velocity and Acceleration
θ angular displacement (radians)
ω angular velocity (radians/s); ω1 = initial, ω2 = final
α angular acceleration (radians/s2)
ω2 = ω1 + α t
θ = ω1 + ω2 x t
2
θ = ω1 t + ½ α t2
ω22 = ω12 + 2 α θ
linear displacement, s = r θ
linear velocity, v = r ω
linear, or tangential acceleration, aT = r α
Page 11
Tangential, Centripetal and Total Acceleration
Tangential acceleration aT is due to angular acceleration α
aT=rα
Centripetal (Centrifugal) acceleration ac is due to change in direction only
ac = v2/r = r ω2
Total acceleration, a, of a rotating point experiencing angular acceleration is the vector sum
of aT and ac
a = aT + ac
FORCE
Vector quantity, a push or pull which changes the shape and/or motion of an object
In SI the unit of force is the newton, N, defined as a kgms2
In Imperial the unit of force is the pound lb
Conversion: 9.81 N = 2.2 lb
Weight
The gravitational force of attraction between a mass, m, and the mass of the Earth
In SI weight can be calculated from
Weight = F = mg , where g = 9.81 m/s2
In Imperial, the mass of an object (rarely used), in slugs, can be calculated from the known weight in pounds
m=Weightgg=32.2fts2
Page 12
Newton's Second Law of Motion
An unbalanced force F will cause an object of mass m to accelerate a, according to:
F = ma (Imperial F = wg a, where w is weight)
Torque Equation
T = I α where T is the acceleration torque in Nm, I is the moment of inertia in kg m2 and α is the angular acceleration in radians/s2
Momentum
Vector quantity, symbol p,
p = mv (Imperial p = wg v, where w is weight)
in SI unit is kgms
Work
Scalar quantity, equal to the (vector) product of a force and the displacement of an object. In simple systems, where W is work, F force and s distance
W = Fs
In SI the unit of work is the joule, J, or kilojoule, kJ
1 J = 1 Nm
In Imperial the unit of work is the ft-lb
Energy
Energy is the ability to do work, the units are the same as for work; J, kJ, and ft-lb
Page 13
Kinetic Energy
Energy due to motion
Ek=12mv2
In Imperial this is usually expressed as Ek=w2gv2 where w is weight
Kinetic Energy of Rotation
22Rωmk21 E= where k is radius of gyration, ω is angular velocity in rad/s
or
2RIω21 E= where I = mk2 is the moment of inertia
CENTRIPETAL (CENTRIFUGAL) FORCE
rmv F2C= where r is the radius
or
FC = m ω2 r where ω is angular velocity in rad/s
Potential Energy
Energy due to position in a force field, such as gravity
Ep = m g h
In Imperial this is usually expressed Ep = w h where w is weight, and h is height above some specified datum
Page 14
Thermal Energy
In SI the common units of thermal energy are J, and kJ, (and kJ/kg for specific quantities)
In Imperial, the units of thermal energy are British Thermal Units (Btu)
Conversions: 1 Btu = 1055 J
1 Btu = 778 ft-lb
Electrical Energy
In SI the units of electrical energy are J, kJ and kilowatt hours kWh. In Imperial, the unit of electrical energy is the kWh
Conversions: 1 kWh = 3600 kJ
1 kWh = 3412 Btu = 2.66 x 106 ft-lb
Power
A scalar quantity, equal to the rate of doing work
In SI the unit is the Watt W (or kW)
1W=1Js
In Imperial, the units are:
Mechanical Power - ft–lbs, horsepower h.p.
Thermal Power - Btus
Electrical Power - W, kW, or h.p.
Conversions: 746 W = 1 h.p.
1 h.p. = 550 ft–lbs
1 kW = 0.948 Btus
Page 15
Pressure
A vector quantity, force per unit area
In SI the basic units of pressure are pascals Pa and kPa
1Pa=1Nm2
In Imperial, the basic unit is the pound per square inch, psi
Atmospheric Pressure
At sea level atmospheric pressure equals 101.3 kPa or 14.7 psi
Pressure Conversions
1 psi = 6.895 kPa
Pressure may be expressed in standard units, or in units of static fluid head, in both SI and Imperial systems
Common equivalencies are:
1 kPa = 0.294 in. mercury = 7.5 mm mercury
1 kPa = 4.02 in. water = 102 mm water
1 psi = 2.03 in. mercury = 51.7 mm mercury
1 psi = 27.7 in. water = 703 mm water
1 m H2O = 9.81 kPa
Other pressure unit conversions:
1 bar = 14.5 psi = 100 kPa
1 kg/cm2 = 98.1 kPa = 14.2 psi = 0.981 bar
1 atmosphere (atm) = 101.3 kPa = 14.7 psi
Page 16
Simple Harmonic Motion
Velocity of P = sm x- R ω22
Acceleration of P = ω2 x m/s2
The period or time of a complete oscillation = ωπ2 seconds
General formula for the period of S.H.M.
T = 2πonacceleratintdisplaceme
Simple Pendulum
T = 2π gL T = period or time in seconds for a double swing
L = length in metres
The Conical Pendulum
R/H = tan θ= Fc/W = ω2 R/g
Page 17
Lifting Machines
W = load lifted, F = force applied
M.A. = effortload = FW
V.R. (velocity ratio) = distance loaddistanceeffort
η = efficiency = V.R.M.A.
1. Lifting Blocks
V.R. = number of rope strands supporting the load block
2. Wheel & Differential Axle
Velocity ratio = 2)r -(r π2Rπ21
= 1r -r 2R2 R
Or, using diameters instead of radii,
Velocity ratio = )d - (d2D1
3. Inclined Plane
V.R. = heightlength
4. Screw Jack
V.R. = threadofpitch leverage of ncecircumfere Page 18
Indicated Power
I.P. = Pm A L N where I.P. is power in W, Pm is mean or "average" effective pressure in Pa, A is piston area in m2, L is length of stroke in m and N is number of power strokes per second
Brake Power
B.P. = Tω where B.P. is brake power in W, T is torque in Nm and ω is angular velocity in radian/second
STRESS, STRAIN and MODULUS OF ELASTICITY
Direct stress = APareaload=
Direct strain = L length originalextensionΔ=
Modulus of elasticity
E = Δ=Δ=APLL/P/Astraindirect stressdirect
Shear stress τ = shearunder areaforce
Shear strain = Lx
Modulus of rigidity
G = strainshear stressshear Page 19
General Torsion Equation (Shafts of circular cross-section)
TJ=τr=GθL
)d (d32π )r - (r 2π J32πdr2π J 4241424144−==== Shaft HollowFor 2. Shaft SolidFor 1.
T = torque or twisting moment in newton metres
J = polar second moment of area of cross-section about shaft axis.
τ = shear stress at outer fibres in pascals
r = radius of shaft in metres
G = modulus of rigidity in pascals
θ = angle of twist in radians
L = length of shaft in metres
d = diameter of shaft in metres
Relationship Between Bending Stress and External Bending Moment
MI=σy=ER
1. For Rectangle
M = external bending moment in newton metres
I = second moment of area in m4
σ = bending stress at outer fibres in pascals
y = distance from centroid to outer fibres in metres
E = modulus of elasticity in pascals
R = radius of currative in metres
I = 12BD3
2. For Solid Shaft
I=πD464
Page 20
THERMODYNAMICS
Temperature Scales
°)32F (95 C−°= °F = 32 C 59+°
°R = °F + 460 (R Rankine) K = °C + 273 (K Kelvin)
Sensible Heat Equation
Q = mcΔT
m is mass
c is specific heat
ΔT is temperature change
Latent Heat
Latent heat of fusion of ice = 335 kJ/kg
Latent heat of steam from and at 100°C = 2257 kJ/kg
1 tonne of refrigeration = 335 000 kJ/day
= 233 kJ/min
Gas Laws
1. Boyle’s Law
When gas temperature is constant
PV = constant or
P1V1 = P2V2
where P is absolute pressure and V is volume
2. Charles’ Law
When gas pressure is constant, constant TV=
or V1T1=V2T2 , where V is volume and T is absolute temperature
Page 21
3. Gay-Lussac's Law
When gas volume is constant, constantTP=
Or 2211TPTP= , where P is absolute pressure and T is absolute temperature
4. General Gas Law
P1V1T1=P2V2T2=constant
P V = m R T where P = absolute pressure (kPa)
V = volume (m3)
T = absolute temp (K)
m = mass (kg)
R = characteristic constant (kJ/kgK)
Also
PV = nRoT where P = absolute pressure (kPa)
V = volume (m3)
T = absolute temperature K
N = the number of kmoles of gas
Ro = the universal gas constant 8.314 kJ/kmol/K
SPECIFIC HEATS OF GASES
Specific Heat at Specific Heat at Ratio of Specific
Constant Pressure Constant Volume Heats
kJ/kgK kJ/kgK γ = cp / cv
GAS or or
kJ/kg oC kJ/kg oC
Air 1.005 0.718 1.40
Ammonia 2.060 1.561 1.32
Carbon Dioxide 0.825 0.630 1.31
Carbon Monoxide 1.051 0.751 1.40
Helium 5.234 3.153 1.66
Hydrogen 14.235 10.096 1.41
Hydrogen Sulphide 1.105 0.85 1.30
Methane 2.177 1.675 1.30
Nitrogen 1.043 0.745 1.40
Oxygen 0.913 0.652 1.40
Sulphur Dioxide 0.632 0.451 1.40
Page 22
Efficiency of Heat Engines
Carnot Cycle η=T1–T2T1 where T1 and T2 are absolute temperatures of heat source and sink
Air Standard Efficiencies
1. Spark Ignition Gas and Oil Engines (Constant Volume Cycle or Otto Cycle)
1) - (γvr1 - 1 η= where rv = compression ratio = volumeclearanceolumecylinder v
γ = volume)(constant heat specificpressure)(constant heat specific
2. Diesel Cycle
)1 - γ(R r)1(R - 1 η1 - γvγ−= where r = ratio of compression
R = ratio of cut-off volume to clearance volume
3. High Speed Diesel (Dual-Combustion) Cycle
[]1) - k(βγ 1) -(k r1 - kβ - 1 η1 - γvγ+=
where rv = volumeclearanceolumecylinder v
k = combustion Vconstant of beginningat pressue absoluten)(combustio heating Vconstant of endat pressue absolute
β = volumeclearancen)(combustio heating Pconstant of endat volume
4. Gas Turbines (Constant Pressure or Brayton Cycle)
⎟⎟⎠⎞⎜⎜⎝⎛−=γ1γpr1 - 1 η
Page 23
where rp = pressure ratio = pressure intake compressorpressure discharge compressor
Page 24
Heat Transfer by Conduction
Q=λAtΔTdwhereQ=heattransferredinjoulesλ=thermalconductivityorcoeficientofheattransferinJ×mm2×s×°CorWm×°CA=areainm2t=timeinsecondsΔT=temperaturedifferencebetweensurfacesin°Cd=thicknessoflayerinm
COEFFICIENTS OF THERMAL CONDUCTIVITY
Material
Coefficient of
Thermal Conductivity
W/m °C
Air
Aluminum
Brass
Brick
Concrete
Copper
Cork
Felt
Glass
Glass, fibre
Iron, cast
Plastic, cellular
Steel
Wood
Wallboard, paper
0.025
206
104
0.6
0.85
380
0.043
0.038
1.0
0.04
70
0.04
60
0.15
0.076
Page 25
Thermal Expansion of Solids
Increase in length = L α (T2 – T1 )
where L = original length
α = coefficient of linear expansion
(T2 – T1 ) = rise in temperature
Increase in volume = V β (T2 – T1 )
Where V = original volume
β = coefficient of volumetric expansion
(T2 – T1 ) = rise in temperature
coefficient of volumetric expansion = coefficient of linear expansion x 3
β = 3α
Page 26
Chemical Heating Value of a Fuel
Chemical Heating Value MJ per kg of fuel = 22O33.7 C + 144 H - + 9.3 S8()
C is the mass of carbon per kg of fuel
H2 is the mass of hydrogen per kg of fuel
O2 is the mass of oxygen per kg of fuel
S is the mass of sulphur per kg of fuel
Theoretical Air Required to Burn Fuel
Air (kg per kg of fuel) = 228O100C + 8 H - + S 38[()]23
Air Supplied from Analysis of Flue Gases
Air in kg per kg of fuel = N233(CO2+CO)×C
C is the percentage of carbon in fuel by mass
N2 is the percentage of nitrogen in flue gas by volume
CO2 is the percentage of carbon dioxide in flue gas by volume
CO is the percentage of carbon monoxide in flue gas by volume
Boiler Formulae
Equivalent evaporation = kJ/kg 2257)h - (h m21s
Factor of evaporation = kJ/kg 2257)h - (h 21
Boiler efficiency = fuel of valuecalorific x m)h - (h mf21s
where = mass flow rate of steam sm
h1 = enthalpy of steam produced in boiler
h2 = enthalpy of feedwater to boiler
= mass flow rate of fuelfm
Page 27
FLUID MECHANICS
Discharge from an Orifice
Let A = cross-sectional area of the orifice = (π/4)d2
and Ac = cross-sectional area of the jet at the vena conrtacta = ((π/4)2cd
then Ac = CcA
or Cc = 2ccddAA⎟⎠⎞⎜⎝⎛=
where Cc is the coefficient of contraction
At the vena contracta, the volumetric flow rate Q of the fluid is given by
Q=areaofthejetatthevenacontracta×actualvelocity=AcvorQ=CcACv2gh
The coefficients of contraction and velocity are combined to give the coefficient of discharge, Cd
i.e.Cd=CcCvandQ=CdA2gh
Typically, values for Cd vary between 0.6 and 0.65
Circular orifice: Q = 0.62 A 2gh
Where Q = flow (m3/s) A = area (m2) h = head (m)
Rectangular notch: Q = 0.62 (B x H) 232gh
Where B = breadth (m) H = head (m above sill)
Triangular Right Angled Notch: Q = 2.635 H5/2
Where H = head (m above sill)
Page 28
Bernoulli’s Theory
H = 2gvwP h 2++
H = total head (metres) w = force of gravity on 1 m3 of fluid (N)
h = height above datum level (metres) v = velocity of water (metres per second)
P = pressure (N/m2 or Pa)
Loss of Head in Pipes Due to Friction
Loss of head in metres = fLdv22g
L = length in metres v = velocity of flow in metres per second
d = diameter in metres f = constant value of 0.01 in large pipes to 0.02 in small pipes
Note: This equation is expressed in some textbooks as
Loss = 4fLdv22g where the f values range from 0.0025 to 0.005
Actual Pipe Dimensions
Page 29
ELECTRICITY
Ohm's Law
I = RE
or E = IR
where I = current (amperes)
E = electromotive force (volts)
R = resistance (ohms)
Conductor Resistivity
R = ρaL
where ρ = specific resistance (or resistivity) (ohm metres, Ω·m)
L = length (metres)
a = area of cross-section (square metres)
Temperature correction
Rt = Ro (1 + αt)
where Ro = resistance at 0ºC (Ω)
Rt = resistance at tºC (Ω)
α = temperature coefficient which has an average value for copper of 0.004 28 (Ω/ΩºC)
R2 = R1)αt (1)αt (112++
where R1 = resistance at t1 (Ω)
R2 = resistance at t2 (Ω)
α Values Ω/ΩºC
copper 0.00428
platinum 0.00385
nickel 0.00672
tungsten 0.0045
aluminum 0.0040
Page 30
Dynamo Formulae
Average e.m.f. generated in each conductor = 2ΦNpZ60c
where Z = total number of armature conductors
c = number of parallel paths through winding between positive and negative brushes where c = 2 (wave winding), c = 2p (lap winding)
Φ = useful flux per pole (webers), entering or leaving the armature
p = number of pairs of poles
N = speed (revolutions per minute)
Generator Terminal volts = EG – IaRa
Motor Terminal volts = EB + IaRa
where EG = generated e.m.f.
EB = generated back e.m.f.
Ia = armature current
Ra = armature resistance
Alternating Current
R.M.S. value of sine curve = 0.707 maximum value
Mean value of sine curve = 0.637 maximum value
Form factor of sinusoidal = 11.10.6370.707 Mean value valueR.M.S.==
Frequency of alternator = 60pN cycles per second
Where p = number of pairs of poles
N = rotational speed in r/min
Page 31
Slip of Induction Motor
100 x field of Speedrotor of speed - field of speed Slip
Inductive Reactance
Reactance of AC circuit (X) = 2πfL ohms
where L = inductance of circuit (henries)
Inductance of an iron cored solenoid = henries10x LμAT256.182
where T = turns on coil
μ = magnetic permeablility of core
A = area of core (square centimetres)
L = length (centimetres)
Capacitance Reactance
Capacitance reactance of AC circuit = πfC21ohms
where C = capacitance (farads)
Total reactance = ohmsfC 2π1- πfL2⎟⎠⎞⎜⎝⎛
Impedence (Z) = 22reactance)(e)(resistanc+
= ohms )fC π21 - fL (2π R22+
Current in AC Circuit
impedance voltsimpressed Current =
Page 32
Power Factor
p.f. = amperes x volts wattstrue
also p.f. = cos Φ, where Φ is the angle of lag or lead
Three Phase Alternators
Star connected
Line voltage = 3 x phase voltage
Line current = phase current
Delta connected
Line voltage = phase voltage
Line current = 3 x phase current
Three phase power
P = 3 EL IL cos Φ
EL = line voltage
IL = line current
cos Φ = power factor
Page 33
Page 34
ION NAMES AND FORMULAE
MONATOMIC POLYATOMIC
Ag+ silver ion BO33- borate ion
Al3+ aluminum ion C2H3O2- acetate ion
Au+ and Au2+ gold ion ClO- hypochlorite ion
Be2+ beryllium ion ClO2- chlorite ion
Ca2+ calcium ion ClO3- chlorate ion
Co2+ and Co3+ cobalt ion ClO4- perchlorate ion
Cr2+ and Cr3+ chromium ion CN- cyanide ion
Cu+ and Cu2+ copper ion CO32- carbonate ion
Fe2+ and Fe3+ iron ion C2O42- oxalate ion
K+ potassium ion CrO42- chromate ion
Li+ lithium ion Cr2O72- dichromate ion
Mg2+ magnesium ion HCO3- hydrogen carbonate or bicarbonate ion
Na+ sodium ion H3O+ hydronium ion
Zn2+ zinc ion HPO42- hydrogen phosphate ion
H2PO4- dihydrogen phosphate ion
HSO3- hydrogen sulphite or bisulphite ion
HSO4- hydrogen sulphate or bisulphate ion
MnO4- permanganate ion
N3- azide ion
NH4+ ammonium ion
NO2- nitrite ion
NO3- nitrate ion
O22- peroxide ion
OCN- cyanate ion
OH- hydroxide ion
PO33- phosphite ion
PO43- phosphate ion
SCN- thiocyanate ion
SO32- sulphite ion
SO42- sulphate ion
S2O32- thiosulphate ion
Page 35
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