1
1
Nome Nomenc ncla latu ture re Roman
a A a D Dh f F F g C c p cv COP e E h h H k k K L m M n n N p P q Q R R R ˜ R s S t T u U U v V
2
acceleration area speed of sound diameter hydraulic diameter Fanning friction factor force view factor acceleration due to gravity heat capacity const pressure pressure specific specific heat heat capacity capacity const const volum volumee specific specific heat heat capaci capacity ty coefficient of performance specific energy energy conve convecti ctive ve heat heat transf transfer er coefficie coefficient nt specific enthalpy enthalpy loss coefficient thermal conductivity bulk modulus length or length scale mass molecular mass number of moles polytropic index Avagadro’s number pressure perimeter specific heat energy heat energy radius thermal resistance specific gas constant universal gas constant (8.3145 103 ) specific entropy entropy time temperature specific internal energy internal energy overal eralll heat heat tran transf sfer er coeffi coeffici cien entt specific volume volume
×
m.s 2 m2 m.s 1 m m N m.s 2 J.K 1 J.kg 1 .K 1 J.kg J.kg 1 .K 1 J.kg 1 J W.m 2 .K 1 J.kg 1 J W.m 1 .K 1 Pa or N.m 2 m kg kg.mol 1 mol 1 Pa or N.m 2 m J.kg 1 J m K.W 1 J.kg 1 .K 1 J.mol 1 .K 1 J.kg 1 .K 1 J.K 1 s K J.kg 1 J W.m W.m 2 .K 1 m3 .kg 1 m3 −
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
m.s J.kg J m
1
−
1
−
Greek
−
−
velocity specific work energy work energy dryness fraction change of elevation
V w W x ∆z
−
−
α β δ η γ µ ν ρ ρ σ τ τ T ω
absorptivity volume expansion coeffi oefficient boundary layer thickness wall roughness effectiveness emisivity efficiency c ratio of specific heats ( cpv ) dynamic viscosity kinematic viscosity density reflectivity Stefan-Boltzmann constant shear streass transmitivity torque angular velocity
m m Pa.s m 2 .s kg.m W.m N.m N.m s 1
X 1,2,3..etc x,y,z,r,θ ∆X δX ˙ X X X ∗ X ◦ X ◦ X X X i X e X H H X C C X f f X g X fg fg
Other location or instantaneous value of X spacial spacial coordinates, coordinates, radius radius and and angle angle finite change of X infinitesimal change of X rate of X a vector X critical value of X stagnation value of X value of X at STP average of X modified value of X inlet value of X exit value of X hot value of X cold value of X value of X at saturated liquid value of X at saturated vapour change in X between X f f and X g
-
1
−
3
−
2
−
.K
2
−
1
−
−
Mate Materi rial al Prope Propert rtie iess
2.1
Viscosi Viscosity ty variation ariation with temperatu temperature re
• Exponential model for liquids:
µ = µ0
where µ0 , B and C are constants.
• For water µ0=2.414×105 Pa.s, B=247.8 K and C =140 =140 K. 2
× 10
B
(T −C )
(1)
4
−
• Poiseuille formula for dynamic viscosity:
µ = µ0
1 1 + AT + BT 2
(2)
◦
where µ0 , A and B are constants and T is the temperature in C.
• For water, the value of µ0 is 0.00179 Pa.s, and the values of constants A and B are 0.033368
◦
1
C
−
2
and 0.000221 C ◦
−
,
respectively.
2.2
Mater Material ial prop propert ertie iess for air air and wate water r
Temp.
Water µ( 10
◦
C 10 20 25 30 40 50 60 70 80 90 100
3
×
3
−
)
Pa.s 1.31 1 0.91 0.8 0.65 0.55 0.47 0.4 0.36 0.32 0.28
ρ
Temp.
k
kg.m 3 1000 998 997 996 992 988 983 978 971 965 958 −
W.m 1 .K 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.68 −
1
−
c p
µ( 10
kJ.kg 1 .K 4.195 4.182 4.178 4.167 4.175 4.178 4.181 4.187 4.194 4.202 4.211 −
Air at 1 atm
1
−
×
◦
C -150 -100 -50 0 20 40 60 80 100 200 400
6
−
Pa.s 8.60 11.8 14.6 17.2 18.2 19.1 20.2 20.9 21.8 25.8 32.7
)
ρ
k
kg.m 3 2.79 1.98 1.53 1.29 1.21 1.13 1.07 1.00 0.95 0.62 0.52 −
W.m 1 .K 0.012 0.016 0.020 0.024 0.026 0.027 0.029 0.030 0.031 0.039 0.052 −
1
−
c p
kJ.kg 1 .K 1.026 1.009 1.005 1.005 1.005 1.005 1.009 1.009 1.009 1.026 1.068 −
1
−
Newt Newton on’s ’s laws laws of mo moti tion on
• Newton’s laws of motion First Every object remains in a state of rest or in uniform motion in a straight line unless acted upon by a (nett) force. Second F = m
×a
Third For every action there is an equal and opposite reaction.
• Equations of linear motion V 2 x2 x2 V 22
4
= V 1 + a∆t
(3)
1 = x1 + V 1 ∆t + a (∆t)2 2 1 = x1 + (V 2 + V 1 ) ∆t 2 2 = V 1 + 2a (x2 x1 )
−
(4) (5) (6)
Flui Fluid d Mec Mechani hanics cs
4.1 4.1
Flui Fluid d Stat Static icss
• Pascal’s law • Force on a submerged plane
dp = ρg dz
y p =
I G + yG A.yG
(7)
(8)
where y p is the distance to the centre of pressure and yG is distance to the centre of gravity, measured along the surface of the plane. I G is the second moment of area about the centroid, A is the area of the submerged plane.
3
4.2 4.2
Flo Flow in pipe pipess
• Continuity ˙ m dm ˙ dt
• Hydraulic mean diameter
= ρV A
(9)
= m ˙ in
(10)
Dh =
4
− m˙ out
×A
(11)
P
• The Hagen-Poiseuille equation V
=
dp − 41µ dx
˙ V
=
pR − π∆8µL
R2
− r2
4
(12) (13)
• Steady flow Energy Equation (SFEE) pin +
2 ρV in
2
+ ρgzin + ρw p = pout +
2 ρV out
+ ρgz out + ρwf + ρwt
2
(14)
where ρwf is the volumetric work lost due to friction, ρw p is the volumetric work supplied by a pump and ρwt is the volumetric work generated by a turbine.
• Darcy’s Equation for losses in long pipes wf = 4 f
L V 2 D 2
(15)
• Fanning friction factors 16 laminar flow Re f = (0.79l 79ln n (ReD ) 1.64) f =
√1f
=
−1.8log10
− 6.9 + Re
(16) 2
−
1 3.71 D
(17)
1.11
(18)
• Typical pipe roughnesses given below: Material Coarse concrete Smooth concrete Drawn tubing Glass, Plastic, Perspex pex Cast Iron Old Sewers Mortar lined steel Rusted steel Forged steel Old water mains
• Loss coefficient for piping network components wf = k
where k is the loss coefficient,
• values of k are given in table below:
4
Roughness (mm) 0.25 0.025 0.0025 0.0025 0.15 3.0 0.1 0.5 0.025 1.0 V 2
2
(19)
Component Sharp Entry Rounded Entry Contraction (50% area) Contraction (50% diameter,based on V 2 ) Expansion (based on V 2 ) 180o elbow 90o elbow 45o elbow Globe valve (open) Angle valve (open) Gate valve (open) Gate valve (25% closed) Gate valve (50% closed) Gate valve (75% closed) Angle valve (open) Swing check valve (open) Ball valve (open) Ball valve (33% closed) Ball valve (66% closed) Diaphragm valve (open) Diaphragm valve (50% closed) Diaphragm valve (75% closed) Water meter
• Moody Diagram:
5
k 0.5 0.25 0.24 0.35
− A2 A1
0.9 0.9 0.4 10 2 0.15 0.25 2.1 17 2 2 17 5.5 200 2.3 4.3 21 7
1
4.3
Conserv Conservatio ation n of linear linear momen momentum tum
• Force on fluid in control volume 4.4 4.4
F = m ˙ out V out out
− m˙ inV inin
(20)
Lift Lift and and drag drag
• Lift force
1 2
(21)
1 2
(22)
F L = C L ρV 2 A
where C L is the coefficient of lift
• Drag force
F D = C D ρV 2 A
where C D is the coefficient of drag
• Coefficients of skin friction drag for laminar flow over flat plate 1.328 Re < 105 C D = √ Re
(23)
• Coefficients of skin friction drag for turbulent flow over flat plate C D = C D =
0.2
Re
0.455 (log(Re))2.58
• Coefficient Coefficientss of form drag around a cylinder cylinder 4.5
0.074
C D =
105 < Re < 107
(24)
107 < Re < 109
24
(25)
(Re < 1)
Re
(26)
Compr Compres essib sible le flow flow
• Isothermal compressible flow in a constant cross section pipe, neglecting change in gravitational potential energy p22
=
p21
+ 2RT
m ˙ A
• Speed of sound a=
• Change in velocity with area of nozzel
2
−
4f L D
RT
K = ρ
∂V = V
• SFEE for isentropic compressible flow
p2 ln p1
m ˙ A
2
(27)
γRT
(28)
1 − ∂A A 1 − M 2
(29)
1 2 1 V 1 + c p T 1 = V 22 + c p T 2 2 2
(30)
• Adiabatic, isentropic, compressible flow γ −1 T 2 p2 ( γ ) = T 1 p1
T 2 = T 1
v1 v2
γ −1
(31)
• Stagnation conditions T T 0
=
p p0
=
1
T = T 0 +
1 + M 2 γ 2 1 1 −
1 + M 2 γ 2 1
−
6
γ γ −1
V 2 2c p
(32) (33)
• Critical conditions T T 0
∗
p p0
∗
=
2 γ + 1
=
(34)
2 γ + 1
γ γ −1
(35)
where T and p are the critical temperature and pressure, respectively.
∗
4.6 4.6
∗
Water ater Hamm Hammer er
• Pressure drop due to water hammer.
∆ p = ρV a
(36)
• Augmented bulk modulus (K ) for non-rigid pipes. 1
1
D = + K K tE
a=
g iving
K ρ
(37)
where D is the internal diameter of the pipe, t is the thickness of the pipe wall and E is the Young’s modulus. pD 2t
σθ =
(38)
where σθ is the hoop stress, t is the thickness of the pipe wall and E is the Young’s modulus.
5
Heat Heat Trans ransfe fer r
5.1
Therm Thermal al expan expansi sion on
• Linear expansion
β
∆L = L1 ∆T 3
• Area expansion
∆A = A1
• Volumetric expansion where
5.2 5.2
β 3
2β ∆T 3
∆V = V 1 β ∆T
(39)
(40)
(41)
is the coefficient of linear expansion, sometimes referred to as α in other texts.
1D heat heat tran transf sfer er
• Conduction
Q˙ = k
• Conduction in thick walled cylinder
A ∆T L
Q˙ = 2 πkL
T 1 ln
• Convection
(42) T 2
− R2 R1
Q˙ = hA∆T
(43)
(44)
where h can be found using the Nusselt number, given in equation 99.
• Resistor analogy for composite surfaces
Q˙ =
∆T R1 + R2 + R3 + . . . + Rn
7
(45)
L kA
=
Rcond, cond,planar
R2 R1
ln Rcond, cond,clyind
=
Rconvect
=
(46)
(47)
2πkL 1
(48)
hA
(49)
• Radiation
Q˙ = σA T 14
where σ is the Stefan-Boltzmann constant, of 5.67051
5.3
8
−
− T 24
W.m
2
−
(50)
.K
4
−
Radia Radiatio tion n heat heat transf transfer er view view factor factorss
• Radiation equation with view factors
4 Q˙ ij = Ai F ij ij σ i T i
• Reciprocity Relation 5.4
×10
− T j4
(51)
Ai F ij ij = Aj F ji ji
(52)
Forced orced conve convectio ction n
• For an isothermal flat plate
1
1
Nu = 0.032ReL2 Pr 3
valid for Re L < 105 and 0.6 < Pr < 60.
4
(53)
1
Nu = 0 .0296ReL5 Pr 3
(54)
valid for 10 8 > ReL > 105 and 0.6 < Pr < 60. 4
1
Nu = 0.037ReL5 Pr 3
(55)
valid for Re L > 108 and 0.6 < Pr < 60.
• For an isothermal horizontal cylinder
1
3 Nu = C Re Rem D Pr
(56)
C = 0 .193, m = 0 .618 for 4000 < Re < 40000 and C = 0 .027, m = 0 .805 for 40000 < Re < 400000.
• Dittus-Boelter equation for forced convection in pipes. 4
5 Nu = 0 .023ReD Prn
(57)
where n = 0 .4 for heating fluid and n = 0 .3 for cooling fluid. Valid for Re
• Log mean temperature difference
∆T LM LM =
∆T 1
≥ 10000 and 0.7 ≤ Pr ≥ 160
− ∆T 2
(58)
T 1 ln ∆ ∆T 2
• Exit temperature ( T e) for constant wall temperature pipe T e = T w + (T i
− T w ) exp
−
where T i is the inlet temperature and tempw is the wall temperature.
8
hA mc ˙ p
(59)
5.5
Heat Heat exc excha hange nger r des design ign
• Capacity rates (m˙ × c p) C min min = min(C H H , C C C )
• Correction factor F
C ∗ =
C max max = max(C H H , C C C )
C min min C max max
(60)
q˙ = F U A∆T LM LM
(61)
where F is a function of P and R: P =
T tube,o tube,o T shell,i shell,i
• Effectiveness-NTU
− T tube,i tube,i − T tube,i tube,i
R=
Q˙ max = C min min (T H,i H,i =
5.6
C tube T shell,i tube shell,i = C shell T tube,o shell tube,o
Q˙ act
− T shell,o shell,o − T tube,i tube,i
(62)
− T C,i C,i )
(63) (64)
Q˙ max
Natura Naturall conve convecti ction on
• Free convection at a vertical wall (Churchill and Chu) 1/4
NuL = 0 .68 + for RaL
≤ 109.
0.67RaL
492 9/16 1 + ( 0.Pr )
(65)
4/9
• For the horizontal surface with top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment.
NuL NuL
1
= 0.54RaL4 1 3
= 0.15RaL
104 < RaL < 107
(66)
107 < RaL < 1011
(67) (68)
• For the horizontal surface with the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment.
NuL
5.7 5.7
1
= 0.27RaL4
105 < RaL < 1010
(69)
Com Combust bustio ion n
• Molar masses of atoms and heats of combustion of fuels Chem Chemic ical al H He C N O
M (g.m (g.mol ol 1 4 12 14 16
1
−
)
Fuel Petrol Diesel Ethanol (C2 H5 OH) Methane (CH4 ) Ethane (C2 H6 )
LCV (MJ.kg 44.4 50 47.8
1
−
)
HCV HCV (MJ.k MJ.kg g 47.3 44.8 29.7 55.5 51.9
1
−
)
• Volume occupied by one mole, from the ideal gas equation ˜ V RT = n p At 1 bar and 25 degrees C, the recommended value is 24.5 litres per mole.
9
(70)
6 6.1
Ther Thermod modyn ynam amic icss Laws Laws of Therm Thermody odynam namics ics
Zeroth If two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal equilibrium with each other. First Energy can neither be created nor destroyed. It can only change from one form to another. Second The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium. Third As temperature approaches absolute zero, the entropy of a system approaches a constant minimum.
6.2
Method Method for Solving Solving Therm Thermodyna odynamics mics Problem Problemss
10
6.3
Conve Convent ntions ions for for Conserv Conservatio ation n of Energy Energy
• In finite form • In rate form 6.4
E 1 + E 2 + E 3 + ... + E n = ∆ E
(71)
dE dt
(72)
˙1 + E ˙2 + E ˙3 + ... + E ˙n = E
Thermody Thermodynam namic ic relation relationss
• Ratio Ratio of specific heats: heats:
γ =
c p cv
(73)
• Enthaply H =
U + pV
= u + pv dh = c p dT
(74) (75) (76)
= cv dT
(77)
h
• Internal energy du
• Gas constant ˜ = 8 .314 kJ.kmol where R
R = c p 1
−
K
− cv
R=
˜ R M
(78)
1
−
• Polytropic processes
pV n = constant
(79)
where n is the polytropic constant. Process Isochoric (const volume) Isothermal (const temperature) Isobaric (const pressure) Isentropic (const entropy)
n n = inf n =1 n =0 n = γ
• Entropy q ds s2
− s2
= sdT =
(80)
dq T
= cv ln
(81) T 2 v2 T 2 p2 + Rln = c p ln + Rln T 1 v1 T 1 p1
11
(82)
• Isentropic expansion and compression γ −1 T 2 p2 ( γ ) = = T 1 p1
v1 v2
(γ −1)
p2 = p1
γ T 2 ( γ −1 ) = T 1
v1 v2
(γ )
v2 = v1
1 1 T 1 ( 1−γ ) p1 ( γ ) = T 2 p2
(83)
The value of the ratio of specific heats, γ , can usually be considered to be 1.4 for air.
6.5 6.5
Gas Laws Laws
• Boyle’s law for a constant temperature (isothermal) process pV = const
p1 V 1 = p2 V 2
(84)
V 1 V 2 = T 1 T 2
(85)
p1 p2 = T 1 T 2
(86)
• Charles’ law for a constant pressure (isobaric) process V = const T
• Gay-Lussac’s law for constant volume (isochoric) process p = const T
• Ideal gas equation: ˜ nRT
= pV = pv = p = pV
(87) (88) (89) (90)
mRT RT ρRT
• Thermodynamic properties of common gasses at STP. Gas Air Carbon dioxide Hydrogen Methane Natural Gas Nitrogen Oxygen
6.6
c p (kJ.kg−1 .K−1 )
cv (kJ.kg−1 .K−1 )
γ
R (kJ.kg−1 .K−1 )
1.005 0.884 14.32 2.22 2.34 1.04 0.919
0.718 0.655 10.16 1.70 1.85 0.743 0.659
1.40 1.289 1.41 1.30 1.27 1.40 1.40
0.287 0.189 4.12 0.518 0.5 0.297 0.260
Therm Thermodyn odynam amic ic devic devices es
• Work done
2
W 12 12 =
(91)
pdV
1
• Heat engine thermal efficiency:
ηth =
w˙ qH ˙
(92)
where qH is the heat energy from the hot source.
• Carnot thermal efficiency:
ηth,car = 1
− T T L
(93)
H H
• COP of a heat pump and refrigerator: C OP HP HP =
q˙H
˙ in W in
where qL is the heat energy from the cold source. 12
C OP R =
q˙L
˙ in W in
(94)
• Carnot COP:
C OP HP HP =
1 1
−
C OP R =
T L T H
1 T H T L
(95)
−1
• Isentropic efficiencies, compressors and turbines ηC =
hout,s hout
− hin − hin
ηT =
hin hin
− hout − hout,s
(96)
• Spark ignition and compression ignition engine thermal efficiency. SPARK : ηth = 1
− 1
1
rvγ −
COMPRESSION : ηth = 1
− 1
1
rvγ −
rcγ γ (rc
−1 − 1)
(97)
where rc is the cut off ration and rv is the compression ratio
7
Dime Dimens nsion ionle less ss Numbers Numbers
• Reynolds number • Nusselt number • Mach number • Prandtl number • Grashof number where β is
1 T
• Biot number
8 .1
(98)
Nu =
hL k
(99)
M=
V a
(100)
c p µ k
(101)
3 gβρ gβ ρ2 (T S T f f ) L µ2
−
(102)
× Pr
(103)
Pr =
Gr =
for an ideal gas.
• Rayleigh number
8
ρU D µ
Re =
Ra = Gr
Bi =
hL k
(104)
Appro Approxim ximate ate Values alues for for Quan Quantit tities ies Mass
10 g 100 g 50 500 0g 1 kg 1.5 1.5 kg 70 kg 10 1000 00 kg 1500 150 0 kg kg
8.2 8.2
a pen mobile phone bot bottle tle of drin drink k house brick bag bag of suga sugarr person mass ma ss of smal smalll car car mass mass of of fami family ly saloon saloon
Len Length gth
8 cm 20 cm 30 cm 2m 30 m 13
finger widt width h A4 shee sheett leng length th A4 shee sheett doo oorr height Owen Owen buil buildi ding ng
8.3 8.3
Volum olume e
5 ml 25 ml 330 ml 568 ml 750 ml 2l 2,500, 2,5 00,000 000 l
8.4 8.4
tea spoon shot of whisky can of coke a pint of beer bot bottle of wine bottle of coke Olymp Olympic ic swim swimmin ming g pool
walki alking ng running spri sprin nter ter reside resident ntial ial speed speed limit limit moto mo torw rwaay speed speed limi limitt Brit Britis ish h trai train n good goo d trai train n speed speed of of soun sound d in air air
8.6 8.6
8.7 8.7
air pet petrol oil water ater alum alumin iniu ium m stee steell mercur mercury y ◦
− −
CO2 /Ar/He/N 2 air Petr Petrol ol water oil honey
Power
10 W 10 100 0W 2.5 kW 10 100 0 kW 5 MW 4,000 4,0 00 MW 40,000 40, 000 MW
energy saving lightbulb elec electr tric ic ligh lightt bulb bulb (inc (incan ande desc scen ent) t) domestic kettle Car Car eng engine ine (13 135 5 bhp bhp) Big wind turbine Drax Drax power power statio station n output output UK power power consum consumpti ption on
14
Cork Cork Styro Styrofoa foam m Fibr Fibreg egla lass ss Genera Generall insula insulatio tion n Paper per Wood (balsa (balsa)) Wood (pin (pine) e) Leat Leathe herr Wood (oak (oak)) Plas Plaste terbo rboar ard d PVC HD Poly olyethe ethene ne Asph Asphal altt
Further urther condu conductiv ctivity ity of solid solidss
0.00 0.004 4 W/ W/mK mK 0.033 0.0 33 W/m W/mK K 0.04 0.04 W/ W/mK mK 0.04 0.0 4 W/m W/mK K 0.0 0.05 W/ W/m mK 0.055 0.0 55 W/m W/mK K 0.12 0.12 W/ W/mK mK 0.14 0.14 W/ W/mK mK 0.15 0.15 W/ W/mK mK 0.17 0.17 W/ W/mK mK 0.2 W/mK 0.5 0.5 W/ W/mK mK 0.75 0.75 W/ W/mK mK 1 W/mK 1.0 1.05 W/ W/m mK 16 W/ W/m mK 35 W/mK 55 W/mK 109 W/mK 25 250 0 W/ W/mK mK 400 W/mK
Visc Viscos osit ity y @ 20 C
1 10 5 Pa.s 2 10 5 Pa.s 0.00 0.000 0 6 Pa.s Pa.s 0.001 Pa.s 0.2 Pa.s 2 Pa.s
× ×
8.9
Dens Densit ity y
1.2 kg/m3 700 kg/m3 800 kg/m3 1,00 1,000 0 kg/m kg/m3 3 2,70 2,700 0 kg/m kg/m3 3 7,80 7,800 0 kg/m kg/m3 3 13,500 13, 500 kg/ kg/m3 m3
Conduc Conductiv tivit ity y of of soli solids ds
0.00 0.004 4 W/ W/mK mK 0.033 0.0 33 W/m W/mK K 0.04 0.04 W/ W/mK mK 0.04 0.0 4 W/m W/mK K 0.0 0.05 W/ W/m mK 0.055 0.0 55 W/m W/mK K 0.12 0.12 W/ W/mK mK 0.14 0.14 W/ W/mK mK 0.15 0.15 W/ W/mK mK 0.17 0.17 W/ W/mK mK 0.2 W/mK 0.5 0.5 W/ W/mK mK 0.75 0.75 W/ W/mK mK
Veloci elocitty
1.5 1.5 m/ m/ss 3 m/s 10 m/s 13 m/s 30 m/ m/ss 45 m/ m/ss 15 150 0 m/ m/ss 33 330 0m m/s /s
8.5 8.5
8.8
Cork Cork Styro Styrofoa foam m Fibr Fibreg egla lass ss Genera Generall insula insulatio tion n Paper per Wood (balsa (balsa)) Wood (pin (pine) e) Leat Leathe herr Wood (oak (oak)) Plas Plaste terbo rboar ard d PVC HD Poly olyethe ethene ne Asph Asphal altt Bricks Glass lass Stai Stainl nles esss Ste Steel Lead Carbon bon Steel Brass Alum Alumin iniu ium m Copper per
9
Spa Space for for you your r not notes es
15
10
Credits
Compiled and edited by Dr Andrew Garrard (
[email protected]) Cover Cover illustrat illustration ion by Jack Jack Good ( www.jackgood.co.uk)
All material within this book is held under copyright c 2011 Andrew Garrard, Sheffield Hallam Univesity. Permission is granted to copy and freely distribute provided credit is given to the original author.
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