Data and Formulae for Mechanical Engineering Students

Data and Formulae for Mechanical Engineering Students

Department of Mechanical Engineering, Imperial College London

September 2009

Contents

A General information 1

B Mathematics and computing 5B.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

B.1.1 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5B.1.2 Quadratic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5B.1.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5B.1.4 Vector algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6B.1.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6B.1.6 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7B.1.7 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8B.1.8 Analytic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9B.1.9 Solid geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10B.1.10 Differential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10B.1.11 Standard Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11B.1.12 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

B.2 Integral calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12B.3 Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14B.4 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

B.4.1 Approximate solution of an algebraic equation . . . . . . . . . . . . . . . 15B.4.2 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15B.4.3 Richardson’s error estimation formula for use with Simpson’s rule . . . . . 16B.4.4 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

B.5 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17B.6 Probabilities for events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

B.6.1 Distribution, expectation and variance . . . . . . . . . . . . . . . . . . . . 17B.6.2 Probability distributions for a continuous random variable . . . . . . . . . 18B.6.3 Discrete probability distributions . . . . . . . . . . . . . . . . . . . . . . . 18B.6.4 Continuous probability distributions . . . . . . . . . . . . . . . . . . . . . . 19B.6.5 System reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

B.7 Bias, standard error and mean square error . . . . . . . . . . . . . . . . . . . . . 19B.7.1 Central limit property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19B.7.2 Confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

C Mechatronics and control 21C.1 Charge, current, voltage and power . . . . . . . . . . . . . . . . . . . . . . . . . . 21C.2 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22C.3 Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

i

CONTENTS

C.4 AC networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23C.4.1 Average and root mean square values . . . . . . . . . . . . . . . . . . . . 23C.4.2 Phasors and complex impedance . . . . . . . . . . . . . . . . . . . . . . . 24C.4.3 Balanced 3 phase a.c supply . . . . . . . . . . . . . . . . . . . . . . . . . 24C.4.4 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24C.4.5 DC machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25C.4.6 Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

C.5 Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26C.6 Step function response and frequency response . . . . . . . . . . . . . . . . . . 26

C.6.1 First-order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26C.6.2 Second-order systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

C.7 Operational amplifier stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

D Solid Mechanics 33D.1 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

D.1.1 Square screw threads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33D.1.2 Flat clutches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33D.1.3 Kinematics of particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33D.1.4 Mass flow problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34D.1.5 Kinematics of rigid bodies with sliding contacts . . . . . . . . . . . . . . . 34D.1.6 Mass moments of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

D.2 Stress analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35D.2.1 Elastic constants of materials . . . . . . . . . . . . . . . . . . . . . . . . . 35D.2.2 Beam theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35D.2.3 Elastic torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38D.2.4 Thin walled pressure vessels . . . . . . . . . . . . . . . . . . . . . . . . . 39

D.3 Two-dimensional stress transformation . . . . . . . . . . . . . . . . . . . . . . . . 39D.4 Yield criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40D.5 Two-dimensional strain transformation . . . . . . . . . . . . . . . . . . . . . . . . 40D.6 Elastic stress-strain relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . 40D.7 Thick-walled cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

E Thermofluids 43E.1 Cross-references to table numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 43E.2 Dimensionless groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44E.3 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45E.4 Continuity and equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 46

E.4.1 Cylindrical polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 46E.4.2 Rectangular Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . 46E.4.3 Vector form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

E.5 Equations for compressible flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 47E.6 Friction factor for flow in circular pipes (Moody diagram) . . . . . . . . . . . . . . 48E.7 Perfect gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50E.8 Heating (or calorific) values of fuels . . . . . . . . . . . . . . . . . . . . . . . . . . 54E.9 Properties of R134a refrigerant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55E.10 Transport properties of air, water and steam . . . . . . . . . . . . . . . . . . . . . 62E.11 Approximate physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 65E.12 Thermodynamic property tables for water/steam (IAPWS-IF97 formulation) . . . 67

Page ii Mechanical Engineering Data & Formulae

LIST OF TABLES

List of Tables

A.1 SI Units and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1A.2 Conversion factors from Imperial to SI units . . . . . . . . . . . . . . . . . . . . . 2A.3 Decimal prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3A.4 Physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3B.1 Some indefinite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13B.2 Some definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13B.3 Standard normal table: values of pdf φ(y) = f (y) and cdf Φ(y) = F (y). . . . . . . 20B.4 Student t table: values tm,p of x for which P (|X | > x) = p, when X is tm. . . . . . 20C.1 Colour codes for resistors etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21C.2 Standard values for components . . . . . . . . . . . . . . . . . . . . . . . . . . . 22C.3 Operational amplifier signal processing stages . . . . . . . . . . . . . . . . . . . 31D.1 Second moments of area for simple cross-sections . . . . . . . . . . . . . . . . . 36D.2 Beams bent about principal axis . . . . . . . . . . . . . . . . . . . . . . . . . . . 37D.3 Torsion of solid non-circular sections . . . . . . . . . . . . . . . . . . . . . . . . . 38E.1 Dimensionless groups for Thermofluids . . . . . . . . . . . . . . . . . . . . . . . 44E.2 Empirical correlations for forced convection . . . . . . . . . . . . . . . . . . . . . 45E.3 Perfect gases (ideal gases with constant specific heats) . . . . . . . . . . . . . . 50E.4 Isentropic compressible flow functions for perfect gas with γ = 1.40 . . . . . . . . 51E.5 Ideal (semi-perfect) gas specific enthalpy h (kJ kg−1, 25 C datum) . . . . . . . . 52E.6 Molar Enthalpy of Formation h0

f (kJ kmol−1 at 25 C and 1 atmosphere) as gas orvapour (g), except where indicated as solid (s) or liquid (l). . . . . . . . . . . . . . 53

E.7 Ideal gas molar enthalpy h (kJ kmol−1, 25 C datum) . . . . . . . . . . . . . . . . 53E.8 Heating (or calorific) values of gas fuels at 25 C. . . . . . . . . . . . . . . . . . . 54E.9 Heating (or calorific) values of liquid fuels at 25 C. . . . . . . . . . . . . . . . . . 54E.10 Saturated Refrigerant 134a — Temperature (−60C to critical point) . . . . . . . 56E.11 Saturated Refrigerant 134a — Pressure (0.2 bar to critical point) . . . . . . . . . 57E.12 Superheated Refrigerant 134a (0.2 bar to 1 bar) . . . . . . . . . . . . . . . . . . 58E.13 Superheated Refrigerant 134a (1.5 bar to 4 bar) . . . . . . . . . . . . . . . . . . 59E.14 Superheated Refrigerant 134a (5 bar to 12 bar) . . . . . . . . . . . . . . . . . . . 60E.15 Superheated Refrigerant 134a (16 bar to 30 bar) . . . . . . . . . . . . . . . . . . 61E.16 Transport properties of dry air at atmospheric pressure . . . . . . . . . . . . . . . 63E.17 Transport properties of saturated water and steam . . . . . . . . . . . . . . . . . 64E.18 Approximate physical properties at 20 C, 1 bar. . . . . . . . . . . . . . . . . . . . 65E.19 Saturated water and steam — Temperature (triple point to 100 C) . . . . . . . . 68E.20 Saturated water and steam — Pressure (triple point to 2 bar) . . . . . . . . . . . 69E.21 Saturated water and steam — Pressure (triple point to 2 bar) . . . . . . . . . . . 70E.22 Saturated water and steam — Pressure (triple point to 2 bar) . . . . . . . . . . . 71E.23 Subcooled water and Superheated Steam (triple point to 0.1 bar) . . . . . . . . . 72E.24 Subcooled water and Superheated Steam (0.1 bar to 1 atmosphere) . . . . . . . 73E.25 Subcooled water and Superheated Steam (2 bar to 8 bar) . . . . . . . . . . . . . 74E.26 Subcooled water and Superheated Steam (10 bar to 40 bar) . . . . . . . . . . . 75E.27 Subcooled water and Superheated Steam (50 bar to 80 bar) . . . . . . . . . . . 76E.28 Subcooled water and Superheated Steam (90 bar to 140 bar) . . . . . . . . . . . 77E.29 Subcooled water and Superheated Steam (160 bar to 220 bar) . . . . . . . . . . 78E.30 Supercritical steam (250 bar to 500 bar) . . . . . . . . . . . . . . . . . . . . . . . 79E.31 Supercritical steam (600 bar to 1000 bar) . . . . . . . . . . . . . . . . . . . . . . 80

Mechanical Engineering Data & Formulae Page iii

LIST OF FIGURES

List of Figures

C.1 Step response of a first-order low pass filter . . . . . . . . . . . . . . . . . . . . . 27C.2 Bode plot for first-order low and high pass filters . . . . . . . . . . . . . . . . . . 28C.3 Step response of a second-order low pass filter . . . . . . . . . . . . . . . . . . . 29C.4 Bode plot for a second-order low pass filter . . . . . . . . . . . . . . . . . . . . . 30E.1 Moody Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49E.2 Psychrometric Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Page iv Mechanical Engineering Data & Formulae

A General information


Table A.1: SI Units and abbreviations

Quantity Unit Unit symbol

Basic unitsLength metre mMass kilogram kgTime second sElectric current ampere AThermodynamic temperature kelvin KLuminous intensity candela cd

Derived unitsAcceleration, linear metre/second2 m s−2

Acceleration, angular radian/second2 rad s−2

Area metre2 m2

Density kilogram/metre3 kg m−3

Force newton N (= kg m s−2)Frequency hertz (Hz = s−1)Impulse, linear newton-second N sImpulse, angular newton-metre-second N m sMoment of force newton-metre N mSecond moment of area metre4 m4

Moment of inertia kilogram-metre2 kg m2

Momentum, linear kilogram-metre/second kg m s−1

Momentum, angular kilogram-metre2/second kg m2 s−1

Power watt W (= J s−1 = N m s−1

Pressure, stress pascal Pa (= N m−2)Stiffness (linear), spring constant newton/metre N m−1

Velocity, linear metre/second m s−1

Velocity, angular radian/second rad s−1

Volume metre3 m3

Work, energy joule J (= N m)

Electrical unitsPotential volt V (= W A−1)Resistance ohm Ω (= V A−1)Charge coulomb C (= A s)Capacitance farad F (= A s V−1)Electric field strength volt/metre V m−1

Electric flux density coulomb/metre2 C m−2

Magnetic unitsMagnetic flux weber Wb (= V s)Inductance henry H (= V s A−1)Magnetic field strength — A m−1

Magnetic flux density — Wb m−2

Mechanical Engineering Data & Formulae Page 1


Table A.2: Conversion factors from Imperial to SI units

To convert from to multiply by

Acceleration foot/second2 (ft/sec2) metre/second2 (m s−2) 0.3048inch/second2 (in/sec2) metre/second2 (m s−2) 0.0254

Area foot2 (ft2) metre2 (m2) 0.092903inch2 (in.2) metre2 (m2) 6.4516 × 10−4

Density pound mass/inch3(

lbm/in3)

kilogram/metre3 (kg m−3) 2.7680 × 104

pound mass/foot3(

lbm/ft3)

kilogram/metre3 (kg m−3) 16.018

Force kip (1000 lb) newton (N) 4.4482 × 103

pound force (lb) newton (N) 4.4482

Length foot (ft) metre (m) 0.3048inch (in) metre (m) 0.0254mile (mi), U.S. statute metre (m) 1.6093 × 103

mile (mi), international nautical metre (m) 1.852 × 103

Mass pound mass (lbm) kilogram (kg) 0.45359slug (lb-sec2 /ft) kilogram (kg) 14.594ton (2000 lbm) kilogram (kg) 907.18

Moment of force pound-foot (lb-ft) newton-metre (N m) 1.3558pound-inch (lb-in.) newton-metre (N m) 0.11298

Moment of inertia pound-foot-second2(lb-ft-sec2) kilogram-metre2 (kg m2) 1.3558

Momentum, linear pound-second (lb-sec) kilogram-metre/second (kg m s−1) 4.4482

Momentum, angular pound-foot-second (lb-ft-sec) newton-metre-second (kg m2 s−1) 1.3558

Power foot-pound/minute (ft-lb/min) watt (W) 0.022597horsepower (550 ft-lb/sec) watt (W) 745.70

Pressure, stress atmosphere (std) (14.7 lb/in2) newton/metre2 (N m−2 or Pa) 1.0133 × 105

pound/foot2(lb/ft2) newton/metre2 (N m−2 or Pa) 47.880pound/inch2(lb/in.2or psi) newton/metre2 (N m−2 or Pa) 6.8948 × 103

Second moment of area inch4 metre4 (m4) 41.623 × 10−8

Stiffness (linear) pound/inch (lb/in.) newton/metre (N m−1) 175.13

Velocity foot/second (ft/sec) metre/second (m s−1) 0.3048knot (nautical mi/hr) metre/second (m s−1) 0.51444mile/hour (mi/hr) metre/second (m s−1) 0.44704mile/hour (mi/hr) kilometre/hour (km h−1) 1.6093

Volume foot3(ft3) metre3 (m3) 0.028317inch3(in.3) metre3 (m3) 1.6387 × 10−5

UK gallon metre3 (m3) 4.546 × 10−3

Work, Energy British thermal unit (BTU) joule (J) 1.0551 × 103

foot-pound force (ft-lb) joule (J) 1.3558kilowatt-hour (kw-h) joule (J) 3.60 × 106

Page 2 Mechanical Engineering Data & Formulae


Table A.3: Decimal prefixes

Multiplication factora Prefix Symbol

1 000 000 000 000 = 1012 tera T1 000 000 000 = 109 giga G

1 000 000 = 106 mega M1 000 = 103 kilo k

100 = 102 hecto a h10 = 10 deka a da

0.1 = 10−1 deci b d0.01 = 10−2 centi c

0.001 = 10−3 milli m0.000 001 = 10−6 micro µ

0.000 000 001 = 10−9 nano n0.000 000 000 001 = 10−12 pico p

aUse prefixes to keep numerical values generally between 0.1 and 1000bThe use of prefixes hecto, deka, deci and centi should be avoided except for certain areas or volumes where

the numbers would otherwise become awkward.

Table A.4: Physical constants

Avogadro’s numbera N 6.022 × 1023 mol−1

Absolute zero of temperature — 0 K = −273.2 C

Boltzmann’s constant k 1.380 × 10−23 J K−1

Characteristic impedance of vacuum Z0 =(µ0

ε0

)1/2

= 120π Ω

Electron volt eV 1.602 × 10−19 J

Electronic charge e 1.602 × 10−19 C

Electronic rest mass me 9.109 × 10−31 kg

Electronic charge to mass ratio(eme

)1.759 × 1011 C kg−1

Faraday’s constanta F 9.65 × 104 C mol−1

Gas constanta R 8.314 J mol−1 K−1

Permeability of free space µ0 4π × 10−7 H m−1

Permittivity of free space ε01

36π× 10−9 F m−1

Planck’s constant h 6.626 × 10−34 J s

Standard gravitational acceleration g 9.807 m s−2

Stefan-Boltzmann constant σ 5.67 × 10−8 J m−2 s−1 K−4

Velocity of light in vacuum c 2.9979 × 108 m s−1

Volume of perfect gas at S.T.P.b — 22.42 × 10−3 m3

aThese are conventional definitions in gram mol units. For SI calculations in kg mol units multiply the values givenby 103

bAt Standard Temperature (0 C) and Pressure (one atmosphere pressure or 1.013 × 105 N m−2)




B Mathematics and computing

B Mathematics and computing

Data and formulae for core course examinations in:

• Mathematics

• Computing

and in other, related, optional courses.

B.1 Algebra

B.1.1 Logarithms

If by = x, y = logb (x) and:log (x1x2) = logx1 + logx2

log(x1

x2

)= logx1 − logx2

log(

1x

)= − logx

logxn = n logx

log 1 = 0

For natural logarithms b = e = 2.718282 and if ey = x,

y = loge (x) = ln (y)

Hencelog10 x = 0.4343 lnx.

B.1.2 Quadratic equations

If ax2 + bx + c = 0, then

x =−b ±

√b2 − 4ac2a

and (b2 > 4ac) for real roots.

B.1.3 Determinants

2nd order: ∣∣∣∣ a1 b1a2 b2

∣∣∣∣ = a1b2 − a2b1

3rd order:∣∣∣∣∣∣a1 a2 a3b1 b2 b3c1 c2 c3

∣∣∣∣∣∣ = +a1b2c3 + a2b3c1 + a3b1c2 − a3b2c1 − a2b1c3 − a1b3c2


B.1 Algebra

B.1.4 Vector algebra

a = (a1i + a2j + a3k) = (a1, a2, a3) etc.

Scalar (dot) product:a.b = a1b1 + a2b2 + a3b3

Vector (cross) product:

a × b =

∣∣∣∣∣∣i j k

a1 a2 a3b1 b2 b3

∣∣∣∣∣∣Scalar triple product:

[a,b,c] = a.b × c = b.c × a = c.a × b =

∣∣∣∣∣∣a1 a2 a3b1 b2 b3c1 c2 c3

∣∣∣∣∣∣Vector triple product:

a × (b × c) = b (a.c) − c (a.b)

B.1.5 Series

Binomial series:

(1 + x)α = 1 + αx +α(α − 1)

2!x2 +

α (α − 1) (α − 2)

3!x3 + . . . (α arbitrary, |x| < 1).

ex = 1 + x +x2

2!+ · · · + x

n

n!+ . . . (|x| <∞)

cosx = 1 − x2

2!+x4

4!− · · · + (−1)n

x2n

(2n)!+ . . . (|x| <∞)

sinx = x − x3

3!+x5

5!− · · · + (−1)n

x2n+1

(2n + 1)!+ . . . (|x| <∞)

tanx = x +x3

3+

2x5

15+

17x7

315+ . . . (−π

2< x <

π2

)

sinhx =ex − e−x

2= x +

x3

3!+x5

5!+x7

7!+ . . . (|x| <∞)

cosh x =ex + e−x

2= 1 +

x2

2!+x4

4!+x6

6!+ . . . (|x| <∞)

ln (1 + x) = x − x2

2+x3

3− · · · + (−1)n

xn+1

(n + 1)+ . . . (−1 < x < 1)


B.1 Algebra

Stirling’s formula for n! when n is large:

n! ∼=(ne

)n√2πn

or

ln (n!) ∼=(n +

12

)lnn − n + 1

2ln (2π)

or

log10(n!) ∼= 0.39909 +(n +

12

)log10 n − 0.43429n

B.1.6 Trigonometry

Definitions:

θA

B

C sinθ =AC

cosθ =BC

tanθ =AB

cscθ =CA

secθ =CB

cotθ =BA

Signs of trigonometric functions in the four quadrants:

(+)

(+)

I

θ

(+)

(+)

IIθ

(+)

(+)IIIθ

(+)

(+)IV

θ

Quadrant: I II III IV

sinθ + + − −cosθ + − − +tanθ + − + −cscθ + + − −secθ + − − +cotθ + − + −

Trigonometrical identitiescos2 θ + sin2 θ = 1

1 + tan2 θ = sec2 θ

sin 2θ = 2 sinθ cosθ

cos 2θ = cos2 θ − sin2 θ = 2 cos2 −1 = 1 − 2 sin2 θ

sinθ2=

√12

(1 − cosθ)

cosθ2=

√12

(1 + cosθ)


B.1 Algebra

a

AC b

Bc d

Sine rule:A

sina=

Bsinb

=C

sinc

Cosine rule: C2 = A2 + B2 − 2AB coscC2 = A2 + B2 + 2AB cosd

sin (a + b) = sina cosb + cosa sinb sin (a − b) = sina cosb − cosa sinbcos (a + b) = cosa cosb − sina sinb cos (a + b) = cosa cosb + sina sinb

sina + sinb = 2 sin(a + b

2

)cos(a − b

2

)sina − sinb = 2 cos

(a + b

2

)sin(a − b

2

)cosa + cosb = 2 cos

(a + b

2

)cos(a − b

2

)cosa − cosb = −2 sin

(a + b

2

)sin(a − b

2

)sin iz = i sinh z sinh iz = i sin zcos iz = cosh z cosh iz = cos z

B.1.7 Geometry

θ1

θ1 = θ2

θ2

When the two inter-secting lines are, re-spectively, perpendic-ular to two other lines,the angles formed byeach pair are equal.

θ sr Circle:

circumference = 2πrArc length s = rθSector area = 1

2r2θ

x h

b y

Similar triangles:xb=h − yh

θ1 θ2

θ1 + θ2 = ½

Every triangle in-scribed within asemicircle is a righttriangle.

h

b

Any triangle:Area = 1

2bhθ1 θ4θ3

θ2Angles of a triangle:θ1 + θ2 + θ3 = 180

θ4 + θ1 + θ2

xa Db

x

Intersecting chords:x2 = abx2 ≈ Db when b D


B.1 Algebra

B.1.8 Analytic geometry

y

x

mb

Straight line:y = b +mx

by

xa

xa+yb= 1

y

x

r

Circle:x2 + y2 = r2

r

b

y

xa

(x − a)2 + (y − b)2 = r2

y

x

a

b Ellipse:(xa

)2+(yb

)2= 1

y

x

b

a

Parabola:

y = b(xa

)2

y

x

b

a

x = a(yb

)2

y

x

a

a

Hyperbola:xy = a2

y

x

b

a

(xa

)2−(yb

)2= 1


B.1 Algebra

B.1.9 Solid geometry

rSphere:volume = 4

3πr3

surface area = 4πr2

hL

r

Right-circular cone:volume = 1

3πr2h

lateral area = πrLL =√r2 + h2

r

θ Spherical wedge:volume = 2

3r3θ B h

Any pyramid or cone:volume = 1

3Bhwhere B = area of base.

B.1.10 Differential calculus

Leibnitz’s rule:

Dn (f g) = f(Dng)+ n (Df )

(Dn−1g

)+n(n − 1)

2!

(D2f)(

Dn−2g)+ · · · +

(Dnf)g

where D =d

dx, f = f (x) and g = g(x)

Taylor’s expansion of f(x) about x = a:

f (x) = f (a) + (x − a)f ′(a) +(x − a)2

2!f ′′(a) + · · · +

(x − a)n

n!f (n)(a) +

(x − a)n+1

(n + 1)!f (n+1)(x)

where a < x < x. Substituting h = x − a gives the following form:

f (a + h) = f (a) + hf ′(a) +h2

2!f ′′(a) + · · · + h

n

n!f (n)(a) + Rn(h)

where Rn(h) =hn+1

(n + 1)!f (n+1) (a + θh) , (0 < θ < 1).

Taylor’s expansion of f(x,y) about the point (a,b):

f (x, y) = f (a, b) +[(x − a)fx + (y − b)fy

]a,b

+12!

[(x − a)2 fxx + 2 (x − a) (y − b) fxy + (y − b)2 fyy

]a,b

+ . . .

Substituting h = x − a and k = y − b gives the following form:

f (a + h, b + k) = f (a, b) +[hfx + kfy

]a,b +

12!

[h2fxx + 2hkfxy + k

2fyy]a,b

+ . . .

Partial differentiation:

If y = Y (x), then f (x, y) = f [x, Y (x)] ≡ F (x) and

dFdx

=∂f∂x

+∂f∂y

dYdx


B.1 Algebra

If x = X (t) and y = Y (t), then f (x, y) = F (t) and

dFdt

=∂f∂x

dXdt

+∂f∂y

dYdt

If x = X (u, v) and y = Y (u, v) then f (x, y) = F (u, v) and

∂F∂u

=∂f∂x

∂x∂u

+∂f∂y

∂y∂u

∂F∂v

=∂f∂x

∂x∂v

+∂f∂y

∂y∂v

Stationary points of f(x,y):

These occur where fx = 0, fy = 0 simultaneously. Let (a, b) be a stationary point: exam-ine

K =[fxxfyy − (fxy )2

]a,b

If:

• K < 0, then (a, b) is a saddle point ;

• K > 0 and fxx(a, b) < 0, then (a, b) is a maximum;

• K > 0 and fxx(a, b) > 0, then (a, b) is a minimum.

Radius of curvature in Cartesian coordinates:

ρxy =

[1 +(

dydx

)2]3/2

d2y

dx2

B.1.11 Standard Differentials

f (x)df (x)

dxxn nxn−1

uv udvdx

+ vdudx

uv

vdudx− udv

dxv2

sinx cosx

cosx − sinx

tanx sec2 x

sinhx coshx

coshx sinhx

tanhx sech2x

loge x = lnx1x

eax aeax


B.2 Integral calculus

B.1.12 Differential equations

The first-order linear equationdydx

+ R (x) y = S (x)

has an integrating factor

λ (x) = exp[∫R (x) dx

],

so thatd

dx(yλ) = Sλ.

P (x, y) dx +Q (x, y) dy = 0

is an exact equation ifdPdy

=dQdx.


An important substitution:

tanθ2= t.

Then

sinθ =2t

(1 + t2)

cosθ =(1 − t2)

(1 + t2)

and

dθ =2

(1 + t2)dt.



Table B.1: Some indefinite integrals

f (x)∫f (x) dx

secx ln (secx + tanx) = ln tan(x

2+π4

)cosecx ln (cosecx − cotx) = ln tan

(x2

)(a2 − x2

)−1/2sin−1

(xa

), (|x| < a)(

a2 + x2)−1/2

sinh−1(xa

)= ln

[x +(a2 + x2

)1/2]− lna = ln

[xa+(

1 +(xa

)2)1/2

](x2 − a2

)−1/2cosh−1

(xa

)= ln

[x +(x2 − a2

)1/2]− lna = ln

[xa+((xa

)2− 1)1/2

], (x ≥ a)

(a2 + x2

)−1 1a

tan−1(xa

)(a2 − x2

)−1 1a

tanh−1(xa

)=

12a

ln(a + xa − x

), (|x| < a)(

x2 − a2)−1 1

2aln(x − ax + a

), (|x| > a)

Table B.2: Some definite integrals

In ≡∫ π/2

0sinn x dx =

∫ π/2

0cosn x dx =

n − 1n

In−2, where I0 =π2

and I1 = 1

Im,n ≡∫ π/2

0sinm x cosn x dx =

m − 1m + n

Im−2,n =n − 1m + n

Im,n−2, (m > 1, n > 1)∫ ∞0e−ax sinbx dx =

b(a2 + b2

) , (a > 0)∫ ∞0e−ax cosbx dx =

a(a2 + b2

) , (a > 0)∫ ∞0e−x

2dx =

√π

2


B.3 Laplace transforms

B.3 Laplace transforms

Function Transform

Definition: f (t) f (s) =∫ ∞0e−stf (t) dt

af (t) + bg(t) af (s) + bg(s)

dfdt

sf (s) − f (0)

d2fdt2

s2f (s) − sf (0) − f ′(0)

e−atf (t) f (s − a)

tf (t) −df (s)

ds∂f (t, a)

∂a∂f (s, a)

∂a∫ t0f (t) dt

f (s)s∫ t

0f (u)g(t − u) du f (s)g(s)

δ(t0), unit impulse at t = t0 1

1, unit step1s

(s > 0)

tn, n = 1,2 . . .n!

sn+1(s > 0)

eat1

s − a (s > a)

e−at1

s + a1

(n − 1)!tn−1e−at

1

(s + a)n

1 − e−at as(s + a)

1(b − a)

(e−at − e−bt

) 1(s + a)(s + b)

1(b − a)

[(c − a)e−at − (c − b)e−bt

] s + c(s + a)(s + b)

1 − b(b − a)

e−at +a

(b − a)e−bt

abs(s + a)(s + b)

e−at

(b − a)(c − a)+

e−bt

(c − a)(a − b)+

e−ct

(a − c)(b − c)1

(s + a)(s + a)(s + b)

c −b(c − a)

(b − a)e−at +

a(c − b)

(b − a)e−bt

ab(s + c)

s(s + a)(s + b)


B.4 Numerical analysis

Function Transform

sinωtω(

s2 +ω2) (s > 0)

cosωts(

s2 +ω2) (s > 0)√

(a2 +ω2)ω

sin (ωt +φ) , φ = tan−1(ωa

) s + a(s2 +ω2

) (s > 0)

e−at sinωtω

(s + a)2 +ω2

e−at cosωts + a

(s + a)2 +ω2

1ω

√(c − a)2 +ω2 e−at sin(ωt +φ), φ = tan−1

( ωc − a

) (s + c)

(s + a)2 +ω2

ωn√1 − ζ2

e−ζωnt sinωn

√(1 − ζ2) t, (ζ < 1)

ω2n

s2 + 2ζωns +ω2n

1

a2 +ω2+

1

ω√a2 +ω2

e−at sin (ωt −φ) , φ = tan−1 −ωa

1

s[(s + a)2 +ω2

]1 − 1√

1 − ζ2e−ζωnt sin

(ωn

√1 − ζ2 t +φ

), φ = cos−1 ζ, ζ < 1

ω2n

s(s2 + 2ζωns +ω

2n

)H(t − T ) (= 0, t < T ; = 1, t > T )

1se−sT (s, T > 0)


B.4.1 Approximate solution of an algebraic equation

• An iterative method for x = φ(x) converges when∣∣φ′(x)

∣∣ < 1 near the root: if a rootoccurs near to x = a take x0 = a and

xn+1 = φ(xn), n = 0,1,2, . . .

• If a root of f (x) = 0 occurs near to x = a, take x0 = a and:

xn+1 = xn −f (xn)

f ′(xn), n = 0,1,2, . . .

(the Newton-Raphson method).

B.4.2 Numerical integration

Write xn = x0 + nh, yn = y(xn). Then:



• Trapezium Rule (1 strip): ∫ x1

x0

y(x) dx ≈ h2

[y0 + y1

]• Simpson’s Rule (2 strips): ∫ x1

x0

y(x) dx ≈ h3

[y0 + 4y1 + y2

]

B.4.3 Richardson’s error estimation formula for use with Simpson’s rule

Let

I =∫ baf (x) dx

and let I1, I2 be two estimates of I obtained using Simpson’s rule with intervals h1 and h2, where

h1 < h2 (i.e. h1 =b − an1

, h2 =b − an2

, where n1, n2 are even). Then a better estimate of I is

given by:

I = I2 +(I2 − I1)[(h1

h2

)4

− 1

] .

If h2 =12h1 then I = I2 +

115 (I2 − I1).

B.4.4 Fourier series

If f (x) is periodic of period 2L, i.e. f (x + 2L) = f (x), then

f (x) =a0

2+∞∑n=1

an cosnπxL

+∞∑n=1

bn sinnπxL

where

an =1L

∫ L−Lf (x) cos

nπxL

dx, n = 0,1,2, . . .

bn =1L

∫ L−Lf (x) sin

nπxL

dx, n = 1,2,3, . . .

If f (x) is an even function of x, i.e. f (−x) = f (x), then

an =2L

∫ L0f (x) cos

nπxL

dx, bn = 0, n = 0,1,2, . . .

If f (x) is an odd function of x, i.e. f (−x) = −f (x), then

bn =2L

∫ L0f (x) sin

nπxL

dx, an = 0, n = 1,2,3, . . .


B.5 Statistics

B.5 Statistics

B.6 Probabilities for events

For events A, B and C:P (A ∪ B) = P (A) + P (B) − P (A ∩ B)

The odds in favour of A are:P (A)

P (A)

Conditional probability :

P (A|B) =P (A ∩ B)

P (B)(if P (B) > 0)

The chain rule:P (A ∩ B ∩ C) = P (A)P (B|A)P (C|A ∩ B)

Bayes’ rule:

P (A|B) =P (A)P (B|A)

P (A)P (B|A) + P (A)P (B|AA and B are independent if

P (B|A) = P (B)

A, B and C are independent if

P (A ∩ B ∩ C) = P (A)P (B)P (C),

and P (A ∩ B) = P (A)P (B), P (B ∩ C) = P (B)P (C) and P (C ∩ A) = P (C)P (A).

B.6.1 Distribution, expectation and variance

The probability distribution for a discrete random variable X is the set px, where

px = P (X = x).

The expectation isE (X ) = µ =

∑x

xpx

From independent observations x1, x2, . . . xn, the sample mean

x =1n

∑k

xk

estimates µ.

The variance isvar(X ) = σ2 = E

(X − µ)2

= E (X 2) − µ2,

whereE (X 2) =

∑x

x2px.


B.6 Probabilities for events

The sample variance:

s2 =1

n − 1

∑k

x2k −

1n

∑j

xj

2

estimates σ2.

The standard deviation is:sd(X ) = σ.

If the value y is observed with frequency ny , then

n =∑y

ny ,∑k

xk =∑y

yny ,∑k

x2k =∑y

y2ny .

For a function g(x) of x,E g(x) =

∑x

g(x)px.

B.6.2 Probability distributions for a continuous random variable

The cumulative distribution function (cdf) is

F (x) = P (X ≤ x) =∫ xx0=−∞

f (x0) dx0

The probability density function (pdf) is

f (x) =dF (x)

dx

µ =∫ ∞−∞

xf (x) dx, σ2 = E (X 2) − µ2

where

E (X 2) =∫ ∞−∞

x2f (x) dx

B.6.3 Discrete probability distributions

Binomial distribution Binomial (n, θ)

px =(nx

)θx (1 − θ)n−x (x = 0,1,2, . . . , n)

µ = nθ, σ2 = nθ(1 − θ).

Poisson distribution Poisson (λ)

px =λxe−λ

x!(x = 0,1,2, . . . ) (with λ > 0)

µ = λ, σ2 = λ.


B.7 Bias, standard error and mean square error

B.6.4 Continuous probability distributions

Uniform distribution Uniform (α,β)

f (x) =

1

β − α(α < x < β)

0 (otherwise).µ =

α + β2

, σ2 =(β − α)2

12

Exponential distribution Exponential (λ)

f (x) =λe−λx (0 < x <∞),

0 (−∞ < x ≤ 0).µ =

1λ, σ2 =

1

λ2.

Normal distribution N(µ, σ2)

f (x) =1√

2πσ2exp−1

2

(x − µσ

)2

(−∞ < x <∞), E (X ) = µ, var(X ) = σ2

Standard normal distribution is N(0,1)

If X is N(µ, σ2), then Y =X − µσ

is N(0,1).

B.6.5 System reliability

For a system of k devices, which operate independently, let

Ri = P (Di ) = P (“device i operates”).

The system reliability, R, is the probability of a path of operating devices.

A system of devices in series fails if any device fails:

R = P (D1 ∩ D2 ∩ · · · ∩ Dk) = R1R2 . . . Rk

A system of devices in parallel operates if any device operates:

R = P (D1 ∪ D2 ∪ · · · ∪ Dk) = 1 − (1 − R1)(1 − R2) . . . (1 − Rk)


If t estimates θ and comes from a distribution having random variable T :

• Bias of t: bias(t) = E (T ) − θ• Standard error of t: se(t) = sd(t)

• Mean square error of t: MSE(t) = E

(t − θ2= se(t)2 + bias(t)2

if x estimates µ, then bias(x) = 0, se(x) =σ√n

, MSE(x) =σ2

n, se(x) =

s√n

B.7.1 Central limit property

If n is fairly large, x is approximately from N

(µ,σ2

n

).



B.7.2 Confidence intervals

If x1, x2, . . . , xn are independent observations from N(µ, σ2) and σ2 is known, then the 95%

confidence interval for µ is(x − 1.96

σ√n, x + 1.96

σ√n

).

If σ2 is estimated then, from the table of t(n−1), we find t0 = t(n−1),0.05. Then the 95% CI for µ is(x − t0

s√n, x + t0

s√n

).

y φ(y) Φ(y) y φ(y) Φ(y) y φ(y) Φ(y) y Φ(y)

0 0.399 0.5 0.9 0.266 0.816 1.8 0.079 0.964 2.8 0.9970.1 0.397 0.540 1.0 0.242 0.841 1.9 0.066 0.971 3.0 0.9980.2 0.391 0.579 1.1 0.218 0.864 2.0 0.054 0.977 0.841 0.80.3 0.381 0.618 1.2 0.194 0.885 2.1 0.044 0.982 1.282 0.90.4 0.368 0.655 1.3 0.171 0.903 2.2 0.035 0.986 1.645 0.950.5 0.352 0.691 1.4 0.150 0.919 2.3 0.028 0.989 1.96 0.9750.6 0.333 0.726 1.5 0.130 0.933 2.4 0.022 0.992 2.326 0.990.7 0.312 0.758 1.6 0.111 0.945 2.5 0.018 0.994 2.576 0.9950.8 0.290 0.788 1.7 0.094 0.955 2.6 0.014 0.995 3.09 0.999

Table B.3: Standard normal table: values of pdf φ(y) = f (y) and cdf Φ(y) = F (y).

p 0.10 0.05 0.02 0.01 p 0.10 0.05 0.02 0.01

m 1 6.31 12.71 31.82 63.66 m 9 1.83 2.26 2.82 3.252 2.92 4.30 6.96 9.92 10 1.81 2.23 2.76 3.173 2.35 3.18 4.54 5.84 12 1.78 2.18 2.68 3.054 2.13 2.78 3.75 4.60 15 1.75 2.13 2.60 2.955 2.02 2.57 3.36 4.03 20 1.72 2.09 2.53 2.856 1.94 2.45 3.14 3.71 25 1.71 2.06 2.48 2.787 1.89 2.36 3.00 3.50 40 1.68 2.02 2.42 2.708 1.86 2.31 2.90 3.36 ∞ 1.645 1.96 2.326 2.576

Table B.4: Student t table: values tm,p of x for which P (|X | > x) = p, when X is tm.


C Mechatronics and control

C Mechatronics and control


• Mechatronics

• Dynamics

• Machine System Dynamics


C.1 Charge, current, voltage and power

q = charge

i = current =dqdt

v = electrical potential (voltage)P = power leaving networkU = energy storedR = resistanceC = capacitanceC = inductance

Subscript and arrow notations

aiaP

b

vab

Potential difference:

vab = va − vb

Power P leaving network between terminals a and b:

P = vabia

Passive components

Resistor: v = iR (Ohm’s law) Power dissipated: P = i2R =v2

RInductor: v = L

didt

Energy stored U = 12Li

2

Capacitor: i = Cdvdt

, q = Cv Energy stored U = 12Cv

2

Table C.1: Colour codes for resistors etc.

Colour Digit

Black 0 Green 5Brown 1 Blue 6Red 2 Violet 7Orange 3 Grey 8Yellow 4 White 9


C.2 Networks

Table C.2: Standard values for components

E3 series E6 series E12 series E24 series

10 10 10 10

1112 12

1315 15 15

1618 18

20

22 22 22 22

2427 27

3033 33 33

3639 39

43

47 47 47 47

5156 56

6368 68 68

7582 82

91

C.2 Networks

Kirchhoff’s voltage law (KVL): ∑(p.d.s around loop) = 0

Kirchhoff’s current law (KCL): ∑(currents into node) = 0

Resistors in series:Rser = R1 + R2 + . . .

Resistors in parallel:1Rpar

=1R1

+1R2

+ . . .

i.e. for two resistors

Rpar =R1R2

R1 + R2


C.3 Transients

Potential divider (with R2 as output resistor):

vout =R2

R1 + R2vin

C.3 Transients

x(t) = instantaneous voltage v or current iX0 = initial value x(0)Xf = final (steady state) value x(∞).τ = time constant.

At time t = 0 a switch operates so that the network of resistors and d.c. voltage sourcesconnected to a capacitor or inductor changes, instantaneously. Then for t ≥ 0:

x (t) = Xf − (Xf − X0) exp(− tτ

)

For a capacitor:

• v remains unchanged through t = 0;

• i → 0 as t→∞;

• τ = RsC

For an inductor:

• i remains unchanged through t = 0;

• v → 0 as t→∞;

• τ = LR

C.4 AC networks

Xm = peak amplitude (or semi-amplitude)Xav = mean value x(∞).Xpp = 2Xm is peak-to-peak amplitudef = frequency (Hz), and ω = 2πf (rad s−1)

T =1f

is period

C.4.1 Average and root mean square values

General definitions for any periodic waveform:

Xav =1T

∫ T0x dt

Xrms =

√1T

∫ T0x2 dt


C.4 AC networks

For a waveform consisting of N samples of equal duration:

Xrms =

√√√√ 1N

N∑n=1

x2n dt

For a sinusoidal waveform x = Xm sin (ωt +φ):

Xrms =1√

2Xm

and for a sinusoidal positive half-cycle:

Xav =2πXm

C.4.2 Phasors and complex impedance

CIVIL: current leads voltage for a capacitor, voltage leads current for an inductor.

Current is common phasor for series circuits, voltage is common phasor for parallel circuits.

Inductive reactance: XL = ωL;

Capacitative reactance XC =1ωC

.

Complex impedance: V = I · Z where V , I , and Z are complex quantities, and

Z = R ± jX

where X is impedance.

C.4.3 Balanced 3 phase a.c supply

Relationships between line voltage VL and current iL, phase voltage VP and current iP for starconnected load

VL =√

3VP, IL = IPand for delta connected load:

VL = VP, IL =√

3IP

C.4.4 Electromagnetism

N = number of coil turnsH = magnetic field strengthl = magnetic flux path lengthΦ = magnetic fluxB = magnetic flux densityµr = relative permeabilityµ0 = permittivity of free spaceA = cross-sectional area of magnetic flux path.L = length of conductor in magnetic field.U = velocity of conductor


C.4 AC networks

Magnetomotive force (m.m.f.) = iN, H =m.m.f.l

B =ΦA

= µrµ0H(tesla)

Reluctance of magnetic path:

S =m.m.f.Φ

=l

µrµ0A

Magnetic force of attraction:

F =B2A2µ0

Force acting on a conductor:

F = BiL

Induced e.m.f.:

E = BLU =dΦdt

C.4.5 DC machines

T0 = shaft torqueKe = e.m.f. constantRa = armature resistanceva = armature voltageia = armature currentω = angular velocity

Torque-speed relationship for a permanent-magnet or shunt-wound d.c. machine:

ω =1Keva −

Ra

K 2e

T0

and T0 = Keia.

C.4.6 Transformers

Φ = peak fluxf = frequencyN1, N2 = primary and secondary turns

Ideal transformer:V1

V2=N1

N2=I2I1

RMS value of induced e.m.f.:

E = 4.44NfΦ


C.5 Communications

C.5 Communications

Information (in bits) communicated by each of N equally probable messages:

I =1

log10 2logN

Information (in bits) communicated by a message of probability P :

I =1

log10 2log(

1P

)

C.6 Step function response and frequency response

θin, θout = input and output variablesτ = time constantωn = natural frequencyζ = damping factorH = gainφ = phase shift

The transfer function for any linear system is generally expressed as a linear function of theLaplace variable s.

C.6.1 First-order systems

Transfer function of first order low pass (lag):

θout

θin=

11 + τs

Figure C.1 shows the time plot for response to a unit step input:

θin =

0 (t < 0)1 (t > 0)

Gain (power ratio) in decibels (dB): |H | = 20 log10

(Vout

Vin

).

Figure C.2 shows the Bode plots for sinusoidal input θin = θin sin (ωt −φ) to first-order low-passand high-pass filters. For active filters (see Table C.3):

Low-pass filter High-pass filter

Passive|H | = 1√

1 + (ωRC)2

φ = − tan−1 (ωRC)

|H | = ωRC√1 + (ωRC)2

φ = 90 − tan−1 (ωRC)

Active|H | =

R2

R1

1√1 + (ωR2C)2

φ = 180 − tan−1 (ωR2C)

|H | =C1

C2

ωRC2√1 + (ωRC2)2

φ = −90 − tan−1 (ωRC)


C.7 Operational amplifier stages

0.1

0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Time, — = ωct

tτ

Input

Figure C.1: Step response of a first-order low pass filter

C.6.2 Second-order systems

Transfer function of a second-order low-pass system:

θout

θin=

1(1 + 2

ζωns +

1

ω2n

s2

)

Unit step and frequency response are shown in Figs. C.3 and C.4.


Table C.3 shows op-amp networks which implement various signal processing operations.



Gai

n (d

B)

–25

–20

–15

–10

–5

0

Low

pas

s ph

ase

shift

(deg

rees

)

–90

0

–10

–20

–30

–40

–50

–60

–70

–80

Hig

h pa

ss p

hase

shi

ft (d

egre

es)

0

90

80

70

60

50

40

30

20

10

0.1 0.2 0.5 2 51 10Frequency, — = ωτ ω

ωc

0.1 0.2 0.5 2 51 10Frequency, — = ωτ ω

ωc

Figure C.2: Bode plot for first-order low and high pass filters



0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Input

ζ = 0

ζ = 0.2

ζ = 0.05

ζ = 0.1

ζ = 0.3

ζ = 0.5

ζ = 0.7 ζ = 1ζ = 1.5

ζ = 2

0 1 2 3 4 5 6 7 8 9 10Time, ωnt

Figure C.3: Step response of a second-order low pass filter



–40

0

–45

–90

–135

–180

–30

–20

–10

0

10

20G

ain

(dB

)P

hase

(deg

rees

)

Frequency, — = ωτ ωωc

0.1 0.2 0.5 2 51 10

Frequency, — = ωτ ωωc

0.1 0.2 0.5 2 51 10

ζ = 0.05ζ = 0.1ζ = 0.2

ζ = 0.3

ζ = 0 ζ = 0

ζ = 0.5ζ = 0.7

ζ = 1.0ζ = 1.5

ζ = 2.0

Figure C.4: Bode plot for a second-order low pass filter



Table C.3: Operational amplifier signal processing stages

Current to voltage Charge to voltage

Inverting amp Non-inverting amp

Summing amp Difference amp

Integrator Differentiator

R1

CR2 R

C1

C2

First order low pass First order high pass




D Solid Mechanics

D Solid Mechanics


• Mechanics

• Stress Analysis

• Materials


D.1 Mechanics

D.1.1 Square screw threads

M = moment required to raise an axially loaded nutW = axial load on nutφ = helix angle of threadµs = static coefficient of frictiondm = mean thread diameter

M =(

tanφ + µs

1 − µs tanφ

)Wdm

2

D.1.2 Flat clutches

T = maximum torque transmittedF = thrustR1, R2 = outer and inner radii for annular clutchµs = static coefficient of friction

For uniform pressure conditions:

T =23µsF

(R3

1 − R32

R21 − R

22

)

For uniform wear conditions:

T = µsF(R1 + R2

2

)

D.1.3 Kinematics of particle

a = acceleration vectors = distance travelledv = tangential velocityt = timeen,et = unit vectors in n-t coordinateser,eθ = unit vectors in r-θ coordinatesρ = instantaneous radius of path curvature


D.1 Mechanics

For normal and tangential components:

a =d2sdt2

et +v2

ρen

For polar components:

a =(r − rθ2

)er +

(rθ + 2r θ

)eθ

D.1.4 Mass flow problems

F = internal force vector exerted from the emitted massa = acceleration vectorm = mass of objectmf = emitted massvf = velocity vector of emitted mass relative to object

ma = −dmf

dtvf = F

D.1.5 Kinematics of rigid bodies with sliding contacts

v = velocity vectora = acceleration vectorvrel = velocity vector relative to rotating body (sliding velocity)arel = acceleration vector relative to rotating body (sliding acceleration)ω = angular velocity vectorα = angular acceleration vectorr = position vector

v = vrel +ω × r

a = arel + 2ω × vrel + α × r +ω × (ω × r )

D.1.6 Mass moments of inertia

M = total mass of bodyG = centre of mass (centre of gravity)IG = Mass moment of inertia about GIaa = Mass moment of inertia about an axis a − a

Body Mass moment of inertia

Rectangular lamina, b × h IG = 112M(b2 + h2)

Circular lamina, radius r IG = 12Mr

2

Uniform slender rod, total length L IG = 112ML

2

Sphere, radius r IG = 25Mr

2


D.2 Stress analysis

Parallel axis theorem:

aaaG

Iaa = IG +Ma2

D.2 Stress analysis

D.2.1 Elastic constants of materials

ρ = DensityE = Young’s modulus, modulus of elasticityG = Shear modulus, modulus of rigidityK = Bulk modulusν = Poisson’s ratioα = Coefficient of linear thermal expansion

Relationships between elastic constants:

G =E

2 (1 + ν)K =

E3 (1 − 2ν)

Some typical values:

ρ E G K ν αkg m−3 GPa GPa GPa ×106 K−1

Mild steel 7850 207 79.6 175 0.3 11Aluminium alloy 2720 68.9 26.5 69 0.3 23Brass 8410 103 38.3 117 0.35 19Titanium alloy 5000 110 42 0.31 11Softwood along grain 9Water 1000 2.2Concrete 2400 13.8 0.1

D.2.2 Beam theory

σ = axial stress at axial position z and vertical distance y from neutral axisτ = shear stress in vertical×axial planed = total depth of beamM = bending moment about neutral axis at zS = shear force at zI = second moment of area about neutral axisR = radius of curvature at zv = vertical deflection at z

Bending about a principal axis :σy=MI=ER

= Ed2vdz2


D.2 Stress analysis

and

τ =S2I

(d2

4− y2

)

Table D.1: Second moments of area for simple cross-sections

y

y

xx

b12 b1

2

d12

d12

Ixx =1

12bd3

Iyy =112b

3d

y

y

xxa

Ixx = Iyy =14πa

4

C

y

xx

(b1 – b2)13

yb1 b2

h23

h13

Ixx =1

36 (b1 + b2)h3

Iyy =136 (b1 + b2)

(b2

1 + b1b2 + b22

)

Parallel axis theorem: xx

aaaG

Area A

Iaa = Ixx + Aa2


D.2 Stress analysis

Table D.2: Beams bent about principal axis

End End Centralslope deflection deflection

M

L

MLEI

ML2

2EI

W

L

WL2

2EIW L3

3EI

L

w per unit length wL3

6EIwL4

8EI

MM

L

ML2EI

ML2

8EI

L

WWL2

16EIW L3

48EI

L

w per unit lengthwL3

24EI5wL4

384EI

End Centralmoment deflection

W

½L ½L

WL8

WL3

192EI

L

w per unit length WL2

12WL4

384EI


D.2 Stress analysis

D.2.3 Elastic torsion

Circular solid and hollow shafts

τ = shear stress at radius rT = applied torqueJ = polar second moment of aread = diameter of circular sectionθ = angle of twist over length L

τr=TJ=GθL

For a solid circular section:

J = 2Ixx =πd4

32

Table D.3: Torsion of solid non-circular sections

Shape of Maximum shear stress, Angle of twist,cross section τmax θ

a

a

Square

4.81Ta3

7.10TLa4G

a

Equilateral triangle

20Ta3

46TLa4G

2a

2b

Ellipse

2πTab2

(a2 + b2

) TLπa3b3G


D.3 Two-dimensional stress transformation

Thin walled tubes of arbitrary cross-section

A = enclosed area to mid-thicknesst = wall thicknesss = distance around perimeter

τ =T

2At

Torsional stiffness:T

θ/l=

4A2G∮ (1t

)ds

Springs

d = wire diameterD = helix diameterδ = deflectionF = force

End deflection of a closed-helix, round wire spring:

δ =8F D3NGd4

Maximum shear stress (torsion only):

τ =8F Dπd3

D.2.4 Thin walled pressure vessels

R = mean radiust = wall thicknessp = internal pressure

Hoop stress in hollow, pressurised cylinder:

σθ = p(Rt

)Stress in hollow, pressurised sphere :

σ = p(R2t

)

D.3 Two-dimensional stress transformation

Sn

Ss

θ

θ

τyx (= τxy)

τxy

σx

σy

Sn =σx + σy

2+σx − σy

2cos 2θ + τxy sin 2θ

Ss =σx − σy

2sin 2θ − τxy cos 2θ


D.4 Yield criteria

Principal stresses:

σ1, σ2 =σx + σy

2±

√(σx − σy2

)2

+ τ2xy

The direction of the principal stresses (and of the normal to the principal planes) to the x axisis θp where:

tan 2θp =2τxyσx − σy

Maximum shear stress: The maximum shear stress is half the difference of the principalstresses and acts on planes at 45 to the principal planes.

D.4 Yield criteria

Y = yield stress in uniaxial tensiont = wall thickness

In a three dimensional stress system having principal stresses σ1, σ2 and σ3 where

σ1 ≥ σ2 ≥ σ3.

Tresca yield criterion:|σ1 − σ3| = Y

Von Mises yield criterion:

(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2 = 2Y 2

D.5 Two-dimensional strain transformation

en =ex + ey

2+ex − ey

2cos 2θ +

γxy2

sin 2θ

es2

=ex − ey

2sin 2θ −

γxy2

cos 2θ

where:

• ex, ey and en are the direct strains acting in the same directions as, respectively, thestresses sx, sy and Sn above;

• γxy and es are the shear strains associated with the stresses τxy and Ss

NOTE: the relevant strain relationships may be obtained from the stress relationships by sub-stituting the appropriate direct stresses by the associated direct strain and shear stresses byone half of the associated shear strain.

D.6 Elastic stress-strain relationships

ex =1E

(σx − ν

(σy + σz

))etc

γxy =1Gτxy etc


D.7 Thick-walled cylinders


For axi-symmetric systems, the circumferential and radial stresses at radius r are, repec-tively:

σθθ = A +Br2

σrr = A −Br2




E Thermofluids

E Thermofluids


• Fluid Mechanics

• Thermodynamics

• Heat Transfer

• Thermodynamics and Energy


E.1 Cross-references to table numbers

Some Tables in this handbook are referred to by different numbers in lecture notes and problemsheets.

External reference Table number in this handbook

Table E1 Table E.1Table E2 Table E.2Table E3 Section E.4Table E4 Section E.5Table E5 Table E.3Table E6 Table E.4Table E7a Table E.5Table E7b Table E.6Table E7c Table E.7Table E8 Tables E.8 and E.9Table E9a Table E.16Table E9b Table E.17Table E10 Table E.18

Table R1 Table E.10Table R2 Table E.11Table R3 Part 1 Table E.12Table R3 Part 2 Table E.13Table R3 Part 3 Table E.14Table R3 Part 4 Table E.15

Table S1 Table E.19Table S2 Tables E.20 to E.22Table S3 Tables E.23 to E.29


E.2 Dimensionless groups

E.2 Dimensionless groups

A = surface areaCp = specific heat at constant pressureD = pipe diameterFD, FL = drag force, lift forceg = gravitational accelerationh = surface heat-transfer coefficientkf, ks = thermal conductivity of fluid, of solidL = reference length4P = pressure dropT∞, Ts = temperature at infinity, at surfaceU = characteristic velocityα = thermal diffusivityβ = coefficient of volumetric thermal expansionε = roughness heightµ = absolute viscosityρ = density

Table E.1: Dimensionless groups for Thermofluids

Parameter Definition

Biot number (Bi)hLks

Coefficient of lift (CL)FL

12ρU

2A

Coefficient of drag (CD)FD

12ρU

2A

Fourier number (Fo)αtL2

Friction factor (f )4P(

LD

)12ρU

2

Grashof number (Gr)βg (T∞ − Ts)L3ρ2

µ2

Nusselt number (Nu)hLkf

Prandtl number (Pr)µCp

kf

Rayleigh number (Ra) Gr · Pr

Reynolds number (Re)ULρµ

Roughness ratioεL

Stanton number (St)h

ρUCp=

NuRe · Pr


E.3 Heat transfer

E.3 Heat transfer

Table E.2: Empirical correlations for forced convection

Correlation Conditions

Laminar flow over a flat plate:

Cf(x) =0.664

Re1/2x

Pr ≥ 0.6

Nu(x) =h(x)xk

= 0.332Re1/2x Pr1/3 Pr ≥ 0.6

Cf =1.328

Re1/2L

Pr ≥ 0.6

Nu = 0.664Re1/2L Pr1/3 Pr ≥ 0.6

Turbulent flow over a flat plate:

Cf(x) =0.0592

Re1/5x

5 × 105 ≤ Rex ≤ 107

Nu(x) = 0.0296Re4/5x Pr

1/3x

0.6 ≤ Pr ≤ 60

5 × 105 ≤ Rex ≤ 107

Cf =0.074

Re1/5L

5 × 105 ≤ ReL ≤ 107

Nu = 0.037Re4/5Pr1/3

0.6 ≤ Pr ≤ 60

5 × 105 ≤ Rex ≤ 107

Mixed flow over a flat plate:

Cf(x) =0.074

Re1/5L

− 1742ReL

5 × 105 ≤ ReL ≤ 107

Nu =(

0.037Re4/5L − 871

)Pr1/3

0.6 ≤ Pr ≤ 60

5 × 105 ≤ Rex ≤ 107

Flat plate with uniform heat flux:

Nu(x) = 0.453Re1/2x Pr1/3 Laminar flow

Nu(x) = 0.0308Re0.8x Pr1/3 Turbulent flow

Fully developed laminar flow in a pipe:

Nu = 3.66 Constant surface temperature

Nu = 4.36 Constant heat flux

Fully developed turbulent flow in a pipe:

NuD = 0.023Re4/5D Prn

(n = 0.4 for heating, n = 0.3 for cooling)

0.7 ≤ Pr ≤ 160

Re > 10000


E.4 Continuity and equation of motion

E.4 Continuity and equation of motion

E.4.1 Cylindrical polar coordinates

Equation of continuity for unsteady flow, variable density:

∂ρ∂t

+1r∂(rρvr )∂r

+1r∂(ρvθ)

∂θ+∂(ρvx)

∂x= 0

Equations of Motion for unsteady flow, variable density: Cauchy form

∂vx∂t

+ vx∂vx∂x

+ vr∂vx∂r

+vθr∂vx∂θ

= −1ρ∂p∂x

+ Fx,int + Fx,ext

∂vr∂t

+ vx∂vr∂x

+ vr∂vr∂r

+vθr∂vr∂θ

−v2θ

r= −1

ρ∂p∂r

+ Fr,int + Fr,ext

∂vθ∂t

+ vx∂vθ∂x

+ vr∂vθ∂r

+vθr∂vθ∂θ

+vθvrr

= − 1ρr∂p∂θ

+ Fθ,int + Fθ,ext

where Fi ,

intext

are the internal (viscous) or external (body) forces per unit mass, as appropriate,

acting in the direction of coordinate i . For example,

Fx,int = ν

[1r∂∂r

(r∂vx∂r

)+

1

r2

∂2vx∂θ2

+∂2vx∂x2

]

E.4.2 Rectangular Cartesian coordinates

Equation of continuity for unsteady flow, variable density

∂ρ∂t

+∂(ρu)

∂x+∂(ρv)

∂y+∂(ρw)

∂z= 0

Equations of motion for unsteady flow, variable density: Cauchy form

∂u∂t

+ u∂u∂x

+ v∂u∂y

+ w∂u∂z

= −1ρ∂p∂x

+ Fx,int + Fx,ext

∂v∂t

+ u∂v∂x

+ v∂v∂y

+ w∂v∂z

= −1ρ∂p∂y

+ Fy,int + Fy,ext

∂w∂t

+ u∂w∂x

+ v∂w∂y

+ w∂w∂z

= −1ρ∂p∂z

+ Fz,int + Fz,ext

Equations of motion for unsteady uniform property flow in two dimensions (the xy plane)only:

∂u∂t

+ u∂u∂x

+ v∂u∂y

= −1ρ∂p∂x

+µρ

(∂2u∂x2

+∂2u∂y2

)︸︷︷︸ + Fx,ext

Fx,int

∂v∂t

+ u∂v∂x

+ v∂v∂y

= −1ρ∂p∂y

+µρ

(∂2v∂x2

+∂2v∂y2

)︸︷︷︸ + Fy,ext

Fy,int


E.5 Equations for compressible flows

The boundary layer (“approximately Couette flow”) form of the equations of motion for strainconfined to the xy plane with uniform properties:

∂u∂t

+ u∂u∂x

+ v∂u∂y

= −1ρ

dpdx

+µρ

(∂2u∂y2

)

E.4.3 Vector form

Equation of continuity for unsteady flow, variable density, in vector form:

∂p∂t

+ ∇ · (ρV ) = 0

where ∇ is the vector-gradient operator expressing the divergence of a vector, in this case V ,the velocity field.

Equations of Motion in vector form for unsteady flow: Cauchy form

∂V∂t

+ (V · ∇)∂V∂x

= −1ρ∇p + Fint + Fext

E.5 Equations for compressible flows

Isentropic compressible flow relations

ρ0 = ρ1

(1 +

γ − 12

Ma21

) 1γ−1

p0 = p1

(1 +

γ − 12

Ma21

) γγ−1

T0 = T1

(1 +

γ − 12

Ma21

)

a0 = a1

(1 +

γ − 12

Ma21

)1/2

Prantl-Meyer function

ν(Ma) =(γ + 1γ − 1

)1/2

tan−1

(γ + 1)

(Ma2 − 1

)γ − 1

1/2 − tan−1

(Ma2 − 1

)1/2

Normal Shock Relations

Ma22 =

1 + 12 (γ − 1) Ma2

1

γMa21 −

12 (γ − 1)

ρ2

ρ1=

(γ + 1) Ma21

(γ − 1) Ma21 + 2


E.6 Friction factor for flow in circular pipes (Moody diagram)

p2

p1= 1 +

2γ(γ + 1)

(Ma2

1 − 1)

T2

T1=

[2γMa2

1 − (γ − 1)] [

(γ − 1) Ma21 + 2

](γ + 1)2 Ma2

1


d = pipe diameter

f = Darcy friction factor =4τw

12ρV

2=

1

(L/d )

4P12ρV

2

L = pipe length

Re = Reynolds Number =ρV dµ

V = fluid bulk mean velocity4P = frictional pressure drop in length Lε = roughness heightµ = absolute or dynamic viscosityρ = fluid densityτw = shear stress at pipe wall



Friction factor f Relative roughnessε/d0.06

0.05

0.04

0.03

0.02

0.01

0

R.I.C. 7/95

103 104 105 106 107 108

laminar turbulenttransition Reynolds number Re

0.02

0.01

0.005

0.002

0.001

0.0005

0.00020.00010.000050.00001smooth

Figure E.1: Moody Diagram


E.7 Perfect gases

E.7 Perfect gases

Ma = Mach numberP , P0 = absolute pressure, stagnation pressureT , T0 = absolute temperature, stagnation temperaturev = specific volumeρ, ρ0 = density, stagnation density

Over a limited range of temperatures and pressures close to ambient values, the followingsubstances can be assumed to behave as perfect gases with properties given by the followingrelationships:

equation of state P v = RT

specific heat at constant volume Cv =dudT

= constant for the particular gas

specific heat at constant pressure Cp =dhdT

= constant for the particular gas

ratio of principal specific heats γ =Cp

Cv= constant for the particular gas

gas constant R = Cp − Cv = constant for the particular gas

and R =RM

where R = universal gas constant = 8.314 kJ kmol−1 K−1.

Table E.3: Perfect gases (ideal gases with constant specificheats)

Gas Chemical Molar mass M Gas constant R Cp Cv γformula kg kmol−1 kJ kg−1 K−1

aira — 28.96 0.287 1.01 0.72 1.40oxygen O2 32.00 0.260 0.92 0.66 1.40nitrogen N2 28.01 0.297 1.04 0.74 1.40atmospheric nitrogenb (AN) 28.17 0.295 1.03 0.74 1.40carbon dioxide CO2 44.01 0.189 0.84 0.65 1.29carbon monoxide CO 28.01 0.297 1.04 0.74 1.40hydrogen H2 2.016 4.12 14.31 10.18 1.41methane CH4 16.04 0.518 2.23 1.71 1.30ethane C2H6 30.07 0.277 1.75 1.47 1.19helium He 4.00 2.08 5.20 3.12 1.67

aComposition of dry air: 21.0% oxygen, 79.0% atmospheric nitrogen by no. of kmol or by volume; 23.2% oxygen,76.8% atmospheric nitrogen by mass.

bAtmospheric nitrogen contains approx. 1% (by no. of kmol or volume) argon and traces of carbon dioxide andother gases, in addition to nitrogen


E.7 Perfect gases

Table E.4: Isentropic compressible flow functions for perfect gas with γ = 1.40

MaPP0

ρρ0

TT0

MaPP0

ρρ0

TT0

0 1 1 1 1 0.5283 0.6339 0.83330.05 0.9983 0.9988 0.9995 1.02 0.5160 0.6234 0.82780.10 0.9930 0.9950 0.9980 1.04 0.5039 0.6129 0.82220.15 0.9844 0.9888 0.9955 1.06 0.4919 0.6024 0.8165

1.08 0.4800 0.5920 0.81080.20 0.9725 0.9803 0.99210.22 0.9668 0.9762 0.9904 1.10 0.4684 0.5817 0.80520.24 0.9607 0.9718 0.9886 1.15 0.4398 0.5562 0.79080.26 0.9541 0.9670 0.9867 1.20 0.4124 0.5311 0.77640.28 0.9470 0.9619 0.9846 1.25 0.3861 0.5067 0.7619

1.30 0.3609 0.4829 0.74740.30 0.9395 0.9564 0.9823 1.35 0.3370 0.4598 0.73290.32 0.9315 0.9506 0.97990.34 0.9231 0.9445 0.9774 1.40 0.3142 0.4374 0.71840.36 0.9143 0.9380 0.9747 1.45 0.2927 0.4158 0.70400.38 0.9052 0.9313 0.9719 1.50 0.2724 0.3950 0.6897

1.55 0.2533 0.3750 0.67540.40 0.8956 0.9243 0.9690 1.60 0.2353 0.3557 0.66140.42 0.8857 0.9170 0.9659 1.65 0.2184 0.3373 0.64750.44 0.8755 0.9094 0.96270.46 0.8650 0.9016 0.9594 1.70 0.2026 0.3197 0.63370.48 0.8541 0.8935 0.9559 1.75 0.1878 0.3029 0.6202

1.80 0.1740 0.2868 0.60680.50 0.8430 0.8852 0.9524 1.85 0.1612 0.2715 0.59360.52 0.8317 0.8766 0.9487 1.90 0.1492 0.2570 0.58070.54 0.8201 0.8679 0.9449 1.95 0.1381 0.2432 0.56800.56 0.8082 0.8589 0.94100.58 0.7962 0.8498 0.9370 2.00 0.1278 0.2300 0.5556

2.10 0.1094 0.2058 0.53130.60 0.7840 0.8405 0.9328 2.20 0.09352 0.1841 0.50810.62 0.7716 0.8310 0.9286 2.30 0.07997 0.1646 0.48590.64 0.7591 0.8213 0.9243 2.40 0.06840 0.1472 0.46470.66 0.7465 0.8115 0.91990.68 0.7338 0.8016 0.9153 2.50 0.05853 0.1317 0.4444

2.60 0.05012 0.1179 0.42520.70 0.7209 0.7916 0.9107 2.70 0.04295 0.1056 0.40680.72 0.7080 0.7814 0.9061 2.80 0.03685 0.09463 0.38940.74 0.6951 0.7712 0.9013 2.90 0.03165 0.08489 0.37290.76 0.6821 0.7609 0.89640.78 0.6691 0.7505 0.8915 3.00 0.02722 0.07623 0.3571

3.20 0.02023 0.06165 0.32810.80 0.6560 0.7400 0.8865 3.40 0.01512 0.05009 0.30190.82 0.6430 0.7295 0.8815 3.60 0.01138 0.04089 0.27840.84 0.6300 0.7189 0.8763 3.80 0.008629 0.03355 0.25720.86 0.6170 0.7083 0.87110.88 0.6041 0.6977 0.8659 4.00 0.006586 0.02766 0.2381

4.50 0.003455 0.01745 0.19800.90 0.5913 0.6870 0.8606 5.00 0.001890 0.01134 0.16670.92 0.5785 0.6764 0.85520.94 0.5658 0.6658 0.8498 6.00 0.000633 0.005194 0.12200.96 0.5532 0.6551 0.8444 8.00 0.000102 0.001414 0.072460.98 0.5407 0.6445 0.8389 10.00 0.000024 0.000495 0.04762

1 0.5283 0.6339 0.8333 ∞ 0 0 0


E.7 Perfect gases

Table E.5: Ideal (semi-perfect) gas specific enthalpy h (kJ kg−1, 25 C datum)for combustion calculations on a mass basis

carbon water atmos. carbondioxide vapour nitrogen nitrogen oxygen air monoxide hydrogen

T (C) CO2 H2O N2 (AN) O2 — CO H2

0 –22.5 –45.7 –25.6 –25.4 –23.1 –24.8 –25.7 –356.8

25 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

100 69.2 139.5 77.5 76.8 70.1 75.2 77.9 1074.5200 165.3 331.0 182.4 180.9 165.5 177.2 183.5 2517.0300 266.0 528.9 289.3 286.8 263.1 281.2 291.2 3970.5400 371.1 733.1 398.0 394.6 362.9 387.1 400.9 5435.3500 480.7 943.6 508.6 504.3 465.0 494.9 512.7 6911.1

600 594.8 1160.5 621.1 615.8 569.2 604.7 626.5 8398.1700 713.3 1383.8 735.4 729.2 675.6 716.4 742.5 9896.3800 836.4 1613.3 851.6 844.4 784.2 830.0 860.4 11405.5900 963.9 1849.2 969.8 961.5 895.0 945.6 980.5 12925.9

1000 1095.9 2091.5 1089.8 1080.4 1008.0 1063.2 1102.6 14457.5

1100 1232.3 2340.1 1211.6 1201.2 1123.2 1182.6 1226.7 16000.21200 1373.3 2595.0 1335.4 1323.9 1240.6 1304.0 1352.9 17554.01300 1518.7 2856.3 1461.0 1448.4 1360.2 1427.4 1481.2 19119.01400 1668.7 3123.9 1588.5 1574.8 1482.0 1552.7 1611.6 20695.11500 1823.1 3397.8 1717.9 1703.1 1606.0 1679.9 1744.0 22282.4

Enthalpy values in Table E.5 have been computed using the approximation

Cp = a + bT

so that

h − hD =∫Cp dT = (T − TD)

[a +

12b (T + TD)

]where datum temperature TD = 298.15 K and datum enthalpy hD(at T = TD) = 0 (this is equiv-alent to using a mean Cp between 298.15 K (25 C) and T ). The values of the constants are

tabulated below; T must be in K, giving h in kJ kg−1. The magnitudes of the maximum andmean errors in h refer to the range 0 to 1500 C.

carbon water atmos. carbondioxide vapour nitrogen nitrogen oxygen air monoxide hydrogen

CO2 H2O N2 (AN) O2 — CO H2

a 0.772 1.647 0.970 0.962 0.861 0.938 0.969 13.95b × 106 448 634 188 186 220 194 206 1114(kJ kg−1 K−2)

max. error 4.0 0.7 0.5 0.6 1.3 0.4 0.5 0.7mean. error 1.4 0.2 0.2 0.2 0.5 0.2 0.2 0.3(%)


E.7 Perfect gases

Table E.6: Molar Enthalpy of Formation h0f (kJ kmol−1 at 25 C and 1 atmosphere) as gas or

vapour (g), except where indicated as solid (s) or liquid (l).

carbon as graphite C(s) 0hydrogen H2 0methane CH4 –74 850ethane C2H6 –84 680propane C3H8 –103 850n-octane C8H18(l) –249 950ethanol C2H5OH(l) –277 690hydrogen peroxide H2O2 –136 310carbon dioxide CO2 –393 520water vapour H2O(g) –241 820liquid water H2O(l) –285 830nitrogen N2 0atmospheric nitrogen (AN) 0oxygen O2 0air — 0carbon monoxide C0 –110 530

Table E.7: Ideal gas molar enthalpy h (kJ kmol−1, 25 C da-tum)

carbon water atmos. carbondioxide vapour nitrogen nitrogen oxygen aira monoxide hydrogen

T (C) CO2 H2O N2 (AN) O2 — CO H2

0 –990 –823 –717 –715 –739 –719 –720 –71925 0 0 0 0 0 0 0 0

100 3 045 2 513 2 170 2 164 2 244 2 179 2 181 2 166200 7 276 5 964 5 110 5 096 5 297 5 133 5 139 5 074300 11 705 9 528 8 103 8 080 8 420 8 143 8 155 8 005400 16 332 13 207 11 148 11 117 11 614 11 210 11 229 10 957500 21 155 17 001 14 245 14 205 14 878 14 333 14 360 13 933600 26 176 20 908 17 396 17 347 18 213 17 512 17 550 16 931700 31 393 24 930 20 599 20 540 21 618 20 747 20 796 19 951800 36 808 29 066 23 855 23 786 25 094 24 038 24 101 22 994900 42 420 33 316 27 163 27 085 28 640 27 385 27 463 26 059

1000 48 229 37 680 30 524 30 436 32 256 30 789 30 883 29 1461100 54 236 42 159 33 938 33 839 35 943 34 249 34 361 32 2561200 60 439 46 751 37 404 37 295 39 700 37 765 37 896 35 3891300 66 840 51 458 40 923 40 803 43 527 41 337 41 489 38 5441400 73 438 56 280 44 495 44 363 47 425 44 965 45 140 41 7211500 80 233 61 215 48 119 47 976 51 393 48 650 48 848 44 921

aN.B. In a reaction equation, nox(O2 + 3.762N2) represents 4.762nox equivalent kmol of air


E.8 Heating (or calorific) values of fuels

E.8 Heating (or calorific) values of fuels

M = molar mass

Cp = mean constant-pressure specific heat for use near 25 Cρ = approximate density

Abbreviations:

GCV gross calorific value (= HHV = HCV)HHV higher heating value (=GCV = HCV = negative of enthalpy of combustion

with liquid H20 in products)LHV lower heating value (=NCV = LCV = negative of enthalpy of combustion

with vapour H20 in products)NCV net calorific value (=LHV = LCV)

Table E.8: Heating (or calorific) values of gas fuels at 25 C.

Gas M HHV or GCV LHV or NCV Cp

kg kmol−1 kJ kg−1 kJ kg−1 kJ kg−1 K−1

hydrogen 2.016 141 800 119 980 14.31methane 16.04 55 500 50 020 2.23ethane 30.07 51 880 47 490 1.75propane 44.10 50 350 46 360 1.67n-butane 58.12 49 500 45 720 1.68n-pentane 72.15 49 020 45 360 1.67n-hexane 86.18 48 680 45 110 1.66carbon monoxide 28.01 10 100 10 100 1.04typical North Sea gasa 17.05 53 510 48 290 2.15

amolar composition: CH4 94.4%, C2H6 3.0%, N2 1.5%, other gases 1.1%. Elemental composition by mass: C73.26%, H 23.90%, O 0.38%, N 2.46%.

Table E.9: Heating (or calorific) values of liquid fuels at 25 C.

Approx. elemental

Liquid composition ρ HHV or GCV LHV or NCV Cp

by mass (%) kg m−3 kJ kg−1 kJ kg−1 kJ kg−1 K−1

n-octane C8H18 C84.1 H15.9 703 47 890 44 420 2.11methanol CH3OH C37.5 H12.6 O49.9 790 22 690 19 960 2.51petrol (gasoline) C85.0 H15.0 740 46 900 43 630 2.06kerosine C86.1 H13.9 770 46 140 43 100 2.02distillate fuel oil C86.8 H13.2 820 45 600 42 720 1.95


E.9 Properties of R134a refrigerant


R134a or HFC134a is a hydrofluorocarbon refrigerant (1,1,1,2-tetrafluoroethane, CH2FCF3)with zero ozone-depleting potential, although it has some global warming potential. It is a sub-stitute for the chlorofluorocarbon refrigerant R12 (banned under the Montreal Protocol 1984) indomestic refrigeration and freezing applications and in the coolers of air conditioning plant.

T , Tsat = temperature, at saturationP = absolute pressurev = specific volume, m3 kg−1

h = specific enthalpy, kJ kg−1

s = specific entropy, kJ kg−1 K−1



Table E.10: Saturated Refrigerant 134a — Temperature (−60C to critical point)

T P vf vg hf hg sf sg

(C) bar (abs) m3 kg−1 m3 kg−1 kJ kg−1 kJ kg−1 kJ kg−1 K−1 kJ kg−1 K−1

−60 0.1587 0.0006795 1.0808 22.92 261.17 0.6828 1.8005−50 0.2942 0.0006923 0.6064 35.33 267.44 0.7396 1.7798

−40 0.5118 0.0007060 0.3609 47.88 273.71 0.7946 1.7632−35 0.6612 0.0007132 0.2838 54.22 276.84 0.8214 1.7562−30 0.8436 0.0007206 0.2257 60.60 279.95 0.8479 1.7500−25 1.064 0.0007284 0.1814 67.03 283.05 0.8740 1.7445

−20 1.327 0.0007364 0.1473 73.51 286.13 0.8998 1.7397−15 1.639 0.0007448 0.1206 80.05 289.18 0.9252 1.7354−10 2.005 0.0007536 0.09954 88.64 292.20 0.9504 1.7316−5 2.432 0.0007627 0.08279 93.29 295.19 0.9753 1.7283

0 2.925 0.0007723 0.06933 100.00 298.10 1.0000 1.7254

5 3.492 0.0007823 0.05842 106.78 301.02 1.0244 1.722810 4.139 0.0007929 0.04951 113.62 303.86 1.0486 1.720515 4.873 0.0008041 0.04218 120.54 306.64 1.0727 1.718520 5.702 0.0008160 0.03610 127.54 309.35 1.0965 1.7167

25 6.634 0.0008285 0.03102 134.61 311.97 1.1202 1.715030 7.675 0.0008419 0.02676 141.77 314.49 1.1437 1.713435 8.835 0.0008563 0.02315 149.02 316.91 1.1671 1.711940 10.12 0.0008718 0.02008 156.37 319.20 1.1904 1.7104

45 11.55 0.0008886 0.01746 163.83 321.34 1.2136 1.708750 13.11 0.0009068 0.01520 171.41 323.31 1.2368 1.7069

60 16.73 0.0009493 0.01154 187.00 326.60 1.2833 1.702480 26.21 0.001077 0.006516 221.07 328.95 1.3798 1.6852

101 40.55 0.001964 0.001964 289.40 289.40 1.5609 1.5609



Table E.11: Saturated Refrigerant 134a — Pressure (0.2 bar to critical point)

P T vf vg hf hg sf sg

bar (abs) C m3 kg−1 m3 kg−1 kJ kg−1 kJ kg−1 kJ kg−1 K−1 kJ kg−1 K−1

0.2 −56.38 0.0006841 0.8703 27.40 263.44 0.7035 1.79250.4 −44.58 0.0006996 0.4547 42.11 270.84 0.7696 1.77030.6 −36.93 0.0007104 0.3109 51.77 275.64 0.8111 1.75880.8 −31.11 0.0007189 0.2373 59.18 279.26 0.8420 1.7513

1.0 −26.36 0.0007262 0.1923 65.28 282.21 0.8669 1.74601.2 −22.31 0.0007327 0.1620 70.51 284.71 0.8879 1.74181.4 −18.75 0.0007385 0.1400 75.14 286.89 0.9062 1.73851.6 −15.58 0.0007438 0.1234 79.29 288.83 0.9223 1.7358

2.0 −10.06 0.0007534 0.09978 86.53 292.16 0.9500 1.73162.5 −4.26 0.0007641 0.08062 94.28 295.62 0.9790 1.72783.0 0.70 0.0007737 0.06766 100.95 298.54 1.0034 1.72503.5 5.07 0.0007825 0.05829 106.87 301.06 1.0247 1.7228

4.0 8.98 0.0007907 0.05119 112.22 303.29 1.0437 1.72105.0 15.80 0.0008060 0.04113 121.66 307.08 1.0765 1.71826.0 21.66 0.0008201 0.03431 129.88 310.23 1.1044 1.7161

8.0 31.45 0.0008460 0.02565 143.86 315.21 1.1505 1.713010.0 39.55 0.0008703 0.02034 155.70 319.00 1.1883 1.710512.0 46.50 0.0008939 0.01675 166.09 321.95 1.2206 1.7082

15.0 55.45 0.0009289 0.01308 179.83 325.23 1.2621 1.704620.0 67.72 0.0009896 0.009318 199.58 328.37 1.3197 1.697530.0 86.38 0.001144 0.005306 233.52 327.42 1.4135 1.6747

40.55 101.00 0.001964 0.001964 289.35 289.35 1.5609 1.5609



Tabl

eE

.12:

Sup

erhe

ated

Ref

riger

ant1

34a

(0.2

bar

to1

bar)

0.2

bar

abs

0.4

bar

abs

0.6

bar

abs

0.8

bar

abs

1.0

bar

abs

(Tsa

t=−5

6.38 C

)(T

sat=−4

4.58 C

)(T

sat=−3

6.93 C

)(T

sat=−3

1.11 C

)(T

sat=−2

6.36 C

)T

( C

)v

hs

vh

sv

hs

vh

sv

hs

T( C

)

Sat

.0.

8703

263.

441.

7925

0.45

4727

0.84

1.77

030.

3109

275.

641.

7588

0.23

7327

9.26

1.75

130.

1923

282.

211.

7460

Sat

.

−50

0.89

7526

7.92

1.81

29−5

0−4

00.

9398

275.

081.

8443

0.46

4727

4.21

1.78

49−4

0−3

00.

9818

282.

411.

8750

0.48

6428

1.67

1.81

620.

3212

280.

901.

7808

0.23

8628

0.13

1.75

49−3

0−2

01.

0236

289.

911.

9053

0.50

7928

9.27

1.84

690.

3359

288.

611.

8119

0.24

9928

7.95

1.78

640.

1982

287.

271.

7662

−20

−10

1.06

5229

7.59

1.93

500.

5291

297.

031.

8769

0.35

0429

6.46

1.84

220.

2610

295.

881.

8171

0.20

7329

5.30

1.79

73−1

0

01.

107

305.

431.

9643

0.55

0230

4.94

1.90

640.

3647

304.

451.

8720

0.27

1930

3.94

1.84

720.

2162

303.

431.

8276

0

101.

148

313.

451.

9931

0.57

1231

3.02

1.93

550.

3789

312.

581.

9013

0.28

2731

2.14

1.87

670.

2250

311.

691.

8573

1020

1.18

932

1.64

2.02

150.

5921

321.

261.

9641

0.39

3032

0.87

1.93

000.

2935

320.

471.

9056

0.23

3732

0.08

1.88

6420

301.

231

330.

012.

0496

0.61

2932

9.66

1.99

220.

4071

329.

311.

9584

0.30

4132

8.96

1.93

410.

2423

328.

601.

9150

3040

1.27

233

8.54

2.07

730.

6337

338.

222.

0200

0.42

1033

7.91

1.98

630.

3147

337.

591.

9621

0.25

0933

7.27

1.94

3240

500.

6544

346.

952.

0475

0.43

5034

6.67

2.01

380.

3252

346.

381.

9897

0.25

9434

6.09

1.97

0950

600.

6751

355.

842.

0746

0.44

8935

5.58

2.04

100.

3357

355.

322.

0169

0.26

7935

5.05

1.99

8260

700.

4627

364.

662.

0678

0.34

6236

4.41

2.04

380.

2763

364.

172.

0251

7080

0.28

4737

3.44

2.05

1880



Tabl

eE

.13:

Sup

erhe

ated

Ref

riger

ant1

34a

(1.5

bar

to4

bar)

1.5

bar

abs

2.0

bar

abs

2.5

bar

abs

3.0

bar

abs

3.5

bar

abs

(Tsa

t=−1

7.13 C

)(T

sat=−1

0.06 C

)(T

sat=−4.2

6 C

)(T

sat=

0.70 C

)(T

sat=

8.98 C

)T

(C

)v

hs

vh

sv

hs

vh

sv

hs

T(

C)

Sat

.0.

1311

287.

891.

7371

0.09

9829

2.16

1.73

160.

0806

295.

621.

7278

0.06

7729

8.54

1.72

500.

0512

303.

291.

7210

Sat

.

−10

0.13

5729

3.79

1.75

990.

0998

292.

221.

7318

−10

00.

1419

302.

131.

7910

0.10

4730

0.77

1.76

380.

0824

299.

371.

7416

0

100.

1481

310.

551.

8212

0.10

9530

9.37

1.79

470.

0864

308.

151.

7732

0.07

0930

6.90

1.75

500.

0515

304.

251.

7244

1020

0.15

4131

9.07

1.85

080.

1142

318.

031.

8247

0.09

0231

6.97

1.80

380.

0742

315.

881.

7862

0.05

4231

3.59

1.75

6820

300.

1600

327.

701.

8798

0.11

8732

6.78

1.85

410.

0940

325.

841.

8336

0.07

7532

4.88

1.81

640.

0568

322.

891.

7880

3040

0.16

5833

6.46

1.90

820.

1232

335.

641.

8828

0.09

7733

4.80

1.86

270.

0806

333.

951.

8458

0.05

9333

2.18

1.81

8140

500.

1716

345.

361.

9362

0.12

7734

4.61

1.91

100.

1013

343.

861.

8911

0.08

3734

3.09

1.87

450.

0617

341.

511.

8475

50

600.

1773

354.

391.

9637

0.13

2135

3.71

1.93

880.

1049

353.

031.

9191

0.08

6835

2.33

1.90

270.

0641

350.

911.

8761

6070

0.18

3136

3.56

1.99

080.

1364

362.

941.

9661

0.10

8536

2.31

1.94

650.

0898

361.

681.

9303

0.06

6536

0.38

1.90

4170

800.

1887

372.

872.

0176

0.14

0837

2.30

1.99

300.

1120

371.

731.

9736

0.09

2837

1.14

1.95

750.

0688

369.

961.

9316

8090

0.19

4438

2.34

2.04

400.

1451

381.

812.

0195

0.11

5538

1.27

2.00

020.

0957

380.

731.

9843

0.07

1037

9.64

1.95

8690

100

0.20

0039

1.95

2.07

010.

1493

391.

452.

0457

0.11

8939

0.95

2.02

650.

0986

390.

452.

0107

0.07

3338

9.44

1.98

5310

0

110

0.15

3640

1.24

2.07

160.

1224

400.

772.

0525

0.10

1540

0.31

2.03

680.

0755

399.

362.

0115

110

120

0.10

4441

0.30

2.06

250.

0777

409.

412.

0374

120

130

0.07

9941

9.59

2.06

3013

0



Tabl

eE

.14:

Sup

erhe

ated

Ref

riger

ant1

34a

(5ba

rto

12ba

r)

5ba

rab

s6

bar

abs

8ba

rab

s10

bar

abs

12ba

rab

s(T

sat=

15.8

0 C

)(T

sat=

21.6

6 C

)(T

sat=

31.4

5 C

)(T

sat=

39.5

5 C

)(T

sat=

46.5

0 C

)T

( C

)v

hs

vh

sv

hs

vh

sv

hs

T( C

)

Sat

.0.

0411

307.

081.

7182

0.03

4331

0.23

1.71

610.

0256

315.

211.

7130

0.02

0331

9.00

1.71

050.

0167

321.

951.

7082

Sat

.

200.

0421

311.

161.

7322

2030

0.04

4332

0.79

1.76

450.

0360

318.

561.

7440

3040

0.04

6533

0.34

1.79

550.

0379

328.

411.

7760

0.02

7032

4.24

1.74

220.

0204

319.

511.

7121

4050

0.04

8533

9.88

1.82

550.

0397

338.

181.

8067

0.02

8633

4.56

1.77

470.

0218

330.

581.

7469

0.01

7232

6.10

1.72

1150

600.

0505

349.

441.

8546

0.04

1434

7.93

1.83

640.

0300

344.

741.

8057

0.02

3134

1.30

1.77

960.

0184

337.

531.

7559

6070

0.05

2435

9.06

1.88

310.

0431

357.

691.

8653

0.03

1435

4.85

1.83

560.

0243

351.

821.

8107

0.01

9534

8.56

1.78

8670

800.

0543

368.

751.

9109

0.04

4736

7.51

1.89

350.

0327

364.

941.

8646

0.02

5436

2.23

1.84

060.

0205

359.

361.

8196

8090

0.05

6237

8.53

1.93

820.

0463

377.

391.

9210

0.03

3937

5.05

1.89

280.

0265

372.

601.

8696

0.02

1537

0.04

1.84

9490

100

0.05

8138

8.41

1.96

510.

0479

387.

361.

9481

0.03

5238

5.21

1.92

040.

0276

382.

971.

8978

0.02

2438

0.65

1.87

8210

0

110

0.05

9939

8.40

1.99

150.

0495

397.

421.

9747

0.03

6439

5.43

1.94

750.

0286

393.

381.

9253

0.02

3439

1.26

1.90

6311

012

00.

0617

408.

512.

0175

0.05

1040

7.60

2.00

100.

0376

405.

751.

9740

0.02

9640

3.84

1.95

220.

0242

401.

891.

9337

120

130

0.06

3541

8.74

2.04

320.

0525

417.

892.

0268

0.03

8841

6.15

2.00

020.

0306

414.

381.

9787

0.02

5141

2.57

1.96

0513

014

00.

0653

429.

112.

0686

0.05

4042

8.30

2.05

230.

0400

426.

672.

0259

0.03

1642

5.01

2.00

480.

0259

423.

321.

9868

140

150

0.04

1243

7.30

2.05

140.

0325

435.

732.

0304

0.02

6843

4.15

2.01

2715

0



Tabl

eE

.15:

Sup

erhe

ated

Ref

riger

ant1

34a

(16

bar

to30

bar)

15ba

rab

s18

bar

abs

22ba

rab

s26

bar

abs

30ba

rab

s(T

sat=

55.4

5 C

)(T

sat=

63.1

3 C

)(T

sat=

71.9

6 C

)(T

sat=

79.6

2 C

)(T

sat=

86.3

8 C

)T

( C

)v

hs

vh

sv

hs

vh

sv

hs

T( C

)

Sat

.0.

0131

325.

231.

7046

0.01

0632

7.42

1.70

060.

0082

632

8.95

1.69

410.

0065

932

8.99

1.68

580.

0053

132

7.42

1.67

47S

at.

600.

0136

331.

031.

7222

6070

0.01

4734

3.14

1.75

800.

0113

336.

801.

7282

7080

0.01

5635

4.70

1.79

120.

0123

349.

461.

7646

0.00

909

341.

131.

7290

0.00

664

329.

721.

6878

8090

0.01

6536

5.94

1.82

250.

0131

361.

451.

7981

0.00

993

354.

661.

7668

0.00

763

346.

451.

7346

0.00

575

335.

431.

6969

9010

00.

0173

376.

991.

8526

0.01

3937

3.05

1.82

960.

0107

367.

281.

8011

0.00

840

360.

681.

7732

0.00

665

352.

831.

7442

100

110

0.01

8138

7.94

1.88

150.

0146

384.

421.

8596

0.01

1337

9.37

1.83

310.

0090

537

3.80

1.80

790.

0073

436

7.52

1.78

3011

012

00.

0189

398.

851.

9096

0.01

5339

5.67

1.88

860.

0120

391.

171.

8634

0.00

965

386.

311.

8402

0.00

793

381.

001.

8178

120

130

0.01

9640

9.76

1.93

710.

0159

406.

851.

9167

0.01

2540

2.78

1.89

260.

0102

398.

451.

8707

0.00

846

393.

831.

8500

130

140

0.02

0342

0.71

1.96

390.

0165

418.

021.

9441

0.01

3141

4.30

1.92

080.

0107

410.

381.

8999

0.00

894

406.

271.

8805

140

150

0.02

1043

1.71

1.99

020.

0171

429.

211.

9708

0.01

3642

5.77

1.94

830.

0112

422.

191.

9281

0.00

940

418.

471.

9096

150


E.10 Transport properties of air, water and steam


Cp = specific heat at constant pressure kJ kg−1 K−1

Cv = specific heat at constant volume kJ kg−1 K−1

k = thermal conductivity W m−1 K−1

T = temperature K or Cx = definition kJ kg−1 K−1

x = definition kJ kg−1 K−1

α = thermal diffusivity m2 s−1

µ = absolute viscosity kg m−1 s−1 (= N s m−2)ν = kinematic viscosity m2 s−1

ρ = density kg m−3



Tabl

eE

.16:

Tran

spor

tpro

pert

ies

ofdr

yai

rat

atm

osph

eric

pres

sure

Spe

cific

heat

Abs

olut

eat

cons

tant

(or

dyna

mic

)K

inem

atic

The

rmal

The

rmal

Pra

ndtl

Tem

pera

ture

Den

sity

pres

sure

visc

osity

visc

osity

cond

uctiv

itydiffu

sivi

tynu

mbe

rTe

mpe

ratu

re

Tρ

Cp

µν=µ ρ

kα

Pr=µC

p

kgβ αν=

GrP

r

L3∆T

T

(C

)(k

gm−3

)(J

kg−1

K−1

)(k

gm−1

s−1)

(m2

s−1)

(Wm−1

K−1

)(m

2s−

1)

(m−3

K−1

)(

C)

−180

3.72

1035

6.50×

10−6

1.75×

10−6

0.00

761.

9×

10−6

0.92

3.2×

1010

−180

−100

2.04

1010

1.16×

10−5

5.69×

10−6

0.01

67.

6×

10−6

0.75

1.3×

109

−100

−50

1.58

210

061.

45×

10−5

9.17×

10−6

0.02

01.

30×

10−5

0.72

3.67×

108

−50

01.

293

1006

1.71×

10−5

1.32×

10−5

0.02

41.

84×

10−5

0.72

1.48×

108

010

1.24

710

061.

76×

10−5

1.41×

10−5

0.02

51.

96×

10−5

0.72

1.25×

108

1020

1.20

510

061.

81×

10−5

1.50×

10−5

0.02

52.

08×

10−5

0.72

1.07×

108

20

301.

165

1006

1.86×

10−5

1.60×

10−5

0.02

62.

23×

10−5

0.72

9.07×

107

3060

1.06

010

082.

00×

10−5

1.89×

10−5

0.02

82.

74×

10−5

0.70

5.71×

107

6010

00.

946

1011

2.18×

10−5

2.30×

10−5

0.03

23.

28×

10−5

0.70

3.48×

107

100

200

0.74

610

252.

58×

10−5

3.46×

10−5

0.03

95.

19×

10−5

0.68

9.53×

106

200

300

0.61

610

452.

95×

10−5

4.79×

10−5

0.04

57.

17×

10−5

0.68

4.96×

106

300

500

0.45

610

933.

58×

10−5

7.85×

10−5

0.05

61.

14×

10−4

0.70

1.42×

106

500

1000

0.27

711

854.

82×

105

1.74×

10−4

0.07

62.

42×

10−4

0.72

1.8×

105

1000



Tabl

eE

.17:

Tran

spor

tpro

pert

ies

ofsa

tura

ted

wat

eran

dst

eam

Spe

cific

heat

atT

herm

alA

bsol

ute

(or

dyna

mic

)P

rand

tlnu

mbe

r

Tem

p.S

peci

ficvo

lum

eco

nsta

ntpr

essu

reco

nduc

tivity

visc

osity

µC

p

kTe

mp.

Tv f

v gC

pfC

pgk f

k gµ

fµ

gP

r fP

r gT

(C

)(m

3kg−1

)(m

3kg−1

)(J

kg−1

K−1

)(J

kg−1

K−1

)(W

m−1

K−1

)(W

m−1

K−1

)(k

gm−1

s−1)

(kg

m−1

s−1)

(C

)

0.01

0.00

1000

206.

042

1718

540.

569

0.01

731.

755×

10−3

8.8×

10−6

13.0

20.

942

0.01

100.

0010

0010

6.3

4193

1860

0.58

70.

0185

1.30

1×

10−3

9.1×

10−6

9.29

0.91

510

200.

0010

0257

.78

4182

1866

0.60

30.

0191

1.00

2×

10−3

9.4×

10−6

6.95

0.91

820

300.

0010

0432

.90

4179

1875

0.61

80.

0198

7.97×

10−4

9.7×

10−6

5.39

0.92

330

400.

0010

0819

.53

4179

1885

0.63

20.

0204

6.51×

10−4

1.01×

10−5

4.31

0.93

040

500.

0010

1212

.04

4181

1899

0.64

30.

0210

5.44×

10−4

1.04×

10−5

3.53

0.93

950

600.

0010

177.

674

4185

1915

0.65

30.

0217

4.62×

10−4

1.07×

10−5

2.96

0.94

760

700.

0010

235.

045

4190

1936

0.66

20.

0224

4.00×

10−4

1.11×

10−5

2.53

0.95

670

800.

0010

293.

409

4197

1962

0.67

00.

0231

3.50×

10−4

1.14×

10−5

2.19

0.96

680

900.

0010

362.

362

4205

1992

0.67

60.

0240

3.11×

10−4

1.17×

10−5

1.93

0.97

690

100

0.00

1043

1.67

442

1620

280.

681

0.02

492.

78×

10−4

1.21×

10−5

1.72

30.

986

100

125

0.00

1065

0.77

0942

5421

470.

687

0.02

722.

19×

10−4

1.33×

10−5

1.35

81.

047

125

150

0.00

1090

0.39

2943

1023

140.

687

0.03

001.

80×

10−4

1.44×

10−5

1.13

31.

110

150

175

0.00

1121

0.21

6843

8925

420.

679

0.03

341.

53×

10−4

1.56×

10−5

0.99

01.

185

175

200

0.00

1156

0.12

7344

9728

430.

665

0.03

751.

33×

10−4

1.67×

10−5

0.90

21.

270

200

225

0.00

1199

0.07

846

4648

3238

0.64

40.

0427

1.18

2×

10−4

1.79×

10−5

0.85

31.

3622

525

00.

0012

510.

0501

148

6737

720.

616

0.04

951.

065×

10−4

1.91×

10−5

0.84

11.

4525

027

50.

0013

170.

0327

852

0245

610.

582

0.05

879.

72×

10−5

2.02×

10−5

0.86

91.

5627

530

00.

0014

040.

0216

757

6258

630.

541

0.07

198.

97×

10−5

2.14×

10−5

0.95

51.

7430

032

50.

0015

280.

0141

968

6184

400.

493

0.09

297.

90×

10−5

2.30×

10−5

1.10

02.

0932

5

350

0.00

1740

0.00

8812

1010

017

150

0.43

70.

1343

6.48×

10−5

2.58×

10−5

1.50

3.29

350

360

0.00

1894

0.00

6962

1460

025

100

0.40

00.

168

5.82×

10−5

2.75×

10−5

2.11

3.89

360

374

0.00

3106

0.00

3106

∞∞

0.24

0.24

4.5×

10−5

4.5×

10−5

∞∞

374


E.11 Approximate physical properties


ρ = density kg m−3

Cp = specific heat at constant pressure J kg−1 K−1

k = thermal conductivity W m−1 K−1

µ = absolute or dynamic viscosity kg m−1 m−1 or N s m−2

σ = surface tension N m−1

Table E.18: Approximate physical properties at 20 C, 1 bar.

Gasesa ρ Cp k µ

air 1.19 1 010 0.025 1.81 × 10−5

oxygen 1.31 910 0.026 2.03 × 10−5

nitrogen 1.15 1 040 0.026 1.76 × 10−5

carbon dioxide 1.80 8 40 0.017 1.47 × 10−5

hydrogen 0.083 14 300 0.18 0.88 × 10−5

helium 0.164 5 230 0.14 1.96 × 10−5

Liquids ρ Cp k µ σ

water 1000 4 190 0.60 1.00 × 10−3 0.073mercury 13 600 140 8.7 1.55 × 10−3 0.51ethanol 790 2 860 0.19 1.20 × 10−3 0.022R134a (25 C) 1 210 1430 0.080 0.21 × 10−3

Solids ρ Cp k Notes

mild steel 7 850 460 52stainless steel 7810 460 16 18% Ni & 8% Craluminium alloy 2720 880 170 Duralumincopper 8 950 380 400brass 8 410 380 120 30% Znpolyethylene 1 000 2 100 0.5 moderately high densityexpanded polystyrene 25 0.04 boardconcrete 2 400 900 1.0 moderately densebrick 1 800 750 0.6 common building brickwood 500 2 500 0.15 pine & dryglass 2 500 800 1.0 window

aNB. Constant Cp values in Table E.3 are averages over a range of temperature. Cp(T ) relationships used in TableE.5 are fits over a different temperature range. Neither will necessarily agree well with the 20 C values tabulatedhere.



Psy

chro

met

ric c

hart

for m

oist

air

at a

tmos

pher

ic p

ress

ure

wet-bu

lb

tempe

rature Tw, °C

rela

tive

hum

idity

φ, %

dry-

bulb

tem

pera

ture

Td,

°C (=

actu

al m

ixtu

re te

mpe

ratu

re)

90

0.90

Mix

ture

spe

cific

vol

ume

v, m

3/k

g

0.85

0.80

0.75

8070

6050

4030

200.

030

0.02

8

0.02

6

0.02

4

0.02

2

0.02

0

0.01

8

0.01

6

0.01

4

0.01

2

0.01

0

0.00

8

0.00

6

0.00

4

0.00

2

0–1

0–5

05

1015

2025

3035

4045

5055

60

–10

–5

0

5

10

15

20

25

30

specific humidity ω (kg water vapour per kg dry air)

Figure E.2: Psychrometric Chart


E.12 Thermodynamic property tables for water/steam (IAPWS-IF97 formulation)


h = specific enthalpy kJ kg−1

P = absolute pressure bar (1 bar = 105 Pa = 105 N m−2)s = specific entropy kJ kg−1 K−1

T = temperature Cu = specific internal energy kJ kg−1

v = specific volume m−3 kg−1

Subscripts:

f = saturated liquid statefg = change between saturated liquid and saturated vapour states at constant pressure

(hfg = hg − hf)g = saturated vapour statesat = saturation state

Properties in these tables were evaluated from the Industrial Formulation 1997 of the Interna-tional Association for the Properties of Water and Steam (IAPWS). There are small differencesbetween these values and those in use in the Department of Mechanical Engineering up to2003 (evaluated from the IAPWS 1984 Formulation for Scientific and General Use and 1986Supplementary Release on Saturation Properties).

Triple point: T = 0.01 C P = 0.006117 barCritical point: T = 373.95 C P = 220.64 bar

u and s are chosen to be zero for saturated liquid at the triple point.

Linear interpolation in the tables is not advisable near the critical point.



Table E.19: Saturated water and steam — Temperature (triple point to 100 C)

T P vf vg uf ug hf hfg hg sf sgC bar (abs) m3 kg−1 m3 kg−1 kJ kg−1 kJ kg−1 kJ kg−1 kJ kg−1 kJ kg−1 kJ kg−1 K−1 kJ kg−1 K−1

0.01 0.006117 0.001000 206.0 0 2374.9 0.000612 2500.9 2500.9 0 9.155

1 0.006571 0.001000 192.4 4.18 2376.3 4.177 2498.6 2502.7 0.0153 9.1292 0.007060 0.001000 179.8 8.39 2377.7 8.392 2496.2 2504.6 0.0306 9.1033 0.007581 0.001000 168.0 12.60 2379.0 12.60 2493.8 2506.4 0.0459 9.0764 0.008135 0.001000 157.1 16.81 2380.4 16.81 2491.4 2508.2 0.0611 9.0515 0.008726 0.001000 147.0 21.02 2381.8 21.02 2489.1 2510.1 0.0763 9.025

6 0.009354 0.001000 137.6 25.22 2383.2 25.22 2486.7 2511.9 0.0913 8.9997 0.01002 0.001000 128.9 29.42 2384.5 29.43 2484.3 2513.7 0.1064 8.9748 0.01073 0.001000 120.8 33.62 2385.9 33.63 2481.9 2515.6 0.1213 8.9499 0.01148 0.001000 113.3 37.82 2387.3 37.82 2479.6 2517.4 0.1362 8.924

10 0.01228 0.001000 106.3 42.02 2388.7 42.02 2477.2 2519.2 0.1511 8.900

11 0.01313 0.001000 99.79 46.21 2390.0 46.22 2474.8 2521.1 0.1659 8.87612 0.01403 0.001001 93.72 50.41 2391.4 50.41 2472.5 2522.9 0.1806 8.85113 0.01498 0.001001 88.07 54.60 2392.8 54.60 2470.1 2524.7 0.1953 8.82814 0.01599 0.001001 82.80 58.79 2394.1 58.79 2467.7 2526.5 0.2099 8.80415 0.01706 0.001001 77.88 62.98 2395.5 62.98 2465.4 2528.4 0.2245 8.780

16 0.01819 0.001001 73.29 67.17 2396.9 67.17 2463.0 2530.2 0.2390 8.75717 0.01938 0.001001 69.01 71.36 2398.3 71.36 2460.6 2532.0 0.2534 8.73418 0.02065 0.001001 65.00 75.55 2399.6 75.55 2458.3 2533.8 0.2678 8.71119 0.02198 0.001002 61.26 79.73 2401.0 79.73 2455.9 2535.7 0.2822 8.68920 0.02339 0.001002 57.76 83.92 2402.4 83.92 2453.5 2537.5 0.2965 8.666

21 0.02488 0.001002 54.49 88.10 2403.7 88.10 2451.2 2539.3 0.3108 8.64422 0.02645 0.001002 51.42 92.29 2405.1 92.29 2448.8 2541.1 0.3250 8.62223 0.02811 0.001003 48.55 96.47 2406.4 96.47 2446.4 2542.9 0.3391 8.60024 0.02986 0.001003 45.86 100.7 2407.8 100.7 2444.1 2544.7 0.3532 8.57825 0.03170 0.001003 43.34 104.8 2409.2 104.8 2441.7 2546.5 0.3673 8.557

26 0.03364 0.001003 40.98 109.0 2410.5 109.0 2439.3 2548.4 0.3813 8.53527 0.03568 0.001004 38.76 113.2 2411.9 113.2 2437.0 2550.2 0.3952 8.51428 0.03783 0.001004 36.68 117.4 2413.2 117.4 2434.6 2552.0 0.4091 8.49329 0.04009 0.001004 34.72 121.6 2414.6 121.6 2432.2 2553.8 0.4230 8.47330 0.04247 0.001004 32.88 125.7 2415.9 125.7 2429.8 2555.6 0.4368 8.452

32 0.04759 0.001005 29.53 134.1 2418.7 134.1 2425.1 2559.2 0.4643 8.41134 0.05325 0.001006 26.56 142.5 2421.4 142.5 2420.3 2562.8 0.4916 8.37236 0.05948 0.001006 23.93 150.8 2424.0 150.8 2415.6 2566.4 0.5187 8.33238 0.06632 0.001007 21.60 159.2 2426.7 159.2 2410.8 2570.0 0.5457 8.29440 0.07384 0.001008 19.52 167.5 2429.4 167.5 2406.0 2573.5 0.5724 8.256

42 0.08209 0.001009 17.67 175.9 2432.1 175.9 2401.2 2577.1 0.5990 8.21844 0.09112 0.001009 16.01 184.2 2434.8 184.3 2396.4 2580.7 0.6255 8.18246 0.1010 0.001010 14.54 192.6 2437.4 192.6 2391.6 2584.2 0.6517 8.14548 0.1118 0.001011 13.21 201.0 2440.1 201.0 2386.8 2587.8 0.6778 8.11050 0.1235 0.001012 12.03 209.3 2442.8 209.3 2382.0 2591.3 0.7038 8.075

52 0.1363 0.001013 10.96 217.7 2445.4 217.7 2377.1 2594.8 0.7296 8.04054 0.1502 0.001014 10.01 226.0 2448.0 226.1 2372.3 2598.4 0.7552 8.00756 0.1653 0.001015 9.145 234.4 2450.7 234.4 2367.4 2601.9 0.7807 7.97358 0.1817 0.001016 8.369 242.8 2453.3 242.8 2362.6 2605.4 0.8060 7.94060 0.1995 0.001017 7.668 251.1 2455.9 251.2 2357.7 2608.8 0.8312 7.908

62 0.2187 0.001018 7.034 259.5 2458.5 259.5 2352.8 2612.3 0.8563 7.87664 0.2394 0.001019 6.460 267.9 2461.1 267.9 2347.9 2615.8 0.8811 7.84566 0.2618 0.001020 5.940 276.2 2463.7 276.3 2343.0 2619.2 0.9059 7.81468 0.2860 0.001022 5.468 284.6 2466.3 284.6 2338.0 2622.7 0.9305 7.78470 0.3120 0.001023 5.040 293.0 2468.9 293.0 2333.1 2626.1 0.9550 7.754

72 0.3400 0.001024 4.650 301.4 2471.4 301.4 2328.1 2629.5 0.9793 7.72574 0.3701 0.001025 4.295 309.7 2474.0 309.8 2323.1 2632.9 1.004 7.69676 0.4024 0.001026 3.971 318.1 2476.5 318.2 2318.1 2636.3 1.028 7.66778 0.4370 0.001028 3.675 326.5 2479.0 326.6 2313.1 2639.7 1.052 7.63980 0.4741 0.001029 3.405 334.9 2481.6 334.9 2308.1 2643.0 1.075 7.611

82 0.5139 0.001030 3.158 343.3 2484.1 343.3 2303.0 2646.4 1.099 7.58484 0.5564 0.001032 2.932 351.7 2486.6 351.7 2297.9 2649.7 1.123 7.55786 0.6017 0.001033 2.724 360.1 2489.0 360.1 2292.8 2653.0 1.146 7.53088 0.6502 0.001035 2.534 368.5 2491.5 368.6 2287.7 2656.3 1.169 7.50490 0.7018 0.001036 2.359 376.9 2494.0 377.0 2282.6 2659.5 1.193 7.478

92 0.7568 0.001037 2.198 385.3 2496.4 385.4 2277.4 2662.8 1.216 7.45394 0.8154 0.001039 2.050 393.7 2498.8 393.8 2272.2 2666.0 1.239 7.42796 0.8777 0.001040 1.914 402.1 2501.2 402.2 2267.0 2669.2 1.262 7.40398 0.9439 0.001042 1.788 410.6 2503.6 410.7 2261.7 2672.4 1.284 7.378100 1.0142 0.001043 1.672 419.0 2506.0 419.1 2256.5 2675.6 1.307 7.354



Table E.20: Saturated water and steam — Pressure (triple point to 2 bar)

P T vf vg uf ug hf hfg hg sf sg

bar (abs) C m3 kg−1 m3 kg−1 kJ kg−1 kJ kg−1 kJ kg−1 kJ kg−1 kJ kg−1 kJ kg−1 K−1 kJ kg−1 K−1

0.006117 0.01 0.001000 206.0 0 2374.9 0.000611783 2500.9 2500.9 0 9.155

0.008 3.76 0.001000 159.6 15.81 2380.1 15.809 2492.0 2507.8 0.0575 9.0570.010 6.97 0.001000 129.2 29.30 2384.5 29.298 2484.4 2513.7 0.1059 8.9750.012 9.65 0.001000 108.7 40.57 2388.2 40.569 2478.0 2518.6 0.1460 8.9080.014 11.97 0.001001 93.90 50.28 2391.4 50.28 2472.5 2522.8 0.1802 8.8520.016 14.01 0.001001 82.75 58.83 2394.2 58.84 2467.7 2526.6 0.2101 8.8040.018 15.84 0.001001 74.01 66.49 2396.7 66.49 2463.4 2529.9 0.2366 8.761

0.020 17.50 0.001001 66.99 73.43 2398.9 73.43 2459.5 2532.9 0.2606 8.7230.022 19.01 0.001002 61.21 79.79 2401.0 79.79 2455.9 2535.7 0.2824 8.6880.024 20.41 0.001002 56.38 85.65 2402.9 85.66 2452.6 2538.2 0.3024 8.6570.026 21.72 0.001002 52.27 91.11 2404.7 91.11 2449.5 2540.6 0.3210 8.6280.028 22.94 0.001002 48.73 96.20 2406.4 96.20 2446.6 2542.8 0.3382 8.601

0.030 24.08 0.001003 45.66 100.99 2407.9 100.99 2443.9 2544.9 0.3543 8.5770.035 26.67 0.001003 39.47 111.83 2411.4 111.84 2437.7 2549.6 0.3907 8.5210.040 28.96 0.001004 34.79 121.40 2414.5 121.40 2432.3 2553.7 0.4224 8.4730.045 31.01 0.001005 31.13 129.98 2417.3 129.98 2427.4 2557.4 0.4507 8.4310.050 32.88 0.001005 28.19 137.76 2419.8 137.77 2423.0 2560.8 0.4763 8.394

0.060 36.16 0.001006 23.73 151.49 2424.3 151.49 2415.2 2566.7 0.5209 8.3290.070 39.00 0.001007 20.53 163.36 2428.1 163.37 2408.4 2571.8 0.5591 8.2750.080 41.51 0.001008 18.10 173.84 2431.4 173.85 2402.4 2576.2 0.5925 8.2270.090 43.76 0.001009 16.20 183.25 2434.5 183.26 2397.0 2580.3 0.6223 8.1860.100 45.81 0.001010 14.67 191.80 2437.2 191.81 2392.1 2583.9 0.6492 8.149

0.11 47.68 0.001011 13.41 199.65 2439.7 199.66 2387.6 2587.2 0.6737 8.1150.12 49.42 0.001012 12.36 206.90 2442.0 206.91 2383.4 2590.3 0.6963 8.0850.13 51.04 0.001013 11.46 213.65 2444.1 213.66 2379.5 2593.1 0.7172 8.0570.14 52.55 0.001013 10.69 220.0 2446.1 220.0 2375.8 2595.8 0.7366 8.0310.15 53.97 0.001014 10.02 225.9 2448.0 225.9 2372.4 2598.3 0.7548 8.007

0.16 55.31 0.001015 9.431 231.5 2449.8 231.6 2369.1 2600.7 0.7720 7.9850.17 56.59 0.001015 8.909 236.9 2451.4 236.9 2366.0 2602.9 0.7882 7.9640.18 57.80 0.001016 8.443 241.9 2453.0 241.9 2363.1 2605.0 0.8035 7.9440.19 58.95 0.001017 8.025 246.8 2454.5 246.8 2360.2 2607.0 0.8181 7.9250.20 60.06 0.001017 7.648 251.4 2456.0 251.4 2357.5 2608.9 0.8320 7.907

0.22 62.13 0.001018 6.994 260.1 2458.7 260.1 2352.5 2612.6 0.8579 7.8740.24 64.05 0.001019 6.446 268.1 2461.2 268.1 2347.8 2615.9 0.8818 7.8440.26 65.84 0.001020 5.979 275.6 2463.5 275.6 2343.4 2619.0 0.9040 7.8170.28 67.52 0.001021 5.578 282.6 2465.7 282.6 2339.2 2621.8 0.9246 7.7910.30 69.10 0.001022 5.229 289.2 2467.7 289.2 2335.3 2624.6 0.9439 7.767

0.35 72.68 0.001024 4.525 304.2 2472.3 304.3 2326.4 2630.7 0.9876 7.7150.40 75.86 0.001026 3.993 317.5 2476.3 317.6 2318.5 2636.1 1.026 7.6690.45 78.71 0.001028 3.576 329.5 2479.9 329.6 2311.3 2640.9 1.060 7.6290.50 81.32 0.001030 3.240 340.4 2483.2 340.5 2304.7 2645.2 1.091 7.5930.55 83.71 0.001032 2.964 350.5 2486.2 350.5 2298.7 2649.2 1.119 7.561

0.60 85.93 0.001033 2.732 359.8 2488.9 359.8 2293.0 2652.9 1.145 7.5310.65 87.99 0.001035 2.535 368.5 2491.5 368.5 2287.7 2656.2 1.169 7.5040.70 89.93 0.001036 2.365 376.6 2493.9 376.7 2282.7 2659.4 1.192 7.4790.75 91.76 0.001037 2.217 384.3 2496.1 384.4 2278.0 2662.4 1.213 7.4560.80 93.49 0.001038 2.087 391.6 2498.2 391.6 2273.5 2665.2 1.233 7.434

0.85 95.13 0.001040 1.972 398.5 2500.2 398.5 2269.3 2667.8 1.252 7.4130.90 96.69 0.001041 1.869 405.0 2502.1 405.1 2265.2 2670.3 1.269 7.3940.95 98.18 0.001042 1.777 411.3 2503.8 411.4 2261.3 2672.7 1.286 7.3761.00 99.61 0.001043 1.694 417.3 2505.5 417.4 2257.5 2674.9 1.303 7.359

1.01325 99.97 0.001043 1.673 418.9 2506.0 419.0 2256.5 2675.5 1.307 7.354

1.1 102.3 0.001045 1.550 428.7 2508.7 428.8 2250.4 2679.2 1.333 7.3271.2 104.8 0.001047 1.428 439.2 2511.6 439.3 2243.8 2683.1 1.361 7.2981.3 107.1 0.001049 1.325 449.0 2514.3 449.1 2237.5 2686.6 1.387 7.2711.4 109.3 0.001051 1.237 458.2 2516.9 458.4 2231.6 2690.0 1.411 7.2461.5 111.4 0.001053 1.159 466.9 2519.2 467.1 2226.0 2693.1 1.434 7.223

1.6 113.3 0.001054 1.091 475.2 2521.4 475.3 2220.7 2696.0 1.455 7.2011.7 115.1 0.001056 1.031 483.0 2523.5 483.2 2215.6 2698.8 1.475 7.1811.8 116.9 0.001058 0.9775 490.5 2525.5 490.7 2210.7 2701.4 1.494 7.1621.9 118.6 0.001059 0.9293 497.6 2527.3 497.8 2206.1 2703.9 1.513 7.1442.0 120.2 0.001061 0.8857 504.5 2529.1 504.7 2201.6 2706.2 1.530 7.127






2.0 120.2 0.001061 0.8857 504.5 2529.1 504.7 2201.6 2706.2 1.530 7.1272.2 123.3 0.001063 0.8101 517.4 2532.4 517.6 2193.0 2710.6 1.563 7.0952.4 126.1 0.001066 0.7467 529.4 2535.4 529.6 2185.0 2714.6 1.593 7.0662.6 128.7 0.001068 0.6928 540.6 2538.2 540.9 2177.4 2718.3 1.621 7.0392.8 131.2 0.001071 0.6463 551.2 2540.8 551.5 2170.3 2721.7 1.647 7.015

3.0 133.5 0.001073 0.6058 561.1 2543.2 561.5 2163.4 2724.9 1.672 6.9923.2 135.7 0.001075 0.5702 570.6 2545.4 570.9 2156.9 2727.9 1.695 6.9703.4 137.8 0.001078 0.5387 579.6 2547.5 580.0 2150.7 2730.6 1.717 6.9503.6 139.9 0.001080 0.5105 588.2 2549.5 588.6 2144.7 2733.3 1.738 6.9313.8 141.8 0.001082 0.4852 596.4 2551.3 596.8 2138.9 2735.7 1.758 6.913

4.0 143.6 0.001084 0.4624 604.3 2553.1 604.7 2133.3 2738.1 1.777 6.8954.2 145.4 0.001085 0.4417 611.9 2554.8 612.3 2127.9 2740.3 1.795 6.8794.4 147.1 0.001087 0.4227 619.2 2556.4 619.7 2122.7 2742.4 1.812 6.8634.6 148.7 0.001089 0.4054 626.2 2557.9 626.7 2117.6 2744.4 1.829 6.8494.8 150.3 0.001091 0.3895 633.0 2559.3 633.6 2112.7 2746.3 1.845 6.834

5.0 151.8 0.001093 0.3748 639.6 2560.7 640.2 2107.9 2748.1 1.861 6.8215.5 155.5 0.001097 0.3426 655.3 2563.9 655.9 2096.5 2752.3 1.897 6.7896.0 158.8 0.001101 0.3156 669.8 2566.8 670.5 2085.6 2756.1 1.931 6.7596.5 162.0 0.001104 0.2926 683.5 2569.4 684.2 2075.4 2759.6 1.963 6.7327.0 165.0 0.001108 0.2728 696.4 2571.8 697.1 2065.6 2762.7 1.992 6.707

7.5 167.8 0.001111 0.2555 708.6 2574.0 709.4 2056.3 2765.6 2.020 6.6848.0 170.4 0.001115 0.2403 720.1 2576.0 721.0 2047.3 2768.3 2.046 6.6628.5 172.9 0.001118 0.2269 731.2 2577.9 732.1 2038.6 2770.8 2.071 6.6419.0 175.4 0.001121 0.2149 741.7 2579.7 742.7 2030.3 2773.0 2.094 6.6219.5 177.7 0.001124 0.2041 751.8 2581.3 752.9 2022.3 2775.2 2.117 6.603

10 179.9 0.001127 0.1943 761.6 2582.8 762.7 2014.4 2777.1 2.138 6.58511 184.1 0.001133 0.1774 780.0 2585.5 781.2 1999.5 2780.7 2.179 6.55212 188.0 0.001139 0.1632 797.1 2587.9 798.5 1985.3 2783.8 2.216 6.52213 191.6 0.001144 0.1512 813.3 2590.0 814.8 1971.7 2786.5 2.251 6.49414 195.0 0.001149 0.1408 828.5 2591.8 830.1 1958.8 2788.9 2.284 6.468

15 198.3 0.001154 0.1317 843.0 2593.5 844.7 1946.3 2791.0 2.315 6.44316 201.4 0.001159 0.1237 856.8 2594.9 858.6 1934.3 2792.9 2.344 6.42017 204.3 0.001163 0.1167 869.9 2596.2 871.9 1922.6 2794.5 2.371 6.39818 207.1 0.001168 0.1104 882.5 2597.3 884.6 1911.4 2796.0 2.398 6.37819 209.8 0.001172 0.1047 894.6 2598.3 896.8 1900.4 2797.3 2.423 6.358

20 212.4 0.001177 0.09958 906.3 2599.2 908.6 1889.8 2798.4 2.447 6.33921 214.9 0.001181 0.09493 917.5 2600.0 920.0 1879.4 2799.4 2.470 6.32122 217.3 0.001185 0.09070 928.4 2600.7 931.0 1869.2 2800.2 2.492 6.30423 219.6 0.001189 0.08681 938.9 2601.3 941.6 1859.3 2800.9 2.514 6.28724 221.8 0.001193 0.08324 949.1 2601.8 952.0 1849.6 2801.5 2.534 6.271

25 224.0 0.001197 0.07995 959.0 2602.2 962.0 1840.1 2802.0 2.554 6.25626 226.1 0.001201 0.07690 968.6 2602.5 971.7 1830.7 2802.5 2.574 6.24127 228.1 0.001205 0.07407 978.0 2602.8 981.2 1821.5 2802.8 2.593 6.22728 230.1 0.001209 0.07143 987.1 2603.0 990.5 1812.5 2803.0 2.611 6.21329 232.0 0.001213 0.06897 996.0 2603.2 999.5 1803.6 2803.2 2.628 6.199

30 233.9 0.001217 0.06666 1004.7 2603.3 1008.4 1794.9 2803.3 2.646 6.18631 235.7 0.001220 0.06450 1013.2 2603.3 1017.0 1786.3 2803.3 2.662 6.17332 237.5 0.001224 0.06247 1021.5 2603.3 1025.5 1777.8 2803.2 2.679 6.16033 239.2 0.001228 0.06056 1029.7 2603.3 1033.7 1769.4 2803.1 2.695 6.14834 240.9 0.001231 0.05876 1037.6 2603.2 1041.8 1761.1 2803.0 2.710 6.136

35 242.6 0.001235 0.05706 1045.5 2603.0 1049.8 1753.0 2802.7 2.725 6.12536 244.2 0.001239 0.05545 1053.1 2602.9 1057.6 1744.9 2802.5 2.740 6.11337 245.8 0.001242 0.05392 1060.6 2602.6 1065.2 1736.9 2802.1 2.755 6.10238 247.3 0.001246 0.05247 1068.0 2602.4 1072.8 1729.0 2801.8 2.769 6.09139 248.9 0.001249 0.05109 1075.3 2602.1 1080.2 1721.2 2801.4 2.783 6.080

40 250.4 0.001253 0.04978 1082.4 2601.8 1087.4 1713.5 2800.9 2.797 6.07041 251.8 0.001256 0.04853 1089.4 2601.4 1094.6 1705.8 2800.4 2.810 6.05942 253.3 0.001259 0.04733 1096.3 2601.1 1101.6 1698.2 2799.9 2.823 6.04943 254.7 0.001263 0.04619 1103.1 2600.6 1108.6 1690.7 2799.3 2.836 6.03944 256.1 0.001266 0.04510 1109.8 2600.2 1115.4 1683.2 2798.7 2.849 6.029

45 257.4 0.001270 0.04406 1116.4 2599.7 1122.1 1675.9 2798.0 2.861 6.020






45 257.4 0.001270 0.04406 1116.4 2599.7 1122.1 1675.9 2798.0 2.861 6.02046 258.8 0.001273 0.04306 1122.9 2599.2 1128.8 1668.5 2797.3 2.874 6.01047 260.1 0.001276 0.04210 1129.3 2598.7 1135.3 1661.2 2796.6 2.886 6.00148 261.4 0.001280 0.04118 1135.7 2598.2 1141.8 1654.0 2795.8 2.898 5.99249 262.7 0.001283 0.04030 1141.9 2597.6 1148.2 1646.8 2795.0 2.909 5.983

50 263.9 0.001286 0.03945 1148.1 2597.0 1154.5 1639.7 2794.2 2.921 5.97451 265.2 0.001290 0.03863 1154.2 2596.4 1160.7 1632.7 2793.4 2.932 5.96552 266.4 0.001293 0.03784 1160.2 2595.7 1166.9 1625.6 2792.5 2.943 5.95653 267.6 0.001296 0.03708 1166.1 2595.1 1173.0 1618.6 2791.6 2.954 5.94854 268.8 0.001300 0.03635 1172.0 2594.4 1179.0 1611.7 2790.7 2.965 5.939

55 270.0 0.001303 0.03564 1177.8 2593.7 1184.9 1604.8 2789.7 2.976 5.93156 271.1 0.001306 0.03496 1183.5 2593.0 1190.8 1597.9 2788.7 2.986 5.92257 272.3 0.001309 0.03430 1189.2 2592.2 1196.6 1591.1 2787.7 2.997 5.91458 273.4 0.001313 0.03366 1194.8 2591.5 1202.4 1584.3 2786.7 3.007 5.90659 274.5 0.001316 0.03305 1200.3 2590.7 1208.1 1577.6 2785.6 3.017 5.898

60 275.6 0.001319 0.03245 1205.8 2589.9 1213.7 1570.8 2784.6 3.027 5.89062 277.7 0.001326 0.03131 1216.6 2588.2 1224.9 1557.5 2782.3 3.047 5.87464 279.8 0.001332 0.03024 1227.3 2586.5 1235.8 1544.2 2780.0 3.067 5.85966 281.9 0.001339 0.02923 1237.7 2584.7 1246.5 1531.1 2777.6 3.085 5.84468 283.9 0.001345 0.02828 1247.9 2582.8 1257.1 1518.1 2775.1 3.104 5.829

70 285.8 0.001352 0.02738 1258.0 2580.9 1267.4 1505.1 2772.6 3.122 5.81572 287.7 0.001358 0.02653 1267.9 2578.9 1277.7 1492.3 2769.9 3.140 5.80074 289.6 0.001365 0.02572 1277.6 2576.9 1287.7 1479.5 2767.2 3.157 5.78676 291.4 0.001371 0.02495 1287.2 2574.8 1297.6 1466.8 2764.4 3.174 5.77278 293.2 0.001378 0.02422 1296.7 2572.6 1307.4 1454.1 2761.5 3.191 5.758

80 295.0 0.001385 0.02353 1306.0 2570.4 1317.1 1441.5 2758.6 3.208 5.74582 296.7 0.001391 0.02286 1315.2 2568.1 1326.6 1429.0 2755.6 3.224 5.73184 298.4 0.001398 0.02223 1324.3 2565.8 1336.0 1416.5 2752.5 3.240 5.71886 300.1 0.001405 0.02163 1333.3 2563.4 1345.3 1404.0 2749.4 3.256 5.70588 301.7 0.001411 0.02105 1342.1 2560.9 1354.5 1391.6 2746.2 3.271 5.692

90 303.3 0.001418 0.02049 1350.9 2558.4 1363.7 1379.2 2742.9 3.287 5.67992 304.9 0.001425 0.01996 1359.6 2555.9 1372.7 1366.9 2739.5 3.302 5.66694 306.5 0.001432 0.01945 1368.1 2553.3 1381.6 1354.5 2736.1 3.317 5.65396 308.0 0.001439 0.01896 1376.6 2550.6 1390.4 1342.2 2732.6 3.331 5.64198 309.5 0.001446 0.01849 1385.0 2547.9 1399.2 1329.9 2729.1 3.346 5.628

100 311.0 0.001453 0.01803 1393.3 2545.1 1407.9 1317.6 2725.5 3.360 5.616105 314.6 0.001470 0.01697 1413.8 2538.0 1429.3 1286.9 2716.1 3.396 5.585110 318.1 0.001489 0.01599 1433.9 2530.5 1450.3 1256.1 2706.4 3.430 5.555115 321.4 0.001507 0.01510 1453.6 2522.6 1470.9 1225.3 2696.2 3.464 5.524120 324.7 0.001526 0.01427 1473.0 2514.4 1491.3 1194.3 2685.6 3.496 5.494

125 327.8 0.001546 0.01350 1492.1 2505.7 1511.5 1163.0 2674.5 3.529 5.464130 330.9 0.001566 0.01279 1511.0 2496.7 1531.4 1131.5 2662.9 3.561 5.434135 333.8 0.001588 0.01212 1529.8 2487.2 1551.2 1099.6 2650.8 3.592 5.404140 336.7 0.001610 0.01149 1548.3 2477.2 1570.9 1067.2 2638.1 3.623 5.373145 339.5 0.001633 0.01090 1566.8 2466.8 1590.5 1034.3 2624.8 3.654 5.342

150 342.2 0.001657 0.01034 1585.3 2455.8 1610.2 1000.7 2610.9 3.684 5.311155 344.8 0.001682 0.009811 1603.8 2444.1 1629.9 966.4 2596.2 3.715 5.279160 347.4 0.001710 0.009308 1622.3 2431.9 1649.7 931.1 2580.8 3.746 5.246165 349.9 0.001738 0.008828 1641.0 2418.9 1669.7 894.9 2564.6 3.776 5.213170 352.3 0.001769 0.008369 1660.0 2405.1 1690.0 857.4 2547.4 3.808 5.179

175 354.7 0.001803 0.007927 1679.2 2390.4 1710.8 818.4 2529.1 3.839 5.143180 357.0 0.001839 0.007499 1698.9 2374.6 1732.0 777.5 2509.5 3.872 5.106185 359.3 0.001880 0.007082 1719.2 2357.4 1754.0 734.4 2488.4 3.905 5.066190 361.5 0.001925 0.006673 1740.3 2338.6 1776.9 688.5 2465.4 3.940 5.025195 363.6 0.001977 0.006267 1762.5 2317.8 1801.1 638.9 2440.0 3.976 4.980

200 365.7 0.002039 0.005858 1786.3 2294.2 1827.1 584.3 2411.4 4.015 4.930205 367.8 0.002114 0.005438 1812.6 2266.7 1855.9 522.3 2378.2 4.059 4.874210 369.8 0.002212 0.004988 1842.9 2232.8 1889.4 448.1 2337.5 4.109 4.806215 371.8 0.002360 0.004463 1882.1 2186.2 1932.8 349.4 2282.2 4.175 4.717220 373.7 0.002750 0.003577 1961.4 2085.5 2021.9 142.3 2164.2 4.311 4.531

220.64 373.95 0.003106 0.003106 2019.0 2019.0 2087.5 0 2087.5 4.412 4.412



Tabl

eE

.23:

Sub

cool

edw

ater

and

Sup

erhe

ated

Ste

am(t

riple

poin

tto

0.1

bar)

0.00

6117

bar

(Tsa

t=

0.01 C

)0.

01ba

r(T

sat=

7.0 C

)0.

05ba

r(T

sat=

32.9 C

)0.

1ba

r(T

sat=

45.8 C

)T

(C

)v

uh

sv

uh

sv

uh

sv

uh

sT

(C

)

0.01

206.

023

74.9

2500

.99.

155

0.00

1000

0.00

0007

0.00

1007

0.00

000.

0010

000.

0000

810.

0050

820.

0000

0.00

1000

0.00

0174

0.01

018

0.00

0001

0.01

2022

1.1

2403

.125

38.4

9.28

813

5.2

2403

.025

38.2

9.06

00.

0010

0283

.92

83.9

20.

2965

0.00

1002

83.9

283

.93

0.29

6520

4023

6.2

2431

.325

75.8

9.41

114

4.5

2431

.225

75.7

9.18

428

.85

2430

.125

74.4

8.43

80.

0010

0816

7.5

167.

50.

5724

4060

251.

324

59.5

2613

.39.

527

153.

724

59.5

2613

.29.

300

30.7

124

58.7

2612

.38.

555

15.3

424

57.8

2611

.28.

233

6080

266.

424

87.9

2650

.89.

637

163.

024

87.8

2650

.89.

410

32.5

724

87.3

2650

.18.

666

16.2

724

86.7

2649

.38.

344

8010

028

1.5

2516

.426

88.6

9.74

117

2.2

2516

.326

88.5

9.51

434

.42

2516

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88.0

8.77

017

.20

2515

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87.4

8.44

910

0

120

296.

625

45.0

2726

.59.

840

181.

425

45.0

2726

.49.

613

36.2

725

44.7

2726

.18.

869

18.1

225

44.3

2725

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548

120

140

311.

725

73.9

2764

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934

190.

725

73.9

2764

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707

38.1

225

73.6

2764

.28.

964

19.0

525

73.3

2763

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643

140

160

326.

826

02.9

2802

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199.

926

02.9

2802

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798

39.9

726

02.7

2802

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055

19.9

826

02.5

2802

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734

160

180

341.

926

32.2

2841

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209.

126

32.2

2841

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885

41.8

226

32.0

2841

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141

20.9

026

31.8

2840

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821

180

200

357.

026

61.6

2880

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.20

218.

426

61.6

2880

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968

43.6

626

61.5

2879

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225

21.8

326

61.3

2879

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905

200

220

372.

126

91.3

2918

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227.

626

91.3

2918

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45.5

126

91.2

2918

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306

22.7

526

91.1

2918

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986

220

240

387.

227

21.3

2958

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236.

827

21.2

2958

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47.3

627

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2957

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384

23.6

727

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2957

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063

240

260

402.

327

51.4

2997

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246.

127

51.4

2997

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49.2

027

51.3

2997

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459

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027

51.2

2997

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260

280

417.

427

81.8

3037

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255.

327

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3037

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51.0

527

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3037

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532

25.5

227

81.6

3036

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212

280

300

432.

528

12.4

3077

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.57

264.

528

12.4

3077

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52.9

028

12.4

3076

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603

26.4

528

12.3

3076

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283

300

320

447.

628

43.3

3117

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273.

728

43.3

3117

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54.7

528

43.3

3117

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672

27.3

728

43.2

3116

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351

320

340

462.

628

74.5

3157

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283.

028

74.5

3157

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56.5

928

74.4

3157

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738

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928

74.3

3157

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418

340

360

477.

729

05.9

3198

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292.

229

05.8

3198

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58.4

429

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3198

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804

29.2

229

05.7

3197

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484

360

380

492.

829

37.5

3238

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301.

429

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3238

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60.2

829

37.4

3238

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867

30.1

429

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3238

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547

380

400

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929

69.4

3280

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310.

729

69.4

3280

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62.1

329

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3280

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929

31.0

629

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3279

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609

400

420

523.

030

01.6

3321

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319.

930

01.6

3321

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63.9

830

01.5

3321

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990

31.9

930

01.5

3321

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670

420

440

538.

130

34.0

3363

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329.

130

34.0

3363

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65.8

230

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3363

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32.9

130

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3363

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729

440

460

553.

230

66.7

3405

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338.

430

66.7

3405

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67.6

730

66.7

3405

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33.8

330

66.6

3405

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787

460

480

568.

330

99.7

3447

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347.

630

99.7

3447

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69.5

230

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3447

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34.7

630

99.6

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844

480

500

583.

431

32.9

3489

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356.

831

32.9

3489

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71.3

631

32.9

3489

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.22

35.6

831

32.9

3489

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900

500

520

598.

531

66.5

3532

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366.

131

66.5

3532

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73.2

131

66.4

3532

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.27

36.6

031

66.4

3532

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954

520

540

613.

632

00.3

3575

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375.

332

00.3

3575

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75.0

632

00.2

3575

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37.5

332

00.2

3575

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540

560

628.

732

34.4

3618

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384.

532

34.4

3618

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76.9

032

34.3

3618

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38.4

532

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3618

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560

580

643.

732

68.7

3662

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393.

732

68.7

3662

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78.7

532

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600

658.

833

03.4

3706

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033

03.4

3706

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80.5

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03.3

3706

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3706

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640

689.

033

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3794

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421.

433

73.5

3794

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84.2

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73.5

3794

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42.1

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3794

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640

680

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234

44.8

3884

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439.

934

44.8

3884

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87.9

834

44.7

3884

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43.9

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44.7

3884

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680

720

749.

435

17.2

3975

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458.

435

17.2

3975

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91.6

735

17.2

3975

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45.8

435

17.1

3975

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720

760

779.

635

90.7

4067

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476.

835

90.7

4067

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95.3

635

90.7

4067

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47.6

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90.7

4067

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760

800

809.

736

65.4

4160

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495.

336

65.4

4160

.711

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99.0

636

65.4

4160

.610

.95

49.5

336

65.3

4160

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.63

800



Tabl

eE

.24:

Sub

cool

edw

ater

and

Sup

erhe

ated

Ste

am(0

.1ba

rto

1at

mos

pher

e)

0.2

bar

(Tsa

t=

60.1 C

)0.

5ba

r(T

sat=

81.3 C

)1

bar

(Tsa

t=

99.6 C

)1.

0132

5ba

r(T

sat=

100.

0 C

)T

(C

)v

uh

sv

uh

sv

uh

sv

uh

sT

(C

)

0.01

0.00

1000

0.00

0360

0.02

036

0.00

000

0.00

1000

0.00

0915

0.05

092

0.00

000

0.00

1000

0.00

1841

0.10

190.

0000

10.

0010

000.

0018

650.

1032

0.00

001

0.01

200.

0010

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83.9

40.

296

0.00

1002

83.9

183

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0.29

650.

0010

0283

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84.0

10.

2965

0.00

1002

83.9

184

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0.29

6520

400.

0010

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7.5

167.

60.

572

0.00

1008

167.

516

7.6

0.57

240.

0010

0816

7.5

167.

60.

5724

0.00

1008

167.

516

7.6

0.57

2440

600.

0010

1725

1.1

251.

20.

831

0.00

1017

251.

125

1.2

0.83

120.

0010

1725

1.1

251.

20.

8312

0.00

1017

251.

125

1.2

0.83

1260

808.

118

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47.7

8.02

00.

0010

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335.

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0.00

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933

5.0

1.07

50.

0010

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335.

01.

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8010

08.

586

2514

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8.12

63.

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7.69

51.

696

2506

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7.36

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2506

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510

0

120

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225

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2724

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227

3.60

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41.3

2721

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798

1.79

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468

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322

3.79

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895

1.88

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67.8

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567

1.86

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2756

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561

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160

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2602

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01.6

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983

2600

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7.98

71.

984

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2631

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40.3

8.50

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170

2630

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8.07

52.

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2628

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7.75

02.

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2628

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7.74

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010

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2661

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79.1

8.58

44.

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2660

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77.8

8.15

92.

172

2658

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75.5

7.83

62.

144

2658

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0

220

11.3

726

90.8

2918

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665

4.54

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89.9

2917

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240

2.26

626

88.4

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917

2.23

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88.4

2915

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911

220

240

11.8

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20.8

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743

4.72

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20.0

2956

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319

2.36

027

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996

2.32

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18.7

2954

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240

260

12.3

027

51.0

2996

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4.91

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50.3

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2.45

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2.42

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280

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5.09

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2.51

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3034

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140

280

300

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12.1

3076

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962

5.28

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539

2.63

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217

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320

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098

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2.82

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340

360

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164

5.84

029

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2.91

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929

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380

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729

37.3

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6.02

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3.01

029

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2.97

129

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477

380

400

15.5

329

69.2

3279

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289

6.20

929

68.8

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866

3.10

329

68.3

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545

3.06

229

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539

400

420

15.9

930

01.4

3321

.29.

350

6.39

430

01.1

3320

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927

3.19

530

00.5

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606

3.15

430

00.5

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600

420

440

16.4

530

33.8

3362

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409

6.57

930

33.5

3362

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986

3.28

830

33.0

3361

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665

3.24

530

33.0

3361

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659

440

460

16.9

230

66.5

3404

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467

6.76

430

66.3

3404

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044

3.38

130

65.8

3403

.98.

723

3.33

630

65.8

3403

.88.

717

460

480

17.3

830

99.5

3447

.19.

524

6.94

930

99.3

3446

.79.

101

3.47

330

98.8

3446

.28.

780

3.42

830

98.8

3446

.18.

774

480

500

17.8

431

32.8

3489

.69.

580

7.13

431

32.6

3489

.29.

156

3.56

631

32.2

3488

.78.

836

3.51

931

32.1

3488

.78.

830

500

520

18.3

031

66.3

3532

.39.

634

7.31

931

66.1

3532

.09.

211

3.65

831

65.7

3531

.58.

891

3.61

031

65.7

3531

.58.

885

520

540

18.7

632

00.1

3575

.49.

688

7.50

331

99.9

3575

.19.

265

3.75

131

99.6

3574

.68.

944

3.70

131

99.6

3574

.68.

938

540

560

19.2

232

34.2

3618

.79.

741

7.68

832

34.0

3618

.49.

317

3.84

332

33.7

3618

.08.

997

3.79

332

33.7

3618

.08.

991

560

580

19.6

932

68.6

3662

.39.

792

7.87

332

68.4

3662

.19.

369

3.93

632

68.1

3661

.69.

049

3.88

432

68.1

3661

.69.

043

580

600

20.1

533

03.2

3706

.29.

843

8.05

833

03.1

3706

.09.

420

4.02

833

02.8

3705

.69.

100

3.97

533

02.8

3705

.69.

094

600

640

21.0

733

73.4

3794

.89.

942

8.42

733

73.2

3794

.69.

519

4.21

333

73.0

3794

.39.

199

4.15

833

73.0

3794

.29.

193

640

680

21.9

934

44.7

3884

.610

.04

8.79

734

44.5

3884

.49.

615

4.39

834

44.3

3884

.19.

295

4.34

034

44.3

3884

.19.

289

680

720

22.9

235

17.1

3975

.410

.13

9.16

635

17.0

3975

.39.

709

4.58

235

16.7

3975

.09.

389

4.52

235

16.7

3975

.09.

383

720

760

23.8

435

90.6

4067

.410

.22

9.53

535

90.5

4067

.39.

800

4.76

735

90.3

4067

.09.

480

4.70

535

90.3

4067

.09.

474

760

800

24.7

636

65.3

4160

.610

.31

9.90

536

65.2

4160

.49.

888

4.95

236

65.0

4160

.29.

568

4.88

736

65.0

4160

.29.

562

800



Tabl

eE

.25:

Sub

cool

edw

ater

and

Sup

erhe

ated

Ste

am(2

bar

to8

bar)

2ba

r(T

sat=

120.

2 C

)4

bar

(Tsa

t=

143.

6 C

)6

bar

(Tsa

t=

158.

8 C

)8

bar

(Tsa

t=

170.

4 C

)T

(C

)v

uh

sv

uh

sv

uh

sv

uh

sT

(C

)

0.01

0.00

1000

0.00

3688

0.20

370.

0000

10.

0010

000.

0073

650.

4074

0.00

003

0.00

1000

0.01

100.

6110

0.00

004

0.00

1000

0.01

470.

8145

0.00

005

0.01

200.

0010

0283

.91

84.1

10.

2965

0.00

1002

83.8

984

.29

0.29

640.

0010

0283

.88

84.4

80.

2964

0.00

1001

83.8

784

.67

0.29

6320

400.

0010

0816

7.5

167.

70.

5724

0.00

1008

167.

516

7.9

0.57

230.

0010

0816

7.5

168.

10.

5722

0.00

1008

167.

416

8.2

0.57

2140

600.

0010

1725

1.1

251.

30.

8311

0.00

1017

251.

125

1.5

0.83

100.

0010

1725

1.0

251.

60.

8309

0.00

1017

251.

025

1.8

0.83

0860

800.

0010

2933

4.9

335.

11.

075

0.00

1029

334.

833

5.2

1.07

50.

0010

2933

4.8

335.

41.

075

0.00

1029

334.

733

5.5

1.07

580

100

0.00

1043

419.

041

9.2

1.30

70.

0010

4341

8.9

419.

31.

307

0.00

1043

418.

841

9.5

1.30

70.

0010

4341

8.8

419.

61.

306

100

120

0.00

1060

503.

650

3.8

1.52

80.

0010

6050

3.5

503.

91.

528

0.00

1060

503.

450

4.1

1.52

70.

0010

6050

3.4

504.

21.

527

120

140

0.93

5325

61.3

2748

.37.

231

0.00

1080

588.

858

9.2

1.73

90.

0010

8058

8.7

589.

41.

739

0.00

1079

588.

658

9.5

1.73

914

016

00.

9843

2592

.827

89.7

7.32

90.

4839

2581

.627

75.2

6.98

30.

3167

2569

.027

59.0

6.76

60.

0011

0267

4.8

675.

71.

943

160

180

1.03

326

23.9

2830

.47.

421

0.50

9426

14.9

2818

.67.

081

0.33

4726

05.2

2806

.06.

872

0.24

7225

94.7

2792

.46.

715

180

200

1.08

126

54.7

2870

.87.

508

0.53

4326

47.3

2861

.07.

172

0.35

2126

39.4

2850

.76.

968

0.26

0926

31.1

2839

.86.

818

200

220

1.12

826

85.4

2911

.07.

591

0.55

8926

79.1

2902

.77.

259

0.36

9026

72.6

2894

.07.

058

0.27

4026

65.7

2885

.06.

911

220

240

1.17

527

16.1

2951

.27.

671

0.58

3127

10.7

2944

.07.

341

0.38

5727

05.2

2936

.67.

143

0.28

6926

99.4

2928

.96.

999

240

260

1.22

227

46.9

2991

.47.

748

0.60

7227

42.2

2985

.17.

419

0.40

2127

37.4

2978

.67.

223

0.29

9527

32.4

2972

.07.

081

260

280

1.26

927

77.8

3031

.77.

822

0.63

1127

73.7

3026

.17.

495

0.41

8327

69.4

3020

.47.

300

0.31

1927

65.1

3014

.67.

159

280

300

1.31

628

08.8

3072

.17.

894

0.65

4928

05.2

3067

.17.

568

0.43

4428

01.4

3062

.17.

374

0.32

4227

97.6

3056

.97.

235

300

320

1.36

328

40.1

3112

.77.

964

0.67

8628

36.8

3108

.27.

638

0.45

0428

33.4

3103

.77.

445

0.33

6328

30.0

3099

.17.

307

320

340

1.41

028

71.5

3153

.48.

031

0.70

2228

68.5

3149

.47.

706

0.46

6328

65.5

3145

.37.

514

0.34

8428

62.4

3141

.17.

377

340

360

1.45

629

03.1

3194

.48.

097

0.72

5729

00.4

3190

.77.

773

0.48

2228

97.6

3187

.07.

581

0.36

0428

94.9

3183

.27.

444

360

380

1.50

329

35.0

3235

.68.

161

0.74

9229

32.5

3232

.27.

837

0.49

8029

30.0

3228

.87.

646

0.37

2429

27.4

3225

.37.

510

380

400

1.54

929

67.1

3277

.08.

223

0.77

2629

64.8

3273

.97.

900

0.51

3729

62.5

3270

.77.

710

0.38

4329

60.1

3267

.67.

573

400

420

1.59

629

99.5

3318

.68.

284

0.79

6029

97.3

3315

.77.

961

0.52

9429

95.2

3312

.87.

771

0.39

6129

93.0

3309

.97.

635

420

440

1.64

230

32.0

3360

.58.

344

0.81

9430

30.1

3357

.88.

021

0.54

5130

28.1

3355

.17.

831

0.40

8030

26.1

3352

.57.

696

440

460

1.68

930

64.9

3402

.68.

402

0.84

2830

63.0

3400

.18.

080

0.56

0830

61.2

3397

.77.

890

0.41

9830

59.3

3395

.27.

755

460

480

1.73

530

98.0

3445

.08.

459

0.86

6130

96.3

3442

.78.

137

0.57

6430

94.5

3440

.47.

948

0.43

1630

92.8

3438

.17.

813

480

500

1.78

131

31.3

3487

.68.

515

0.88

9431

29.7

3485

.58.

193

0.59

2031

28.1

3483

.38.

004

0.44

3331

26.5

3481

.27.

869

500

520

1.82

831

65.0

3530

.58.

570

0.91

2631

63.5

3528

.58.

248

0.60

7631

62.0

3526

.58.

059

0.45

5131

60.4

3524

.57.

924

520

540

1.87

431

98.9

3573

.78.

624

0.93

5931

97.5

3571

.88.

302

0.62

3231

96.0

3569

.98.

113

0.46

6831

94.6

3568

.07.

979

540

560

1.92

032

33.0

3617

.18.

676

0.95

9132

31.7

3615

.48.

355

0.63

8732

30.4

3613

.68.

166

0.47

8532

29.0

3611

.88.

032

560

580

1.96

732

67.5

3660

.88.

728

0.98

2432

66.2

3659

.28.

407

0.65

4232

65.0

3657

.58.

218

0.49

0232

63.7

3655

.88.

084

580

600

2.01

333

02.2

3704

.88.

779

1.00

633

01.0

3703

.28.

458

0.66

9832

99.8

3701

.78.

269

0.50

1932

98.6

3700

.18.

135

600

640

2.10

633

72.4

3793

.68.

879

1.05

233

71.4

3792

.28.

558

0.70

0833

70.3

3790

.88.

369

0.52

5233

69.3

3789

.48.

235

640

680

2.19

834

43.8

3883

.48.

975

1.09

834

42.9

3882

.28.

654

0.73

1834

41.9

3881

.08.

466

0.54

8534

41.0

3879

.78.

332

680

720

2.29

135

16.3

3974

.49.

068

1.14

535

15.4

3973

.38.

748

0.76

2735

14.6

3972

.28.

560

0.57

1735

13.7

3971

.18.

426

720

760

2.38

335

89.9

4066

.59.

159

1.19

135

89.1

4065

.58.

839

0.79

3735

88.3

4064

.58.

651

0.59

5035

87.6

4063

.58.

517

760

800

2.47

636

64.7

4159

.89.

248

1.23

736

63.9

4158

.98.

927

0.82

4636

63.2

4157

.98.

739

0.61

8236

62.5

4157

.08.

606

800



Tabl

eE

.26:

Sub

cool

edw

ater

and

Sup

erhe

ated

Ste

am(1

0ba

rto

40ba

r)

10ba

r(T

sat=

179.

9 C

)20

bar

(Tsa

t=

212.

4 C

)30

bar

(Tsa

t=

233.

9 C

)40

bar

(Tsa

t=

250.

4 C

)T

(C

)v

uh

sv

uh

sv

uh

sv

uh

sT

(C

)

0.01

0.00

1000

0.01

827

1.01

80.

0000

70.

0009

990.

0360

12.

034

0.00

013

0.00

0999

0.05

321

3.04

90.

0001

90.

0009

980.

0698

94.

063

0.00

024

0.01

200.

0010

0183

.86

84.8

60.

2963

0.00

1001

83.8

085

.80

0.29

610.

0010

0083

.74

86.7

40.

2959

0.00

1000

83.6

887

.68

0.29

5720

400.

0010

0716

7.4

168.

40.

5720

0.00

1007

167.

316

9.3

0.57

170.

0010

0716

7.2

170.

20.

5713

0.00

1006

167.

117

1.1

0.57

0940

600.

0010

1725

1.0

252.

00.

8307

0.00

1016

250.

825

2.8

0.83

020.

0010

1625

0.6

253.

70.

8296

0.00

1015

250.

425

4.5

0.82

9160

800.

0010

2933

4.7

335.

71.

075

0.00

1028

334.

433

6.5

1.07

40.

0010

2833

4.2

337.

31.

073

0.00

1027

334.

033

8.1

1.07

380

100

0.00

1043

418.

741

9.8

1.30

60.

0010

4241

8.4

420.

51.

306

0.00

1042

418.

242

1.3

1.30

50.

0010

4141

7.9

422.

01.

304

100

120

0.00

1060

503.

350

4.3

1.52

70.

0010

5950

2.9

505.

11.

526

0.00

1059

502.

650

5.8

1.52

50.

0010

5850

2.2

506.

51.

524

120

140

0.00

1079

588.

558

9.6

1.73

90.

0010

7958

8.1

590.

31.

738

0.00

1078

587.

759

0.9

1.73

70.

0010

7758

7.3

591.

61.

736

140

160

0.00

1102

674.

767

5.8

1.94

20.

0011

0167

4.2

676.

41.

941

0.00

1100

673.

767

7.0

1.94

00.

0011

0067

3.2

677.

61.

939

160

180

0.19

4425

83.0

2777

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586

0.00

1127

761.

476

3.7

2.13

80.

0011

2676

0.8

764.

22.

137

0.00

1125

760.

276

4.7

2.13

518

020

00.

2060

2622

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28.3

6.69

50.

0011

5685

0.3

852.

62.

330

0.00

1155

849.

585

3.0

2.32

90.

0011

5484

8.8

853.

42.

327

200

220

0.21

7026

58.6

2875

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793

0.10

2226

17.3

2821

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387

0.00

1189

940.

394

3.8

2.51

70.

0011

8893

9.3

944.

12.

515

220

240

0.22

7626

93.4

2921

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884

0.10

8526

60.2

2877

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497

0.06

823

2619

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24.6

6.22

80.

0012

2810

32.7

1037

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700

240

260

0.23

7927

27.4

2965

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968

0.11

4426

99.7

2928

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595

0.07

289

2667

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86.4

6.34

60.

0517

826

30.1

2837

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138

260

280

0.24

8027

60.7

3008

.77.

048

0.12

0027

37.1

2977

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685

0.07

716

2710

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42.2

6.44

90.

0554

926

80.9

2902

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259

280

300

0.25

8027

93.7

3051

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125

0.12

5527

73.2

3024

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769

0.08

118

2750

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94.3

6.54

10.

0588

727

26.2

2961

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364

300

320

0.26

7828

26.5

3094

.47.

198

0.13

0828

08.5

3070

.26.

847

0.08

502

2789

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44.2

6.62

70.

0620

227

68.2

3016

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458

320

340

0.27

7628

59.3

3136

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268

0.13

6028

43.2

3115

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922

0.08

874

2826

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92.4

6.70

70.

0650

228

08.1

3068

.16.

544

340

360

0.28

7328

92.1

3179

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337

0.14

1128

77.6

3159

.96.

994

0.09

235

2862

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39.5

6.78

20.

0679

028

46.5

3118

.16.

624

360

380

0.29

7029

24.9

3221

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403

0.14

6229

11.8

3204

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063

0.09

590

2898

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85.8

6.85

40.

0707

028

83.9

3166

.76.

699

380

400

0.30

6629

57.8

3264

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467

0.15

1229

45.8

3248

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129

0.09

938

2933

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31.6

6.92

30.

0734

329

20.6

3214

.46.

771

400

420

0.31

6229

90.8

3307

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529

0.15

6229

79.8

3292

.27.

193

0.10

2829

68.5

3277

.06.

990

0.07

611

2956

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61.4

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044

00.

3257

3024

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49.8

7.59

00.

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906

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0.33

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317

0.10

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116

0.08

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3028

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00.

3447

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35.7

7.70

70.

1708

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480

500

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0.11

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236

0.08

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0

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0.36

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490

0.11

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43.6

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293

0.08

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3135

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7.15

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054

00.

3730

3193

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66.2

7.87

40.

1853

3186

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56.6

7.54

50.

1227

3178

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47.0

7.34

90.

0914

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71.5

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207

540

560

0.38

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27.7

3610

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927

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0.12

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0.09

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00.

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3256

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317

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0.13

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0.09

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3279

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0

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131

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52.1

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0.43

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228

0.21

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35.2

3872

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0.14

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30.4

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711

0.10

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25.5

3859

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573

680

720

0.45

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12.8

3970

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322

0.22

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08.5

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998

0.15

1635

04.1

3958

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806

0.11

3434

99.7

3953

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669

720

760

0.47

5835

86.8

4062

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413

0.23

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82.8

4057

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090

0.15

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78.8

4052

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899

0.11

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74.8

4047

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762

760

800

0.49

4436

61.8

4156

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502

0.24

736

58.1

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179

0.16

4236

54.5

4147

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989

0.12

2936

50.8

4142

.57.

852

800



Tabl

eE

.27:

Sub

cool

edw

ater

and

Sup

erhe

ated

Ste

am(5

0ba

rto

80ba

r)

50ba

r(T

sat=

263.

9 C

)60

bar

(Tsa

t=

275.

6 C

)70

bar

(Tsa

t=

285.

8 C

)80

bar

(Tsa

t=

295.

0 C

)T

(C

)v

uh

sv

uh

sv

uh

sv

uh

sT

(C

)

0.01

0.00

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0.08

604

5.07

40.

0002

90.

0009

970.

1016

76.

085

0.00

034

0.00

0997

0.11

679

7.09

40.

0003

80.

0009

960.

1313

98.

101

0.00

042

0.01

200.

0010

0083

.62

88.6

10.

2955

0.00

0999

83.5

589

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0.29

520.

0009

9983

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90.4

80.

2950

0.00

0998

83.4

391

.42

0.29

4820

400.

0010

0616

6.9

172.

00.

5705

0.00

1005

166.

817

2.8

0.57

010.

0010

0516

6.7

173.

70.

5697

0.00

1004

166.

617

4.6

0.56

9340

600.

0010

1525

0.3

255.

30.

8286

0.00

1014

250.

125

6.2

0.82

800.

0010

1424

9.9

257.

00.

8275

0.00

1014

249.

725

7.8

0.82

7060

800.

0010

2733

3.8

338.

91.

072

0.00

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333.

533

9.7

1.07

10.

0010

2633

3.3

340.

51.

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333.

134

1.3

1.07

080

100

0.00

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417.

642

2.8

1.30

30.

0010

4041

7.3

423.

51.

302

0.00

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417.

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4.3

1.30

20.

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3941

6.7

425.

01.

301

100

120

0.00

1058

501.

950

7.2

1.52

30.

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1.5

507.

91.

523

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1057

501.

250

8.6

1.52

20.

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5650

0.8

509.

31.

521

120

140

0.00

1077

586.

859

2.2

1.73

40.

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6.4

592.

91.

733

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586.

059

3.5

1.73

20.

0010

7558

5.6

594.

21.

731

140

160

0.00

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767

8.1

1.93

80.

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2.1

678.

71.

936

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671.

667

9.3

1.93

50.

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1.1

679.

91.

934

160

180

0.00

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676

5.2

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40.

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2375

9.0

765.

72.

133

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476

6.2

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2275

7.8

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82.

130

180

200

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085

3.8

2.32

50.

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5284

7.3

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22.

324

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685

4.6

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5.9

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12.

321

200

220

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4.4

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7.6

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72.

511

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794

5.0

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5.8

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32.

507

220

240

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37.7

2.69

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2510

30.4

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37.9

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2210

28.3

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692

240

260

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34.8

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7311

27.0

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34.5

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6911

24.2

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876

260

280

0.04

227

2646

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58.1

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0332

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06.0

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928

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36.3

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2812

25.2

1235

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280

300

0.04

535

2698

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25.6

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92.1

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300

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2745

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20.9

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187

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2693

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63.6

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320

340

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80.

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68.1

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289

0.03

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2745

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85.5

6.18

00.

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077

340

360

0.05

319

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95.6

6.49

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11.9

3072

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380

0.03

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2793

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47.0

6.27

80.

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73.3

3020

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184

360

380

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555

2869

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46.8

6.57

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53.6

3126

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465

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816

2837

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04.4

6.36

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20.3

3081

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279

380

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0.05

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2907

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96.6

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93.6

3178

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543

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2879

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64.5

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366

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45.3

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32.6

3228

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0.04

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2919

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6.52

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329

06.8

3194

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440

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225

2982

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93.3

6.78

80.

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70.9

3278

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688

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337

2959

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63.0

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00.

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47.6

3247

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440

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439

3018

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40.7

6.85

30.

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08.5

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2998

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13.1

6.66

90.

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329

87.5

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460

480

0.06

650

3055

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87.7

6.91

70.

0548

930

45.9

3375

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0.04

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3036

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62.5

6.73

60.

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630

26.6

3349

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661

480

500

0.06

858

3091

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34.5

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82.9

3422

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3074

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11.3

6.80

00.

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65.2

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3127

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331

19.8

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6.86

10.

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531

03.5

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789

520

540

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268

3164

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27.5

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50.

0601

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56.7

3517

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002

0.05

121

3149

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07.6

6.92

10.

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41.5

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850

540

560

0.07

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3200

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74.0

7.15

20.

0618

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93.5

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059

0.05

271

3186

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55.4

6.97

90.

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79.3

3546

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560

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0.07

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3236

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20.4

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30.3

3611

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115

0.05

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3223

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7.03

60.

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17.0

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966

580

600

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3273

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66.8

7.26

00.

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632

67.2

3658

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169

0.05

566

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50.6

7.09

10.

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632

54.7

3642

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022

600

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0.08

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3346

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59.9

7.36

50.

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41.2

3752

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275

0.05

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3335

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45.6

7.19

70.

0510

433

30.1

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130

640

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0.08

657

3420

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53.5

7.46

50.

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15.7

3847

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376

0.06

143

3410

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40.8

7.29

90.

0535

734

05.8

3834

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232

680

720

0.09

045

3495

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47.6

7.56

20.

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734

90.9

3941

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473

0.06

426

3486

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36.2

7.39

70.

0560

734

81.9

3930

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331

720

760

0.09

431

3570

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42.4

7.65

50.

0784

235

66.8

4037

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567

0.06

706

3562

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32.2

7.49

20.

0585

535

58.6

4027

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427

760

800

0.09

815

3647

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37.9

7.74

60.

0816

436

43.4

4133

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658

0.06

985

3639

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28.7

7.58

40.

0610

136

36.0

4124

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519

800



Tabl

eE

.28:

Sub

cool

edw

ater

and

Sup

erhe

ated

Ste

am(9

0ba

rto

140

bar)

90ba

r(T

sat=

303.

3 C

)10

0ba

r(T

sat=

311.

0 C

)12

0ba

r(T

sat=

324.

7 C

)14

0ba

r(T

sat=

336.

7 C

)T

(C

)v

uh

sv

uh

sv

uh

sv

uh

sT

(C

)

0.01

0.00

0996

0.14

559.

107

0.00

046

0.00

0995

0.15

9110

.11

0.00

049

0.00

0994

0.18

4712

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0.00

055

0.00

0993

0.20

8414

.11

0.00

059

0.01

200.

0009

9883

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92.3

50.

2946

0.00

0997

83.3

193

.29

0.29

440.

0009

9683

.19

95.1

50.

2939

0.00

0996

83.0

797

.01

0.29

3520

400.

0010

0416

6.5

175.

50.

5689

0.00

1003

166.

317

6.4

0.56

850.

0010

0316

6.1

178.

10.

5678

0.00

1002

165.

917

9.9

0.56

7040

600.

0010

1324

9.6

258.

70.

8265

0.00

1013

249.

425

9.5

0.82

590.

0010

1224

9.1

261.

20.

8249

0.00

1011

248.

726

2.9

0.82

3860

800.

0010

2533

2.9

342.

11.

070

0.00

1024

332.

634

2.9

1.06

90.

0010

2333

2.2

344.

51.

068

0.00

1023

331.

734

6.1

1.06

680

100

0.00

1039

416.

442

5.8

1.30

00.

0010

3841

6.2

426.

51.

299

0.00

1038

415.

642

8.1

1.29

80.

0010

3741

5.1

429.

61.

296

100

120

0.00

1055

500.

551

0.0

1.52

00.

0010

5550

0.2

510.

71.

519

0.00

1054

499.

551

2.1

1.51

70.

0010

5349

8.8

513.

51.

516

120

140

0.00

1074

585.

259

4.8

1.73

00.

0010

7458

4.8

595.

51.

729

0.00

1073

583.

959

6.8

1.72

70.

0010

7158

3.1

598.

11.

725

140

160

0.00

1096

670.

768

0.5

1.93

30.

0010

9567

0.2

681.

11.

932

0.00

1094

669.

268

2.3

1.92

90.

0010

9366

8.2

683.

51.

927

160

180

0.00

1121

757.

276

7.3

2.12

90.

0011

2075

6.6

767.

82.

127

0.00

1118

755.

476

8.9

2.12

50.

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1775

4.3

769.

92.

122

180

200

0.00

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845.

185

5.5

2.31

90.

0011

4884

4.4

855.

92.

318

0.00

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843.

085

6.8

2.31

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4484

1.6

857.

72.

312

200

220

0.00

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934.

994

5.6

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60.

0011

8193

4.1

945.

92.

504

0.00

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932.

494

6.5

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00.

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7693

0.7

947.

22.

497

220

240

0.00

1221

1027

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38.2

2.69

00.

0012

1910

26.1

1038

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688

0.00

1216

1024

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38.6

2.68

30.

0012

1310

22.0

1039

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679

240

260

0.00

1267

1122

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2.87

30.

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6511

21.5

1134

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871

0.00

1262

1118

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34.0

2.86

60.

0012

5811

16.3

1133

.92.

861

260

280

0.00

1325

1223

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35.3

3.05

90.

0013

2312

21.6

1234

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056

0.00

1317

1218

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33.9

3.05

00.

0013

1212

14.8

1233

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044

280

300

0.00

1402

1331

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44.3

3.25

30.

0013

9813

29.1

1343

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248

0.00

1390

1324

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40.9

3.24

00.

0013

8213

19.6

1339

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231

300

320

0.02

271

2629

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33.9

5.83

50.

0192

725

89.9

2782

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713

0.00

1494

1442

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60.3

3.44

40.

0014

8014

35.2

1455

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432

320

340

0.02

486

2695

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19.6

5.97

70.

0214

926

67.2

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878

0.01

621

2598

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93.5

5.67

20.

0120

025

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429

340

360

0.02

672

2752

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92.5

6.09

40.

0233

327

29.3

2962

.66.

007

0.01

812

2678

.428

95.9

5.83

70.

0142

326

17.2

2816

.45.

661

360

380

0.02

840

2802

.430

58.0

6.19

60.

0249

527

83.6

3033

.16.

117

0.01

971

2742

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79.1

5.96

60.

0158

726

96.1

2918

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819

380

400

0.02

996

2849

.131

18.8

6.28

70.

0264

428

33.0

3097

.46.

214

0.02

111

2798

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51.9

6.07

60.

0172

427

60.9

3002

.25.

946

400

420

0.03

144

2893

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76.1

6.37

10.

0278

328

79.2

3157

.56.

302

0.02

239

2849

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18.2

6.17

30.

0184

628

17.7

3076

.16.

054

420

440

0.03

284

2935

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31.1

6.45

00.

0291

529

23.1

3214

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383

0.02

358

2897

.131

80.1

6.26

10.

0195

828

69.5

3143

.66.

150

440

460

0.03

420

2976

.632

84.4

6.52

30.

0304

129

65.4

3269

.56.

459

0.02

471

2942

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38.8

6.34

30.

0206

229

18.0

3206

.76.

237

460

480

0.03

551

3016

.733

36.3

6.59

30.

0316

330

06.6

3322

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531

0.02

579

2985

.832

95.2

6.41

80.

0216

029

64.1

3266

.56.

318

480

500

0.03

680

3056

.233

87.3

6.66

00.

0328

130

46.9

3375

.16.

599

0.02

683

3028

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50.0

6.49

00.

0225

530

08.4

3324

.16.

393

500

520

0.03

805

3095

.134

37.6

6.72

40.

0339

730

86.6

3426

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665

0.02

784

3069

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03.4

6.55

80.

0234

530

51.5

3379

.86.

464

520

540

0.03

928

3133

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87.2

6.78

60.

0351

031

25.9

3476

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728

0.02

882

3109

.934

55.8

6.62

40.

0243

330

93.5

3434

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532

540

560

0.04

049

3172

.135

36.5

6.84

60.

0362

131

64.8

3526

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789

0.02

978

3150

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07.4

6.68

60.

0251

931

34.9

3487

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597

560

580

0.04

168

3210

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85.4

6.90

40.

0373

032

03.5

3576

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847

0.03

072

3189

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58.4

6.74

70.

0260

231

75.7

3540

.16.

659

580

600

0.04

286

3248

.436

34.2

6.96

00.

0383

832

42.1

3625

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905

0.03

165

3229

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09.0

6.80

60.

0268

432

16.1

3591

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719

600

640

0.04

518

3324

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31.2

7.06

90.

0404

933

19.0

3723

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014

0.03

346

3307

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09.2

6.91

80.

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432

96.1

3694

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834

640

680

0.04

746

3400

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28.0

7.17

30.

0425

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95.8

3821

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119

0.03

523

3385

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08.5

7.02

40.

0300

033

75.5

3795

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942

680

720

0.04

971

3477

.439

24.8

7.27

20.

0446

134

72.9

3919

.07.

219

0.03

697

3463

.839

07.5

7.12

60.

0315

134

54.6

3895

.87.

045

720

760

0.05

193

3554

.540

21.9

7.36

80.

0466

335

50.4

4016

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316

0.03

868

3542

.240

06.4

7.22

30.

0330

135

33.9

3995

.97.

144

760

800

0.05

413

3632

.241

19.4

7.46

10.

0486

236

28.5

4114

.77.

409

0.04

037

3621

.041

05.4

7.31

80.

0344

836

13.4

4096

.07.

239

800



Tabl

eE

.29:

Sub

cool

edw

ater

and

Sup

erhe

ated

Ste

am(1

60ba

rto

220

bar)

160

bar

(Tsa

t=

347.

4 C

)18

0ba

r(T

sat=

357.

0 C

)20

0ba

r(T

sat=

365.

7 C

)22

0ba

r(T

sat=

373.

7 C

)T

(C

)v

uh

sv

uh

sv

uh

sv

uh

sT

(C

)

0.01

0.00

0992

0.23

0216

.11

0.00

061

0.00

0991

0.25

0118

.09

0.00

063

0.00

0990

0.26

8020

.08

0.00

062

0.00

0989

0.28

4222

.05

0.00

061

0.01

200.

0009

9582

.95

98.8

70.

2930

0.00

0994

82.8

310

0.7

0.29

260.

0009

9382

.71

102.

60.

2921

0.00

0992

82.5

910

4.4

0.29

1620

400.

0010

0116

5.6

181.

70.

5662

0.00

1000

165.

418

3.4

0.56

540.

0009

9916

5.2

185.

20.

5646

0.00

0998

165.

018

6.9

0.56

3940

600.

0010

1024

8.4

264.

50.

8228

0.00

1009

248.

126

6.2

0.82

180.

0010

0824

7.7

267.

90.

8207

0.00

1008

247.

426

9.6

0.81

9760

800.

0010

2233

1.3

347.

61.

065

0.00

1021

330.

934

9.2

1.06

40.

0010

2033

0.4

350.

81.

062

0.00

1019

330.

035

2.4

1.06

180

100

0.00

1036

414.

543

1.1

1.29

50.

0010

3541

4.0

432.

61.

293

0.00

1034

413.

443

4.1

1.29

20.

0010

3341

2.9

435.

61.

290

100

120

0.00

1052

498.

151

5.0

1.51

40.

0010

5149

7.5

516.

41.

512

0.00

1050

496.

851

7.8

1.51

00.

0010

4949

6.2

519.

21.

509

120

140

0.00

1070

582.

359

9.5

1.72

30.

0010

6958

1.5

600.

81.

721

0.00

1068

580.

760

2.1

1.71

90.

0010

6758

0.0

603.

41.

718

140

160

0.00

1091

667.

368

4.7

1.92

50.

0010

9066

6.3

685.

91.

923

0.00

1089

665.

468

7.2

1.92

00.

0010

8766

4.5

688.

41.

918

160

180

0.00

1115

753.

277

1.0

2.12

00.

0011

1475

2.0

772.

12.

117

0.00

1112

750.

977

3.2

2.11

50.

0011

1174

9.8

774.

22.

112

180

200

0.00

1143

840.

385

8.6

2.30

90.

0011

4183

8.9

859.

52.

306

0.00

1139

837.

686

0.4

2.30

30.

0011

3783

6.3

861.

32.

300

200

220

0.00

1174

929.

094

7.8

2.49

40.

0011

7292

7.4

948.

52.

490

0.00

1170

925.

894

9.2

2.48

70.

0011

6892

4.3

949.

92.

484

220

240

0.00

1211

1019

.910

39.3

2.67

50.

0012

0810

18.0

1039

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671

0.00

1205

1016

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40.1

2.66

80.

0012

0310

14.1

1040

.62.

664

240

260

0.00

1254

1113

.711

33.8

2.85

60.

0012

5111

11.3

1133

.82.

851

0.00

1247

1108

.911

33.8

2.84

70.

0012

4411

06.5

1133

.92.

842

260

280

0.00

1307

1211

.512

32.5

3.03

80.

0013

0212

08.4

1231

.83.

032

0.00

1298

1205

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31.3

3.02

60.

0012

9312

02.4

1230

.83.

021

280

300

0.00

1375

1315

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37.2

3.22

40.

0013

6813

11.0

1335

.63.

216

0.00

1361

1306

.913

34.1

3.20

90.

0013

5513

03.0

1332

.83.

202

300

320

0.00

1467

1428

.514

51.9

3.42

00.

0014

5614

22.2

1448

.43.

409

0.00

1445

1416

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45.3

3.39

90.

0014

3514

10.9

1442

.53.

390

320

340

0.00

1616

1561

.415

87.3

3.64

50.

0015

9115

50.1

1578

.73.

625

0.00

1569

1540

.115

71.5

3.60

80.

0015

5115

31.2

1565

.33.

593

340

360

0.01

106

2538

.727

15.6

5.46

20.

0046

2421

45.8

2233

.94.

658

0.00

2365

1834

.418

76.1

4.10

10.

0020

4717

68.2

1803

.93.

987

360

380

0.01

288

2642

.228

48.3

5.66

80.

0104

225

77.4

2764

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505

0.00

8258

2494

.026

59.2

5.31

40.

0041

2521

83.2

2279

.44.

700

380

400

0.01

428

2719

.029

47.5

5.81

80.

0119

126

71.8

2886

.35.

688

0.00

9950

2617

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16.8

5.55

20.

0082

5525

54.2

2735

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405

400

420

0.01

548

2783

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30.9

5.94

00.

0131

227

45.7

2981

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828

0.01

120

2704

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28.5

5.71

60.

0095

8826

59.0

2869

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602

420

440

0.01

655

2840

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05.0

6.04

50.

0141

728

08.9

3064

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945

0.01

225

2775

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20.3

5.84

70.

0106

527

39.3

2973

.65.

749

440

460

0.01

753

2892

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73.0

6.13

90.

0151

228

65.6

3137

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047

0.01

317

2837

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00.6

5.95

80.

0115

628

07.2

3061

.65.

871

460

480

0.01

845

2941

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36.7

6.22

50.

0159

929

17.9

3205

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138

0.01

401

2893

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73.4

6.05

60.

0123

828

67.5

3139

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976

480

500

0.01

932

2988

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97.3

6.30

50.

0168

129

67.1

3269

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222

0.01

479

2945

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41.2

6.14

50.

0131

429

22.8

3211

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070

500

520

0.02

016

3033

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55.6

6.37

90.

0175

930

14.1

3330

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300

0.01

553

2994

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05.2

6.22

60.

0138

429

74.5

3279

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156

520

540

0.02

096

3076

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12.1

6.44

90.

0183

330

59.5

3389

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373

0.01

623

3041

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66.4

6.30

30.

0145

130

23.7

3342

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236

540

560

0.02

174

3119

.434

67.3

6.51

60.

0190

531

03.6

3446

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443

0.01

690

3087

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25.6

6.37

40.

0151

430

71.0

3404

.16.

310

560

580

0.02

250

3161

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21.4

6.58

10.

0197

531

46.8

3502

.46.

509

0.01

755

3132

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83.0

6.44

30.

0157

531

16.8

3463

.46.

381

580

600

0.02

324

3202

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74.6

6.64

20.

0204

331

89.3

3557

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572

0.01

818

3175

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39.2

6.50

80.

0163

531

61.6

3521

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448

600

640

0.02

467

3284

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79.2

6.75

90.

0217

432

72.7

3664

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692

0.01

940

3260

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48.7

6.63

00.

0174

832

48.6

3633

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573

640

680

0.02

607

3365

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82.2

6.87

00.

0230

133

54.7

3768

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804

0.02

056

3344

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55.5

6.74

50.

0185

633

33.5

3742

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690

680

720

0.02

742

3445

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84.1

6.97

50.

0242

434

36.0

3872

.36.

911

0.02

169

3426

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60.5

6.85

30.

0196

134

17.2

3848

.66.

799

720

760

0.02

875

3525

.539

85.5

7.07

50.

0254

435

17.1

3975

.07.

012

0.02

279

3508

.639

64.4

6.95

50.

0206

335

00.1

3953

.86.

903

760

800

0.03

006

3605

.740

86.6

7.17

10.

0266

235

98.1

4077

.27.

109

0.02

387

3590

.440

67.7

7.05

30.

0216

235

82.6

4058

.27.

002

800



Tabl

eE

.30:

Sup

ercr

itica

lste

am(2

50ba

rto

500

bar)

250

bar

300

bar

400

bar

500

bar

T(

C)

vu

hs

vu

hs

vu

hs

vu

hs

T(

C)

0.01

0.00

0988

0.30

4925

0.00

056

0.00

0986

0.33

0829

.90.

0004

30.

0009

810.

3511

39.6

-0.0

0008

0.00

0977

0.33

2349

.17

-0.0

0087

0.01

200.

0009

9182

.41

107.

180.

2909

0.00

0989

82.1

211

1.8

0.28

970.

0009

8581

.52

120.

90.

2872

0.00

098

80.9

313

00.

2845

2040

0.00

0997

164.

618

9.5

0.56

270.

0009

9516

4.1

193.

90.

5607

0.00

0991

163

202.

60.

5568

0.00

0987

161.

921

1.3

0.55

2840

600.

0010

0624

6.9

272.

10.

8181

0.00

1004

246.

127

6.2

0.81

560.

001

244.

628

4.6

0.81

050.

0009

9624

3.1

292.

90.

8054

6080

0.00

1018

329.

435

4.8

1.05

90.

0010

1632

8.3

358.

81.

056

0.00

1011

326.

336

6.8

1.05

0.00

1007

324.

437

4.7

1.04

480

100

0.00

1031

412.

143

7.9

1.28

80.

0010

2941

0.8

441.

71.

284

0.00

1024

408.

344

9.3

1.27

70.

0010

240

5.9

456.

91.

2710

0

120

0.00

1047

495.

252

1.4

1.50

60.

0010

4549

3.6

525

1.50

20.

0010

449

0.6

532.

21.

494

0.00

1035

487.

753

9.4

1.48

612

014

00.

0010

6557

8.8

605.

41.

715

0.00

1062

576.

960

8.8

1.71

0.00

1057

573.

361

5.6

1.70

10.

0010

5256

9.8

622.

41.

692

140

160

0.00

1085

663.

169

0.2

1.91

50.

0010

8266

0.8

693.

31.

910.

0010

7665

6.5

699.

61.

899

0.00

107

652.

470

61.

889

160

180

0.00

1108

748.

277

5.9

2.10

80.

0011

0574

5.5

778.

72.

102

0.00

1098

740.

578

4.4

2.09

10.

0010

9173

5.6

790.

22.

079

180

200

0.00

1135

834.

486

2.7

2.29

60.

0011

383

1.2

865.

12.

289

0.00

1122

825.

287

0.1

2.27

60.

0011

1581

9.6

875.

32.

263

200

220

0.00

1164

921.

995

1.1

2.47

90.

0011

5991

8.2

953

2.47

10.

0011

591

1.1

957.

12.

456

0.00

1141

904.

496

1.5

2.44

122

024

00.

0011

9910

11.3

1041

.32.

658

0.00

1193

1006

.810

42.6

2.64

90.

0011

8199

8.4

1045

.62.

632

0.00

1171

990.

510

492.

615

240

260

0.00

1239

1103

.111

34.1

2.83

50.

0012

3110

97.6

1134

.62.

825

0.00

1217

1087

.411

36.1

2.80

50.

0012

0410

78.1

1138

.32.

786

260

280

0.00

1287

1198

.112

30.2

3.01

20.

0012

7711

91.3

1229

.63

0.00

1259

1178

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29.1

2.97

60.

0012

4311

67.5

1229

.72.

954

280

300

0.00

1346

1297

.413

31.1

3.19

20.

0013

3212

88.7

1328

.73.

176

0.00

1308

1273

.113

25.4

3.14

70.

0012

8812

59.3

1323

.73.

121

300

320

0.00

1421

1403

.214

38.7

3.37

60.

0014

0113

91.5

1433

.53.

355

0.00

1368

1371

.314

263.

319

0.00

1341

1354

.214

21.2

3.28

932

034

00.

0015

2615

19.3

1557

.53.

573

0.00

1493

1502

.315

47.1

3.54

40.

0014

4314

74.8

1532

.53.

496

0.00

1405

1452

.815

23.1

3.45

734

036

00.

0018

5817

14.6

1750

.43.

897

0.00

1715

1663

.317

04.5

3.81

10.

0015

8116

05.1

1660

.13.

713

0.00

1507

1568

.416

37.2

3.65

136

038

00.

0027

4919

87.1

2052

.84.

350.

0021

6518

67.2

1922

4.14

50.

0018

0517

57.9

1816

.63.

962

0.00

1656

1698

.317

68.3

3.86

338

040

00.

0052

8423

72.9

2510

.35.

030.

0029

3720

93.3

2180

.34.

518

0.00

2102

1913

.819

87.5

4.21

30.

0018

3818

29.3

1908

.64.

072

400

420

0.00

7579

2580

2769

.45.

420.

0045

6623

71.7

2512

.34.

999

0.00

2518

2077

.621

74.1

4.47

50.

0020

6519

62.6

2056

.84.

282

420

440

0.00

8697

2679

.628

97.1

5.60

10.

0062

2825

6227

48.9

5.34

20.

0031

3822

54.3

2381

.24.

761

0.00

2355

2099

.822

13.5

4.49

644

046

00.

0096

1727

58.8

2999

.25.

743

0.00

7193

2668

2883

.85.

528

0.00

4149

2447

.426

13.3

5.08

40.

0027

4122

42.4

2379

.64.

7246

048

00.

0104

228

26.6

3087

.15.

861

0.00

7992

2752

.229

925.

674

0.00

495

2579

.227

77.2

5.30

50.

0032

7723

91.7

2556

.84.

957

480

500

0.01

114

2887

.431

65.9

5.96

40.

0086

928

24.1

3084

.85.

796

0.00

5625

2681

.729

06.7

5.47

50.

0038

8925

2827

22.5

5.17

650

0

520

0.01

181

2943

.232

38.5

6.05

70.

0093

228

88.1

3167

.75.

901

0.00

6213

2766

.930

15.4

5.61

30.

0044

1726

36.5

2857

.45.

348

520

540

0.01

244

2995

.733

06.6

6.14

20.

0098

9929

46.7

3243

.75.

996

0.00

674

2841

.131

10.7

5.73

20.

0048

9627

28.4

2973

.25.

492

540

560

0.01

303

3045

.633

71.3

6.22

0.01

044

3001

.633

14.8

6.08

30.

0072

2129

07.8

3196

.75.

837

0.00

5332

2808

.830

75.4

5.61

756

058

00.

0135

930

93.6

3433

.56.

294

0.01

095

3053

.633

82.3

6.16

30.

0076

6929

69.3

3276

5.93

10.

0057

3428

8131

67.7

5.72

658

060

00.

0141

431

40.2

3493

.76.

364

0.01

144

3103

.534

46.9

6.23

70.

0080

8930

26.9

3350

.46.

017

0.00

6109

2947

.232

52.6

5.82

560

0

640

0.01

518

3230

.236

09.7

6.49

40.

0123

731

98.9

3569

.96.

375

0.00

8869

3134

.134

88.8

6.17

20.

0067

9630

67.4

3407

.25.

998

640

680

0.01

617

3317

.437

21.5

6.61

40.

0132

432

90.1

3687

.26.

501

0.00

9589

3234

3617

.66.

310.

0074

2231

76.9

3548

6.14

968

072

00.

0171

134

02.8

3830

.66.

726

0.01

406

3378

.638

00.5

6.61

70.

0102

633

29.4

3740

6.43

60.

0080

0432

79.5

3679

.66.

284

720

760

0.01

803

3487

.239

37.9

6.83

20.

0148

634

65.5

3911

.36.

727

0.01

091

3421

.738

57.9

6.55

20.

0085

5233

77.4

3805

6.40

876

080

00.

0189

235

70.9

4044

6.93

20.

0156

335

51.4

4020

.26.

830.

0115

235

11.9

3972

.86.

661

0.00

9074

3472

.339

266.

523

800



Tabl

eE

.31:

Sup

ercr

itica

lste

am(6

00ba

rto

1000

bar)

600

bar

700

bar

800

bar

1000

bar

T(

C)

vu

hs

vu

hs

vu

hs

vu

hs

T(

C)

0.01

0.00

0972

0.27

7758

.63

-0.0

0193

0.00

0968

0.19

0267

.97

-0.0

0323

0.00

0964

0.07

2577

.22

-0.0

0476

0.00

0957

-0.2

439

95.4

3-0

.008

440.

01

200.

0009

7780

.35

138.

90.

2818

0.00

0973

79.7

614

7.9

0.27

90.

0009

6979

.19

156.

70.

2761

0.00

0962

78.0

417

4.2

0.27

2040

0.00

0983

160.

921

9.9

0.54

890.

0009

815

9.9

228.

40.

5449

0.00

0976

158.

923

70.

5409

0.00

0969

157

253.

90.

5329

4060

0.00

0992

241.

630

1.1

0.80

040.

0009

8924

0.2

309.

40.

7955

0.00

0985

238.

831

7.6

0.79

050.

0009

7823

6.2

334

0.78

0860

800.

0010

0332

2.5

382.

71.

038

0.00

0999

320.

739

0.6

1.03

20.

0009

9631

8.9

398.

51.

026

0.00

0988

315.

541

4.4

1.01

580

100

0.00

1016

403.

546

4.5

1.26

30.

0010

1240

1.3

472.

11.

257

0.00

1008

399.

147

9.8

1.25

0.00

139

549

51.

237

100

120

0.00

103

484.

854

6.7

1.47

80.

0010

2648

2.1

554

1.47

0.00

1022

479.

556

1.3

1.46

30.

0010

1447

4.6

576

1.44

912

014

00.

0010

4756

6.5

629.

31.

683

0.00

1042

563.

363

6.2

1.67

40.

0010

3756

0.2

643.

21.

666

0.00

1028

554.

465

7.2

1.65

140

160

0.00

1065

648.

571

2.4

1.87

90.

0010

664

4.8

718.

91.

870.

0010

5464

1.2

725.

51.

861

0.00

1045

634.

473

8.9

1.84

316

018

00.

0010

8573

179

6.2

2.06

80.

0010

7972

6.7

802.

22.

058

0.00

1074

722.

580

8.4

2.04

80.

0010

6371

4.7

820.

92.

028

180

200

0.00

1108

814.

288

0.7

2.25

10.

0011

0180

9.1

886.

22.

239

0.00

1095

804.

389

1.8

2.22

80.

0010

8379

5.3

903.

52.

207

200

220

0.00

1133

898.

296

6.2

2.42

80.

0011

2589

2.3

971

2.41

50.

0011

1888

6.7

976.

12.

402

0.00

1104

876.

398

6.7

2.37

922

024

00.

0011

6198

3.2

1052

.82.

60.

0011

5297

6.3

1056

.92.

586

0.00

1143

969.

810

61.3

2.57

20.

0011

2895

7.9

1070

.72.

546

240

260

0.00

1193

1069

.411

412.

769

0.00

1182

1061

.411

44.1

2.75

20.

0011

7210

53.9

1147

.72.

737

0.00

1154

1040

.211

55.6

2.70

826

028

00.

0012

2911

57.3

1231

2.93

40.

0012

1611

47.8

1232

.92.

916

0.00

1204

1139

.112

35.4

2.89

80.

0011

8311

23.3

1241

.62.

866

280

300

0.00

127

1247

1323

.33.

098

0.00

1254

1235

.913

23.7

3.07

70.

0012

412

25.7

1324

.93.

057

0.00

1215

1207

.413

28.9

3.02

230

0

320

0.00

1318

1339

.214

18.3

3.26

10.

0012

9813

25.9

1416

.73.

237

0.00

128

1313

.814

16.3

3.21

40.

0012

512

92.7

1417

.73.

174

320

340

0.00

1374

1434

.215

16.7

3.42

40.

0013

4914

18.1

1512

.53.

395

0.00

1327

1403

.815

09.9

3.36

90.

0012

913

79.1

1508

.13.

324

340

360

0.00

1457

1541

.116

23.1

3.60

40.

0014

1915

19.1

1613

.73.

565

0.00

1388

1500

.416

07.4

3.53

20.

0013

3914

69.6

1600

.63.

477

360

380

0.00

1569

1657

.817

40.6

3.79

40.

0015

0916

27.2

1722

.83.

742

0.00

1464

1602

.517

10.7

3.7

0.00

1398

1563

.516

96.5

3.63

238

040

00.

0017

0117

75.1

1864

.63.

982

0.00

1613

1735

.618

36.8

3.91

60.

0015

4917

04.7

1818

3.86

30.

0014

6216

57.5

1795

.23.

782

400

420

0.00

1856

1893

.419

94.1

4.16

70.

0017

3118

44.5

1955

4.08

60.

0016

4518

07.1

1928

.54.

023

0.00

1532

1751

.318

963.

929

420

440

0.00

2042

2013

.321

28.8

4.35

20.

0018

6819

54.2

2076

.94.

255

0.00

1754

1909

.920

41.9

4.18

10.

0016

0918

45.1

1998

.74.

073

440

460

0.00

227

2135

.422

68.6

4.53

90.

0020

2820

6522

02.2

4.42

20.

0018

7720

13.2

2158

4.33

70.

0016

9419

39.1

2103

.34.

214

460

480

0.00

2555

2260

.124

13.7

4.73

0.00

2219

2177

2330

.94.

591

0.00

202

2117

.222

76.4

4.49

20.

0017

8920

33.2

2209

.44.

354

480

500

0.00

2922

2388

2564

.64.

926

0.00

2449

2290

.524

62.9

4.76

10.

0021

8622

2223

97.1

4.64

70.

0018

9521

27.6

2316

.84.

491

500

520

0.00

3361

2512

.327

13.9

5.11

90.

0027

3224

05.5

2598

.34.

933

0.00

2382

2327

.525

19.8

4.80

10.

0020

1522

22.1

2425

.34.

627

520

540

0.00

3762

2617

.228

42.9

5.28

0.00

3067

2518

.727

33.4

5.10

30.

0026

1624

33.9

2644

.84.

957

0.00

215

2316

.825

34.8

4.76

254

056

00.

0041

4227

09.6

2958

.15.

420.

0033

7826

17.6

2854

.15.

250.

0028

8525

38.2

2769

5.11

0.00

2305

2411

.726

45.3

4.89

656

058

00.

0044

9927

92.1

3062

5.54

30.

0036

8327

07.3

2965

.15.

382

0.00

3135

2631

.228

825.

244

0.00

2484

2506

.827

56.8

5.03

158

060

00.

0048

3428

66.9

3157

5.65

30.

0039

7527

89.3

3067

.55.

50.

0033

8427

17.4

2988

.15.

367

0.00

2672

2597

.828

65.1

5.15

860

0

640

0.00

5447

3000

.433

27.3

5.84

40.

0045

229

34.8

3251

.25.

706

0.00

3861

2872

.131

815.

583

0.00

3026

2761

.230

63.8

5.38

164

068

00.

0060

0331

19.6

3479

.86.

007

0.00

5018

3063

.134

14.4

5.88

10.

0043

0630

08.5

3352

.95.

768

0.00

3376

2907

.732

45.3

5.57

568

072

00.

0065

1832

29.5

3620

.66.

152

0.00

5478

3180

.235

63.6

6.03

40.

0047

231

32.1

3509

.75.

929

0.00

3712

3041

.734

135.

748

720

760

0.00

733

33.2

3753

.26.

283

0.00

5909

3289

.537

03.1

6.17

20.

0051

0832

46.7

3655

.36.

073

0.00

4031

3165

.535

68.6

5.90

176

080

00.

0074

5734

32.7

3880

.26.

403

0.00

6317

3393

.638

35.8

6.29

80.

0054

7633

55.2

3793

.36.

204

0.00

4336

3281

.637

15.2

6.04

800



Acknowledgements

The Editor wishes to express his sincere gratitude to all of the many Mechanical EngineeringDepartment staff, past and present, who have helped to compile this work.

Perfect gas properties in Table E.3 are based on data from Technical Data on Fuel (7th edition),ed. J.W. Rose and J.R. Cooper, Scottish Academic Press, 1977.

Enthalpies of ideal (but not perfect) gases in Table E.5 are approximate values obtained fromfitting quadratic relationships to enthalpy-temperature data from Rose and Cooper, TechnicalData on Fuel ; the coefficients of the corresponding linear specific heat relationships and themean and maximum errors are also given. This allows a choice between tabular and ana-lytical methods for solving combustion problems, with consistent results. The particular formof functional relationship is one, which enables combustion product temperature to be foundwithout iteration. Tables E.8 and E.9, for heating (or calorific) values (negative of enthalpiesof combustion) of simple compounds and some typical fuels, is based on data from the samesource.

The Moody diagram of Figure E.1 was plotted using the equation of Colebrook and White forturbulent flow in rough pipes and Prandtl’s equation for turbulent flow in smooth pipes, as quotedin Engineering Sciences Data Unit document ESDU 66027.

The psychrometric chart in Figure E.2, for standard atmospheric pressure, is based on thatpublished in 1970 by the Institution of Heating and Ventilating Engineers.

The R134a data of Tables E.10 to E.15 have been condensed from much more detailed ta-bles in the ICI Chemicals & Polymers Ltd publication Thermodynamic Properties of KLEA134a.Other physical properties of R134a, together with the equations from which the thermodynamicproperties were computed, are in Physical Property Data KLEA134a, ICI Chemicals & Poly-mers Ltd, Runcorn, 1993. The refrigerants business ICI Klea was acquired in 2001 by INEOSFluor.

The typescript was prepared in LATEX 2εand set in Helvetica and Micropress HV Math.

Patrick Leevers, July 2009


Index

Aluminium, 35Aluminium alloy, 65Amplifier

operational, 27

Beam, 35Bias, 19binomial series, 6Biot number, 44Bit, 26Bode plot, 26Boundary layer, 47Brass, 35, 65Brick, 65

Calorific value, 54Capacitor, 21Charge, 21Chords

intersecting, 8Circle, 9Clutch, 33Combustion data, 52Compressible flow, 47Concrete, 35, 65Cone, 10Confidence, 20Connection

delta, 24star, 24

Constantphysical, 3time, 23

ContinuityEquation of, 46

Convectionforced, 45

Cosine rule, 8Couette flow, 47Cross product, 6Current, 21Curvature

radius, 11Cylinder

pressurised, 39thick-walled, 41

Determinant, 5Difference amp, 27

Differential, 11Differential equation, 12Differentiation

partial, 10Differentiator, 27Distribution, 17

binomial, 18normal, 19Poisson, 18

Dot product, 6Drag

coefficient of, 44

Electromagnetism, 24Ellipse, 9Expansion

thermal, 35Expectation, 17Extremum, 11

Filterhigh-pass, 26low-pass, 26

Fluxmagnetic, 24

Forcemagnetic, 24

Fourier number, 44Fourier series, 16Friction factor, 44, 48

Geometry, 8Glass, 65Grashof number, 44

Heating value, 54Hyperbola, 9

Impedance, 24Inductor, 21Information, 26Integral, 12

definite, 12indefinite, 12

Integrationnumerical, 15

Integrator, 27Isentropic flow, 47

Kinematics, 33

82

INDEX

Kirchhoff’s laws, 22

Laplace transform, 14Leibnitz’s rule, 10Lift

coefficient of, 44Logarithm, 5

natural, 5

Magnetomotive force, 24Modulus

Shear, 35shear, 35Young’s, 35

Moody diagram, 48Motor

d.c., 25

Networka.c., 23

Nusselt number, 44

Op-amp, 27

Parabola, 9Parallel axis theorem, 35, 36Perfect gas, 50Permeability, 24Permittivity, 24Phasor, 24Poisson’s ratio, 35polyethylene, 65Polystyrene

expanded, 65Prandtl number, 44Prantl-Meyer function, 47Prefix, 3Principal stress, 40Psychrometric chart, 67Pyramid, 10

Quadratic, 5

Rayleigh number, 44Refrigerant, 55Reliability, 19Reluctance, 24Resistor, 21Reynolds number, 44Richardson’s formula, 16Root mean square value, 23

Screw, 33Semi-perfect gas, 52

Shear stressmaximum, 40

Shock, 47SI units, 1Sine rule, 8Specific heat, 65Sphere, 10Sphere, hollow pressurised, 39Spring, 39Standard deviation, 18Standard error, 19Stanton number, 44Stationary point, 11Statistics, 17Steam tables, 67Steel

mild, 35, 65stainless, 65

Stirling’s formula, 7Stress

principal, 40Student t table, 20Summing amp, 27Surface tension, 65

Taylor’s expansion, 10Thermal conductivity, 65Titanium, 35Torsion, 38Transformer, 25Transient, 23Triple product

scalar, 6vector, 6

Twist, 38

Unsteady flow, 46

Variance, 17Vector, 6Viscosity

absolute, 65dynamic, 65

Voltage, 21

Waveform, 23Wedge, spherical, 10Wood, 35, 65

Yieldcriterion of, 40


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