KFUPM CHEMICAL ENGINEERING DATA BOOK
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This document contains a data book compiled by the Department of Chemical Engineering at King Fahd University of Petroleum and Minerals. The data book provides concise summaries of key physical constants, unit conversions, dimensionless groups, the periodic table, kinetics and reaction engineering concepts, and other core chemical engineering topics. It aims to be a one-stop reference for important formulas and relationships for students to more efficiently study and prepare for exams. The document encourages efficient use as a study aid and requests it be returned to the department after exams.
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KFUPM CHEMICAL ENGINEERING DATA BOOK
- 1. KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS DEPARTMENT OF CHEMICAL ENGINEERING THE DATA BOOK Contents Page Physical Constants........................................................................................................2 Greek Alphabet.............................................................................................................2 Symbol Use...................................................................................................................2 Unit Conversion............................................................................................................3 Dimensionless Groups..................................................................................................4 Periodic Table...............................................................................................................5 Kinetics & Reactors......................................................................................................6 Fundamental Geometry.................................................................................................8 Averaging......................................................................................................................9 Polynomials ..................................................................................................................9 Trigonometric Formulae...............................................................................................9 Differentiation.............................................................................................................10 Integration...................................................................................................................11 Differential Equations.................................................................................................12 Stationary Points.........................................................................................................13 Numerical Methods.....................................................................................................15 Dynamics & Control...................................................................................................16 Laplace Transforms ....................................................................................................16 Vectors & Matrices.....................................................................................................18 Thermodynamics ........................................................................................................19 Transport Processes ....................................................................................................24 Fluid Mechanics..........................................................................................................24 Heat Transport ............................................................................................................28 Mass Transport ...........................................................................................................34 Separation Processes...................................................................................................40 Physical Properties Of Air & Water ...........................................................................41 Ammonia Properties ...................................................................................................49 Bessel Functions .........................................................................................................50 Gaussian Error Function.............................................................................................50
- 2. 1 Dear student: Please return this hardcopy to your instructor after the end of the exam. Dear instructor: If you are not teaching this course next semester, kindly return this hardcopy to the CHE secretary. Preface The aim of compiling this Data Book is to unify all core material into one place. Many a time, undergraduate students ask which formulae to commit to memory, as they approach exams. This Data Book contains all the key relations used frequently in core courses that tend to be longer, comprise subtle but important details, or are essential and hence necessitate repeat exposure for effective learning. Such a resource is a synopsis of the entire chemical engineering degree, therefore yielding a bird’s eye view of how all the component parts work collectively with the benefit of minimal text often seen in conventional textbooks. The idea of the Data Book has proven to work effectively at other esteemed universities, with subject matter organized according to individual topics. While no compilation is perfect, we have aimed to strike a balance such that not every conceivable relation need be listed, thereby facilitating the student’s ability to efficiently look up the needed information. We hope such a Data Book will serve as a useful study aid during their time within the department. Prepared by R.S.M. Chrystie et al. (2022) © Acknowledgement We are very grateful to all the chemical engineering faculty in assisting the compilation of this degree- wide formula book.
- 3. 2 PHYSICAL CONSTANTS Avogadro’s constant 𝑁𝐴 6.022 × 1023 mol-1 Boltzmann’s constant 𝑘𝐵 1.381 × 10-23 J/K Charge on electron 𝑒 1.602 × 10-19 C Gravitational acceleration 𝑔 9.81 m/s2 Mass of electron 𝑚𝑒 9.110 × 10-31 kg Planck’s constant ℎ 6.626 × 10-34 J s Standard pressure 𝑃𝑜 1.013 × 105 Pa Standard temperature 𝑇𝑜 273.15 K Stefan-Boltzmann constant 𝜎 5.670 × 10-8 W/(m2 K4 ) Universal gas constant 𝑅 8.314 J/(mol K) Velocity of light in vacuum 𝑐 2.998 × 108 m/s Volume of an ideal gas at STP 𝑉𝑜 2.241 × 10-2 m3 /mol GREEK ALPHABET Α 𝛼 alpha Ν 𝜈 nu Β 𝛽 beta Ξ 𝜉 xi Γ 𝛾 gamma Ο 𝜊 omicron Δ 𝛿 delta Π 𝜋 pi Ε 𝜀 epsilon Ρ 𝜌 rho Ζ 𝜁 zeta Σ 𝜎 sigma Η 𝜂 eta Τ 𝜏 tau Θ 𝜃 theta Υ 𝜐 upsilon Ι 𝜄 iota Φ 𝜙 phi Κ 𝜅 kappa Χ 𝜒 chi Λ 𝜆 lambda Ψ 𝜓 psi Μ 𝜇 mu Ω 𝜔 omega SYMBOL USE Alternative notation in textbooks Beware of similar symbols h Planck´s constant h heat transfer coefficient h height h fluid head H specific enthalpy h interval width hfg latent heat H Henry´s constant CONTEXT IS IMPORTANT Specific/molar value (J/kg) or (J/mol) Absolute value (J) Partial value (J/mol) 𝐻 𝐻 𝐻 𝐻𝑡 𝐻𝑖 ℎ 𝐻 Our convention in this book Every effort is made here to use a consistent and user-friendly set of symbols.
- 4. 3 UNIT CONVERSION Mechanical Quantity SI Additional (non-SI) Name Symbol Definition Name Symbol Definition Force Newton N kg m s-2 dyne dyn g cm s-2 Torque N m Work, Energy Joule J N m erg erg 10-7 J Power Watt W J s-1 Pressure Pascal Pa N m−2 bar bar 105 Pa Stress Pascal Pa N m−2 Dynamic viscosity Pa s poise P g cm-1 s-1 Kinematic viscosity m2 s-1 stokes St cm2 s-1 Thermal Quantity SI Additional (non-SI) Name Symbol Definition Name Symbol Definition Temperature Kelvin K rankine °R 𝑇𝑅 = 1.8𝑇𝐾 Energy, Work, Heat Joule J N m kilowatt- hour kWh 3.6 MJ Specific heat capacity J kg-1 K-1 Specific entropy J kg-1 K-1 Thermal conductivity W m-1 K-1 Heat transfer coefficient W m-2 K-1 Surface tension N m-1 Electrical Quantity SI Additional (non-SI) Name Symbol Definition Name Symbol Definition Energy Joule J N m kilowatt- hour electronvolt kWh eV 3.6 MJ 0.1602 aJ Power Watt W J s-1 Current Ampere A Charge Coulomb C A s Potential, e.m.f. Volt V Resistance Ohm Ω V A-1
- 5. 4 DIMENSIONLESS GROUPS Drag Coefficient 𝐶𝐷 = 2𝐹𝐷 𝜌𝑣2𝐴 Nusselt 𝑁𝑢 = ℎ𝐿 𝑘 Bond 𝐵𝑜 = ∆𝜌𝑔𝐿2 𝜎 Peclet (Heat) 𝑃𝑒 = 𝑅𝑒𝑃𝑟 = 𝑐𝑝𝜌𝑣𝐿 𝑘 Flow Coefficient 𝐶𝑄 = 𝑉̇ 𝑁𝐷3 Peclet (Mass) 𝑃𝑒ʹ = 𝑅𝑒𝑆𝑐 = 𝑣𝐿 𝔇 Fourier 𝐹𝑜 = 𝑘𝑡 𝜌𝑐𝑝𝐿2 Power Coefficient 𝑁𝑃 = ℙ 𝜌𝑁3𝐷5 Fanning Friction 𝐶𝑓 = 2𝜏𝑤 𝜌𝑣2 Prandtl 𝑃𝑟 = 𝜇𝑐𝑝 𝑘 Froude 𝐹𝑟 = 𝑣 √𝑔ℎ Rayleigh 𝑅𝑎 = 𝐺𝑟𝑃𝑟 = 𝐿3 𝜌𝑔Δ𝜌𝑐𝑝 𝜇𝑘 Grashof 𝐺𝑟 = 𝐿3 𝜌𝑔Δ𝜌 𝜇2 Reynolds 𝑅𝑒 = 𝜌𝑣𝐿 𝜇 Head Coefficient 𝐶𝐻 = 𝑔ℎ 𝑁2𝐷2 Richardson 𝑅𝑖 = 𝐺𝑟 𝑅𝑒2 = 𝑔𝐿Δ𝜌 𝜌𝑣2 j-factor (Heat) 𝑗𝐻 = 𝑆𝑡 𝑃𝑟2 3 ⁄ Schmidt 𝑆𝑐 = 𝜇 𝜌𝔇 j-factor (Mass) 𝑗𝐷 = 𝑆𝑡 ʹ 𝑆𝑐2 3 ⁄ Sherwood 𝑆ℎ = 𝑘𝑐𝐿 𝔇 Lewis 𝐿𝑒 = 𝑆𝑐 𝑃𝑟 = 𝑘 𝜌𝑐𝑝𝔇 Specific Speed 𝑁𝑆 = 𝑁√𝑉̇ (𝑔ℎ)3 4 ⁄ Mach 𝑀 = 𝑣 𝑎 Stanton 𝑆𝑡 = 𝑁𝑢 𝑅𝑒𝑃𝑟 = ℎ 𝜌𝑣𝑐𝑝 Morton 𝑀𝑜 = 𝜇4 𝑔∆𝜌 𝜌2𝜎3 Modified Stanton 𝑆𝑡ʹ = 𝑆ℎ 𝑅𝑒𝑆𝑐 = 𝑘𝑐 𝑣 Weber 𝑊𝑒 = 𝜌𝑣2 𝐿 𝜎
- 7. 6 KINETICS & REACTORS Species balancing Conversion of mole and mass fractions: Most used formula in CHE: “IN + GEN = OUT + ACC” 𝐹𝑗𝑖𝑛 + 𝐺𝑗 = 𝐹𝑗𝑜𝑢𝑡 + 𝑑𝑁𝑗 𝑑𝑡 Where: 𝐺𝑗 = 𝑟𝑗 𝑉 for species 𝑗, noting: 𝐺𝑗 > 0 (production) 𝐺𝑗 < 0 (consumption) Single generic reaction: A + b a B ⇌ c a C + d a D with forward rate 𝑘𝑓 and reverse rate 𝑘𝑟 The basis here assumes A to be the limiting reactant. Fractional conversion (based on A): 𝑋𝐴 ≡ moles A reacted moles A fed ⟹ 𝐹𝐴 = 𝐹𝐴0(1 − 𝑋𝐴) 𝐹𝑗 = 𝐹𝑗0 + 𝐹𝐴0𝑋𝐴𝜈𝑗 𝑤𝐴 = 𝑥𝐴𝑀𝐴 𝑥𝐴𝑀𝐴 + 𝑥𝐵𝑀𝐵 + ⋯ 𝑥𝐴 = 𝑤𝐴 𝑀𝐴 ⁄ 𝑤𝐴 𝑀𝐴 ⁄ + 𝑤𝐵 𝑀𝐵 ⁄ + ⋯ 𝑤𝐴: mass fraction of A 𝑥𝐴: mole fraction of A 𝑀𝐴: molar mass of A where: − 𝑟𝐴 = 𝑟𝑗 𝜈𝑗 𝜈𝐵 = − 𝑏 𝑎 𝜈𝐴 = −1 𝜈𝐶 = 𝑐 𝑎 𝜈𝐷 = 𝑑 𝑎
- 8. 7 Extent of reaction (dimensionless): 𝜉 ≡ moles 𝑗 reacted stoichiometry ⟹ 𝐹𝑗 = 𝐹𝑗0 + 𝜉𝜈𝑗 Reaction thermodynamics 𝑘(𝑇 ) = 𝔸𝑒− 𝐸𝑎 𝑅𝑇 ln 𝐾𝑝(𝑇 ) 𝐾𝑝(𝑇1) = 𝛥𝐻𝑟 𝑜 𝑅 1 𝑇1 − 1 𝑇 Arrhenius Van't Hoff 𝐾𝑐 = 𝑘𝑓 𝑘𝑟 𝐾𝑐 = 𝐶𝐶 𝑐 𝑎 ⁄ 𝐶𝐷 𝑑 𝑎 ⁄ 𝐶𝐴 𝐶𝐵 𝑏 𝑎 ⁄ 𝐾𝑝 = 𝑃𝐶 𝑐 𝑎 ⁄ 𝑃𝐷 𝑑 𝑎 ⁄ 𝑃𝐴 𝑃𝐵 𝑏 𝑎 ⁄ 𝛿 = 𝑐 𝑎 + 𝑑 𝑎 − 𝑏 𝑎 − 1 𝐾𝑝 = 𝐾𝑐(𝑅𝑇)𝛿 −𝑟𝐴 = 𝑘𝑓 𝐶𝐴 𝛼 𝐶𝐵 𝛽 − 𝑘𝑟𝐶𝐶 𝛾 𝐶𝐷 𝜀 Rate Law, whose overall order is 𝛼 + 𝛽 (forward direction) Net reaction rate for A Energy balance for open systems: 𝑄̇ + 𝑊̇𝑠 = 𝛥𝐻̇ + 𝑑𝐸𝑠𝑦𝑠 𝑑𝑡 Where: 𝛥𝐻̇ = 𝜉̇ 𝛥𝐻𝑟(𝑇 ) + ∑ 𝐹𝑗 𝐻𝑗 − ∑ 𝐹𝑗0 𝐻𝑗0 With: 𝛥𝐻𝑟(𝑇) = 𝛥𝐻𝑟 𝑜 𝑇𝑟𝑒𝑓 + 𝑇 − 𝑇𝑟𝑒𝑓 𝜈𝑗𝑐𝑝,𝑗
- 9. 8 FUNDAMENTAL GEOMETRY Cylindrical coordinates Spherical coordinates (𝑟, 𝜃, 𝜙) 𝜃 is the azimuthal angle 𝜙 is the zenith angle 𝑥 = 𝑟 cos 𝜃 sin 𝜙 𝑦 = 𝑟 sin 𝜃 sin 𝜙 𝑟 = √𝑥2 + 𝑦2 + 𝑧2 𝑧 = 𝑟 cos 𝜙 Basic shapes (𝑟, 𝜃, 𝑧) 𝜃 is the azimuthal angle Note: both coordinate systems here follow the convention used in mathematics 𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃 𝑟 = √𝑥2 + 𝑦2 𝑧 = 𝑧 Total surface area: 𝟒𝝅𝒓𝟐 𝟐𝝅𝒓𝟐 + 𝟐𝝅𝒓𝒉 𝝅𝒓(𝒓 + 𝑳) Volume: 𝟒 𝟑 𝝅𝒓𝟑 𝝅𝒓𝟐 𝒉 𝟏 𝟑 𝝅𝒓𝟐 𝒉 A differential volume element in cylindrical coordinates A differential volume element in spherical coordinates
- 10. 9 AVERAGING Weighted average (discrete) Ordinary average (continuous) 𝑥 = ∑ 𝑤𝑖𝑥𝑖 𝑛 𝑖=1 ∑ 𝑤𝑖 𝑛 𝑖=1 𝑓 = 1 𝑏 − 𝑎 𝑓(𝑥) 𝑑𝑥 𝑏 𝑎 Note the weighted average becomes the ordinary average when: 𝑤𝑖 = 1 for all 𝑖 POLYNOMIALS 𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏) 𝑎3 + 𝑏3 = (𝑎 + 𝑏) 𝑎2 − 𝑎𝑏 + 𝑏2 𝑎3 − 𝑏3 = (𝑎 − 𝑏) 𝑎2 + 𝑎𝑏 + 𝑏2 𝐴𝒙2 + 𝐵𝒙 + 𝐶 = 0 𝒙 = −𝐵 ± √𝐵2 − 4𝐴𝐶 2𝐴 Partial fraction decomposition 𝒇(𝒙) = 𝑷 (𝒙) 𝑸(𝒙) with the order of 𝑃 < 𝑄 (see example below) 2𝑥6 − 4𝑥5 + 5𝑥4 − 3𝑥3 + 𝑥2 + 3𝑥 (𝑥 − 1)3 𝑥2 + 1 2 = 𝐴 𝑥 − 1 + 𝐵 (𝑥 − 1)2 + 𝐶 (𝑥 − 1)3 + 𝐷𝑥 + 𝐸 𝑥2 + 1 + 𝐹𝑥 + 𝐺 𝑥2 + 1 2 Solve A G by comparing coefficients of 𝑥 TRIGONOMETRIC FORMULAE sin(𝐴 ± 𝐵) = sin 𝐴 cos 𝐵 ± cos 𝐴 sin 𝐵 sin 𝐴 cos 𝐵 = 1 2 [sin(𝐴 + 𝐵) + sin(𝐴 − 𝐵)] cos(𝐴 ± 𝐵) = cos𝐴 cos 𝐵 ∓ sin𝐴 sin 𝐵 cos 𝐴 cos 𝐵 = 1 2 [cos(𝐴 + 𝐵) + cos(𝐴 − 𝐵)] tan(𝐴 ± 𝐵) = tan 𝐴 ± tan 𝐵 1 ∓ tan 𝐴 tan 𝐵 sin 𝐴 sin 𝐵 = 1 2 [cos(𝐴 − 𝐵) − cos(𝐴 + 𝐵)] sin 𝐴 + sin 𝐵 = 2 sin 𝐴 + 𝐵 2 cos 𝐴 + 𝐵 2 sin 𝐴 − sin 𝐵 = 2 cos 𝐴 + 𝐵 2 sin 𝐴 − 𝐵 2
- 11. 10 cos 𝐴 + cos 𝐵 = 2 cos 𝐴 + 𝐵 2 cos 𝐴 − 𝐵 2 cos 𝐴 − cos 𝐵 = −2 sin 𝐴 + 𝐵 2 sin 𝐴 − 𝐵 2 sinh𝑥 = −𝑖 sin 𝑖𝑥 = 1 2 (𝑒𝑥 − 𝑒−𝑥 ) cosh 𝑥 = cos 𝑖𝑥 = 1 2 (𝑒𝑥 + 𝑒−𝑥 ) DIFFERENTIATION For functions 𝑢(𝑥) and 𝑣(𝑥): Product Rule Quotient Rule 𝑑(𝑢𝑣) 𝑑𝑥 = 𝑣 𝑑𝑢 𝑑𝑥 + 𝑢 𝑑𝑣 𝑑𝑥 𝑑 𝑑𝑥 𝑢 𝑣 = 𝑣 𝑑𝑢 𝑑𝑥 − 𝑢 𝑑𝑣 𝑑𝑥 𝑣2 Chain rule When 𝑥, 𝑦, 𝑧,… are functions of 𝑢, 𝑣, 𝑤,… 𝜕𝜙 𝜕𝑢 𝑣,𝑤,… = 𝜕𝜙 𝜕𝑥 𝜕𝑥 𝜕𝑢 𝑣,𝑤,… + 𝜕𝜙 𝜕𝑦 𝜕𝑦 𝜕𝑢 𝑣,𝑤,… + 𝜕𝜙 𝜕𝑧 𝜕𝑧 𝜕𝑢 𝑣,𝑤,… + ⋯ Total derivative For any function 𝜙(𝑥, 𝑦, 𝑧, … ) 𝑑𝜙 = 𝜕𝜙 𝜕𝑥 𝑑𝑥 + 𝜕𝜙 𝜕𝑦 𝑑𝑦 + 𝜕𝜙 𝜕𝑧 𝑑𝑧 + ⋯ If 𝑓(𝑥, 𝑦)𝑑𝑥 + 𝑔(𝑥, 𝑦)𝑑𝑦 = 𝑑𝜙 for some function 𝜙(𝑥, 𝑦), then 𝜕𝑓 𝜕𝑦 = 𝜕𝑔 𝜕𝑥 in which 𝜕𝜙 𝜕𝑥 means 𝜕𝜙 𝜕𝑥 𝑦,𝑧,… (i.e. with 𝑦, 𝑧,…kept constant)
- 12. 11 INTEGRATION Integration by parts 𝑢(𝑥) 𝑑𝑣(𝑥) 𝑑𝑥 𝑑𝑥 = 𝑢(𝑥)𝑣(𝑥) − 𝑣(𝑥) 𝑑𝑢(𝑥) 𝑑𝑥 𝑑𝑥 for a product of two functions 𝑢(𝑥) and 𝑣(𝑥) Logarithmic integrals 1 1 + 𝜀𝑥 𝑑𝑥 = 1 𝜀 ln(1 + 𝜀𝑥) + ℂ 1 + 𝜀𝑥 (1 − 𝑥)2 𝑑𝑥 = (1 + 𝜀)𝑥 1 − 𝑥 − 𝜀 ln 1 1 − 𝑥 + ℂ (1 + 𝜀𝑥)2 (1 − 𝑥)2 𝑑𝑥 = 2𝜀(1 + 𝜀) ln(1 − 𝑥) + 𝜀2 𝑥 + (1 + 𝜀)2 𝑥 1 − 𝑥 + ℂ Trigonometric & hyperbolic integrals cos 𝑥 𝑑𝑥 = sin 𝑥 + ℂ cosh 𝑥 𝑑𝑥 = sinh 𝑥 + ℂ tan 𝑥 𝑑𝑥 = − ln(cos 𝑥) + ℂ tanh 𝑥 𝑑𝑥 = ln(cosh 𝑥) + ℂ cosec 𝑥 𝑑𝑥 = ln tan 𝑥 2 + ℂ cosech 𝑥 𝑑𝑥 = ln tanh 𝑥 2 + ℂ sec 𝑥 𝑑𝑥 = ln(tan 𝑥 + sec 𝑥) + ℂ sech 𝑥 𝑑𝑥 = 2 tan−1 (𝑒𝑥 ) + ℂ cot 𝑥 𝑑𝑥 = ln(sin 𝑥) + ℂ coth 𝑥 𝑑𝑥 = ln(sinh 𝑥) + ℂ sec2 𝑥 𝑑𝑥 = tan 𝑥 + ℂ sech2 𝑥 𝑑𝑥 = tanh 𝑥 + ℂ tan 𝑥 sec 𝑥 𝑑𝑥 = sec 𝑥 + ℂ tanh 𝑥 sech 𝑥 𝑑𝑥 = − sech 𝑥 + ℂ cot 𝑥 cosec 𝑥 𝑑𝑥 = −cosec 𝑥 + ℂ coth 𝑥 cosech 𝑥 𝑑𝑥 = −cosech 𝑥 + ℂ 1 √𝑎2 − 𝑥2 𝑑𝑥 = sin−1 𝑥 𝑎 + ℂ 1 √𝑎2 + 𝑥2 𝑑𝑥 = sinh−1 𝑥 𝑎 + ℂ 1 𝑎2 + 𝑥2 𝑑𝑥 = 1 𝑎 tan−1 𝑥 𝑎 + ℂ 1 𝑎2 − 𝑥2 𝑑𝑥 = 1 𝑎 tanh−1 𝑥 𝑎 + ℂ 1 𝑎 + 𝑏𝑥4 𝑑𝑥 = 𝑘 4𝑎 ln 𝑥 + 𝑘 𝑥 − 𝑘 + 2 tan−1 𝑥 𝑘 + ℂ valid for 𝑎𝑏 < 0, where 𝑘 = − 𝑎 𝑏 4
- 13. 12 Standard substitutions If the integrand is a function of: Substitute 𝑎2 − 𝑥2 or √𝑎2 − 𝑥2 𝑥 = 𝑎 sin 𝜃 or 𝑥 = 𝑎 cos 𝜃 𝑎2 + 𝑥2 or √𝑎2 + 𝑥2 𝑥 = 𝑎 tan 𝜃 or 𝑥 = 𝑎 sinh𝜃 𝑥2 − 𝑎2 or √𝑥2 − 𝑎2 𝑥 = 𝑎 sec 𝜃 or 𝑥 = 𝑎 cosh 𝜃 If the integral is of the form: Substitute 𝑑𝑥 (𝑎𝑥 + 𝑏)√𝑝𝑥 + 𝑞 𝑝𝑥 + 𝑞 = 𝑢2 𝑑𝑥 (𝑎𝑥 + 𝑏)√𝑝𝑥2 + 𝑞𝑥 + 𝑟 𝑎𝑥 + 𝑏 = 1 𝑢 DIFFERENTIAL EQUATIONS First-order linear ODE A first-order linear ODE of the form: 𝑑𝑦 𝑑𝑥 + 𝑃(𝑥)𝑦 = 𝑄(𝑥) can be solved by using the integrating factor 𝑒∫ , such that: 𝑑 𝑑𝑥 𝑦𝑒∫ 𝑃 𝑑𝑥 = 𝑄(𝑥)𝑒∫ 𝑃 𝑑𝑥 Second-order linear ODE A second-order linear ODE with constant coefficients of the form: 𝑎 𝑑2 𝑦 𝑑𝑥2 + 𝑏 𝑑𝑦 𝑑𝑥 + 𝑐𝑦 = 𝑓(𝑥) can be solved by adding together the complementary function with the particular solution, such that: 𝑦(𝑥) = 𝑦𝐶𝐹 + 𝑦𝑃𝑆
- 14. 13 First, solve the auxiliary equation, such that: 𝑎 𝑚2 + 𝑏 𝑚 + 𝑐 = 0 Whose roots are 𝑚1 and 𝑚2 Root 𝒚𝑪𝑭 Real 𝑚1 ≠ 𝑚2 𝐴𝑒𝑚1𝑥 + 𝐵𝑒𝑚2𝑥 Real 𝑚1 = 𝑚2 (𝐴 + 𝐵𝑥)𝑒𝑚1𝑥 𝑚 = 𝑝 ± 𝑞𝑖 𝑒𝑝𝑥 (𝐴 cos 𝑞𝑥 + 𝐵 sin 𝑞𝑥) Second, find a form of yPS according to RHS of the ODE: 𝒇(𝒙) 𝒚𝑷𝑺 𝑘 (a constant) 𝐶 Linear in 𝑥 𝐶𝑥 + 𝐷 Quadratic in 𝑥 𝐶𝑥2 + 𝐷𝑥 + 𝐸 𝑘 sin 𝑝𝑥 or 𝑘 cos 𝑝𝑥 𝐶 cos 𝑝𝑥 + 𝐷 sin 𝑝𝑥 𝑘𝑒𝑝𝑥 𝐶𝑒𝑝𝑥 Sum of the above Sum of the above Product of the above Product of the above Note: If suggested form of yPS already appears in the complementary function, then multiply suggested form by 𝑥. STATIONARY POINTS Unconstrained Stationary points occur for 𝑓(𝑥, 𝑦) where ∇𝑓 = 0 i.e. where 𝜕𝑓 𝜕𝑥 = 0 and 𝜕𝑓 𝜕𝑦 = 0 simultaneously
- 15. 14 Let (𝑎, 𝑏) be the stationary point and define: 𝑓𝑥𝑥 = 𝜕2 𝑓 𝜕𝑥2 𝑎,𝑏 𝑓𝑦𝑦 = 𝜕2 𝑓 𝜕𝑦2 𝑎,𝑏 𝑓𝑥𝑦 = 𝜕2 𝑓 𝜕𝑥𝜕𝑦 𝑎,𝑏 If 𝑓𝑥𝑦 2 − 𝑓𝑥𝑥𝑓𝑦𝑦 < 0 and 𝑓𝑥𝑥 < 0 then 𝑓(𝑥, 𝑦) has a maximum at (𝑎, 𝑏) If 𝑓𝑥𝑦 2 − 𝑓𝑥𝑥𝑓𝑦𝑦 < 0 and 𝑓𝑥𝑥 > 0 then 𝑓(𝑥, 𝑦) has a minimum at (𝑎, 𝑏) If 𝑓𝑥𝑦 2 − 𝑓𝑥𝑥𝑓𝑦𝑦 > 0 then 𝑓(𝑥, 𝑦) has a saddle point at (𝑎, 𝑏) If 𝑓𝑥𝑦 2 − 𝑓𝑥𝑥𝑓𝑦𝑦 = 0 then the nature of the turning point depends on higher order derivatives Constrained Lagrange’s method of undetermined multipliers: Stationary points for 𝑓(𝑥, 𝑦) along the line ℎ(𝑥, 𝑦) = 0 are coincident with the stationary points for 𝐿(𝑥, 𝑦, 𝜆), where: 𝐿(𝑥, 𝑦, 𝜆) = 𝑓(𝑥, 𝑦) − 𝜆 ℎ(𝑥, 𝑦) i.e. where 𝜕𝐿 𝜕𝑥 = 0, 𝜕𝐿 𝜕𝑦 = 0 and 𝜕𝐿 𝜕𝜆 = ℎ(𝑥, 𝑦) = 0 simultaneously
- 16. 15 NUMERICAL METHODS Solving equations Newton Raphson 𝑥𝑛+1 = 𝑥𝑛 − 𝑓(𝑥𝑛) 𝑓 (𝑥𝑛) Linear regression Straight line through scatterplot: 𝑦 = 𝑎0 + 𝑎1𝑥 Where: 𝑎1 = 𝑛 ∑ 𝑥𝑖𝑦𝑖 − ∑ 𝑥𝑖 ∑ 𝑦𝑖 𝑛 ∑ 𝑥𝑖 2 − (∑ 𝑥𝑖)2 and 𝑎0 = 𝑦 − 𝑎1𝑥 Numerical differentiation 𝑦𝑛 are values of 𝑦 at equal intervals of 𝑥 with width ℎ 𝑑𝑦 𝑑𝑥 𝑛 = 𝑦𝑛+1 − 𝑦𝑛−1 2ℎ + 𝑂(ℎ2 ) 𝑑𝑦 𝑑𝑥 𝑛 = −3𝑦𝑛 + 4𝑦𝑛+1 − 𝑦𝑛+2 2ℎ + 𝑂(ℎ2 ) 𝑑2 𝑦 𝑑𝑥2 𝑛 = 𝑦𝑛+1 − 2𝑦𝑛 + 𝑦𝑛−1 ℎ2 + 𝑂(ℎ2 ) Differential equations Problem: 𝑑𝑦(𝑥) 𝑑𝑥 = 𝑓(𝑥, 𝑦), 𝑦0 = 𝑦(𝑥0) Euler method: 𝑦𝑖+1 = 𝑦𝑖 + ℎ 𝑓(𝑥𝑖, 𝑦𝑖) Numerical integration Trapezium Rule (N is the number of intervals) 𝑦 𝑑𝑥 𝑥𝑁 𝑥0 = ℎ 2 (𝑦0 + 2𝑦1 + ⋯ + 2𝑦𝑛 + ⋯ + 2𝑦𝑁−1 + 𝑦𝑁) Simpsons’s Rule (N is the number of intervals, which must be even) 𝑦 𝑑𝑥 𝑥𝑁 𝑥0 = ℎ 3 𝑦0 + 4𝑦1 + 2𝑦2 + 4𝑦3 + ⋯ + 2𝑦𝑛−1 + 4𝑦𝑛 + 2𝑦𝑛+1 + ⋯ + 4𝑦𝑁−1 + 𝑦𝑁
- 17. 16 DYNAMICS & CONTROL For a feedback loop, the ideal PID (Proportional-Integral-Derivative) control law is given by: 𝑝(𝑡) − 𝑝 = 𝐾𝑐 𝑒(𝑡) + 1 𝜏𝐼 𝑒(𝜃) 𝑑𝜃 𝑡 0 + 𝜏𝐷 𝑑𝑒(𝑡) 𝑑𝑡 Name 𝐺(𝑠) Amplitude Ratio Phase Shift 𝐴𝑅(𝜔) 𝜙(𝜔) First order lag 1 𝜏𝑠 + 1 1 √𝜏2𝜔2 + 1 − tan−1 𝜏𝜔 First order lead 𝜏𝑠 + 1 √𝜏2𝜔2 + 1 tan−1 𝜏𝜔 Integrator 1 𝜏𝐼𝑠 1 𝜏𝐼𝜔 − 𝜋 2 Differentiator 𝜏𝐷𝑠 𝜏𝐷𝜔 𝜋 2 Dead time 𝑒−𝑡𝑑𝑠 1 −𝑡𝑑𝜔 LAPLACE TRANSFORMS 𝓛{𝑦(𝑡)} ≡ 𝑌 (𝑠) = 𝑦(𝑡)𝑒−𝑠𝑡 𝑑𝑡 ∞ 0 Simple functions 𝒚(𝒕) 𝒀 (𝒔) 𝒚(𝒕) 𝒀 (𝒔) 1 1 𝑠 cos(𝜔𝑡 + 𝜃) 𝑠 cos 𝜃 − 𝜔 sin 𝜃 𝑠2 + 𝜔2 𝑡 1 𝑠2 𝑡𝑒−𝛼𝑡 1 (𝑠 + 𝛼)2 𝑡𝑛 𝑛! 𝑠𝑛+1 𝑒−𝛼𝑡 sin 𝜔𝑡 𝜔 (𝑠 + 𝛼)2 + 𝜔2 𝑒−𝛼𝑡 1 𝑠 + 𝛼 𝑒−𝛼𝑡 cos 𝜔𝑡 𝑠 + 𝛼 (𝑠 + 𝛼)2 + 𝜔2
- 18. 17 Derivatives & Integrals Step & Impulse Functions General Properties sin(𝜔𝑡 + 𝜃) 𝑠 sin 𝜃 + 𝜔 cos 𝜃 𝑠2 + 𝜔2 erfc 𝑘 2√𝑡 𝑒−𝑘√𝑠 𝑠 𝒚(𝒕) 𝒀 (𝒔) 𝒚(𝒕) 𝒀 (𝒔) 𝑑𝑥(𝑡) 𝑑𝑡 𝑠𝑋(𝑠) − 𝑥(0) 𝑥(𝜏) 𝑑𝜏 𝑡 0 𝑋(𝑠) 𝑠 𝑡 𝑥(𝑡) − 𝑑𝑋(𝑠) 𝑑𝑠 𝑥(𝑡) 𝑡 𝑋(𝜎) 𝑑𝜎 ∞ 𝑠 𝑑𝑛 𝑥(𝑡) 𝑑𝑡𝑛 𝑠𝑛 𝑋(𝑠) − 𝑠𝑛−1 𝑥(0) − 𝑠𝑛−2 𝑥ʹ (0) − ⋯ − 𝑥(𝑛−1) (0) 𝑥1(𝜏)𝑥2(𝑡 − 𝜏)𝑑𝜏 𝑡 0 𝑋1(𝑠)𝑋2(𝑠) 𝒚(𝒕) 𝒀 (𝒔) 𝒚(𝒕) 𝒀 (𝒔) 𝐻(𝑡) 1 𝑠 𝐻(𝑡 − 𝜏) 𝑒−𝑠𝜏 𝑠 𝛿(𝑡) 1 𝛿(𝑡 − 𝜏) 𝑒−𝑠𝜏 𝒚(𝒕) 𝒀 (𝒔) 𝒚(𝒕) 𝒀 (𝒔) 𝐻(𝑡 − 𝜏)𝑥(𝑡 − 𝜏) 𝑒−𝑠𝜏 𝑋(𝑠) 𝑒−𝛼𝑡 𝑥(𝑡) 𝑋(𝑠 + 𝛼) 𝑥(𝛼𝑡) 𝑋(𝑠 𝛼 ⁄ ) 𝛼 𝛿(𝑡 − 𝜏) 𝑒−𝑠𝜏 lim 𝑡→0 𝑥(𝑡) = lim 𝑠→∞ 𝑠𝑋(𝑠) lim 𝑡→∞ 𝑥(𝑡) = lim 𝑠→0 𝑠𝑋(𝑠)
- 19. 18 VECTORS & MATRICES Matrix properties (AB…X)T = XT …BT AT Reversal rule (AB…X)-1 = X-1 …B-1 A-1 (if inverses exist) 𝐀 𝒙 = 𝒃 can be solved for vector 𝒙 if A is square, A ≠ 0, and 𝑑𝑒𝑡A ≠ 0 Vector algebra Cross product (right hand rule) 𝒄 = 𝒂 × 𝒃 = 𝒙̂ 𝒚̂ 𝒛̂ 𝑎𝑥 𝑎𝑦 𝑎𝑧 𝑏𝑥 𝑏𝑦 𝑏𝑧 Scalar triple product 𝒂 ∙ (𝒃 × 𝒄) = 𝑎𝑥 𝑎𝑦 𝑎𝑧 𝑏𝑥 𝑏𝑦 𝑏𝑧 𝑐𝑥 𝑐𝑦 𝑐𝑧 = (𝒃 × 𝒄) ∙ 𝒂 (𝒂 × 𝒃) ∙ 𝒄 (𝒄 × 𝒂) ∙ 𝒃 Vector triple product 𝒂 × (𝒃 × 𝒄) = (𝒂 ∙ 𝐜)𝒃 − (𝒂 ∙ 𝐛)𝒄 (𝒂 × 𝒃) × 𝒄 = (𝒂 ∙ 𝒄)𝒃 − (𝒃 ∙ 𝒄)𝒂 Vector calculus 𝑇 (𝑥, 𝑦, 𝑧) denotes a scalar function, and 𝒒(𝑥, 𝑦, 𝑧) a vector function 𝒒(𝑥, 𝑦, 𝑧) = 𝑞𝑥(𝑥, 𝑦, 𝑧) 𝒙̂ + 𝑞𝑦(𝑥, 𝑦, 𝑧) 𝒚̂ + 𝑞𝑧(𝑥, 𝑦, 𝑧) 𝒛̂ = 𝑞𝑥 𝑞𝑦 𝑞𝑧 Gradient grad 𝑇 = ∇𝑇 = 𝜕𝑇 𝜕𝑥 𝒙̂ + 𝜕𝑇 𝜕𝑦 𝒚̂ + 𝜕𝑇 𝜕𝑧 𝒛̂ 𝐊 = R K R𝐓 where transformation matrix R rotates property matrix K into another coordinate system yielding 𝐊
- 20. 19 Divergence div 𝒒 = ∇ ∙ 𝒒 = 𝜕𝑞𝑥 𝜕𝑥 + 𝜕𝑞𝑦 𝜕𝑦 + 𝜕𝑞𝑧 𝜕𝑧 Curl curl 𝒒 = ∇ × 𝒒 = 𝜕𝑞𝑧 𝜕𝑦 − 𝜕𝑞𝑦 𝜕𝑧 𝒙̂ + 𝜕𝑞𝑥 𝜕𝑧 − 𝜕𝑞𝑧 𝜕𝑥 𝒚̂ + 𝜕𝑞𝑦 𝜕𝑥 − 𝜕𝑞𝑥 𝜕𝑦 𝒛̂ Laplacian ∇ ∙ ∇𝑇 = ∇2 𝑇 = 𝜕2 𝑇 𝜕𝑥2 + 𝜕2 𝑇 𝜕𝑦2 + 𝜕2 𝑇 𝜕𝑧2 ∇ ∙ ∇𝒒 = ∇2 𝒒 = 𝜕2 𝒒 𝜕𝑥2 + 𝜕2 𝒒 𝜕𝑦2 + 𝜕2 𝒒 𝜕𝑧2 Advection operator (𝒒 ∙ ∇) 𝒃 = 𝑞𝑥 𝜕𝒃 𝜕𝑥 + 𝑞𝑦 𝜕𝒃 𝜕𝑦 + 𝑞𝑧 𝜕𝒃 𝜕𝑧 Surface integral 𝑄̇ = 𝒒 ∙ 𝑑𝑺 total flux of 𝒒 through surface 𝑆 𝑆 THERMODYNAMICS Equations of state Ideal gas law where 𝑅̂ = 𝑅 𝑀𝑅 ⁄ ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 𝑃𝑉 𝑡 = 𝑛𝑅𝑇 𝑐 𝑃 𝑉 𝑡 = 𝑚𝑅̂𝑇 𝑐 𝑃𝑉 = 𝑅̂𝑇 𝑐 𝑃 = 𝜌𝑅̂𝑇 Van der Waals 𝑃 = 𝑅𝑇 𝑉 − 𝑏 − 𝑎 𝑉 2 𝑎 = 27 64 𝑅2 𝑇𝑐 2 𝑃𝑐 𝑏 = 1 8 𝑅𝑇𝑐 𝑃𝑐
- 21. 20 Update Formulae Liquid root 𝑉 𝐿 Initial guess: 𝑉 = 𝑏 𝑉𝑛+1 = 𝑏 + 𝑉𝑛 2 𝑅𝑇 + 𝑃(𝑏 − 𝑉𝑛) 𝑎 Vapour root 𝑉 𝑉 Initial guess: 𝑉 = 𝑅𝑇 𝑃 𝑉𝑛+1 = 𝑅𝑇 𝑃 + 𝑏 − 𝑎 𝑃 𝑉𝑛 − 𝑏 𝑉𝑛 2 Perfect & ideal gases Specific heat relationship 𝑐𝑝 𝑖𝑔 − 𝑐𝑣 𝑖𝑔 = 𝑅̂ Change in internal energy 𝑈2 − 𝑈1 = 𝑐𝑣 𝑖𝑔 (𝑇2 − 𝑇1) Change in enthalpy 𝐻2 − 𝐻1 = 𝑐𝑝 𝑖𝑔 (𝑇2 − 𝑇1) 𝛽 ≡ 1 𝑉 𝜕𝑉 𝜕𝑇 𝑃 𝜅 ≡ − 1 𝑉 𝜕𝑉 𝜕𝑃 𝑇 Volume expansivity Isothermal compressibility ln 𝑃𝑠𝑎𝑡 = 𝐴 − 𝐵 𝑇 + 𝐶 𝑑𝑃𝑠𝑎𝑡 𝑑𝑇 = ∆𝐻𝑣𝑎𝑝 𝑇 ∆𝑉𝑣𝑎𝑝 Antoine Clapeyron
- 22. 21 Heat capacity ratio 𝛾 ≡ 𝑐𝑝 𝑖𝑔 𝑐𝑣 𝑖𝑔 Speed of sound 𝑎 = 𝛾𝑅̂𝑇 For Isentropic changes 𝑃 𝑉 𝛾 = 𝑐𝑜𝑛𝑠𝑡. 𝑇 𝑉 𝛾−1 = 𝑐𝑜𝑛𝑠𝑡. 𝑇 𝑃 (𝛾−1) 𝛾 ⁄ ⁄ = 𝑐𝑜𝑛𝑠𝑡. Fundamental relations ∆𝑈 = 𝑄 + 𝑊 ∆𝐻̇ = 𝑄̇ + 𝑊̇𝑠 Closed systems Open systems (steady state) 𝑐𝑣 ≡ 𝜕𝑈 𝜕𝑇 𝑣 𝑐𝑝 ≡ 𝜕𝐻 𝜕𝑇 𝑝 𝐻 ≡ 𝑈 + 𝑃𝑉 𝐴 ≡ 𝑈 − 𝑇𝑆 𝐺 ≡ 𝐻 − 𝑇𝑆 Enthalpy Helmholtz Gibbs 𝑑𝑈 = 𝑇𝑑𝑆 − 𝑃𝑑𝑉 𝑑𝐺 = 𝑉𝑑𝑃 − 𝑆𝑑𝑇 𝑑𝐻 = 𝑇𝑑𝑆 + 𝑉𝑑𝑃 𝑑𝐴 = −𝑆𝑑𝑇 − 𝑃𝑑𝑉 Maxwell relations: Idealized reversible displacement work: 𝑊𝑟𝑒𝑣 = − 𝑃 𝑑𝑉 𝑉2 𝑉1 𝜕𝑇 𝜕𝑃 𝑆 = 𝜕𝑉 𝜕𝑆 𝑃 𝜕𝑇 𝜕𝑉 𝑆 = − 𝜕𝑃 𝜕𝑆 𝑉 𝜕𝑃 𝜕𝑇 𝑉 = 𝜕𝑆 𝜕𝑉 𝑇 𝜕𝑉 𝜕𝑇 𝑃 = − 𝜕𝑆 𝜕𝑃 𝑇
- 23. 22 Real pure fluids Compressibility factor: 𝑍 ≡ 𝑃𝑉 𝑅𝑇 = 𝑉 𝑉 𝑖𝑔 Residual properties: 𝑀𝑅 ≡ 𝑀 − 𝑀𝑖𝑔 where: 𝑀 is either 𝑉 , 𝑈, 𝐻, 𝑆, 𝐴, 𝑜𝑟 𝐺 Fugacity of pure species: Residual Gibbs-Free Energy: 𝐺𝑅 = 𝑅𝑇 ln 𝜙 with 𝜙 = 𝑓 𝑃 𝐺𝑅 = 𝐴𝑅 + 𝑃𝑉 − 𝑅𝑇 (1 + ln 𝑍) with 𝐴𝑅 = − 𝑃 − 𝑅𝑇 𝑉 𝑑𝑉 𝑉 ∞ Mixtures 𝑦𝑖𝑃 = 𝑥𝑖𝛾𝑖𝑃𝑖 𝑠𝑎𝑡 𝛼12 = 𝑦1 𝑥1 ⁄ 𝑦2 𝑥2 ⁄ Modified Raoult's Law Relative volatility 𝑃 = 𝑥𝑖𝛾𝑖𝑃𝑖 𝑠𝑎𝑡 𝑁 𝑖=1 𝑃 = 𝑦𝑖 𝛾𝑖𝑃𝑖 𝑠𝑎𝑡 𝑁 𝑖=1 −1 Bubble point Dew point For all phases ln 𝑓 𝑃 = 1 𝑅𝑇 𝑉 − 𝑅𝑇 𝑃 𝑑𝑃 𝑃 0 For a pure liquid ln 𝑓𝑙𝑖𝑞 𝑓𝑠𝑎𝑡 = 1 𝑅𝑇 𝑉 𝑑𝑃 𝑃 𝑃𝑠𝑎𝑡
- 24. 23 𝑀𝑖 ≡ 𝜕𝑀𝑡 𝜕𝑛𝑖 𝑇,𝑃,𝑛𝑗≠𝑖 𝑀 = 𝑥𝑖𝑀𝑖 𝑁 𝑖=1 Partial property Summation Gibbs-Duhem (𝑐𝑜𝑛𝑠𝑡. 𝑇 , 𝑃) 𝑥𝑖 𝑑𝑀𝑖 𝑁 𝑖=1 = 0 Properties at infinite dilution: lim 𝑥𝑖→0 𝑀𝑖 = 𝑀𝑖 ∞ lim 𝑥𝑖→0 𝛾𝑖(𝑇 , 𝑃, 𝑥𝑖) = 𝛾𝑖 ∞ (𝑇 , 𝑃) Excess properties: 𝑀𝐸 ≡ 𝑀 − 𝑀𝑖𝑑 where: 𝑀 is either 𝑉 , 𝑈, 𝐻, 𝑆, 𝐴, 𝑜𝑟 𝐺 Activity coefficient: 𝑅𝑇 ln 𝛾𝑖 = 𝐺𝑖 𝐸
- 25. 24 TRANSPORT PROCESSES Overview Navier-Stokes: 𝜌 𝐷𝒗 𝐷𝑡 ≡ 𝜌 𝜕𝒗 𝜕𝑡 + 𝒗 ∙ ∇𝒗 = −∇𝑃 + 𝜇∇2 𝒗 + 𝒇𝑏 Energy Equation: 𝜌 𝐷𝑇 𝐷𝑡 ≡ 𝜌 𝜕𝑇 𝜕𝑡 + 𝒗 ∙ ∇𝑇 = 𝑘 𝑐𝑝 ∇2 𝑇 + 𝑒̇𝑔𝑒𝑛 𝑐𝑝 Continuity: 𝜕𝜌 𝜕𝑡 + ∇ ∙ (𝜌𝒗) = 0 Mass Transport: 𝜌 𝐷𝐶𝑖 𝐷𝑡 ≡ 𝜌 𝜕𝐶𝑖 𝜕𝑡 + 𝒗 ∙ ∇𝐶𝑖 = 𝜌𝔇∇2 𝐶𝑖 + 𝜌𝑟𝑖 FLUID MECHANICS Basic relations 𝑀𝑒𝑎𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦: 𝑣 = 1 𝜌𝐴𝑐 𝜌𝑣 𝑑𝐴𝑐 𝐴𝑐 𝜏𝑤 = 𝜇 𝜕𝑣 𝜕𝑦 𝑦=0 = 𝐶𝑓 𝜌𝑣2 2 𝐶𝑓 = 𝑓 4 𝐷ℎ = 4𝐴𝑐 𝑝 𝐶𝑓 = Fanning friction factor 𝑓 = Darcy friction factor 𝐷ℎ = Hydraulic diameter 𝑝 = Wetted perimeter Internal flow in pipes 𝑓 = 64 𝑅𝑒 1 √𝑓 ≈ −1.8 log10 6.9 𝑅𝑒 + 𝜀 𝐷 ⁄ 3.7 1.11 Laminar smooth & rough Turbulent relative roughness 𝜀 𝐷 ⁄ Power of a pump / fan: ℙ = ∆𝑃𝐿 𝑉̇ ℎ𝐿 = 𝛥𝑃𝐿 𝜌𝑔 = 𝑓 𝐿 𝐷 𝑣2 2𝑔
- 26. 25 Conservation laws Momentum balance: Mass flow: 𝑭𝑛𝑒𝑡 ≡ 𝑭 = 𝛽𝑖𝑚̇𝑖𝒗𝑖 𝑜𝑢𝑡 − 𝛽𝑖𝑚̇𝑖𝒗𝑖 𝑖𝑛 𝑚̇ = 𝜌𝑉̇ = 𝜌𝑣𝐴𝑐 Vector form, with 𝛽 as the momentum flux correction factor Mechanical energy balance: 𝑃1 𝜌𝑔 + 𝛼1 𝑣1 2 2𝑔 + 𝑧1 + ℎ𝑝𝑢𝑚𝑝 = 𝑃2 𝜌𝑔 + 𝛼2 𝑣2 2 2𝑔 + 𝑧2 + ℎ𝑡𝑢𝑟𝑏𝑖𝑛𝑒 + ℎ𝑙𝑜𝑠𝑠 With 𝛼 as the kinetic energy correction factor Parallel flow across plate Boundary layer Friction factor Condition Laminar 𝛿 = 4.91𝑥 𝑅𝑒𝑥 1 2 ⁄ 𝐶𝑓 = 1.33 𝑅𝑒𝐿 1 2 ⁄ 𝑅𝑒𝐿 < 5 × 105 Turbulent 𝛿 = 0.38𝑥 𝑅𝑒𝑥 1 5 ⁄ 𝐶𝑓 = 0.074 𝑅𝑒𝐿 1 5 ⁄ 5 × 105 ≤ 𝑅𝑒𝐿 < 107 Combined result 𝐶𝑓 = 0.074 𝑅𝑒𝐿 1 5 ⁄ − 1742 𝑅𝑒𝐿 5 × 105 ≤ 𝑅𝑒𝐿 < 107 Rough turbulent regime 𝐶𝑓 = 1.89 − 1.62 log10 𝜀 𝐿 −2.5 102 < 𝐿 𝜀 < 106
- 27. 26 Flow inside packed bed Ergun equation: Packed-bed Reynolds number: ℎ𝐿 = 3𝑓𝐹 (1 − 𝜀)𝐿𝑣𝑜 2 𝑔𝜀3𝑑𝑝 𝑅𝑒 = 𝜌𝑣0𝑑𝑝 (1 − 𝜀)𝜇 𝑓𝐹 = 1 3 [(150 / 𝑅𝑒) + 1.75] (𝑃1 − 𝑃2)𝑑𝑝𝜀3 𝜌𝑣𝑜 2 𝐿(1 − 𝜀) = 150 𝑅𝑒 + 1.75 + 𝑑𝑝𝜀3 𝑔(𝑧2 − 𝑧1) 𝑣𝑜 2 𝐿(1 − 𝜀) Navier-Stokes Navier-Stokes and Continuity Equations for an Incompressible Newtonian Fluid 𝑥-component 𝜌 𝜕𝑣𝑥 𝜕𝑡 + 𝑣𝑥 𝜕𝑣𝑥 𝜕𝑥 + 𝑣𝑦 𝜕𝑣𝑥 𝜕𝑦 + 𝑣𝑧 𝜕𝑣𝑥 𝜕𝑧 = − 𝜕𝑃 𝜕𝑥 + 𝜇 𝜕2 𝑣𝑥 𝜕𝑥2 + 𝜕2 𝑣𝑥 𝜕𝑦2 + 𝜕2 𝑣𝑥 𝜕𝑧2 + 𝜌𝑔𝑥 𝑦-component 𝜌 𝜕𝑣𝑦 𝜕𝑡 + 𝑣𝑥 𝜕𝑣𝑦 𝜕𝑥 + 𝑣𝑦 𝜕𝑣𝑦 𝜕𝑦 + 𝑣𝑧 𝜕𝑣𝑦 𝜕𝑧 = − 𝜕𝑃 𝜕𝑦 + 𝜇 𝜕2 𝑣𝑦 𝜕𝑥2 + 𝜕2 𝑣𝑦 𝜕𝑦2 + 𝜕2 𝑣𝑦 𝜕𝑧2 + 𝜌𝑔𝑦 𝑧-component 𝜌 𝜕𝑣𝑧 𝜕𝑡 + 𝑣𝑥 𝜕𝑣𝑧 𝜕𝑥 + 𝑣𝑦 𝜕𝑣𝑧 𝜕𝑦 + 𝑣𝑧 𝜕𝑣𝑧 𝜕𝑧 = − 𝜕𝑃 𝜕𝑧 + 𝜇 𝜕2 𝑣𝑧 𝜕𝑥2 + 𝜕2 𝑣𝑧 𝜕𝑦2 + 𝜕2 𝑣𝑧 𝜕𝑧2 + 𝜌𝑔𝑧 Continuity 𝜕𝑣𝑥 𝜕𝑥 + 𝜕𝑣𝑦 𝜕𝑦 + 𝜕𝑣𝑧 𝜕𝑧 = 0 Cylindrical coordinates (𝑟, 𝜃, 𝑧) 𝑟-component 𝜌 𝜕𝑣𝑟 𝜕𝑡 + 𝑣𝑟 𝜕𝑣𝑟 𝜕𝑟 + 𝑣𝜃 𝑟 𝜕𝑣𝑟 𝜕𝜃 − 𝑣𝜃 2 𝑟 + 𝑣𝑧 𝜕𝑣𝑟 𝜕𝑧 = − 𝜕𝑃 𝜕𝑟 + 𝜇 𝜕 𝜕𝑟 1 𝑟 𝜕(𝑟𝑣𝑟) 𝜕𝑟 + 1 𝑟2 𝜕2 𝑣𝑟 𝜕𝜃2 − 2 𝑟2 𝜕𝑣𝜃 𝜕𝜃 + 𝜕2 𝑣𝑟 𝜕𝑧2 + 𝜌𝑔𝑟 𝜃-component 𝜌 𝜕𝑣𝜃 𝜕𝑡 + 𝑣𝑟 𝜕𝑣𝜃 𝜕𝑟 + 𝑣𝜃 𝑟 𝜕𝑣𝜃 𝜕𝜃 + 𝑣𝜃𝑣𝑟 𝑟 + 𝑣𝑧 𝜕𝑣𝜃 𝜕𝑧 = − 1 𝑟 𝜕𝑃 𝜕𝜃 + 𝜇 𝜕 𝜕𝑟 1 𝑟 𝜕(𝑟𝑣𝜃) 𝜕𝑟 + 1 𝑟2 𝜕2 𝑣𝜃 𝜕𝜃2 + 2 𝑟2 𝜕𝑣𝑟 𝜕𝜃 + 𝜕2 𝑣𝜃 𝜕𝑧2 + 𝜌𝑔𝜃 [see Fundamental Geometry (page 8) for coordinate system] Cartesian coordinates (𝑥, 𝑦, 𝑧)
- 28. 27 𝑧-component 𝜌 𝜕𝑣𝑧 𝜕𝑡 + 𝑣𝑟 𝜕𝑣𝑧 𝜕𝑟 + 𝑣𝜃 𝑟 𝜕𝑣𝑧 𝜕𝜃 + 𝑣𝑧 𝜕𝑣𝑧 𝜕𝑧 = − 𝜕𝑃 𝜕𝑧 + 𝜇 1 𝑟 𝜕 𝜕𝑟 𝑟 𝜕𝑣𝑧 𝜕𝑟 + 1 𝑟2 𝜕2 𝑣𝑧 𝜕𝜃2 + 𝜕2 𝑣𝑧 𝜕𝑧2 + 𝜌𝑔𝑧 Continuity 1 𝑟 𝜕(𝑟𝑣𝑟) 𝜕𝑟 + 1 𝑟 𝜕𝑣𝜃 𝜕𝜃 + 𝜕𝑣𝑧 𝜕𝑧 = 0 Spherical coordinates (𝑟, 𝜃, 𝜙) 𝑟-component 𝜌 𝜕𝑣𝑟 𝜕𝑡 + 𝑣𝑟 𝜕𝑣𝑟 𝜕𝑟 + 𝑣𝜙 𝑟 𝜕𝑣𝑟 𝜕𝜙 + 𝑣𝜃 𝑟 sin 𝜙 𝜕𝑣𝑟 𝜕𝜃 − 𝑣𝜙 2 + 𝑣𝜃 2 𝑟 = − 𝜕𝑃 𝜕𝑟 + 𝜇 ∇2 𝑣𝑟 − 2𝑣𝑟 𝑟2 + 2 𝑟2 𝜕𝑣𝜙 𝜕𝜙 − 2 𝑟2 𝑣𝜙 cot 𝜙 − 2 𝑟2 sin 𝜙 𝜕𝑣𝜃 𝜕𝜃 + 𝜌𝑔𝑟 𝜃-component 𝜌 𝜕𝑣𝜃 𝜕𝑡 + 𝑣𝑟 𝜕𝑣𝜃 𝜕𝑟 + 𝑣𝜙 𝑟 𝜕𝑣𝜃 𝜕𝜙 + 𝑣𝜃 𝑟 sin 𝜙 𝜕𝑣𝜃 𝜕𝜃 + 𝑣𝑟𝑣𝜃 𝑟 − 𝑣𝜃𝑣𝜙 cot 𝜙 𝑟 = − 1 𝑟 sin 𝜙 𝜕𝑃 𝜕𝜃 + 𝜇 ∇2 𝑣𝜃 − 𝑣𝜃 𝑟2 sin2 𝜙 + 2 𝑟2 sin 𝜙 𝜕𝑣𝑟 𝜕𝜃 + 2 cos 𝜙 𝑟2 sin2 𝜙 𝜕𝑣𝜙 𝜕𝜃 + 𝜌𝑔𝜃 𝜙-component 𝜌 𝜕𝑣𝜙 𝜕𝑡 + 𝑣𝑟 𝜕𝑣𝜙 𝜕𝑟 + 𝑣𝜙 𝑟 𝜕𝑣𝜙 𝜕𝜙 + 𝑣𝜃 𝑟 sin 𝜙 𝜕𝑣𝜙 𝜕𝜃 + 𝑣𝑟𝑣𝜙 𝑟 − 𝑣𝜃 2 cot 𝜙 𝑟 = − 1 𝑟 𝜕𝑃 𝜕𝜙 + 𝜇 ∇2 𝑣𝜙 + 2 𝑟2 𝜕𝑣𝑟 𝜕𝜙 − 𝑣𝜙 𝑟2 sin2 𝜙 + 2 cos 𝜙 𝑟2 sin2 𝜙 𝜕𝑣𝜃 𝜕𝜃 + 𝜌𝑔𝜙 where: ∇2 = 1 𝑟2 𝜕 𝜕𝑟 𝑟2 𝜕 𝜕𝑟 + 1 𝑟2 sin2 𝜙 𝜕2 𝜕𝜃2 + 1 𝑟2 sin 𝜙 𝜕 𝜕𝜙 sin 𝜙 𝜕 𝜕𝜙 Continuity 1 𝑟2 𝜕(𝑟2 𝑣𝑟) 𝜕𝑟 + 1 𝑟 sin 𝜙 𝜕𝑣𝜃 𝜕𝜃 + 1 𝑟 sin𝜙 𝜕 𝑣𝜙 sin 𝜙 𝜕𝜙 = 0
- 29. 28 HEAT TRANSPORT Fundamental laws Energy conservation with no work done: 𝑄 = 𝑚𝑐𝑣∆𝑇 𝑄̇ = 𝑚̇𝑐𝑝∆𝑇 Closed system Open system Heat transfer mechanisms: 𝑄̇𝑐𝑜𝑛𝑑 = −𝑘𝐴 𝑑𝑇 𝑑𝑥 𝑄̇𝑐𝑜𝑛𝑣 = ℎ𝐴𝑠(𝑇𝑠 − 𝑇∞) Fourier’s Law Newton’s Law Note: all above equations assume constant properties Fundamental definition of heat transfer coefficient (local): ℎ = −𝑘𝑓𝑙𝑢𝑖𝑑 (𝜕𝑇 𝜕𝑦 ⁄ )𝑦=0 𝑇𝑠 − 𝑇∞ Stefan-Boltzmann’s Law with net radiation, surrounded by infinite surface: 𝑄̇𝑟𝑎𝑑 = 𝜀𝜎𝐴𝑠 𝑇𝑠 4 − 𝑇𝑠𝑢𝑟𝑟 4 Thermal circuit modelling Applies to conduction, convection or radiation, with: 𝑸̇ = ∆𝑻 𝑹𝐭𝐨𝐭 ⁄ 𝑅𝑡𝑜𝑡 = 𝑅1 + 𝑅2 + 𝑅3
- 30. 29 1 𝑅𝑡𝑜𝑡 = 1 𝑅1 + 1 𝑅2 + 1 𝑅3 Heat conduction equation [see Fundamental Geometry (page 8) for coordinate system] Cartesian coordinates (𝑥, 𝑦, 𝑧) Cylindrical coordinates (𝑟, 𝜃, 𝑧) 1 𝑟 𝜕 𝜕𝑟 𝑘𝑟 𝜕𝑇 𝜕𝑟 + 1 𝑟2 𝜕 𝜕𝜃 𝑘 𝜕𝑇 𝜕𝜃 + 𝜕 𝜕𝑧 𝑘 𝜕𝑇 𝜕𝑧 + 𝑒̇𝑔𝑒𝑛 = 𝜌𝑐𝑝 𝜕𝑇 𝜕𝑡 Spherical coordinates (𝑟, 𝜃, 𝜙) Finned surfaces Standard fins of constant cross-section, 𝐴𝑐, and perimeter, 𝑝 Convective Tip Adiabatic 𝑇 (𝑥) − 𝑇∞ 𝑇𝑏 − 𝑇∞ = cosh 𝑚(𝐿 − 𝑥) + ℎ 𝑚𝑘 sinh 𝑚(𝐿 − 𝑥) cosh 𝑚𝐿 + ℎ 𝑚𝑘 sinh 𝑚𝐿 𝑇 (𝑥) − 𝑇∞ 𝑇𝑏 − 𝑇∞ = cosh 𝑚(𝐿 − 𝑥) cosh 𝑚𝐿 Given Temperature Infinite Fin 𝑇 (𝑥) − 𝑇∞ 𝑇𝑏 − 𝑇∞ = 𝑇𝐿−𝑇∞ 𝑇𝑏−𝑇∞ sinh 𝑚𝑥 + sinh 𝑚(𝐿 − 𝑥) sinh 𝑚𝐿 𝑇 (𝑥) − 𝑇∞ 𝑇𝑏 − 𝑇∞ = 𝑒−𝑚𝑥 Fin parameter 𝑚 = ℎ𝑝 𝑘𝐴𝑐 Corrected fin length 𝐿𝑐 = 𝐿 + 𝐴𝑐 𝑝 𝜕 𝜕𝑥 𝑘 𝜕𝑇 𝜕𝑥 + 𝜕 𝜕𝑦 𝑘 𝜕𝑇 𝜕𝑦 + 𝜕 𝜕𝑧 𝑘 𝜕𝑇 𝜕𝑧 + 𝑒̇𝑔𝑒𝑛 = 𝜌𝑐𝑝 𝜕𝑇 𝜕𝑡 1 𝑟2 𝜕 𝜕𝑟 𝑘𝑟2 𝜕𝑇 𝜕𝑟 + 1 𝑟2 sin2 𝜙 𝜕 𝜕𝜃 𝑘 𝜕𝑇 𝜕𝜃 + 1 𝑟2 sin 𝜙 𝜕 𝜕𝜙 𝑘 sin 𝜙 𝜕𝑇 𝜕𝜙 + 𝑒̇𝑔𝑒𝑛 = 𝜌𝑐𝑝 𝜕𝑇 𝜕𝑡
- 31. 30 Lumped analysis 𝐵𝑖 = ℎ𝐿 𝑘𝑠𝑜𝑙𝑖𝑑 where 𝐿 = 𝑉 𝐴𝑠 𝐹𝑜 = 𝛼𝑡 𝐿2 With thermal diffusivity 𝛼 = 𝑘 𝜌𝑐𝑝 ⁄ Note: the characteristic length used in 𝐵𝑖 is normally based on 𝐿 above – definitions may differ in charts! External flow Internal flow 𝐵𝑢𝑙𝑘 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒: 𝑇 = 1 𝑚̇ 𝑐𝑝 𝜌𝑣 𝑐𝑝𝑇𝑑𝐴𝑐 𝐴𝑐 Note: Exit and inlet temperatures are based on the bulk- mean value, 𝑇 𝑇𝑒(𝑥) = 𝑇𝑠 − (𝑇𝑠 − 𝑇𝑖)𝑒−ℎ𝐴𝑠(𝑥) 𝑚̇𝑐𝑝 ⁄ 𝑓𝑜𝑟 𝑇𝑠 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Entry lengths for developing flow 𝐿ℎ,𝑙𝑎𝑚𝑖𝑛𝑎𝑟 𝐷 ≈ 0.05 𝑅𝑒 𝐿𝑡,𝑙𝑎𝑚𝑖𝑛𝑎𝑟 𝐷 ≈ 0.05 𝑅𝑒 𝑃𝑟 𝐿ℎ,𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 ≈ 𝐿𝑡,𝑡𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 ≈ 10𝐷 Boiling & Condensation 𝑞̇𝑏𝑜𝑖𝑙 = ℎ(𝑇𝑠 − 𝑇𝑠𝑎𝑡) = ℎ∆𝑇𝑒𝑥𝑐𝑒𝑠𝑠 𝑚̇𝑒𝑣𝑎𝑝 = 𝑄̇𝑏𝑜𝑖𝑙 ℎ𝑓𝑔 ⁄ 𝑞̇𝑐𝑜𝑛𝑑 = ℎ(𝑇𝑠𝑎𝑡 − 𝑇𝑠) 𝑚̇𝑐𝑜𝑛𝑑 = 𝑄̇𝑐𝑜𝑛𝑑 ℎ𝑓𝑔 ∗ Modified latent heat: ℎ𝑓𝑔 ∗ = ℎ𝑓𝑔 + 0.68𝑐𝑝𝑙(𝑇𝑠𝑎𝑡 − 𝑇𝑠) + 𝑐𝑝𝑣(𝑇𝑣 − 𝑇𝑠𝑎𝑡) ℎ = 1 𝐿 ℎ𝑥 𝑑𝑥 𝐿 0 with 𝑁𝑢𝑥 = ℎ𝑥𝑥 𝑘 Note: symbol ℎ is conventionally used instead of ℎ for the average h.t.c. Relative boundary layer thickness: 𝛿 𝛿𝑡 ≈ 𝑃𝑟 1 3
- 35. 34 Heat exchangers Log-mean temperature difference: ∆𝑇𝐿𝑀 = ∆𝑇𝑒 − ∆𝑇𝑖 ln ∆𝑇𝑒 ∆𝑇𝑖 where ∆𝑇𝑒 = 𝑇𝑠 − 𝑇𝑒 and ∆𝑇𝑖 = 𝑇𝑠 − 𝑇𝑖 Overall heat transfer coefficient: where 𝐴𝑖 = 𝜋𝐷𝑖𝐿 and 𝐴𝑜 = 𝜋𝐷𝑜𝐿 are the areas of the inner and outer surfaces, and 𝑅𝑓,𝑖 and 𝑅𝑓,𝑜 are the fouling factors at those surfaces. MASS TRANSPORT Fundamental laws Mass transfer mechanisms: 𝐽𝐴,𝑥 = − 𝔇𝐴𝐵 𝑑𝐶𝐴 𝑑𝑥 𝑁𝐴 = 𝑘𝑐 𝐶𝐴,𝑠 − 𝐶𝐴,∞ Fickian diffusion Convection Fundamental definition of mass transfer coefficient (local): 𝑘𝑐 = −𝔇𝐴𝐵 (𝜕𝐶𝐴 𝜕𝑦 ⁄ )𝑦=0 𝐶𝐴,𝑠 − 𝐶𝐴,∞ Note: The following framework is based on the assumption of binary mixtures. For multicomponent systems refer to the literature. 1 𝑈𝐴𝑠 ≡ 1 𝑈𝑖𝐴𝑖 ≡ 1 𝑈𝑜𝐴𝑜 = 1 ℎ𝑖𝐴𝑖 + 𝑅𝑓,𝑖 𝐴𝑖 + ln(𝐷𝑜 𝐷𝑖 ⁄ ) 2𝜋𝑘𝐿 + 𝑅𝑓,𝑜 𝐴𝑜 + 1 ℎ𝑜𝐴𝑜
- 36. 35 Mass Diffusivity 𝔇𝐴𝐵 = 0.001858 𝑇 3 2 1 𝑀𝐴 + 1 𝑀𝐵 1 2 ⁄ 𝑃𝜎𝐴𝐵 2 𝛺𝐷(𝑇 ) Hirschfelder’s equation for binary gas phase diffusivity 𝔇𝐴𝐵𝜇𝐵 𝑇 = 7.4 × 10−8 (𝛷𝐵𝑀𝐵)1 2 ⁄ 𝑉𝐴 0.6 Wilke-Chang correlation for binary liquid phase diffusivity 𝔇𝐾𝐴 = 4850 𝑑𝑝𝑜𝑟𝑒 𝑇 𝑀𝐴 Knudsen diffusivity in porous solid 𝔇1−𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = 1 𝑦′2 𝔇12 + 𝑦′3 𝔇13 + ⋯ + 𝑦′𝑛 𝔇1𝑛 Diffusivity of minor species in a gas mixture with 𝑦′2 = 𝑦2 𝑦2 + 𝑦3 + ⋯ + 𝑦𝑛 Parameter 𝜎𝐴𝐵 and 𝛺𝐷(𝑇 ) is the collision diameter and collision integral respectively, whose values are given elsewhere. Parameter 𝑉𝐴 is the molar volume of solute A at its normal boiling point, and 𝛷𝐵 is the association parameter for solvent B - whose values are given elsewhere. Mass transfer between fluid phases Concentration units Flux equation Units of 𝒌 (in SI) Liquid Film Mole concentration 𝑁𝐴 = 𝑘𝑐𝐿 𝐶𝐴𝐿,𝑖 − 𝐶𝐴𝐿 m/s Mole fraction 𝑁𝐴 = 𝑘𝑥 𝑥𝐴,𝑖 − 𝑥𝐴 mol/(m2 .s) Gas Film Partial pressure 𝑁𝐴 = 𝑘𝑝 𝑃𝐴 − 𝑃𝐴,𝑖 mol/(m2 .Pa.s) Mole fraction 𝑁𝐴 = 𝑘𝑦 𝑦𝐴 − 𝑦𝐴,𝑖 mol/(m2 .s) Mole concentration 𝑁𝐴 = 𝑘𝑐𝐺 𝐶𝐴𝐺 − 𝐶𝐴𝐺,𝑖 m/s
- 37. 36 Liquid phase: 𝑘𝑐𝐿 = 𝑘𝑥 𝐶𝐿 Where 𝐶𝐿 is the total molar concentration in the liquid phase. Gas phase (ideal): 𝑘𝑝 = 𝑘𝑦 𝑃 = 𝑘𝑐𝐺 𝑅𝑇 Where 𝑃 is the total pressure. Overall mass transfer coefficient for linear equilibrium (Henry’s law) Units Equilibrium relation Liquid phase Gas phase Concentrations and pressure 𝑃𝐴 = 𝐻 𝐶𝐴 1 𝐾𝐿 = 1 𝐻 𝑘𝑝 + 1 𝑘𝑐𝐿 1 𝐾𝐺 = 1 𝑘𝑝 + 𝐻 𝑘𝑐𝐿 Mole fractions 𝑦𝐴 = 𝑚 𝑥𝐴 1 𝐾𝑥 = 1 𝑚 𝑘𝑦 + 1 𝑘𝑥 1 𝐾𝑦 = 1 𝑘𝑦 + 𝑚 𝑘𝑥 External flow Internal flow Entry length for developing flow: 𝐿𝑐,𝑙𝑎𝑚𝑖𝑛𝑎𝑟 𝐷 ≈ 0.05 𝑅𝑒 𝑆𝑐 𝑘𝑐 = 1 𝐿 𝑘𝑐,𝑥 𝑑𝑥 𝐿 0 with 𝑆ℎ𝑥 = 𝑘𝑐,𝑥 𝑥 𝔇𝐴𝐵 Note: symbol 𝑘𝑐 is conventionally used instead of 𝑘𝑐 for the average m.t.c. Relative boundary layer thickness: 𝛿 𝛿𝑐 ≈ 𝑆𝑐 1 3 Mean (Bulk) mass fraction: 𝑤 = 1 𝑚̇ 𝜌𝑣 𝑤 𝑑𝐴𝑐 𝐴𝑐
- 38. 37 Mass transport equation 𝜕𝐶𝐴 𝜕𝑡 + ∇ ∙ 𝑵𝑨 = 𝑟𝐴 Gradient of a scalar quantity (𝒚𝑨, 𝑪𝑨, …): [see Fundamental Geometry (page 8) for coordinate system] Cartesian coordinates (𝑥, 𝑦, 𝑧) ∇𝑦𝐴 = 𝜕𝑦𝐴 𝜕𝑥 𝒙̂ + 𝜕𝑦𝐴 𝜕𝑦 𝒚̂ + 𝜕𝑦𝐴 𝜕𝑧 𝒛̂ Cylindrical coordinates (𝑟, 𝜃, 𝑧) ∇𝑦𝐴 = 𝜕𝑦𝐴 𝜕𝑟 𝒓̂ + 1 𝑟 𝜕𝑦𝐴 𝜕𝜃 𝜽̂ + 𝜕𝑦𝐴 𝜕𝑧 𝒛̂ Spherical coordinates (𝑟, 𝜃, 𝜙) ∇𝑦𝐴 = 𝜕𝑦𝐴 𝜕𝑟 𝒓̂ + 1 𝑟 sin 𝜙 𝜕𝑦𝐴 𝜕𝜃 𝜽̂ + 1 𝑟 𝜕𝑦𝐴 𝜕𝜙 𝝓̂ Divergence of a vector quantity (𝑵𝑨, 𝒗, …): Cartesian coordinates (𝑥, 𝑦, 𝑧) ∇ ∙ 𝑵𝑨 = 𝜕𝑁𝐴,𝑥 𝜕𝑥 + 𝜕𝑁𝐴,𝑦 𝜕𝑦 + 𝜕𝑁𝐴,𝑧 𝜕𝑧 Cylindrical coordinates (𝑟, 𝜃, 𝑧) ∇ ∙ 𝑵𝑨 = 1 𝑟 𝜕 𝜕𝑟 𝑟𝑁𝐴,𝑟 + 1 𝑟 𝜕𝑁𝐴,𝜃 𝜕𝜃 + 𝜕𝑁𝐴,𝑧 𝜕𝑧 Spherical coordinates (𝑟, 𝜃, 𝜙) ∇ ∙ 𝑵𝑨 = 1 𝑟2 𝜕 𝜕𝑟 𝑟2 𝑁𝐴,𝑟 + 1 𝑟 sin 𝜙 𝜕𝑁𝐴,𝜃 𝜕𝜃 + 1 𝑟 sin 𝜙 𝜕 𝜕𝜙 𝑁𝐴,𝜙 sin 𝜙 Equivalent forms of molar flux for a binary system Restriction Diffusive Diffusive + Convective Const. 𝑇 Const. 𝑃 Liquid (ideal solution) 🗸 𝑱𝐴 = −𝐶𝐿𝔇𝐴𝐵∇𝑥𝐴 𝑵𝐴 = −𝐶𝐿𝔇𝐴𝐵∇𝑥𝐴 + 𝑥𝐴(𝑵𝐴 + 𝑵𝐵) Gas (ideal) 🗸 🗸 𝑱𝐴 = −𝐶𝐺𝔇𝐴𝐵∇𝑦𝐴 𝑵𝐴 = −𝐶𝐺𝔇𝐴𝐵∇𝑦𝐴 + 𝑦𝐴(𝑵𝐴 + 𝑵𝐵) 🗸 𝑱𝐴 = −(𝔇𝐴𝐵 𝑅𝑇 ⁄ )∇𝑃𝐴 𝑵𝐴 = −(𝔇𝐴𝐵 𝑅𝑇 ⁄ )∇𝑃𝐴 + 𝑦𝐴(𝑵𝐴 + 𝑵𝐵) 🗸 🗸 𝑱𝐴 = −𝔇𝐴𝐵∇𝐶𝐴 𝑵𝐴 = −𝔇𝐴𝐵∇𝐶𝐴 + 𝑦𝐴(𝑵𝐴 + 𝑵𝐵) Where mass fluxes are related via: 𝒋𝑨 = 𝑀𝐴 𝑱𝑨 and 𝒏𝐴 = 𝑀𝐴 𝑵𝐴
- 41. 40 SEPARATION PROCESSES Flashing & Distillation Rachford-Rice (𝑦𝑖 − 𝑥𝑖) = 𝐶 𝑖=1 (𝐾𝑖 − 1)𝑧𝑖 1 + (𝐾𝑖 − 1) 𝑉 𝐹 = 0 𝐶 𝑖=1 Fenske Modified Fenske 𝑁𝑚𝑖𝑛 = ln (𝐹𝑅𝐿𝐾)𝑑𝑖𝑠𝑡(𝐹𝑅𝐻𝐾)𝑏𝑜𝑡 1 − (𝐹𝑅𝐿𝐾)𝑑𝑖𝑠𝑡 1 − (𝐹𝑅𝐻𝐾)𝑏𝑜𝑡 ln 𝛼𝑎𝑣𝑔 𝑁𝐹,𝑚𝑖𝑛 = ln ⎣ ⎢ ⎢ ⎡ 𝑥𝐿𝐾 𝑥𝐻𝐾 𝑑𝑖𝑠𝑡 𝑧𝐿𝐾 𝑧𝐻𝐾 ⎦ ⎥ ⎥ ⎤ ln 𝛼𝑎𝑣𝑔 𝛼𝑎𝑣𝑔 = 𝛼𝑓𝑒𝑒𝑑𝛼𝑑𝑖𝑠𝑡𝛼𝑏𝑜𝑡 3 𝑁𝐹,𝑚𝑖𝑛 𝑁𝑚𝑖𝑛 = 𝑁𝐹 𝑁 First Underwood Second Underwood 𝐹(1 − 𝑞) = 𝑉𝑚𝑖𝑛 − 𝑉𝑚𝑖𝑛 = 𝛼𝑖−𝑟𝑒𝑓 𝐹𝑧𝑖 𝛼𝑖−𝑟𝑒𝑓 − 𝜙 𝐶 𝑖=1 𝑉𝑚𝑖𝑛 = 𝛼𝑖−𝑟𝑒𝑓 𝐷𝑥𝑖,𝑑𝑖𝑠𝑡 𝛼𝑖−𝑟𝑒𝑓 − 𝜙 𝐶 𝑖=1 Kirkbride correlation log10 𝑁𝐹 − 1 𝑁 − 𝑁𝐹 = 0.26 log10 𝐵 𝐷 𝑧𝐻𝐾 𝑧𝐿𝐾 𝑥𝐿𝐾,𝑏𝑜𝑡 𝑥𝐻𝐾,𝑑𝑖𝑠𝑡 2 O’Connell correlation 𝐸𝑜 = 0.52782 − 0.27511 log10(𝛼𝜇) + 0.044923 log10(𝛼𝜇) 2 Absorption & Stripping Kremser Flooding velocity 𝑦𝐵 − 𝑦𝐵 ∗ 𝑦𝑇 − 𝑦𝑇 ∗ = 𝐴𝑁 𝑢𝑓𝑙𝑜𝑜𝑑 = 𝐶𝑠𝑏,𝑓 𝜎 20 0.2 𝜌𝐿 − 𝜌𝑉 𝜌𝑉 with the absorption factor, 𝐴 ≡ 𝐿 𝑚𝐺
- 42. 41 PHYSICAL PROPERTIES OF AIR & WATER Air Average molar mass 𝑀𝑅 = 29 g/mol Specific gas constant 𝑅̂ = 287 J/(kg K) Specific heat capacities at 298 K 𝑐𝑝 = 1005 J/(kg K) 𝑐𝑣 = 718 J/(kg K) 𝛾 = 1.40 Composition Mole % Mass % O2 21.0 23.1 N2 78.1 75.6 Ar 0.9 1.3 Viscosities and Thermal Conductivity at absolute pressure of 1 bar 𝑻 0 20 40 60 80 100 C 𝜇 1.71 1.81 1.90 2.00 2.09 2.18 × 10−5 Pa s 𝜈 1.32 1.50 1.69 1.88 2.09 2.30 × 10−5 m2 /s 𝑘 0.024 0.025 0.027 0.028 0.029 0.031 W/(m K) Water Specific heat capacity at 298 K 𝑐𝑝 = 4187 J/(kg K) Surface tension with air at 298 K 𝜎 = 0.073 N/m Viscosities, Thermal Conductivity and Vapour Pressure 𝑻 0 20 40 60 80 100 C 𝜇 1.79 1.01 0.656 0.469 0.357 0.284 × 10−3 Pa s 𝜈 1.79 1.01 0.661 0.477 0.367 0.296 × 10−6 m2 /s 𝑘 0.57 0.60 0.63 0.65 0.67 0.68 W/(m K) 𝑃𝑠𝑎𝑡 0.61 2.34 7.38 19.9 47.4 101.3 kPa
- 43. 42 TRANSPORT PROPERTIES The below values are correct for a pressure of 1 atm = 1.01325 bar, but may be used with sufficient accuracy at other reasonable pressures. Air 𝑇 𝑐𝑝 𝑘 𝜇 / 10-6 𝑃𝑟 = 𝜇𝑐𝑝 𝑘 ⁄ C kJ/(kg K) W/(m K) Pa s -100 1.01 0.016 12 0.75 0 1.01 0.024 17 0.72 100 1.02 0.032 22 0.70 200 1.03 0.039 26 0.69 300 1.05 0.045 30 0.69 400 1.07 0.051 33 0.70 500 1.10 0.056 36 0.70 600 1.12 0.061 39 0.71 700 1.14 0.066 42 0.72 800 1.16 0.071 44 0.73 This table may be used with reasonable accuracy for values of 𝑐𝑝, 𝑘, 𝜇 and 𝑃𝑟 of N2, O2 and CO. Steam 𝑇 𝑐𝑝 𝑘 𝜇 / 10-6 𝑃𝑟 = 𝜇𝑐𝑝 𝑘 ⁄ C kJ/(kg K) W/(m K) Pa s 100 2.028 0.0245 12.1 0.986 200 1.979 0.0331 16.2 0.968 300 2.010 0.0434 20.4 0.946 400 2.067 0.0548 24.6 0.928 500 2.132 0.0673 28.8 0.912 600 2.201 0.0805 32.9 0.898 700 2.268 0.0942 36.8 0.887 800 2.332 0.1080 40.6 0.876
- 44. 43 THERMODYNAMIC DATA FOR WATER & STEAM Source of data The following data tables have been sourced from IAPWS – the international association of the properties of water and steam website: www.iapws.org The arbitrary datum chosen is that saturated liquid water at the triple point has internal energy 𝑈 = 0 and entropy 𝑆 = 0 Triple point data Temperature = 273.16 K (0.01 ℃) Pressure = 0.00611 bar Phase Specific volume Specific enthalpy Specific entropy m3 /kg kJ/kg kJ/(kg K) Ice 0.0010905 −333.5 −1.221 Water 0.0010002 0.0062 0.0 Steam 206 2500.9 9.156 Critical point data Temperature = 647.1 K (374 ℃) Pressure = 220.64 bar Density = 322 kg/m3
- 45. 44 SATURATED STEAM TABLES Temperatures from the triple point 100 ℃
- 46. 45 SATURATED STEAM TABLES Temperatures from 100 ℃ critical point
- 50. 49 AMMONIA PROPERTIES ♱ The arbitrary datum chosen for this table is that saturated liquid ammonia at −40 ℃ has enthalpy 𝐻 = 0 and entropy 𝑆 = 0
- 51. 50 BESSEL FUNCTIONS GAUSSIAN ERROR FUNCTION