Black oils Correlations Comparative Study

Size: px
Start display at page:

Download "Black oils Correlations Comparative Study"

Transcription

1 Reservoir Technologies Blck oils Correltions Comprtive Study Dr. Muhmmd Al-Mrhoun, Mnging Director Sturdy, 26 April, 2014 Copyright 2008, NExT, All rights reserved

2 Blck oils Correltions Introduction to correltions Types of correltions Evlution of empiriclly derived PVT properties for Middle Est crude oils 2

3 Introduction to correltions PVT properties re obtined from lbortory experiment using oil representtive smples However, vlues of reservoir liquid nd gs properties re often needed when lbortory detiled PVT dt re not vilble Therefore, correltions re used to estimte those properties Correltion re bsed on esily obtined dt like R s, g, P, T, API 3

4 Introduction to correltions PVT properties depend on pressure, temperture, nd chemicl compositions For the development of correltion, geologicl condition is considered importnt becuse the chemicl composition of crude oil differs from region to region To ccount for regionl chrcteristics, PVT correltions need to be modified for their ppliction by reclculting the correltion constnts for the region of interest 4

5 Why we need correltions? They re useful in mking estimtes for experimentl design s check ginst lbortory results In estimting properties when smpling is impossible or uneconomicl In generliztion of properties - it is impossible to run experiments on ll possible reservoir or surfce conditions 5

6 Types of correltions Grphs Nomogrphs Equtions 6

7 Grphs Correltion chrt for totl formtion volume fctor by Stnding

8 Nomogrphs Nomogrph correltion for bubble point pressure by Al-Mrhoun

9 Equtions Correltion for bubble point oil formtion volume fctor by Vsquez & Beggs 1980 B ob R s s T 2 60 API / g T 60 / R API g 9

10 Blck-oil PVT correltions 1. Oil density 2. Bubble point pressure 3. Solution gs-oil rtio 4. Bubble point oil FVF 5. Totl FVF 6. Isotherml oil compressibility 7. Understurted oil viscosity 8. Bubble point oil viscosity 9. Ded oil viscosity 10.Surfce tension 10

11 Oil density The oil density is defined s the mss per unit volume t specified pressure nd temperture. o m v o o The reltive density of oil is defined s: o o w 11

12 Oil density The reltive density of oil t ny other temperture T could be clculted using ot 1 o x10 3 ( T 60) In the petroleum industry, it is common to express grvity in terms of oil API grvity, or: pi O 12

13 Oil density Oil density is required t vrious pressures nd t reservoir temperture for reservoir engineering clcultions. An eqution for oil density t Pb in eqution form is expressed s ob o 2.18x10 B ob 4 R s g 13

14 Oil density Above bubble point pressure, incresed pressure will compress the liquid nd increse its density. For the cse of P > P b, the oil density is clculted from Correltion for clculting verge oil compressibility C o t vrious conditions is presented lter o ob e c o (PP b ) 14

15 Reservoir Pressure Bubble point pressure Bubble point pressure is the pressure t which the first bubble of gs evolves s the pressure decreses 1-Phse 2-Phse 60% 40% CP 20% 0% 1-Phse Reservoir Temperture 15

16 Bubble point pressure. Stnding (1947) P b 1 R γ s g 2 e 3 T 4γ pi where 1 = 18 2 = = E-3 4 = E-3 16

17 Bubble point pressure.. Vsquez nd Beggs (1980) P b 1 R s g 2 e 3 pi ( T 460) where Coefficient pi 30 pi

18 Bubble point pressure Al-Mrhoun (1988) P b 1 R s g o T where 1 = E-3 2 = = = =

19 Sttisticl ccurcy of P b correltions ER EA E mx STD R Correltion Stnding (1947) Vsquez & Beggs (1980) Al-Mrhoun (1988) Modified Correltion Stnding (1947) Vsquez & Beggs (1980) Al-Mrhoun (1988)

20 Absolute Averge Percent Error Absolute error of P b correltions Correltion Modified Correltion Stnding Vsquez& Beggs Al-Mrhoun 20

21 Avg Absolute Reltive Error % Absolute error versus API grvity Stnding corr Vsquez & Beggs corr Mrhoun corr Stnding modified Vsquez & Beggs modified Mrhoun modified 10 0 API<20 20<API<25 25<API<30 30<API<35 35<API<40 API>40 (12) (48) (177) (191) (74) (28) 21

22 Solution GOR, SCF STB Solution gs-oil rtio Solution Gs- Oil Rtio is the rtio of gs evolves from solution to oil. It is usully expressed in units of scf/stb Pressure, psi Typicl solution GOR curve 22

23 Solution gs-oil rtio. Stnding (1947) R s γ 1 g P b 2 e 2 T 4γ pi where 1 = E-3 2 = = E-3 4 = E-3 23

24 Solution gs-oil rtio.. Vsquez nd Beggs (1980) R s γ 1 g P b 2 e 3 pi ( T 460) where Coefficient pi 30 pi

25 Solution gs-oil rtio Al-Mrhoun (1988) R s 1 g b o T 5 2 P where 1 = E+3 2 = = = =

26 Sttisticl ccurcy of R s correltions ER EA E mx STD R Correltion Stnding (1947) Vsquez & Beggs (1980) Al-Mrhoun (1988) Modified Correltion Stnding (1947) Vsquez & Beggs (1980) Al-Mrhoun (1988)

27 Absolute Averge Percent Error Absolute error of R s correltions 20 Correltion Modified Correltion Stnding Vsquez& Beggs Al-Mrhoun 27

28 Avg Absolute Reltive Error % Absolute error versus API grvity Stnding corr Vsquez & Beggs corr Mrhoun corr Stnding modified Vsquez & Beggs modified Mrhoun modified 10 0 API<20 20<API<25 25<API<30 30<API<35 35<API<40 API>40 (12) (48) (177) (191) (74) (28) 28

29 Oil FVF Oil formtion volume fctor Oil Formtion Volume Fctor is the volume t reservoir conditions occupied by one stock tnk brrel of oil plus its solution gs Pressure, psi Typicl oil FVF curve 29

30 Oil formtion volume fctor. Stnding (1947) B ob 1 R [ ( / ) 2 s g o 4 T ] 3 5 where 1 = = = = =

31 Oil formtion volume fctor.. Vsquez nd Beggs (1980) B ob R s T 60 / s 2 pi g T 60 / R pi g where Coefficient pi 30 pi E E E E E E-9 31

32 Oil formtion volume fctor Al-Mrhoun (1992) B ob 1 1 R s 2 R s g / o 3 R s T 601 T 60 o 4 where 1 = E-3 2 = E-3 3 = E-6 4 = E-3 32

33 Sttisticl ccurcy of B ob correltions ER EA E mx STD R Correltion Stnding (1947) Vsquez & Beggs (1980) Al-Mrhoun (1992) Modified Correltion Stnding (1947) Vsquez & Beggs (1980) Al-Mrhoun (1992)

34 Absolute Averge Percent Error Absolute error of B ob correltions Correltion Modified Correltion Stnding Vsquez& Beggs Al-Mrhoun 34

35 Avg Absolute Reltive Error % Absolute error versus API grvity Stnding corr Vsquez & Beggs corr Mrhoun corr Stnding modified Vsquez & Beggs modified Mrhoun modified API<20 20<API<25 25<API<30 30<API<35 35<API<40 API>40 (12) (48) (177) (191) (74) (28) 35

36 Physicl trends of correltions Trend tests re to check whether the performnce of correltion follows physicl behvior or not: Trend tests on predicted vlues 36

37 Oil FVF Correltion with two equtions Modeling physicl properties with two equtions might produce non-physicl trend Stnding Mrhoun 1.25 Vsquez & Beggs Oil API Grvity 37

38 Oil FVF Correltion with non-physicl constrint Restriction of correltion model gives non-physicl trend 1.45 Stnding 1.4 Mrhoun Vsquez & Beggs Gs Reltive Density (Air=1.0) 38

39 Pb, psi Correltion with limited dt Correltion development for limited dt will give good fit, but might led to non-physicl trend Stnding Vzquez Mrhoun Dokl & Osmn Reservoir Temperture deg F 39

40 Bo, Bt Two-phse formtion volume fctor The two-phse formtion volume fctor is the volume of oil plus the volume of gs evolved converted to reservoir conditions per stock tnk brrel. B t B o P b B t =B o Reservoir Pressure Typicl totl FVF curve B t B o B g ( R R sb s ) 40

41 Totl formtion volume fctor. Stnding (1947) log F C B t log( 1 R T s 2.9x g F R s C o ) 7 6 log P where 1 = = = = = = =

42 Totl formtion volume fctor.. Glso (1980) ln B t 1 2 ln F 3 ln F 2 F R T s 4 g 5 P 6 C o C 2.9x R s where 1 = = = = = E-3 6 =

43 Totl formtion volume fctor Al-Mrhoun (1992) nd d B B ( p / p ) t 4 1 ( T ln o ob 460) 5 ( p / p 2 b ) b ln g 6 d ln( 3 p o / p b ) where 1 = E-3 4 = = = = =

44 Sttisticl ccurcy of B t correltions ER EA E mx STD R Correltion Stnding (1947) Glso (1980) Al-Mrhoun (1992) Modified Correltion Stnding (1947) Glso (1980) Al-Mrhoun (1992)

45 Absolute Averge Percent Error Absolute error of B t correltions Correltion Modified Correltion Stnding Glso Al-Mrhoun 45

46 Avg Absolute Reltive Error % Absolute error versus API grvity Glso corr Mrhoun corr Glso modified Mrhoun modified 10 0 API<20 20<API<25 25<API<30 30<API<35 35<API<40 API>40 (93) (363) (1918) (1810) (860) (294) 46

47 Isotherml oil compressibility It is defined s the unit chnge of volume with pressure t constnt temperture. C o is used in the clcultion of oil density nd FVF bove Pb s shown. c o 1 V V P T op ob e c o (PP b ) B o B ob e c o (P b P) Typicl C o curve bove P b 47

48 Isotherml oil compressibility To clculte understurted oil density or FVF bove bubble point pressure the verge oil compressibility is used C o c o p 1 p b P P b c o (p)dp To void the clcultion involved, C o cn be clculted t verge pressure P s follows: c o c where P o P ( P) 2 P b 48

49 Isotherml oil compressibility. Vsquez & Beggs (1980) C O ( 2 Rs 3 T 4 g 5 1 pi ) / p where 1 = E-3 2 = 50 E-6 3 = E-3 4 = E-3 5 = E-3 49

50 Isotherml oil compressibility.. Petrosky & Frshd (1993) C o R 1 s 2 g 3 4 pi T 5 P 6 where 1 = E-6 4 = = = = =

51 Isotherml oil compressibility Al-Mrhoun (2003) ln c / ( P P ) / /( T o 1 2 ob 3 b ob 4 nd ob ( 4 o 2.18x 10 R ) / where 1 = = = E-6 4 = s g B ob 3 460) 51

52 Sttisticl ccurcy of C o correltions ER EA E mx STD R Correltion Vsquez & Beggs (1980) Petrosky & Frshd (1993) Al-Mrhoun (2003) Modified Correltion Vsquez & Beggs (1980) Petrosky & Frshd (1993) Al-Mrhoun (2003)

53 Absolute Averge Percent Error Absolute error of C o correltions Correltion Modified Correltion Vsquez & Beggs Petrosky & Frshd Al-Mrhoun 53

54 Avg Absolute Reltive Error % Absolute error versus API grvity Vsquez & Beggs corr Petrosky & Frshd corr Vsquez & Beggs modified Petrosky & Frshd modified Mrhoun correltion 20 0 API<20 20<API<25 25<API<30 30<API<35 35<API<40 API>40 (60) (312) (1066) (1260) (535) (179) 54

55 Oil compressibility below P b It is defined s the unit chnge of volume with pressure t constnt temperture Below bubble point, volume occupied by gs evolved from the oil during differentil chnge in pressure must be tken into ccount in the clcultion of oil compressibility The defining eqution is: c o Typicl C o curve below P b 1 B o B p o T B g R p s T 55

56 Oil compressibility below P b. McCin, Rollins nd Villen (1988) ln c o 1 2 ln P 3 ln P b 4 ln T 5 ln pi 6 ln R sb where 1 = = = = = =

57 Oil compressibility below P b.. Al-Mrhoun (2009) Al-Mrhoun (2003) developed n eqution to estimte co bove Pb ln c / ( P P ) / /( T 460) 3 o 1 2 ob 3 b ob 4 This eqution cn be used for one point estimtion of co t ny sturtion pressure provided oil reltive density is correct: 4 o 2.18x10 Rs g ln c / ob ob 5 2 ob B ob 57

58 Oil compressibility below P b.. Any point below the originl P b is new sturtion pressure for new fluid of different composition, then ln c / op 5 2 op op o 2.18x10 B op 4 R s g By combining equtions, the co t sturtion pressure cn be clculted in term of co t the originl Pb nd reltive live oil densities s follows: 1 1 ln ln ( ) 2 = cop cob 2 op ob 58

59 Oil Viscosity Oil viscosity Oil viscosity is mesure of the resistnce to flow exerted by fluid. In eqution form, reltion between sher stress nd rte of ngulr deformtion of flow of fluids Four viscosity types Ded Oil Viscosity Oil Viscosity below P b Oil Viscosity t P b Oil Viscosity bove P b o dv / dy Pressure P b Typicl viscosity curve 59

60 Oil viscosity bove P b. Bel (1946) ob ( P P )( 2 4 b 1 ob 3 ob ) where 1 = 24 E-6 2 = = 38 E-6 4 =

61 Oil viscosity bove P b.. Lbedi (1992) o m p p ob b ln m ln ln p 1 2 pi 3 od 4 b where 1 = = E-3 3 = =

62 Oil viscosity bove P b Al-Mrhoun (2004) ln o ln ob 2 ob p p b nd ob ( 4 o 2.18x 10 R ) / s g B ob where α = E-3 62

63 Sttisticl ccurcy of correltions ER EA E mx STD R Correltion Bel (1946) Lbedi (1992) Al-Mrhoun (2003) Modified Correltion Bel (1946) Lbedi (1992) Al-Mrhoun (2003)

64 Absolute Averge Percent Error Absolute error of correltions Correltion Modified Correltion Bel Lbedi Al-Mrhoun 64

65 Avg Absolute Reltive Error % Absolute error versus API grvity 10 Bel corr Lbedi Corr 8 Bel modified Lbedi modified 6 Mrhoun correltion API<20 20<API<25 25<API<30 30<API<35 35<API<40 API>40 (26) (225) (727) (839) (292) (107) 65

66 Oil viscosity t P b. Chew nd Connlly (1959) ob nd od e e 3 6 R s R s where 1 = = = = = E-3 6 = E-3 66

67 Oil viscosity t P b.. Beggs nd Robinson (1975) ob where od 1 4 ( R s ( R s 2 5 ) ) 3 6 where 1 = = = = = =

68 Oil viscosity t P b Lbedi (1992) ln ob ln pi od 4 ln p b where 1 = = = =

69 Sttisticl ccurcy of ob correltions ER EA E mx STD R Correltion Chew & Connlly (1959) Beggs & Robinson (1975) Lbedi (1992) Modified Correltion Chew & Connlly (1959) Beggs & Robinson (1975) Lbedi (1992)

70 Absolute Averge Percent Error Absolute error of ob correltions 45 Correltion 48 Modified Correltion Chew & Connlly Beggs & Robinson Lbedi 70

71 Avg Absolute Reltive Error % Absolute error versus API grvity Chew & Connlly corr Beggs & Robinson corr Lbedi corr Chew & Connlly modified Beggs & Robinson modified Lbedi modified 30 0 API<20 20<API<25 25<API<30 30<API<35 35<API<40 API>40 (4) (29) (90) (115) (42) (16) 71

72 Ded oil viscosity. Beggs nd Robinson (1975) ln(ln( 1)) ln od 1 2 pi 3 T where 1 = = =

73 Ded oil viscosity.. Glso (1980) ln od 1 2 ln T 3 ln(ln pi ) 4 (ln T)ln(ln pi ) where 1 = = = =

74 Ded oil viscosity Lbedi (1992) ln ln od 1 2 pi 3 ln T where 1 = = =

75 Sttisticl ccurcy of od correltions ER EA E mx STD R Correltion Beggs & Robinson (1975) Glso (1980) Lbedi (1992) Modified Correltion Beggs & Robinson (1975) Glso (1980) Lbedi (1992)

76 Absolute Averge Percent Error Absolute error of od correltions Correltion Modified Correltion Beggs & Robinson Glso Lbedi 76

77 Avg Absolute Reltive Error % Absolute error versus API grvity Beggs & Robinson corr Glso corr Lbedi corr Beggs & Robinson modified Glso modified Lbedi modified 25 0 API<20 20<API<25 25<API<30 30<API<35 35<API<40 API>40 (4) (29) (90) (115) (42) (16) 77

78 Interfcil tension pure substnce The force exerted on the boundry lyer between liquid phse nd vpor phse per unit length Sugden (1924) P ch L v M P ch ( L M = Surfce tension for pure substnces = Prchor = Density of the liquid = Density of the vpor = Moleculr mss V ) 4 78

79 Prchor, P Interfcil tension prchors Prchor is function expressing the reltionship between the surfce tension, density, nd moleculr mss Moleculr Weight Prchors for computing interfcil tension of norml prffin hydrocrbons 79

80 Interfcil tension hydrocrbon mixture Ktz et l. (1943) 1 A 4 n o 62.4M ( Pch ) i ( Axi By i ) i1 L B g 62.4M o = density of oil phse, lb/ft 3 M L = pprent moleculr mss of oil phse g = density of gs phse, lb/ft 3 M g = pprent moleculr mss of gs phse x i = mole frction of component i in oil phse y i = mole frction of component i in gs phse n = totl number of component in the system g 80

81 References 1. Stnding, M.B.: A Pressure-Volume-Temperture Correltion for Mixtures of Cliforni Oils nd Gses, Drill. & Prod. Prct, API (1947), pp Vsquez, M.E. nd Beggs, H.D.: Correltions for Fluid Physicl Property Prediction, JPT (June 1980) Al-Mrhoun, M.A.: PVT Correltions for Middle Est Crude Oils, JPT (My 1988) Al-Mrhoun, M.A.: New Correltion for Formtion Volume Fctor of Oil nd Gs Mixtures, JCPT (Mrch 1992) Glso, O.: Generlized Pressure-Volume Temperture Correltions, JPT (My 1980), Petrosky, G.E. Jr. nd Frshd, F.F.: Pressure-Volume-Temperture Correltions for Gulf of Mexico, pper SPE 26644, presented t the 1993 SPE Annul Technicl Conference nd Exhibition, Houston, Oct Al-Mrhoun, M.A.: The Coefficient of Isotherml Compressibility of Blck Oils, pper SPE presented t the 2003 SPE Middle Est Oil Show nd Conference, Bhrin, June

82 References 8. Bel, C.: The Viscosity of Air, Wter, Nturl Gs, Crude Oil nd its Associted Gses t Oil Field Temperture nd Pressures, Trns., AIME (1946) 165, pp Lbedi, R.: Improved Correltions for Predicting the Viscosity of Light Crudes, J. Pet. Sce. Eng. (Aug. 1992) Al-Mrhoun, M.A.: "Evlution of empiriclly derived PVT properties for Middle Est crude oils, Journl of Petroleum Science nd Engineering, 42 (2004) Chew, J. nd Connlly, C.A. Jr.: A Viscosity Correltion for Gs-Sturted Crude Oils, Trns., AIME (1959) 216, pp Beggs, H.D. nd Robinson, J.R.: Estimting the Viscosity of Crude Oil System, JPT (Sept. 1980) McCin. W.D. Jr., Rollins, J.B., nd Villen. A.J.: The Coefficient of Isotherml Compressibility of Blck Oils t Pressures below the Bubblepoint, SPEFE (Sept. 1988) ; Trns., AIME Al-Mrhoun, M.A.: The Oil Compressibility below Bubble Point Pressure Revisited Formultions nd Estimtions, pper SPE presented t 16th SPE Middle Est Oil Show & Conference, Bhrin, Mrch

Psychrometric Applications

Psychrometric Applications Psychrometric Applictions The reminder of this presenttion centers on systems involving moist ir. A condensed wter phse my lso be present in such systems. The term moist irrefers to mixture of dry ir nd

More information

The International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O

The International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O IAPWS R-7 The Interntionl Assocition for the Properties of Wter nd Stem Lucerne, Sitzerlnd August 7 Relese on the Ioniztion Constnt of H O 7 The Interntionl Assocition for the Properties of Wter nd Stem

More information

Part I: Basic Concepts of Thermodynamics

Part I: Basic Concepts of Thermodynamics Prt I: Bsic Concepts o Thermodynmics Lecture 4: Kinetic Theory o Gses Kinetic Theory or rel gses 4-1 Kinetic Theory or rel gses Recll tht or rel gses: (i The volume occupied by the molecules under ordinry

More information

Rel Gses 1. Gses (N, CO ) which don t obey gs lws or gs eqution P=RT t ll pressure nd tempertures re clled rel gses.. Rel gses obey gs lws t extremely low pressure nd high temperture. Rel gses devited

More information

CHEMICAL KINETICS

CHEMICAL KINETICS CHEMICAL KINETICS Long Answer Questions: 1. Explin the following terms with suitble exmples ) Averge rte of Rection b) Slow nd Fst Rections c) Order of Rection d) Moleculrity of Rection e) Activtion Energy

More information

CHM Physical Chemistry I Chapter 1 - Supplementary Material

CHM Physical Chemistry I Chapter 1 - Supplementary Material CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion

More information

Module 2: Rate Law & Stoichiomtery (Chapter 3, Fogler)

Module 2: Rate Law & Stoichiomtery (Chapter 3, Fogler) CHE 309: Chemicl Rection Engineering Lecture-8 Module 2: Rte Lw & Stoichiomtery (Chpter 3, Fogler) Topics to be covered in tody s lecture Thermodynmics nd Kinetics Rection rtes for reversible rections

More information

Terminal Velocity and Raindrop Growth

Terminal Velocity and Raindrop Growth Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion,

More information

PETE 310. Lecture # 15 Properties of Black Oils Definitions (pages )

PETE 310. Lecture # 15 Properties of Black Oils Definitions (pages ) PETE 310 Lecture # 15 Properties of Black Oils Definitions (pages 224-240) PETROLEUM ENGINEERING 310 Please adhere strictly to the rules indicated in this test Disable your cell phone (no text messages

More information

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es

More information

Predicting the Dead Oil Viscosity of Reservoir Fluids: a case study of the Niger Delta

Predicting the Dead Oil Viscosity of Reservoir Fluids: a case study of the Niger Delta Predicting the Dead Oil Viscosity of Reservoir Fluids: a case study of the Niger Delta Abstract Kamilu Folorunsho Oyedeko 1* and Ubong William Ulaeto 1 1.Chemical & Polymer Engineering, Lagos State University,

More information

Applications of Bernoulli s theorem. Lecture - 7

Applications of Bernoulli s theorem. Lecture - 7 Applictions of Bernoulli s theorem Lecture - 7 Prcticl Applictions of Bernoulli s Theorem The Bernoulli eqution cn be pplied to gret mny situtions not just the pipe flow we hve been considering up to now.

More information

Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Lecture 33. Psychrometric Properties of Moist Air

Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Lecture 33. Psychrometric Properties of Moist Air Deprtment of Mechnicl Engineering ME 3 Mechnicl Engineering hermodynmics Lecture 33 sychrometric roperties of Moist Air Air-Wter Vpor Mixtures Atmospheric ir A binry mixture of dry ir () + ter vpor ()

More information

Simulations of the irradiation and temperature dependence of the efficiency of tandem photoelectrochemical water-splitting systems

Simulations of the irradiation and temperature dependence of the efficiency of tandem photoelectrochemical water-splitting systems SUPPORTING INFORMATION FOR Simultions of the irrdition nd temperture dependence of the efficiency of tndem photoelectrochemicl wter-splitting systems Sophi Hussener, Shu Hu, Chengxing Xing, Adm Z. Weber,

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching

More information

Estimation of Global Solar Radiation at Onitsha with Regression Analysis and Artificial Neural Network Models

Estimation of Global Solar Radiation at Onitsha with Regression Analysis and Artificial Neural Network Models eserch Journl of ecent Sciences ISSN 77-5 es.j.ecent Sci. Estimtion of Globl Solr dition t Onitsh with egression Anlysis nd Artificil Neurl Network Models Abstrct Agbo G.A., Ibeh G.F. *nd Ekpe J.E. Fculty

More information

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

More information

4. CHEMICAL KINETICS

4. CHEMICAL KINETICS 4. CHEMICAL KINETICS Synopsis: The study of rtes of chemicl rections mechnisms nd fctors ffecting rtes of rections is clled chemicl kinetics. Spontneous chemicl rection mens, the rection which occurs on

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

MAT187H1F Lec0101 Burbulla

MAT187H1F Lec0101 Burbulla Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is

More information

Predict Global Earth Temperature using Linier Regression

Predict Global Earth Temperature using Linier Regression Predict Globl Erth Temperture using Linier Regression Edwin Swndi Sijbt (23516012) Progrm Studi Mgister Informtik Sekolh Teknik Elektro dn Informtik ITB Jl. Gnesh 10 Bndung 40132, Indonesi 23516012@std.stei.itb.c.id

More information

First Law of Thermodynamics. Control Mass (Closed System) Conservation of Mass. Conservation of Energy

First Law of Thermodynamics. Control Mass (Closed System) Conservation of Mass. Conservation of Energy First w of hermodynmics Reding Problems 3-3-7 3-0, 3-5, 3-05 5-5- 5-8, 5-5, 5-9, 5-37, 5-0, 5-, 5-63, 5-7, 5-8, 5-09 6-6-5 6-, 6-5, 6-60, 6-80, 6-9, 6-, 6-68, 6-73 Control Mss (Closed System) In this section

More information

Effects of dry density on soil water characteristic curve of clay

Effects of dry density on soil water characteristic curve of clay 5th Interntionl Conference on Civil, Architecturl nd Hydrulic Engineering (ICCAHE 2016) Effects of dry density on soil wter chrcteristic curve of cly Hu Mengling, byo Hilin, cren Jinxi School of Architecture

More information

The Thermodynamics of Aqueous Electrolyte Solutions

The Thermodynamics of Aqueous Electrolyte Solutions 18 The Thermodynmics of Aqueous Electrolyte Solutions As discussed in Chpter 10, when slt is dissolved in wter or in other pproprite solvent, the molecules dissocite into ions. In queous solutions, strong

More information

The Properties of Stars

The Properties of Stars 10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product

More information

HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016

HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016 HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M

More information

1. a) Describe the principle characteristics and uses of the following types of PV cell: Single Crystal Silicon Poly Crystal Silicon

1. a) Describe the principle characteristics and uses of the following types of PV cell: Single Crystal Silicon Poly Crystal Silicon 2001 1. ) Describe the principle chrcteristics nd uses of the following types of PV cell: Single Crystl Silicon Poly Crystl Silicon Amorphous Silicon CIS/CIGS Gllium Arsenide Multijunction (12 mrks) b)

More information

DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE

DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE Determintion RevAdvMterSci of mechnicl 0(009) -7 properties of nnostructures with complex crystl lttice using DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING

More information

13: Diffusion in 2 Energy Groups

13: Diffusion in 2 Energy Groups 3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

More information

3.1 Exponential Functions and Their Graphs

3.1 Exponential Functions and Their Graphs . Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

More information

Driving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d

Driving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d Interntionl Industril Informtics nd Computer Engineering Conference (IIICEC 15) Driving Cycle Construction of City Rod for Hybrid Bus Bsed on Mrkov Process Deng Pn1,, Fengchun Sun1,b*, Hongwen He1, c,

More information

The Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases.

The Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases. Section 1 Section 2 The module clcultes nd plots isotherml 1-, 2- nd 3-metl predominnce re digrms. ccesses only compound dtbses. Tble of Contents Tble of Contents Opening the module Section 3 Stoichiometric

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

Conservation Law. Chapter Goal. 5.2 Theory

Conservation Law. Chapter Goal. 5.2 Theory Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very

More information

Determination of the activation energy of silicone rubbers using different kinetic analysis methods

Determination of the activation energy of silicone rubbers using different kinetic analysis methods Determintion of the ctivtion energy of silicone rubbers using different kinetic nlysis methods OU Huibin SAHLI ohmed BAIEE Thierry nd GELIN Jen-Clude FETO-ST Institute / Applied echnics Deprtment, 2 rue

More information

!"#$ Reservoir Fluid Properties. State of the Art and Outlook for Future Development. Dr. Muhammad Al-Marhoun

!#$ Reservoir Fluid Properties. State of the Art and Outlook for Future Development. Dr. Muhammad Al-Marhoun Society of Petroleum Engineers SPE 2001 2002 Distinguished Lecturer Program 4 July 2002 Reservoir Fluid Properties State of the Art and Outlook for Future Development Dr. Muhammad Al-Marhoun King Fahd

More information

INTRODUCTION. The three general approaches to the solution of kinetics problems are:

INTRODUCTION. The three general approaches to the solution of kinetics problems are: INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information

Travelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing

Travelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing Applied Mthemtics E-Notes 8(8) - c IN 67-5 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge

More information

G. MATEESCU 1 A. MATEESCU 1 C. SAMOILĂ 2 D. URSUŢIU 2

G. MATEESCU 1 A. MATEESCU 1 C. SAMOILĂ 2 D. URSUŢIU 2 PRELIMINARY EXPERIMENTS OF THE NEW FACILITY AND TECHNOLOGY FOR VACUUM DRYING AND THERMAL POLIMERIZATION OF THE TURBOGENERATORS STATOR BARS INSULATION (INTEPOL) G. MATEESCU 1 A. MATEESCU 1 C. SAMOILĂ 2

More information

SPE Improved Permeability Prediction Relations for Low Permeability Sands

SPE Improved Permeability Prediction Relations for Low Permeability Sands SPE 07954 Imroved Permebility Prediction Reltions for Low Permebility Snds Frncois-Andre Florence, Texs A&M University T.A. Blsingme, Texs A&M University Dertment of Petroleum Engineering Texs A&M University

More information

Intro to Nuclear and Particle Physics (5110)

Intro to Nuclear and Particle Physics (5110) Intro to Nucler nd Prticle Physics (5110) Feb, 009 The Nucler Mss Spectrum The Liquid Drop Model //009 1 E(MeV) n n(n-1)/ E/[ n(n-1)/] (MeV/pir) 1 C 16 O 0 Ne 4 Mg 7.7 14.44 19.17 8.48 4 5 6 6 10 15.4.41

More information

MULTIPHASE, MULTICOMPONENT COMPRESSIBILITY IN GEOTHERMAL RESERVOIR ENGINEERING. L. Macias-Chapa and H. J. Ramey, Jr.

MULTIPHASE, MULTICOMPONENT COMPRESSIBILITY IN GEOTHERMAL RESERVOIR ENGINEERING. L. Macias-Chapa and H. J. Ramey, Jr. PROCEEDNGS. Twelfth Workshop on Geotherml Reservoir Engineering Stnford University, Stnford, Cliforni. Jnury 2022, 1987 SGPTR109 MULTPHASE, MULTCOMPONENT COMPRESSBLTY N GEOTHERMAL RESERVOR ENGNEERNG L.

More information

MATH SS124 Sec 39 Concepts summary with examples

MATH SS124 Sec 39 Concepts summary with examples This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples

More information

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation 1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview

More information

Research on the Quality Competence in Manufacturing Industry

Research on the Quality Competence in Manufacturing Industry Reserch on the Qulity Competence in Mnufcturing Industry Xioping M, Zhijun Hn Economics nd Mngement School Nnjing University of Science nd Technology Nnjing 210094, Chin Tel: 86-25-8431-5400 E-mil: hnzhij4531@sin.com

More information

PVT Concepts (Reservoir Fluids)

PVT Concepts (Reservoir Fluids) PVT Concepts (Reservoir Fluids) Thomas A. Blasingame, Ph.D., P.E. Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116 (USA) +1.979.845.2292 t-blasingame@tamu.edu Orientation

More information

CBE 291b - Computation And Optimization For Engineers

CBE 291b - Computation And Optimization For Engineers The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b - Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn

More information

Heat flux and total heat

Heat flux and total heat Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce

More information

MATH 144: Business Calculus Final Review

MATH 144: Business Calculus Final Review MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

More information

Effect of soil profile modulus distribution on pile head lateral stiffness

Effect of soil profile modulus distribution on pile head lateral stiffness Proc. 18 th NZGS Geotechnicl Symposium on Soil-Structure Interction. d. CY Chin, Aucklnd Michel Pender University of Aucklnd, New Zelnd eywords: pile hed stiffness, effect of pile shft size, soil modulus

More information

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of

More information

Acceptance Sampling by Attributes

Acceptance Sampling by Attributes Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

More information

Measuring Electron Work Function in Metal

Measuring Electron Work Function in Metal n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd

More information

Vadose Zone Hydrology

Vadose Zone Hydrology Objectives Vdose Zone Hydrology 1. Review bsic concepts nd terminology of soil physics. 2. Understnd the role of wter-tble dynmics in GW-SW interction. Drcy s lw is useful in region A. Some knowledge of

More information

Tests for the Ratio of Two Poisson Rates

Tests for the Ratio of Two Poisson Rates Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson

More information

1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1.2. Linear Variable Coefficient Equations. y + b ! = a y + b  Remark: The case b = 0 and a non-constant can be solved with the same idea as above. 1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

More information

UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2007

UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2007 EXAMINATIONS: SUBJECT, COURSE AND CODE: HYDROLOGY 20 DURATION: HOURS TOTAL MARKS: 00 Internl Exminer : Ms ML Wrburton : Prof RE Schulze : Ms KT Chetty : Mr MJC Horn Externl Exminer : Prof PJT Roberts STUDENTS

More information

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory 2. Diffusion-Controlled Reaction

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory 2. Diffusion-Controlled Reaction Ch. 4 Moleculr Rection Dynmics 1. Collision Theory. Diffusion-Controlle Rection Lecture 17 3. The Mteril Blnce Eqution 4. Trnsition Stte Theory: The Eyring Eqution 5. Trnsition Stte Theory: Thermoynmic

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More information

13.4 Work done by Constant Forces

13.4 Work done by Constant Forces 13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

A027 Uncertainties in Local Anisotropy Estimation from Multi-offset VSP Data

A027 Uncertainties in Local Anisotropy Estimation from Multi-offset VSP Data A07 Uncertinties in Locl Anisotropy Estimtion from Multi-offset VSP Dt M. Asghrzdeh* (Curtin University), A. Bon (Curtin University), R. Pevzner (Curtin University), M. Urosevic (Curtin University) & B.

More information

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present? University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the three-phse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the

More information

ME 309 Fluid Mechanics Fall 2006 Solutions to Exam3. (ME309_Fa2006_soln3 Solutions to Exam 3)

ME 309 Fluid Mechanics Fall 2006 Solutions to Exam3. (ME309_Fa2006_soln3 Solutions to Exam 3) Fll 6 Solutions to Exm3 (ME39_F6_soln3 Solutions to Exm 3) Fll 6. ( pts totl) Unidirectionl Flow in Tringulr Duct (A Multiple-Choice Problem) We revisit n old friend, the duct with n equilterl-tringle

More information

Math Calculus with Analytic Geometry II

Math Calculus with Analytic Geometry II orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

More information

Department of Physical Pharmacy and Pharmacokinetics Poznań University of Medical Sciences Pharmacokinetics laboratory

Department of Physical Pharmacy and Pharmacokinetics Poznań University of Medical Sciences Pharmacokinetics laboratory Deprtment of Physicl Phrmcy nd Phrmcoinetics Poznń University of Medicl Sciences Phrmcoinetics lbortory Experiment 1 Phrmcoinetics of ibuprofen s n exmple of the first-order inetics in n open one-comprtment

More information

1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x

1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x I. Dierentition. ) Rules. *product rule, quotient rule, chin rule MATH 34B FINAL REVIEW. Find the derivtive of the following functions. ) f(x) = 2 + 3x x 3 b) f(x) = (5 2x) 8 c) f(x) = e2x 4x 7 +x+2 d)

More information

Time Truncated Two Stage Group Sampling Plan For Various Distributions

Time Truncated Two Stage Group Sampling Plan For Various Distributions Time Truncted Two Stge Group Smpling Pln For Vrious Distributions Dr. A. R. Sudmni Rmswmy, S.Jysri Associte Professor, Deprtment of Mthemtics, Avinshilingm University, Coimbtore Assistnt professor, Deprtment

More information

LECTURE 14. Dr. Teresa D. Golden University of North Texas Department of Chemistry

LECTURE 14. Dr. Teresa D. Golden University of North Texas Department of Chemistry LECTURE 14 Dr. Teres D. Golden University of North Texs Deprtment of Chemistry Quntittive Methods A. Quntittive Phse Anlysis Qulittive D phses by comprison with stndrd ptterns. Estimte of proportions of

More information

AP Calculus. Fundamental Theorem of Calculus

AP Calculus. Fundamental Theorem of Calculus AP Clculus Fundmentl Theorem of Clculus Student Hndout 16 17 EDITION Click on the following link or scn the QR code to complete the evlution for the Study Session https://www.surveymonkey.com/r/s_sss Copyright

More information

Chapter 13 Lyes KADEM [Thermodynamics II] 2007

Chapter 13 Lyes KADEM [Thermodynamics II] 2007 Gs-Vpor Mixtures Air is mixture of nitrogen nd oxygen nd rgon plus trces of some other gses. When wtervpor is not included, we refer to it s dry ir. If wter-vpor is included, we must properly ccount for

More information

Modelling of chemical vapour deposition of carbon based on detailed surface chemistry

Modelling of chemical vapour deposition of carbon based on detailed surface chemistry Modelling of chemicl vpour deposition of cron sed on detiled surfce chemistry Aijun Li *, Koyo Noring, Günter Schoch, Sven Lichtenerg, Olf Deutschmnn Institut für Technische Chemie und Polymerchemie, Universität

More information

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available.

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available. DEPARTMENT OF PHYSICS AND ASTRONOMY Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner grph pper is vilble. Spring Semester 2015-2016 PHYSICS 1 HOUR Answer questions

More information

On the Uncertainty of Sensors Based on Magnetic Effects. E. Hristoforou, E. Kayafas, A. Ktena, DM Kepaptsoglou

On the Uncertainty of Sensors Based on Magnetic Effects. E. Hristoforou, E. Kayafas, A. Ktena, DM Kepaptsoglou On the Uncertinty of Sensors Bsed on Mgnetic Effects E. ristoforou, E. Kyfs, A. Kten, DM Kepptsoglou Ntionl Technicl University of Athens, Zogrfou Cmpus, Athens 1578, Greece Tel: +3177178, Fx: +3177119,

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Carbon foam impregnated with Phase Change Material (PCM) as a thermal barrier

Carbon foam impregnated with Phase Change Material (PCM) as a thermal barrier Crbon fom impregnted with Phse Chnge Mteril (PCM) s therml brrier Osm Meslhy 1, Khlid Lfdi 1,2, nd Ahmed Elgfy 1 1 University of Dyton Reserch Institute, 300 College Prk, Dyton OH. 45469 USA 2 AFRL/MLBC,

More information

Damage of Houses and Residential Areas by Niigata Prefecture Earthquakes (Part2)

Damage of Houses and Residential Areas by Niigata Prefecture Earthquakes (Part2) Proceedings of the Eighteenth () Interntionl Offshore nd Polr Engineering Conference Vncouver, BC, Cnd, July -11, Copyright by The Interntionl Society of Offshore nd Polr Engineers (ISOPE) ISBN 97-1-53-7-

More information

4 The dynamical FRW universe

4 The dynamical FRW universe 4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which

More information

Math 116 Final Exam April 26, 2013

Math 116 Final Exam April 26, 2013 Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems

More information

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht

More information

Improved Dead Oil Viscosity Model

Improved Dead Oil Viscosity Model Improved Dead Oil Viscosity Model Ulaeto Ubong and Oyedeko K.F.K Department of Chemical & Polymer Engineering, Lagos State University Abstract This research work developed two new dead oil viscosity correlations:

More information

Sample Problems for the Final of Math 121, Fall, 2005

Sample Problems for the Final of Math 121, Fall, 2005 Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.

More information

INVESTIGATION OF BURSA, ESKIKARAAGAC USING VERTICAL ELECTRICAL SOUNDING METHOD

INVESTIGATION OF BURSA, ESKIKARAAGAC USING VERTICAL ELECTRICAL SOUNDING METHOD INVESTIGATION OF BURSA, ESKIKARAAGAC USING VERTICAL ELECTRICAL SOUNDING METHOD Gökçen ERYILMAZ TÜRKKAN, Serdr KORKMAZ Uludg University, Civil Engineering Deprtment, Burs, Turkey geryilmz@uludg.edu.tr,

More information

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C. A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

More information

du = C dy = 1 dy = dy W is invertible with inverse U, so that y = W(t) is exactly the same thing as t = U(y),

du = C dy = 1 dy = dy W is invertible with inverse U, so that y = W(t) is exactly the same thing as t = U(y), 29. Differentil equtions. The conceptul bsis of llometr Did it occur to ou in Lecture 3 wh Fiboncci would even cre how rpidl rbbit popultion grows? Mbe he wnted to et the rbbits. If so, then he would be

More information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

More information

The Atwood Machine OBJECTIVE INTRODUCTION APPARATUS THEORY

The Atwood Machine OBJECTIVE INTRODUCTION APPARATUS THEORY The Atwood Mchine OBJECTIVE To derive the ening of Newton's second lw of otion s it pplies to the Atwood chine. To explin how ss iblnce cn led to the ccelertion of the syste. To deterine the ccelertion

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

Mechanism of Roughness-induced CO 2 Microbubble Nucleation in Polypropylene Foaming

Mechanism of Roughness-induced CO 2 Microbubble Nucleation in Polypropylene Foaming Electronic Supplementry Mteril (ESI) for Physicl Chemistry Chemicl Physics. This journl is the Owner Societies 2017 Supporting Informtion Mechnism of Roughness-induced CO 2 Microbubble Nucletion in Polypropylene

More information

Shear Degradation and Possible viscoelastic properties of High Molecular Weight Oil Drag Reducer Polymers

Shear Degradation and Possible viscoelastic properties of High Molecular Weight Oil Drag Reducer Polymers ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 3, 2005 Sher Degrdtion nd Possible viscoelstic properties of High Moleculr Weight Oil Drg Reducer Polymers A.A. Hmoud, C. Elissen, C. Idsøe nd T.

More information

Population Dynamics Definition Model A model is defined as a physical representation of any natural phenomena Example: 1. A miniature building model.

Population Dynamics Definition Model A model is defined as a physical representation of any natural phenomena Example: 1. A miniature building model. Popultion Dynmics Definition Model A model is defined s physicl representtion of ny nturl phenomen Exmple: 1. A miniture building model. 2. A children cycle prk depicting the trffic signls 3. Disply of

More information

COMPARISON OF DIFFERENT PROCEDURES TO PREDICT UNSATURATED SOIL SHEAR STRENGTH. S.K. Vanapalli and D.G. Fredlund 1

COMPARISON OF DIFFERENT PROCEDURES TO PREDICT UNSATURATED SOIL SHEAR STRENGTH. S.K. Vanapalli and D.G. Fredlund 1 COMPARISON OF DIFFERENT PROCEDURES TO PREDICT UNSATURATED SOIL SHEAR STRENGTH S.K. Vnplli nd D.G. Fredlund 1 Abstrct: Severl procedures hve been proposed in the recent yers to predict the sher strength

More information

MATHEMATICAL SIMULATION OF GEOTHERMAL HEAT TRANSFER IN HOT DRY ROCK UNDERGROUND HEAT EXCHANGERS

MATHEMATICAL SIMULATION OF GEOTHERMAL HEAT TRANSFER IN HOT DRY ROCK UNDERGROUND HEAT EXCHANGERS SCIENIFIC PROCEEDINGS 009, Fculty of Mechnicl Engineering, SU in Brtislv MAHEMAICAL SIMULAION OF GEOHERMAL HEA RANSFER IN HO DRY ROCK UNDERGROUND HEA EXCHANGERS Kristín KÁZMÉROVÁ, Michl MASARYK Slovk University

More information

Vadose Zone Hydrology

Vadose Zone Hydrology Objectives Vdose Zone Hydrology. Review bsic concepts nd terminology of soil physics. 2. Understnd the role of wter-tble dynmics in GW-SW interction. Wter storge in unsturted soil Minerl surfces hve uneven

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information