Report 1994:17 (English)

Equations and formulas for air and aircontaminants

A literature review

Lars Olander

ii

ForewordThis publication has been compiled because I have missed such a publication. Naturallythis means that the selection has been influenced by my views. However, in most cases Ihave included a formula rather than exclude it. To make the number of pages limitedthe explanations are as short as possible. The result is that this formula compilation cannot be used as a text book but only as a reference book or as a guideline to the literaturein one specific field.

For this, the third edition (in English) dr. techn. Y. Jin has done a lot of work to checkand complete formulas and literature references. I have not had the opportunity to makethe formulas available in a form direct usable for computers. To make calculations it isnecessary to transfer actual formula to a suitable program. To facilitate using andsearching in this compilation a diskette with all the formulas is included. The formulasare there written in WordPerfect 5.1 (DOS/Windows).

This revised version has been transferred to Word (Microsoft © Word 97), probably theEquation Editor must be installed to read the formulas. Since the equation editors inWord and WordPerfect do not agree on how to treat different symbols, there could besome difficulties with differentiating some symbols.

iii

Contents

Page nrIntroduction 1

Books of tables and reference books 3Text books 4

1 Properties of air and water vapor 71.1 The Ideal-Gas Law 81.2 Beattie-Bridgeman equation of state 91.3 Specific heat capacity for ideal gases 91.4 Temperature variation of specific heat capacity, viscosity and diffusion

coefficient 91.5 Air properties (at 100 kPa) 111.6 Water properties (at 100 kPa) 121.7 van der Waal's equation of state for water vapor 121.8 Water vapor pressure between 275 and 647 K 131.9 Mixture of air and water vapor: Density and vapor pressure 131.10 Specific enthalpy of air 151.11 Calculation of humid air's properties 151.12 Psychrometer formula 161.13 Air pressure variation with height 161.14 Connections between sight lenght and contaminant concentration 17

2 Basic flow equations 182.1 Navier - Stokes' equation 182.2 Euler's equations for frictionless flow 192.3 Bernoulli's equation 192.4 Equation of continuity 202.5 Dimensionsless numbers 21

3 Flow generation 263.1 Theoretical total pressure change for fans 263.2 Flow variations for fans 273.3 Pressure variations for fans 273.4 Power dependence for fans 273.5 Efficiency for fans 273.6 Air flow rate through critical orifice 273.7 Temperature increase of air in fans and ducts 29

4 Equations for pipe flow 304.1 General equations 31

iv

4.2 Laminar flow (Re <2300), smooth pipes 324.3 Turbulent flow (Re <80000), smooth pipes 324.4 Turbulent flow, smooth pipes, Prandtl's universal velocity distribution 324.5 Turbulent flow, rough pipe 334.6 Turbulent flow, transition smooth - rough pipes after Colebrook 334.7 Velocity distribution, laminar flow 344.8 Velocity distribution, turbulent flow 344.9 Cirkular and rectangular ducts with identical properties 354.10 Boundary layer thickness for pipe flow 354.11 Heat transfer in pipes 364.12 Deposition of particles in ducts with turbulent flow 384.13 Leakage from flexible ducts 384.14 Leakage from ducts 384.15 Ducts not made of sheet metal 39

5 Measuring 405.1 Pitot tube use in circular ducts 415.2 Orifice plate 425.3 Correction for pressure drop when measuring flow rate through terminal device435.4 Rotameter 435.5 Bag method 445.6 Tracer gas measurements 455.7 Kata thermometer 465.8 Hot wire anemometer 475.9 Sampling of aerosols in ducts 475.10 Sampling of aerosols in calm air 485.11 Tests of laboratory fume hoods 49

6 Air jets 516.1 Circular, isothermal free jets' velocity distribution and flow rate - after Baturin526.2 Circular, isothermal free jets' velocity distribution and flow rate - after Nielsen526.3 Plane, isothermal free jets' velocity distribution and flow rate - after Baturin 536.4 Plane, isothermal free jets' velocity distribution and flow rate - after Nielsen 546.5 Circular (radial) isothermal jets' velocity distribution - after Nielsen 546.6 Flow rate in jets - after Eck 556.7 Velocity distribution across plane, isothermal free jet 566.8 Velocity distribution across circular and plane isothermal free jet 566.9 Temperature and concentration distribution along and across free jets 566.10 Temperature distribution along circular free jet - after Baturin 576.11 Temperature distribution along plane jet 586.12 Temperature change for cold, free jet 586.13 Velocity and temperature decrease for vertical rising jets and bouyant plumes 586.14 Vertical air jets' throw lengths in room and horisontal jets' change of height 60

7 Contaminant generation 627.1 Solubility of gases in liquids 667.2 Evaporation from horisontal surfaces 68

v

7.3 Evaporation from liquid baths 717.4 Evaporation from water baths 717.5 Evaporation from water surfaces 727.6 Evaporation from surfaces 737.7 Evaporation from open vessels 747.8 Leakage from vessels and pipes under pressure 757.9 Heat generation from electrical motors to the surroundings 757.10 Fibre generation from new fibre filters 767.11 Ozon generation from electrostatic filters 767.12 Corrosion of ducts 777.13 Vaporization of additives from plastic folie (PVC) 777.14 Vaporization of F-11 from polyurethan plates 777.15 Grinding machines 777.16 Falling powders' dust generation 797.17 Air flow generated by falling powder 807.18 Dust generation from pressure vessels containing powder 807.19 Particle generation from gas shielded welding 817.20 Airborne droplets from release of liquids under pressure 817.21 Vaporization of oil spill 817.22 Vaporization of organic solvents from water surfaces 837.23 Evaporation of solvents 847.24 Evaporation of liquid spills 948 Heat and contaminants from man 958.1 Man's heat balance 968.2 Fanger's comfort equation 968.3 Perception of thermal climate 978.4 Perception of heat 988.5 Perception of draught 998.6 Particle generation from man 1008.7 Contaminant generation from man 1018.8 Heat losses at low temperatures 101

9 Contaminants 1029.1 Contaminants in rooms 1039.2 Particle deposition on surfaces 1039.3 Ozone in rooms 1049.4 Resuspension 1049.5 Permeability of water vapor through color layers 1049.6 Heights of welding plumes in stable conditions with temperature gradient 1059.7 Life-times for water drops 1069.8 Vaporization of drops in air 1069.9 Diesel exhausts in mines 107

10 Contaminant concentrations - air flow rates in rooms 10810.1 Ideal steady-state, total mixing 11010.2 Time dependent total mixing 11010.3 Correction for non-ideal mixing 110

vi

10.4 Time dependent total mixing with incoming concentration 11010.5 Ozone in room with copying machines 11010.6 Concentration in rooms of Radon (222) 11110.7 Concentration in rooms of Thoron (Rn 220) 11110.8 Radon concentration in room 11210.9 Time dependent contaminant generation 11210.10 Ideal mixing, separate recirculation system, separate local exhaust with outlet

outside the room and with capture efficiency α 11310.11 Age of air 11510.12 Tracer gas measurements 11510.13 Ventilation efficiency, definition 11610.14 Air exchange efficiency, definition 11610.15 Transport efficiency for air cooling systems - ATF 116

11 Contaminant concentrations - air flow with recirculation 11711.1 Central recirculation, total mixing 11811.2 Central recirculation, total mixing, steady-state 11811.3 Local recirculation (local exhaust), total mixing 11811.4 Local recirculation, total mixing 11911.5 Local recirculation, staedy-state, total mixing 11911.6 Local recirculation, steady-state, total mixing 11911.7 Local and central recirculation, steady-state 11911.8 Central recirculation 12011.9 Contaminant concentration in occupied zone and from room air cleaner at recirculation 12011.10 2-zone model with local exhaust and with leakage and recirculation 12111.11 Room air cleaner's effect on contaminant concentration 121

12 Leakage: Flow rates and concentrations 12212.1 Wind pressure for free flow 12212.2 General infiltration equation 12312.3 Theoretical natural draught 12312.4 Flow rates from thermal differences 12412.5 Air lock, steady-state concentration, total mixing 12412.6 Contaminant concentration, recirculation and leakage, total mixing 12512.7 Contaminant concentration, leakage, total mixing 127

13 Properties of mixtures 12913.1 General equation for density of mixtures 12913.2 Density of vapor-air-mixture 13013.3 Concentration conversion 13013.4 Viscosity of mixtures 13113.5 Viscosity of vapor-air at different temperatures 13113.6 Ions on particles at steady-state 13213.7 Ions in air 13213.8 Energy from electrostatic discharges 133

vii

14 Convective flow rates and velocities 13414.1 Criteria for draught (Rydberg) 13514.2 Cold draught from windows 13514.3 Air velocities from cold draught 13614.4 Heat exchange between room air and surfaces I 13714.5 Heat exchange between room air and surfaces II 13814.6 Heat exchange between room air and vertical, plane surfaces 13814.7 Heat transfer from different surfaces through bouyancy 13814.8 Natural ventilation 14014.9 Influence of wind velocity and temperature difference on natural ventilation 14114.10 Air velocity in plume above point heat source 14114.11 Air flow rate in plume above point heat source 14114.12 Air flow rate in plumes above hot sources 14214.13 Air flow rate at upper edge of vertical surface of hot body 14214.14 Air flow rate into hood close above heat source 14314.15 Air flow rate into hood above heat source 14414.16 Air flow rate above horizontal surface 144

15 Local exhausts 14515.1 Velocity distribution from a point sink 14615.2 Centerline velocity for tube end (free hood), circular or rectangular with a

length-width ratio less than 5 14615.3 Centerline velocity from a flanged circular hood 14715.4 Centerline velocity from a slot (aspect ratio larger than 5) 14715.5 Centerline velocity from a flanged slot 14715.6 Capturing hood above bath 14815.7 Capture efficiency 14815.8 Capture efficiency for hood above emission source 14815.9 Exhaust flow rate for contained process with heat generation 14915.10 Push-pull-system for surface treatment 14915.11 Push-pull-system 150

16 Air cleaning 15116.1 Filtration efficiency for fibrous filters 15216.2 Filtration efficiency for electrostatic filters 15416.3 Efficiency for cyclones 15516.4 Efficiency for venturi precipitators 15716.5 Efficiency for wet scrubbers 15816.6 Efficiency for settlement chambers 16016.7 Pressure loss in fibrous filter 16116.8 Costs for fibrous filters 16316.9 Absorption of gases in moving drops 16316.10 Adsorption of gases in materials 16416.11 Cleaning from gases by condensation 16416.12 Break-through for solvents in breathing masks 16417 Outside dispersion 16617.1 Gaussian plume model 167

viii

17.2 Exhausts of hot gases from smoke stacks (Oak Ridge model) 16817.3 Exhausts of cold gases from smoke stacks (Sutton's model) 16817.4 Concentration from exhaust of cold gas from smoke stacks 16817.5 Lowest exhaust height 16917.6 Demands on dilution of exhausts from buildings 16917.7 Exhausts from roof or on lee-side 17017.8 Concentrations from exhausts 17117.9 Velocity and concentration distribution for bouyant plumes in homogenous

surroundings 17117.10 Particle transport in convection plumes 17217.11 Dispersion of traffic contaminants 17417.12 Road tunnel ventilation 175

18 Summary 176

1

Introduction

The introduction has not been translated since it only dscribes the reason for thecompilation of these formulas. It also describes why certain areas are not covered in thisreport. Since the introduction is included in the Swedish version, anyone interestedcould look there.

Books of tables and reference books

Arbeitsmappe Heizung, Lüftung, Klimatechnik. Düsseldorf, VDI-Verlag 1968-1971.

ASHRAE Handbook 1986-1989 (4 vol.). Fundametals, Equipment, HVAC Systems andApplications, Refrigeration (SI-edition) American Society of Heating, Refrigerating and Air-Conditioning Engineer. New York 1986-1989.

Charlesworth PS: Air exchange rate and airtightness measurement techniques - An applicationguide. International Energy Agency, Air Infiltration and Ventilation Centre, Coventry, 1988.

Dittes W, Goettling O, Wolf H: Arbeitsplatzluftreinhaltung. Schriftenreihe der Bundesanstaltfür Arbeitsschutz, Fb Nr 438, Dortmund 1985.

Industrial Ventilation. A manual of recommended practice. 21th edition. AmericanConference of Governmental Industrial Hygienists. 1992.

Liddament MW: Air infiltration and calculation techniques - An application guide.International Energy Agency, Air Infiltration and Ventilation Centre, Coventry, 1986.

Lide DR (Ed.): CRC Handbook of Chemistry and Physics. 73th edition. 1993.

Perry RH, Green DW and Maloney JO (Eds): Perry's Chemical Engineer's Handbook. 6thedition. McGraw Hill, New York 1984.

Rietschel/Raiss: Heiz- und Klimatechnik. 15:e upplagan. Springer-Verlag, Berlin. ErsterBand: Grundlagen, Systeme, Ausführung. 1968. Zweiter Band: Verfahren und Unterlagen zurBerechnung. 1970.

Rohsenow WM and Hartnett JP (Ed.): Handbook of Heat Transfer, McGraw Hill, New York1973.

Teasler R: Klimatdata för Sverige. Statens råd för byggnadsforskning. 1972.

VVS-handboken, Tabeller och diagram. Förlags AB VVS. Stockholm 1974.

Text books

2

Basic theory and measurements

Beckwiht TG, Marangoni RD, and Lienhard JH: Mechanical Measurements (5th edition),Addison-Wesley, New York 1993.

Bird RB, Steward WE and Lightfood EN: Transport Phenomena, Wiley & Sons, New York1960.

Doebelin EO: Measurement systems - Application and design. McGraw-Hill, New York,1966.

Schlichting H: Boundary-Layer Therory (7th editon), McGraw Hill, New York 1979.

Ventilation and contaminant control

Alden L, Kane JM: Design of Industrial Ventilation Systems. 5th edition. Industrial Press,New York 1982.

Awbi HB: Ventilation of Buildings. Chapmann & Hall, London 1991.

Baturin VV: Fundamentals of Industrial Ventilation. Pergamon Press 1972.

British Occupational Hygiene Society, Working Group on Ventilation Design: ControllingAirborne Contaminants in the Workplace. B.O.H.S. Technical Guide No 7, Science ReviewsLtd, Leeds 1987.

Brown RC: Air filtration. Pergamon Press, Oxford 1993.

Davies CN: Air filtration. Academic Press, London 1973.

Burgess WA, Ellenbecker MJ, Treitman RD: Ventilation for control of the work environment.John Wiley & Sons, New York 1989.

Dorman RG: Dust Control and Air Cleaning. Pergamon Press, Oxford 1974.

Goodfellow HD: Advanced Design of Ventilation Systems for Contaminant Control.Chemical Engineering Monographs Vol. 23. Elsevier, Amsterdam 1985.

Goodfellow, H.D. (Ed.): Ventilation '85. Proceedings of the 1st International Symposium onVentilation for Contaminant Control, October 1-3 1985, Toronto, Canada. Elsevier,Amsterdam 1986

Heinsohn RJ: Industrial Ventilation. John Wiley & Sons, New York 1991.

Hemeon WCL: Plant and process ventilation. The Industrial Press, New York 1963.

Hughes, R.T., Goodfellow, H.D., Rajhans, G.S. (Eds): Ventilation '91. Proceedings of the 3rdInternational Symposium on Ventilation for Contaminant Control, September 16-20, 1991,

3

Cincinnati, Ohio, USA. American Conference of Governmental Industrial Hygienists,Cincinnati, Ohio USA 1993.

Jansson, A., Olander, L. (Eds): Ventilation '94. Proceedings of the 4th InternationalSymposium on Ventilation for Contaminant Control, held in Stockholm, September 5-9,1994. Arbete och HΣlsa 1994:18 (2 vols). National Institute of Occupational Health, Solna,Sweden 1994.

Licht W: Air Pollution Control Engineering. 2nd Ed. Marcel Dekker, New York 1988.

McDermott HJ: Handbook of Ventilation for Contaminant Control. Ann Arbor Science 1976.

McQuiston FC and Parker JD: Heating, Ventilating and Air Conditioning - Analysis andDesign (3rd edition), John Wiley & Sons, News York 1988.

Mⁿrmann H: Lufttechnische Anlagen für gewerbliche Betriebe. Carl Marhold, Berlin 1980.

Olander L: Ventilation. Studentlitteratur 1982.

Vincent, J.H. (Ed.): Ventilation '88. Proceedings of the 2d International Symposium onVentilation for Contaminant Control, 20-23 September 1988, London, England, UK.Pergamon Press, Oxford 1989

Aerosols

A bibliography of Aerosol Science and Technology. Aerosol Science and Technology, vol.14,sid 1-4, 1991.

Aerosol Measurement Workshop: Aerosol Measurement. Eds. Lundgren, Harris, Marlow,Lippmann, Clark, Durham. University Presses of Florida 1979.

Calvert S, Englund HM (Eds): Handbook of Air Pollution Technology, John Wiley & Son,New York 1984.

Committee on Medical and Biologic Effects of Environmental Pollutants, Subcommittee onAirborne Particles: Airborne Particles. University Park Press, Baltimore 1979.

Friedlander SK: Smoke, Dust and Haze Fundamentals of Aerosol Behavior. Wiley-Interscience, New York 1977.

Fuchs NA: The Mechanics of Aerosols. Pergamon Press 1964. (Reprint Dover 1989)

Heskett HE: Fine Particles in Gaseous Media. 2nd Ed. Lewis Publishers, Michigan 1986

Hinds WC: Aerosol Technology. Properties, Behavior, and Measurement of AirborneParticles. Wiley-Interscience, New York 1982.

Israel G: Aerosols. Formation and Recetivity. Proceedings Second International AerosolConference, September 1986. Pergamon Press, Oxford 1986.

4

Liu BYH (Ed.): Fine particles. Aerosol Generation, Measurement, Sampling and Analysis.Academic Press, New York 1976.

Liu BYH, Pui DYH, Fissan HJ (Eds): Aerosols. Science, Technology and IndustrialApplications of Airborne Particles. First International Conference, Elsevier 1984.

Marple VA, Liu BYH (Eds): Aerosols in the Mining and Industrial Work Environments. 3Vol. Ann Arbor Science 1983.

Reist PC: Introduction to Aerosol Science. MacMillan 1984.

Shaw DT (Ed.): Fundamentals of Aerosol Science. Wiley-Interscience, New York 1978.

Shaw DT (Ed.): Recent Developments in Aerosol Science. Wiley-Interscience, New York1978.

Willeke K, Baron PA (Eds): Aerosol Measurement. Van Nostrand Reinhold, New York 1993.

5

1 Properties of air and water vaporThis chapter includes some properties dependence of pressure, temperature, humidity etc anddata for the most important properties for air and water vapor. First is the ideal-gas law (1),which is usable for air at normal temperatures. For extreme pressures or temperatures anequation of state (2) can be used. For ideal gases there exist a number of connections (3),which can be used for air. Some properties variation with temperature are presented in 4. In 5are given figures for air and in 6 for water and water vapor. For water vapor can the ideal-gaslaw be used for approximative calculations (1). If more accurate values are needed an equationof state is used (7,8). Mixtures of air and water vapor are frequent and formulas are presentedin 9. In 11 and 12 are formulas to be used when measuring water vapor in air. In 10 are someformulas for the variation of heat content with temperature and humidity. Air pressurevariation with height are given in 13. Some equations for connection between contaminantconcentration and sight length end this chapter (14).

If the pressure is not given or if it is not a part of the formulas, normal pressure i.e. 1.013 bar(=101.3 kPa) is presumed.

6

1.1 The Ideal-Gas Law

T R Mm

= V por T R n = V por T R = v p ••••••••

p = pressurev = molecular volumeR = gas constant (8,31441 J/mol,K = 1,9872 cal/K,mol = 0,08205 lit,atm/K,mol = 62,4 lit,mm Hg/K,mol)T = absolute temperatureV = volumen = number of molsm = massM = molecular weight.(Normally used for air, also when some water vapor or contaminants are present.)

1.2 Beattie-Bridgeman equation of state(Air from - 145 °C to + 200 °C)

p = pressure, atmv = molecular volume, lit/molR = gas constant 0,08205 lit╖atm/mol╖KT = temperature, K.

(To be used at extreme pressures or temperaturs, or when more accurate values thanfrom (1) are needed).

1.3 Specific heat capacities for ideal gases

Cp = specific heat capacity at constant pressureCv = specific heat capacity at constant volumea = air velocity

•

••

•

••••

v0,01931

1 1,3012

T v10 4,34

1 v

0,01101 + 1 0,04611 + v T R = v p

3

42

R + C = C vp

T R = / p = a ••• κρκ

7

κ = isentrop exponent = Cp/Cvρ = densityR, T, p see 1.2.

1.4 Temperature variation of specific heat capacity, viscosity (see also13.4) and diffusion coefficient

a)

Cp in cal/mol,K in the temperature interval 300-1500 K.

b)

η = viscosity in the interval 273-673 K, N,s/m2

T = temperature, K.

c) Another expression is

η = viscosity, kg/m,sT = temperature, K.

d)

η = viskosity at temperature Tη0 = viskosity at temperature T0.

e) In small intervals this can be simplified to

T 10 0,2656 T 10 1,762 + 6,386 = C 26 3 p ••••

T123,6 +1

T 10 150,3 = 8 ••η

110 + TT 10 1.45 =

2/36 ••η

110 + T110 + T )

TT( = 0

0

23

0

•ηη

8

where 0.5 < ω < 1 depends on interval.

f)

D(H2O in air) = diffusion coefficient for water vapor in air, m2/hp = total pressure, kPa/m2

T0 = 273 K.

g) For p = 760 mm Hg this will be

h) Another empirical formula for water vapor in air up to 1350 K is

D = diffusion coefficient for water vapor in air, mm2/sp = pressure, kPaT = temperature, K.

)TT( =

00

ω

ηη

)TT(

p805 = air)in OH( D

0

1.80

2 •

/s)cm( )273T( 0.216 = D 2

1.80

•

)245 + T

T( )p

0.926( = D2.5

•

9

1.5 Air properties (at 100 kPa)

Molecular weight M = 28.962458 g/molDensity at 0°C (dry air) ρ = 1.2929 kg/m3

at 15°C, 0% R.H, 105 Pa 1.2094 kg/m3

at 15°C, 50% R.H, 105 Pa 1.2055 kg/m3

at 15°C, 100% R.H, 105 Pa 1.2017 kg/m3

at 20°C, 0% R.H, 105 Pa 1.1887 kg/m3

at 20°C, 50% R.H, 105 Pa 1.1834 kg/m3

at 20°C, 100% R.H, 105 Pa 1.1783 kg/m3

Heat conductivity at 18°C λ = 0.025 W/m╖°CSpecific heat capacity at 0°C Cp = 1.00 kJ/kg,°C Cp = 29.0 kJ/kmol,KViscosity at 0°C η = 17.0 ⋅ 10-6 kg/s,mViscosity at 20°C η = 18.192 ⋅ 10-6 kg/s,mCritical temperature Tc = 132.5 KCritical pressure Pc = 36 barMelting point Ts = 60.1 KBoiling point Tk = 80.2 KDensity at boiling point ρ = 880 kg/m3

Components of dry atmospheric air

N2 78.084 vol %O2 20.946 "Ar 0.934 "CO2 0.033 " (variabel)Ne 18.18 ppmHe 5.24 "Kr 1.14 "H2 0.5 "Xe 0.087 "CH4 2 "N2O 0.5 "O3 0.01 " (variabel)Rn 6 ⋅10-14 " (variabel)

Mean free path

l = mean free path of air molecules, m (15-25°C, 0-100% RH)η = air viscosity, kg/m,sρ = air density, kg /m3

P = air pressure, Pa

P 8

I = l

•••

ρπη

10

I = constant = 0.4987445.1.6 Water properties (at 100 kPa)

Molecular weight M = 18.0152 g/molDensity at 0°C ρ = 999.84 kg/m3 at 20°C = 998.205 kg/m3

Heat conductivity at 20°C λ = 0.598 W/m,°CSpecific heat capacity at 0°C Cp = 4.218 kJ/kg,K

at 20-100°C Cp = 4.18 kJ/kg,KViscosity at 0°C η = 1792⋅10-6 kg/s,m

at 20°C = 1002⋅10-6 kg/s,mMelting point ts = + 0°CMelting heat Qs = 334 kJ/kgBoiling point tk = + 100°CVaporization enthalpy Qk = 2257 kJ/kgDensity at boiling point (1.013 bar) ρ = 958.35 kg/m3

Critical temperature Tc = 647.4 KCritical pressure Pc = 221.3 bar

Diffusion coefficient for water vapor in air at 0°C D = 0.216 cm2/s

at 20°C D = 0.245 cm2/s

Schmidt number for water vapor in air

Lewis number for water vapor in air

1.7 van der Waal's equation of state (water vapor)

p in atmospheresv in cm3/mol

C)(20 0.617C)(0 0.616

{ = D

= Sc°°ν

C)20 (0 0.866 = PrSc = Le °−

T R = 30.42) (v )v10 5.454 + (p

2

6

•••

11

T in KR = 82.054 atm,cm3/mol,K.

1.8 Water vapor pressure between 275 and 647 K

x =3.647

T 1−

T = temperature, Kpvp = vapor pressure, bar.

1.9 Mixture of air and water vapor: Density and vapor pressurea)

This can also be written in the following way:

ρ = density for humid air, kg/m3

ρt = density for dry air, kg/m3

P = total pressure, mm HgPv = partial pressure of water vapor, mm HgT = absolute temperature, K.

b)

pv = partial pressure for water vapor, in mbar, with water content x(kg H2O / kg dry air)

x1x 1.23303 x 2.77580 x 1.45838 x+ 7.76451

= 221.2

pln

631.5vp ••••

)Tp( 0.176 )

TP v−( 0.465 = ρ

•−••

760p 0.3783 P

T

273.13 = vtρρ

1013

18x +

29118x

= pv •

12

(If 1013 is changed to 760 the partial pressure is expressed in mm Hg.)

c)

ρ, ρt and x see above.

(0.622 is MH2O/Mair = 0.62198)x, pv, P see above

d)

x = water content, kg/m3

td = dew point, °C.

The partial pressure of water vapor for a specific dew point can be calculated by usingdew point temperature instead of air temperature in e or f.

e) Dry temperature above 0 °°°°C

f) Dry temperature below or equal to 0 °°°°C

pm = saturation pressure for water vapor, in mbar, at absolute temperature T.

x)+ (1 = t •ρρ

p Pp 622 . 0 =x

v

v

−•

10 10 4.8 =x 38t3 d••

0.78613974 + 1 10 10 0.42873 +

+ 10 1 10 1.50474 +

273.16T log 5.02808

T273.16 1 10.79586 = ) p ( log

T273.16 1 4.769663

1 273.16

T 8.29692 4

m

−••

−••

−

−•

−•−

−•−−

78613974 0. + 16 . 273

T 1 876817 0. +

T16 . 273 log 56654 3.

T16 . 273 096936 9. = ) p ( log m

−•

•−

−

13

g)

h)

pm = saturation pressure for water vapor, kPat = air temperature, °C.

i) Air humidity variation with barometric pressure

φ = ptr / ps equal to relative humidityptr = vapor pressure in air with temperature ttrps = saturation vapor pressure at temperature ttr , mm Hg pf = vapor pressure in air at temperature tf , mm Hgtf = wet temperature, °Cttr = temperature (dry thermometer for air), °Cb = barometric pressure, mm Hg∆t = ttr - tf 1510 = 755/k, where 755 = initial barometric pressure, mm Hgk = 0.5 mm Hg/K for water-air = 0.4 mm Hg/K for ice-air.

1.10 Specific enthalpy of aira)

h = specific enthalpy, kJ/kgt = temperature, °C

Cp = specific heat capacity for dry air 1.0 kJ/kg,°CCp (H2O) = specific heat capacity for water vapor 1.9 (1.8516) kJ/kg,°Cx = water content kg H2O/kg dry airr = specific vaporization enthalpy for water at 0°C = 2500 kJ/kg.

C) 26 (0 e 0.62796 = p t 10 6.5557m

2 °−• •• −

C) 50 (26 e 83721 . 0 = p t 10 4169 5.m

2 °−• •• −

p1510)t / (b p

s

f ∆• = ϕ

r x + O)H( C x t + C t =h 2pp ••••

14

b)

1.11 Calculation of humid air's properties

By using dry temperature (tt), wet temperature (tv) and barometric pressure (P) it ispossible to calculate relative humidity (_), absolute humidity (x) and specific enthalpy(i):

1 Calculate saturation pressure for water vapor (pm) by using 1.9.e.

2 Calculate absolute humidity for saturated state (xm with 1.9.c). (Put pσ = pm and xm isthe result.)

3 Absolute humidity (x) is calculated from

or from

(tt , tv in °C, x and xm in kg/kg).

4 Relative humidity is calculated in % from φ = x/xm*100

5 Specific enthalpy is calculated from equation 1.10.b.

Dew point, partial pressure and density is calculated by using equations 1.9.d and1.9.b.

If the starting point is relative humidity instead of the wet temperature the followingcalculations are done:

1 Calculate the saturation pressure for water vapor (pm) with 1.9.e.

2 Calculate partial pressure for water vapor (pv) from

x t) 1.9 + (2500 + t =h ••

C 0 > tfor 2500 t 86 . 1 t 19 . 4

2500) t 27 . (2 x + )t t( 1.005 =x xtv

vmtv °−•−•

−••−

C 0 tfor 2833 t 86 . 1 t 11 . 2

2833) t 25 . (0 x + )t t( 005 , 1 =x ttv

vmvt °≤−•−•

−••−

15

3 Calculate absolute humidity for the saturated state (xm) with 1.9.c.

3 Calculate absolute humidity (x) from

5 Specific enthalpy is calculated from equation 1.10.

1.12 Psychrometer formulaa)

pv = partial pressure for water vaporpm = saturation pressure for water vapor

P = total pressure (pv, pm and P in the same units)tt = dry temperature °Ctv = wet temperature °CA = psychrometer constantThermodynamical constant A = 6.53*10-4/°CAssmann psychrometer A = 6.62*10-4/°CAir velocities larger than 5.5 m/sA = 6.5*10-4/°CAir velocity 0 m/s A = 12*10-4/°CNatural ventilated thermometer A = 7.9*10-4/°C

For p = 101.3 ⋅ 103 Pa the Assmann psychrometer will give Pv = pm - 67.1 (tt - tv) [Pa]

1.13 Air pressure variation with height

100p

= p mv

•ϕ

px p =x

m

mv •

)t t( A P p = p vtmv −••−

m) 1524 (0 e 86425 . 101 = p h 10 24087 . 1 4 −• ••− −

m) 3048 (1525 e 12563 . 102 = p h 10 25184 . 1 4 −• ••− −

16

p = barometric pressure, kPah = height over sea level, m.

1.14 Connection between sight length and contaminant concentrationa)

V = visual sight lenght, kmσ = atmospheric extinktion coefficient, km-1 (σ = σM + σR)σR = from Rayleigh-scattering, in normal clean atmosphere = 1.167 ⋅ 10-2

σM = from Mie-scattering, is not included in normal atmosphere

Both σm and σR are influenced of contaminant concentration and humidity.

b) One value for σ (0.55 µm) is

M = contaminant concentration, µg/m3

σ0.55 µm = back-scattering coefficient, m-1, for wavelength 0.55 µm.

A combination of the two equation above results in

L = sight length, mM = contaminant concentration, µg/m3.

c) Diminishing sight length from tobacco smoke.

C = concentration of tobacco smoke in air µg/m3, σ0.55 µm see above.0.20 = constant with maximum expected systematic error and two standard deviations

equal to + 0,11 and - 0,06, respectivly. A standard deviation = 0.03 * σ0.55 µm hasbeen given to between 0.1 * 10-4 and 3 * 10-4 m-1 for different degrees of cleanair, in industrial cities = 3 - 15 * 10-4 m-1.

σ3.91 = V

σ µ m 0.55 5 10 8 . 3 = M ••

L10 .8 1 = M

6•

σ µ m0.55 20 . 0 = C •