- Email: [email protected]
Computation of Physical Properties of Water
Appendix B: Computation of Physical Properties of Water
The basis for physical properties of water that are used in this book will be presented in this appendix. The following abbreviations will be used in this appendix: Parameter
Abbreviation
Units or Value
Temperature Temperature Salinity Chlorinity Density of water Acceleration due to gravity
t T S CL ρ g
Celsius ( C) Kelvin ( C 1 273.15) g/kg g/kg kg/m3 9.80665 m/s2
Density of Water (ρ) in kg/m3 For 040 C and 0.543 g/kg, the 1 atm equation of state of seawater is based on Millero and Poisson (1981): ρ 5 ρo 1 AS 1 BS3=2 1 CS2
(B-1)
A 5 A 0 1 A 1 t 1 A2 t 2 1 A 3 t 3 1 A 4 t 4
(B-2)
B 5 B 0 1 B 1 t 1 B2 t 2 1 B 3 t 3
(B-3)
C 5 C1
(B-4)
ρo 5 D0 1 D1 t 1 D2 t2 1 D3 t3 1 D4 t5
(B-5)
270
Appendix B
Equation (B-1) As Value A0 A1 A2 A3 A4
Equation (B-2) Bs Value
18.24493 3 1021 B0 24.0899 3 1023 B1 17.6438 3 1025 B2 28.2467 3 1027 15.3875 3 1029
Equation (B-3) Cs Value
25.72466 3 1023 C0 4.8314 3 1024 11.0227 3 1024 21.6546 3 1026
Equation (B-4) Ds Value D0 D1 D2 D3 D4 D5
999.842594 16.793952 3 1022 29.095290 3 1023 11.001685 3 1024 1.120083 3 1026 16.536332 3 1029
For NaCl brines over the range of 035 C and 0260 g/kg, Eq. (22) in Sherwood et al. (1991) is used: A0 5 0.999792 A1 5 6.92234 3 1025 A2 5 28.15399 3 1026 A3 5 14.25067 3 1028 A4 5 7.68124 3 1024 A5 5 21.46445 3 1027 A6 5 1.60452 3 1028 A7 5 24.10220 3 1026 A8 5 4.04168 3 1028 A9 5 7.64930 3 10211 A10 5 1.38698 3 1027 A11 5 21.79894 3 1029
ρ51000:0 ½A01A1 t 1A2 t2 1A3 t3 1A4 S1A5 S2 1A6 S5=2 1A7 tS1A8 t2 S 1A9 t3 S1A10 tS3=2 1A11 t2 S3=2
(B-6)
The initial equation has units of g/mL and is multiplied by 1000 to convert to kg/m3.
Specific Weight of Water (γ) in kN/m3 The specific weight of water is based on the density of water (Millero and Poisson, 1981) and the value of the acceleration of gravity (g) listed in the first table in this Appendix: γ5
ρg 1000
(B-7)
Appendix B
271
Hydrostatic Head of Water in mmHg/m and kPa/m The hydrostatic head of water is based on the density of water (Millero and Poisson, 1981) and the appropriate pressure units: Hydrostatic head ðmmHg=mÞ 5 Hydrostatic head ðkPa=mÞ 5
760 mmHg ð101; 325 Pa=ρgÞ
(B-8)
101:325 kPa ð101; 325 Pa=ρgÞ
(B-9)
Dynamic Viscosity of Water (μ) in N s/m2 Dynamic viscosity of water is based on Korson et al. (1969) and Millero (1974): μ20 5 1:0020 3 1023 ð20 C in freshwaterÞ
μt 5 μ20 10
1:1709ð20 2 tÞ 2 0:001827ðt 2 20Þ2 t 1 89:93
(B-10) (B-11)
CL 5 ðS 2 0:03Þ=1:805
(B-12)
CLV 5 ðCLÞðρ=1000:0Þ
(B-13)
A 5 0:000366 1 5:185 3 1025 ðt 2 5:0Þ
(B-14)
B 5 0:002756 1 3:300 3 1025 ðt 2 5:0Þ
(B-15)
μ 5 μt ½1:0 1 AðCLV Þ1=2 1 BðCLV Þ
(B-16)
Note that the table values are given in N s/m2 3 1013. For example, the viscosity at 20 C and 20 g/kg is 1.0424 3 1023 N s/m2.
Kinematic Viscosity of Water (ν) in m2/s The kinematic viscosity of water is based on the dynamic viscosity (Korson et al. 1969; Millero, 1974) and the density of water (Millero and Poisson, 1981): ν5
μ ρ
(B-17)
272
Appendix B
Note that the table values are given in m2/s 3 1016. For example, the kinematic viscosity at 20 C and 20 g/kg is 1.0286 3 1026 m2/s.
Heat Capacity of Water (Cp) in kJ/(kg K) Heat capacity of water is based on Millero et al. (1973): C op 5 4:2174 2 3:720283 3 1023 t 1 1:412855 3 1024 t2 2 2:654387 3 1026 t3 1 2:093236 3 1028 t4 (B-18) CL 5 S=1:80655
(B-19)
A 5 2ð13:81 2 0:1938t 1 :0025t2 Þ=ð1000:0Þ
(B-20)
B 5 ð0:43 2 :0099t 1 0:0001t2 Þ=ð1000:0Þ
(B-21)
Cp 5 Cpo 1 AðCLÞ 1 BðCLÞ3=2
(B-22)
Latent Heat of Vaporization (LHV) of Water in MJ/kg Latent heat of vaporization of water is based on Brooker (1967): LHV ðMJ=kgÞ 5 2:502535259 2 0:00238576424t
(B-23)
Surface Tension of Water (σ) in N/m Surface tension of water in N/m is based on Riley and Skirrow (1975): σ ðN=mÞ5
75:64 2 0:144t 1 0:0221S 1000
(B-24)
Vapor Pressure of Freshwater Water (Pwv) Four different equations are used to compute the vapor pressure of water for the solubility relationships in this book:
Appendix B
273
Solubility Relationship
Source of Vapor Pressure
Gas
Reference
Reference
O2 O2 (brine) N2 and Ar CO2
Benson and Krause (1984) Sherwood et al. (1991) Hamme and Emerson (2004) Weiss (1974), Weiss and Price (1980)
Green and Carritt (1967) Sherwood et al. (1991) Ambrose and Lawrenson (1972) Equation fitted to Goff and Gratch (1946) and Robinson (1954)
The relationship developed by Ambrose and Lawrenson (1972) is the most complex and is preferred for typical estuarine and oceanic salinities. The relationship developed by Weiss (1974) is the simplest and useful for spreadsheet applications. The Sherwood et al. (1991) equation was developed for NaCl brines and may be used for real brines if location-specific information is not available. The original vapor pressure equations were used in the individual gas solubility equations to try to reproduce the author’s results. The use of other vapor pressure relationships may change the solubility parameters in the last decimal place. The water vapor equations given below are given in their original units.
Vapor Pressure of Water (Pwv) in atm (Green and Carritt, 1967) A 5 5.370 3 1024 B 5 18.1973 C 5 1.0 2373.16/T D 5 3.1813 3 1027 E 5 26.1205 F 5 1.0 2 T/373.16 G 5 1.8726 3 1022 H 5 8.03945 X 5 5.02802 Y 5 373.16/T
Pwv 5 fð1:0 2 A 3 SÞ 3 expðB 3 C 1 D 3 ð1:0 2 expðE 3 FÞÞ 2G 3 ð1:0 2 expðH 3 CÞÞ 1 X 3 lnðYÞÞg
(B-25)
Vapor Pressure of Water (Pwv) in atm (Sherwood et al., 1991) A1 5 148.4171 A2 5 26821.5 A3 5 25.0903 A4 5 25.8785 3 1024 A5 5 21.2276 3 1028 A6 5 26.93 3 1029
lnðPwv Þ 5 A1 1 A2 ð1=TÞ 1 A3 lnðTÞ 1 A4 S 1 A5 S2 1 A6 S3
(B-26)
274
Appendix B
Vapor Pressure of Water (Pwv) in kPa (Ambrose and Lawrenson, 1972) The vapor pressure of freshwater is represented by the Chebyshev polynomial: T log10 Powv 5 12a0 1
hX11
a E ðxÞ k51 k k
i (B-27)
where x5
2T 2 921 375
(B-28)
T 5 temperature in Kelvin measured on the International Practical Temperature Scale of 1968.
The coefficients of the Chebyshev polynomial are: a0 5 2794.0144 a1 5 1430.6181 a2 5 218.2465 a3 5 7.6875 a4 5 20.0328 a5 5 0.2728
a6 5 0.1371 a7 5 0.0629 a8 5 0.0261 a9 5 0.0200 a10 5 0.0117 a11 5 0.0067
The actual Chebyshev polynomials are equal to: E0 ðxÞ 5 1 E1 ðxÞ 5 x E2 ðxÞ 5 2x2 21 E3 ðxÞ 5 4x3 23x E4 ðxÞ 5 8x4 28x2 11 E5 ðxÞ 5 16x5 220x3 15x E6 ðxÞ 5 32x6 248x4 118x2 21 E7 ðxÞ 5 64x7 2112x5 156x3 27x E8 ðxÞ 5 128x8 2256x6 1160x4 232x2 11 E9 ðxÞ 5 256x9 2576x7 1 432x5 2120x3 1 9x
Appendix B
275
E10 ðxÞ 5 512x10 2 1280x8 1 1120x6 2 400x4 1 50x2 21 E11 ðxÞ 5 1024x11 2 2816x9 1 2816x7 2 1232x5 1 220x3 2 11x The vapor pressure of seawater (DOE, 1994) is related to that of pure water by: Pswv 5 Powv expð20:018φ
X B
mB =m3 Þ
(B-29)
For seawater, X B
mB =m3 5
31:998S 5ψ 1000 2 1:005S
(B-30)
or Pswv 5 Powv expð20:018φψÞ
(B-31)
The value of the osmotic coefficient (Millero, 1974) of seawater (φ) is equal to 2 3 4 ψ ψ ψ ψ φ 5 0:90799 2 0:08992 2 0:073958 2 0:00221 1 0:18458 2 2 2 2 (B-32)
Vapor pressure of water (Pwv) in atm (Weiss and Price, 1980): Pwv 5 expð24:4543 2 67:4509ð100=TÞ 2 4:8489 lnðT=100Þ 2 0:000544SÞ (B-33)
The basis for physical properties of water that are used in this book will be presented in this appendix. The following abbreviations will be used in this appendix: Parameter
Abbreviation
Units or Value
Temperature Temperature Salinity Chlorinity Density of water Acceleration due to gravity
t T S CL ρ g
Celsius ( C) Kelvin ( C 1 273.15) g/kg g/kg kg/m3 9.80665 m/s2
Density of Water (ρ) in kg/m3 For 040 C and 0.543 g/kg, the 1 atm equation of state of seawater is based on Millero and Poisson (1981): ρ 5 ρo 1 AS 1 BS3=2 1 CS2
(B-1)
A 5 A 0 1 A 1 t 1 A2 t 2 1 A 3 t 3 1 A 4 t 4
(B-2)
B 5 B 0 1 B 1 t 1 B2 t 2 1 B 3 t 3
(B-3)
C 5 C1
(B-4)
ρo 5 D0 1 D1 t 1 D2 t2 1 D3 t3 1 D4 t5
(B-5)
270
Appendix B
Equation (B-1) As Value A0 A1 A2 A3 A4
Equation (B-2) Bs Value
18.24493 3 1021 B0 24.0899 3 1023 B1 17.6438 3 1025 B2 28.2467 3 1027 15.3875 3 1029
Equation (B-3) Cs Value
25.72466 3 1023 C0 4.8314 3 1024 11.0227 3 1024 21.6546 3 1026
Equation (B-4) Ds Value D0 D1 D2 D3 D4 D5
999.842594 16.793952 3 1022 29.095290 3 1023 11.001685 3 1024 1.120083 3 1026 16.536332 3 1029
For NaCl brines over the range of 035 C and 0260 g/kg, Eq. (22) in Sherwood et al. (1991) is used: A0 5 0.999792 A1 5 6.92234 3 1025 A2 5 28.15399 3 1026 A3 5 14.25067 3 1028 A4 5 7.68124 3 1024 A5 5 21.46445 3 1027 A6 5 1.60452 3 1028 A7 5 24.10220 3 1026 A8 5 4.04168 3 1028 A9 5 7.64930 3 10211 A10 5 1.38698 3 1027 A11 5 21.79894 3 1029
ρ51000:0 ½A01A1 t 1A2 t2 1A3 t3 1A4 S1A5 S2 1A6 S5=2 1A7 tS1A8 t2 S 1A9 t3 S1A10 tS3=2 1A11 t2 S3=2
(B-6)
The initial equation has units of g/mL and is multiplied by 1000 to convert to kg/m3.
Specific Weight of Water (γ) in kN/m3 The specific weight of water is based on the density of water (Millero and Poisson, 1981) and the value of the acceleration of gravity (g) listed in the first table in this Appendix: γ5
ρg 1000
(B-7)
Appendix B
271
Hydrostatic Head of Water in mmHg/m and kPa/m The hydrostatic head of water is based on the density of water (Millero and Poisson, 1981) and the appropriate pressure units: Hydrostatic head ðmmHg=mÞ 5 Hydrostatic head ðkPa=mÞ 5
760 mmHg ð101; 325 Pa=ρgÞ
(B-8)
101:325 kPa ð101; 325 Pa=ρgÞ
(B-9)
Dynamic Viscosity of Water (μ) in N s/m2 Dynamic viscosity of water is based on Korson et al. (1969) and Millero (1974): μ20 5 1:0020 3 1023 ð20 C in freshwaterÞ
μt 5 μ20 10
1:1709ð20 2 tÞ 2 0:001827ðt 2 20Þ2 t 1 89:93
(B-10) (B-11)
CL 5 ðS 2 0:03Þ=1:805
(B-12)
CLV 5 ðCLÞðρ=1000:0Þ
(B-13)
A 5 0:000366 1 5:185 3 1025 ðt 2 5:0Þ
(B-14)
B 5 0:002756 1 3:300 3 1025 ðt 2 5:0Þ
(B-15)
μ 5 μt ½1:0 1 AðCLV Þ1=2 1 BðCLV Þ
(B-16)
Note that the table values are given in N s/m2 3 1013. For example, the viscosity at 20 C and 20 g/kg is 1.0424 3 1023 N s/m2.
Kinematic Viscosity of Water (ν) in m2/s The kinematic viscosity of water is based on the dynamic viscosity (Korson et al. 1969; Millero, 1974) and the density of water (Millero and Poisson, 1981): ν5
μ ρ
(B-17)
272
Appendix B
Note that the table values are given in m2/s 3 1016. For example, the kinematic viscosity at 20 C and 20 g/kg is 1.0286 3 1026 m2/s.
Heat Capacity of Water (Cp) in kJ/(kg K) Heat capacity of water is based on Millero et al. (1973): C op 5 4:2174 2 3:720283 3 1023 t 1 1:412855 3 1024 t2 2 2:654387 3 1026 t3 1 2:093236 3 1028 t4 (B-18) CL 5 S=1:80655
(B-19)
A 5 2ð13:81 2 0:1938t 1 :0025t2 Þ=ð1000:0Þ
(B-20)
B 5 ð0:43 2 :0099t 1 0:0001t2 Þ=ð1000:0Þ
(B-21)
Cp 5 Cpo 1 AðCLÞ 1 BðCLÞ3=2
(B-22)
Latent Heat of Vaporization (LHV) of Water in MJ/kg Latent heat of vaporization of water is based on Brooker (1967): LHV ðMJ=kgÞ 5 2:502535259 2 0:00238576424t
(B-23)
Surface Tension of Water (σ) in N/m Surface tension of water in N/m is based on Riley and Skirrow (1975): σ ðN=mÞ5
75:64 2 0:144t 1 0:0221S 1000
(B-24)
Vapor Pressure of Freshwater Water (Pwv) Four different equations are used to compute the vapor pressure of water for the solubility relationships in this book:
Appendix B
273
Solubility Relationship
Source of Vapor Pressure
Gas
Reference
Reference
O2 O2 (brine) N2 and Ar CO2
Benson and Krause (1984) Sherwood et al. (1991) Hamme and Emerson (2004) Weiss (1974), Weiss and Price (1980)
Green and Carritt (1967) Sherwood et al. (1991) Ambrose and Lawrenson (1972) Equation fitted to Goff and Gratch (1946) and Robinson (1954)
The relationship developed by Ambrose and Lawrenson (1972) is the most complex and is preferred for typical estuarine and oceanic salinities. The relationship developed by Weiss (1974) is the simplest and useful for spreadsheet applications. The Sherwood et al. (1991) equation was developed for NaCl brines and may be used for real brines if location-specific information is not available. The original vapor pressure equations were used in the individual gas solubility equations to try to reproduce the author’s results. The use of other vapor pressure relationships may change the solubility parameters in the last decimal place. The water vapor equations given below are given in their original units.
Vapor Pressure of Water (Pwv) in atm (Green and Carritt, 1967) A 5 5.370 3 1024 B 5 18.1973 C 5 1.0 2373.16/T D 5 3.1813 3 1027 E 5 26.1205 F 5 1.0 2 T/373.16 G 5 1.8726 3 1022 H 5 8.03945 X 5 5.02802 Y 5 373.16/T
Pwv 5 fð1:0 2 A 3 SÞ 3 expðB 3 C 1 D 3 ð1:0 2 expðE 3 FÞÞ 2G 3 ð1:0 2 expðH 3 CÞÞ 1 X 3 lnðYÞÞg
(B-25)
Vapor Pressure of Water (Pwv) in atm (Sherwood et al., 1991) A1 5 148.4171 A2 5 26821.5 A3 5 25.0903 A4 5 25.8785 3 1024 A5 5 21.2276 3 1028 A6 5 26.93 3 1029
lnðPwv Þ 5 A1 1 A2 ð1=TÞ 1 A3 lnðTÞ 1 A4 S 1 A5 S2 1 A6 S3
(B-26)
274
Appendix B
Vapor Pressure of Water (Pwv) in kPa (Ambrose and Lawrenson, 1972) The vapor pressure of freshwater is represented by the Chebyshev polynomial: T log10 Powv 5 12a0 1
hX11
a E ðxÞ k51 k k
i (B-27)
where x5
2T 2 921 375
(B-28)
T 5 temperature in Kelvin measured on the International Practical Temperature Scale of 1968.
The coefficients of the Chebyshev polynomial are: a0 5 2794.0144 a1 5 1430.6181 a2 5 218.2465 a3 5 7.6875 a4 5 20.0328 a5 5 0.2728
a6 5 0.1371 a7 5 0.0629 a8 5 0.0261 a9 5 0.0200 a10 5 0.0117 a11 5 0.0067
The actual Chebyshev polynomials are equal to: E0 ðxÞ 5 1 E1 ðxÞ 5 x E2 ðxÞ 5 2x2 21 E3 ðxÞ 5 4x3 23x E4 ðxÞ 5 8x4 28x2 11 E5 ðxÞ 5 16x5 220x3 15x E6 ðxÞ 5 32x6 248x4 118x2 21 E7 ðxÞ 5 64x7 2112x5 156x3 27x E8 ðxÞ 5 128x8 2256x6 1160x4 232x2 11 E9 ðxÞ 5 256x9 2576x7 1 432x5 2120x3 1 9x
Appendix B
275
E10 ðxÞ 5 512x10 2 1280x8 1 1120x6 2 400x4 1 50x2 21 E11 ðxÞ 5 1024x11 2 2816x9 1 2816x7 2 1232x5 1 220x3 2 11x The vapor pressure of seawater (DOE, 1994) is related to that of pure water by: Pswv 5 Powv expð20:018φ
X B
mB =m3 Þ
(B-29)
For seawater, X B
mB =m3 5
31:998S 5ψ 1000 2 1:005S
(B-30)
or Pswv 5 Powv expð20:018φψÞ
(B-31)
The value of the osmotic coefficient (Millero, 1974) of seawater (φ) is equal to 2 3 4 ψ ψ ψ ψ φ 5 0:90799 2 0:08992 2 0:073958 2 0:00221 1 0:18458 2 2 2 2 (B-32)
Vapor pressure of water (Pwv) in atm (Weiss and Price, 1980): Pwv 5 expð24:4543 2 67:4509ð100=TÞ 2 4:8489 lnðT=100Þ 2 0:000544SÞ (B-33)