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Estimation of binary diffusion coefficients in supercritical water - A mini review Stephan Kraft, and Frédéric Vogel Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b00382 • Publication Date (Web): 22 Mar 2017 Downloaded from http://pubs.acs.org on April 6, 2017

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Industrial & Engineering Chemistry Research

Estimation of binary diusion coecients in supercritical water A mini review Stephan Kraft † and Frédéric Vogel ∗ ‡ ,

†Bioenergy 2020+ GmbH, Wiener Strasse 49, A-7540 Güssing, Austria ‡Paul Scherrer Institut, Division Energy and Environment, Laboratory for Bioenergy and

Catalysis, CH-5232 Villigen PSI, Switzerland E-mail: [email protected]

Abstract Diusion coecients of various solutes in supercritical water, which were either measured or retrieved from Molecular Dynamics simulations, were reviewed. Diusion coecients of molecules relevant for supercritical water processes were calculated with correlations reported in the literature and compared to the values of reference data. For conditions well above the critical point of water the simple Stokes-Einstein equation predicts the diusion coecients with an accuracy better than 20%. For conditions near the critical point the Wilke-Chang correlation gives the most accurate results. Diusion coecients for typical molecules occurring in supercritical water processes such as O2 , N2 , CO, CO2 , or CH4 are estimated to be in the range of 60 · 10−9 m2 /s at 673 K and 30 MPa. For H2 , for which no experimental data are available, much higher diusion coecients in the range of 250 · 10−9 m2 /s seem plausible. The data set of binary diusion coecients in supercritical water, either determined experimentally or by Molecular Dynamics simulations, should be extended signicantly to include more solutes, as well as higher temperatures and pressures. 1

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Introduction The critical temperature, pressure, and density of water 1 are 647.096 K, 22.064 MPa and 322 kg/m3 , respectively. Properties such as viscosity, density or specic heat capacity change dramatically around the critical point. Supercritical water (SCW) is used in various processes since it is widely available and not toxic. Peterson et al. 2 provided an extensive review of sub- and supercritical water technologies. Supercritical water gasication (SCWG) is the conversion of organic substances in water to a gas near or above the critical point. The aim of the supercritical water oxidation (SCWO) is the destruction of hazardous organic wastes by complete oxidation in supercritical water. 35 The advantage of processes with supercritical water is the almost complete conversion of the organic feed at relatively low residence times of seconds to minutes, depending on the temperature and the presence of catalysts. More recently supercritical water has been used as a working uid in conventionally red power plants 6 and was proposed for nuclear power plants of the 4th generation. 7 Hereby, greater thermal eciencies are possible. Nevertheless, the high temperatures, pressures and the corrosive characteristics of water under these conditions are challenges for all SCW processes. For the design of SCW processes knowledge of the transport properties is required. Diusion coecients are necessary to calculate the mass transport through the boundary layer and within a porous catalyst particle, as well as to calculate dispersion coecients for non-ideal plug ow. The diusion coecient plays an important role in diusion-controlled chemical reactions or mass transfer processes. Diusion coecients are typically in the order of 10−9 m2 /s for liquids and about 10−5 m2 /s for gases at ambient conditions. 8 The viscosity of SCW is at least one order of magnitude smaller compared to liquid water. 1 According to the Stokes-Einstein equation, at supercritical conditions the diusion coecient is thus about one order of magnitude larger, 9 in the order of 10−8 to 10−7 m2 /s. Diusion coecients can be either determined by experimental methods or calculated by 2


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Molecular Dynamics simulations. Both methods are either expensive and/or time consuming and/or require extensive knowledge of the methodology. s an alternative, many correlations to estimate the diusion coecient have been published. If these equations are able to predict the diusion coecients in SCW with sucient accuracy the design of chemical reactors and processes is simplied. For SCWO and SCWG the main present species, which arise from the degradation process, 2 are O2 , N2 , H2 , CO, CO2 , and CH4 . Therefore, the calculation of the diusion coecient of these species in SCW is of general interest. In this paper binary diusion coecients for these solutes in supercritical water were compiled and calculated with the Stokes-Einstein equation, with correlations that are derived from it and other equations proposed in literature. The results were compared with measured diusion coecients and results from Molecular Dynamics simulations. Finally, diusion coecients are estimated for a range of conditions typical for SCW processes to provide a starting point for mass transfer calculations.

Correlations for the Diusion Coecient In this section several correlations to calculate diusion coecients are discussed. First, the Stokes-Einstein equation, which is of a simple form, is chosen, Eq. (1). Based on the Stokes-Einstein equation other correlations were developed (Wilke-Chang, Scheibel, ReddyDoraiswamy and Lusis-Ratcli, Eqs. (2) to (5)) and are also used in this study. The equation of He and Yu Eq. (6) was included since it was developed for supercritical conditions. van Bennekom 10 used the equation by Fuller et al., Eq. (8), and corrected this equations for higher pressures with the approach by Riazi and Whitson, Eq. (9). The equation by Kallikragas et al. 11 was chosen since it also enables calculating diusion coecients for the OH radical, H2 , and O2 in water under supercritical conditions. However, no other correlation was found that was designed specically for supercritical conditions. He and Yu 12 developed their equation for organic solvents under supercritical conditions. The other correlations were

3



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originally developed for the diusivity at ambient conditions. Since these correlations are often used in literature and have a simple structure they are also included in this study. The simplest equation for the calculation of diusion coecients is the Stokes-Einstein equation (1). It is based on the assumption of the movement of large spherical molecules through a dilute solution that is assumed to be a continuum: 8

D12 =

kT 6πη2 r1

(1)

D12 is the diusion coecient of the solute "1" in the solvent "2". k is the StefanBoltzmann constant, η2 the viscosity of the solvent at temperature T , and r1 is the molecular radius of the solute. Based on the Stokes-Einstein equation several correlations were developed for supercritical conditions. The following equations by Wilke-Chang, Scheibel, Reddy-Doraiswamy and Lusis-Ratcli were developed for dierent solvents at supercritical conditions and are used in the present study to calculate diusion coecients in SCW. Wilke-Chang: 13 −15 T

D12 = 7.4 · 10

√

βM2

(2)

υ10.6 η2

with β = 2.6 for water. Scheibel 14

D12 = D12 =

8.2 · 10−15 T 1/3

η 2 υ1 8.2 · 10−14 T 1/3

η 2 υ1

"

1+

D12 = Ω υ2 υ2

2/3 #

for υ1 ≥ 2.5υ2 (3)

for υ1 < 2.5υ2

Reddy-Doraiswamy 15

with Ω = 10−14 for

3υ2 υ1

√ T M2

(4)

1/3 1/3

η 2 υ1 υ2

≤ 1.5 and Ω = 8.5 · 10−15 for

4

υ2 υ1

> 1.5.


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Lusis-Ratcli 16

D12 =

8.52 · 10−15 T

"

1.40

1/3

η 2 υ1

υ2 υ1

1/3

υ2 + υ1

# (5)

In Eq. (2) to Eq. (5) υ1 and υ2 are the molar volumes of solute 1 in solvent 2 in [cm 3 /mol] at their normal boiling points at 1 atmosphere. The diusion coecient D12 is given in [m2 /s], the temperature T in [K], the dynamic viscosity η in [kg/(m s)] and the molar mass of the solvent M2 in [g/mol]. Additionally, He and Yu 12 developed an equation for the estimation of the diusion coecient at supercritical conditions. He-Yu:

r −9

D12 = Φ · 10

T exp M1

0.3887 Vr2 − 0.23

(6)

with

Tc2 υc2 Φ = 14.882 + 0.005909 + 2.0821 · 10−6 M2

Tc2 υc2 M2

2

(7)

Vr2 is the reduced volume of the solvent: Vr2 = υ2,T,p /υc2 , where υ2,T,p is the molar volume at the temperature T and the pressure p, for which the diusion coecient should be calculated, and υc2 is the critical molar volume of the solvent. Tc2 is the critical temperature of the solvent in [K]. It should be noted that the validation of Eq. (6) was conducted with many solvents except water. van Bennekom 10 used an equation originally introduced by Fuller et al.: 17

D12 where

1 ζ1 and

P

P

2 ζ2

1/2 10−3 T 1.75 M11 + M12 = P P 2 p [( 1 ζ1 )1/3 + ( 2 ζ2 )1/3 ]

(8)

are the special atomic diusion volumes of the solute and solvent,

respectively, according to Fuller et al., 17 p is the pressure. Eq. (8) was originally developed for the calculation of binary gas phase diusion coecients. Riazi and Whitson 18 developed a correlation to estimate diusion coecients at high densities: 5



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ρD12 =a (ρD12 )◦

η η◦

b+cpr

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(9)

with a = 1.07, b = −0.27 − 0.38ω , c = −0.05 + 0.1ω , pr = p/pc . ρ, η and ω are the mixture density, viscosity and acentric factor, respectively. pc is the critical pressure. The superscript

◦

indicates ambient conditions. It should be noted that Eq. (9) was originally

developed for the prediction of binary diusion coecients in hydrocarbon systems at high pressures up to 40.0 MPa. In the present study Eq. (9) is used in combiation with Eq. (8) at supercritical conditions. That is, D12 in the term (ρD12 )◦ is calculated with Eq. (8). Kallikragas et al. 11 used the following correlation for the diusion coecient:

(10)

D12 = a1 T2a + ρ(b1 T −2 + b2 T −1 + b3 + b4 T ) + ρ2 (ln ρ) (c1 T −2 + c2 T −1 + c3 + c4 T ) + ρ2 (d1 T −2 + d2 T −1 + d3 + d4 T )

For the calculation of the diusion coecient of H 2 in supercritical water coecients for Eq. (10) are given in Table 1. The authors performed Molecular Dynamics simulations for H2 O, OH radical, O2 and H2 in water over a temperature range of 298 to 973 K and a density range of 100 to 1000 kg/m 3 . They tted the parameters in Eq. (10) to their results. Table 1: Values for the coecients in Eq. (10) for H 2 in supercritical water. 11

a1 a2 b1 b2 b3 b4 c1

116.211 0.0538917 1.0 372312 1794.21 1.42193 1.0

c2 c3 c4 d1 d2 d3 d4

6

476427 1880.99 1.62233 1.0 377905 1654.15 1.39764


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Diusion Coecients of various solutes in supercritical water Diusion coecients in supercritical solvents were measured mainly for supercritical extraction with CO2 or organic solvents. Data for diusion coecients in near-critical water were measured for various solutes such as CO 2 , 19,20 inorganic nitrate species, 21 CH4 22 or hydroquinone. 23 Magalhaes et al. 24 give a comprehensive report on the errors of various equations for diusion coecients over a wide range of temperatures and pressures. In Table 2 reported values of diusion coecients for various solutes in supercritical water are listed. Furthermore, the self-diusion coecient of water is also included for comparison. The temperature, pressure and/or density at which the measurements or simulations were performed are given. One of the most popular techniques to measure diusion coecients is the Taylor dispersion technique. This method was used by Goemans et al. 21 as well as Plugatyr and Svishchev. 25 Flarsheim et al. 23 and Liu et al. 26 used an electrochemical cell. Such measurements can also be performed with optical methods such as dynamic light scattering, 27 the laser induced grating technique 28 or a Mach-Zehnder interferometer. 29 For their measurements of the self-diusion coecients of water Lamb et al. 30 used NMR measurements. Furthermore, Molecular Dynamics simulations were successfully applied to calculate Diffusion coecients in supercritical water using the SPC/E water model. 9,25,3138 The thermodynamic equation of state for the SPC/E model of water was published by Plugatyr and Svishchev. 39 The values for the diusion coecients in SCW are generally in the order of 10 · 10−9 to

100 · 10−9 m2 /s for most solutes, Table 2. A diusion coecient of 122 · 10−9 m2 /s was obtained by Molecular Dynamics simulations for the OH radical by Svishchev and Plugatyr. 36 They argued that under supercritical conditions water has a low polarity and the OH radical has lesser ability to form hydrogen 7



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bonds. Therefore, the mobility and the diusion coecient increase. It can be seen from Table 2 that smaller molecules such as O 2 or N2 show higher diusion coecients than larger molecules such as phenol. This is also reected by the Stokes-Einstein equation (1) where the diusion coecient is inversely proportional to the radius of the molecule. Kuge et al. 29 measured diusion coecients for N 2 which are considerably higher than the measured diusion coecients for molecules of similar size (like O 2 ) at similar conditions. Kuge's value is also much higher than the value obtained by Xiao et al. 37 The authors did not compare their results to values from literature. From the limited data presented in their study no plausible reason can be found for the high values and they must therefore be considered as awed.

Results and Discussion Comparison of calculated and measured values Diusion coecients for some solutes relevant to processes with SCW are selected from Table 2 and compared to the values calculated with equations (1) to (10). The physical properties for the calculations were taken from Kleiber et al. 41 and NIST. 42 For the Stokes-Einstein equation the van der Waals radius was applied to calculate D12 . The van der Waals radii given by Bondi 43 and Edward 44 were used, Table 3. The relative deviation of the calculated values from the values from literature, rdrel , is dened as

rdrel =

value from literature − calculated value value from literature

(11)

and is shown in Figure 1. If rdrel is > 0 the correlation underpredicts the literature value, if rdrel < 0 the calculated value is larger than the one given in the reference.

8


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Table 2: Binary diusion coecients D12 of various solutes in supercritical water.

D12 T p ρ data source 2 3 [10 m /s] [K] [MPa] [kg/m ] 28a NaNO3 15.8d 673 30 Laser induced grating technique e 13.0 723 40.9 b i 37 NaCl 57 703 30 Molecular Dynamics simulations O2 c 90 703 30 N2 c 80 703 30 b 38 NaClc 57.1 703 30 Molecular Dynamics simulations − b,f 40 Cl 3539 673 400 Molecular Dynamics simulations − a Cl 41.5 673 400 23a I− 49 648 481 Electrochemical cell a,g − 26 I 61.7 658 27 449 Electrochemical cell Hydroquinone 31.7 658 27 449 32b,c Na+ 38.5 673 400 Molecular Dynamics simulations − Cl 43.5 673 400 31b Li+ 37.83 673 350 Molecular Dynamics simulations + Na 36.88 673 350 K+ 31.91 673 350 + Rb 30.26 673 350 + Cs 28 673 350 F− 33.21 673 350 Cl− 32.08 673 350 − Br 31.73 673 350 I− 30.34 673 350 9b,c Benzene 63 673 300 Molecular Dynamics simulations 36b OH radical 122 683 300 Molecular Dynamics simulations Phenol 40 733 300 a,c 25 Phenol 34.71 673 30 Taylor dispersion technique w Phenolb,c 32.45 673 400 Molecular Dynamics simulations 33b O2 82.9 647 426 Molecular Dynamics simulations 34b Methane 78 647 426 Molecular Dynamics simulations 35b,c O2 81 673 426 Molecular Dynamics simulations a,h 29 N2 660 673 27.6 Mach-Zehnder Interferometry 27a PSLj 0.05 647 30.3 Dynamic Light Scattering CGk 0.15 647 30.3 Dynamic Light Scattering a,l 30 H2 O 89.7 673 31.4 NMR measurements 251 773 31.4 a experimental; b Molecular Dynamics simulation; c innite dilution; d 0.25 M solution; e 1.0 M solution; f dierent models; g 0.2 M NaHSO4 solution; h xN2 = 0.05; i 0.437 M solution; j polystyrene sulfate; k colloidal gold l self-diusion coecient ref.

solute

−9

9



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Table 3: Van der Waals radii for the molecules used in the present study. molecule

H2 O2 N2 CO CO2 CH4 Benzene Phenol

r1 [10−10 m] 1.35 1.73 1.84 1.86 1.98 1.89 2.68 2.77

ref.

44 43 43 43 43 43 43 44

The literature values for Eq. (11) were taken from the following references: Refs. 37 and 33 for O2 , Ref. 37 for N2 , Ref. 25 for phenol, Ref. 9 for benzene, and Ref. 34 for methane. The reference values from the literature and the calculated ones are listed in Table 4. Table 4: Calculated values of the binary diusion coecients in supercritical water, D12 in [10−9 m2 /s] solute ref. T [K] p [MPa] value from literature Eq. (1) Eq. (2) Eq. (3) Eq. (4) Eq. (5) Eq. (6) Eq. (8) & (9)

*

O2 Methane Benzene Phenol 33 34 9 25 647.15 647.15 673.15 673.15 22.3* 22.3* 28.8* 30 83 78 63 34.71 55.2 50.5 47.6 40.5 89.4 74.5 56.9 48.6 75.2 67.9 52.9 45.2 68.5 61.9 60.5 52.4 79.3 67.0 56.2 48.3 75.5 106.8 65.2 52.7 20.8 21.1 14.8 11.3 calculated from given density data

O2 37 703.15 30 90 94.7 153.3 128.9 117.5 135.9 129.0 60.0

N2 37 703.15 30 80 89.0 134.7 120.0 109.4 120.5 137.9 59.9

When pressures were not given in studies where Molecular Dynamics simulations were performed they were calculated based on Wagner and Pruss 1 using the temperature and density, p = p(T, ρ). The rst two bars in Figure 1 correspond to conditions near the critical point (O 2 and methane). The last 4 bars (benzene, phenol, O 2 , and N2 ) correspond to conditions which are above the critical point. 10


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80% O2, Ohmori et al. (2002)

60%

Methane, Ohmori et al. (2003) Benzene, Nieto-Draghi et al. (2004) Phenol, Plugatyr et al. (2011) O2, Xiao et al. (2011)

40%

N2, Xiao et al. (2011)

20%

rdrel [-]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0%

-20%

StokesEinstein Eq. (1)

Wilke-Chang Eq. (2)

Scheibel Eq. (3)

ReddyDoraiswamy Eq. (4)

LusisRatcliff Eq. (5)

He-Yu Eq. (6)

Fuller & Riazi Eq. (8) & (9)

-40%

-60%

-80%

Figure 1: Relative deviations of the calculated binary diusion coecients for typical solutes in SCW processes. The references indicate the source of the reference data for the calculation of the relative deviation.

11



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As it can be seen from Figure 1 the Stokes-Einstein equation (1) predicts the measured values near the critical point with a relative deviation of more than +30%, whereas predictions are better for conditions well above the critical point with relative deviations of below 20%, except for benzene. Eqs. (2) to (5), which are based on the Stokes-Einstein equation, underpredict the diusion coecients in the near-critical region with only one exception (Wilke-Chang for O2 ) and give relative deviations around or below +20%. However, they produce larger errors at conditions above the critical point, except for benzene. Relative deviations are consistently larger than 30%. The equation by Fuller et al. (8) combined with the correction by Riazi and Whitson (9) underpredicts all values by more than 20%. However, the predictions are better for O 2 and N2 at supercritical conditions. Interestingly, all correlations (2) to (6) predict the diusion coecient of benzene with a relative deviation of below ± 20%. The reason why benzene is estimated particularly well is not known. The diusion coecient of phenol, which is of similar size, is predicted with higher relative deviations. Under supercritical conditions water behaves as a low polarity solvent. 36 Therefore, the hydrogen bonds with the OH group of phenol are supposed to be weak. Probably this interaction which the water molecules is still strong enough to hinder the free movement of phenol, as opposed to the fully symmetrical benzene molecule. The equation of He and Yu, Eq. (6), produces relative deviations below ±10% for O2 at near-critical conditions and for benzene. For the other cases the relative deviation is between 35% and 75%. For O2 Eq. (10) by Kallikragas et al. 11 can also be used. The value of O 2 in the nearcritical region can be reproduced with a relative deviation of 9%. The value of O 2 in the supercritical region is calculated with a much higher relative deviation of 141%. The density of water at 703 K and 30 MPa used for the calculation is 177 kg/m 3 . This is close to the lower limit of the range of validity of Eq. (10) which might explain the large deviation.

12


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For SCWG and SCWO the typical conditions are above the critical point, so the StokesEinstein equation can be used to estimate the diusion coecients with acceptable accuracy. Its advantages are its simple structure and its small number of parameters. However, data of the van der Waals radii have to be provided. For conditions near the critical point, the Stokes-Einstein equation produces inaccurate results, so its use is not recommended for these conditions. In this case the Wilke-Chang equation gives the best results for both methane and O 2 . Note that these ndings are based on a very small number of experimental data points. A much broader data set is needed for a more generalized recommendation, i.e. for other or larger molecules and/or other conditions. The inuence of inaccurately measured molar volumes of the solute on the diusion coefcients were also examined. This was done to determine how uncertainties in the properties would aect the calculated diusion coecients. The molar volume of the solute was chosen because it is the parameter for which the largest uncertainties may occur. The relative deviation for the molar volume, rdrel,vol , is dened as follows:

rdrel,vol =

inaccurate volume − correct volume correct volume

(12)

The calculation was done for the value of phenol measured by Plugatyr et al. 25 and the result is shown in Figure 2. This sensitivity analysis shows that a relative deviation in the molar volume can aect the diusion coecients signicantly. For example, for the WilkeChang equation, the diusion coecient varies from over 70 · 10−9 m2 /s to below 40 · 10−9 m2 /s when the relative deviation for the molar volume, rdrel,vol , used for the calculation is varied from 50% to +50%. The molar volume will probably not change within such a wide range, but the result shows that the input parameters for the correlation can have a strong inuence on the calculated diusion coecient.

13



80 diffusion coefficient [10-9 m²/s]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

70 60 50 40 30 20 10 0 -50%

Wilke-Chang, Eq. (2) Scheibel, Eq. (3) Reddy-Doraiswamy, Eq. (4) Lusis-Ratcliff, Eq. (5)

-25%

0%

25%

50%

rdrel,vol [-]

Figure 2: Inuence of relative deviations of measured molar volumes at p = 30 MPa and T = 673 K on the calculated binary diusion coecients for phenol in water.

Inuence of pressure and temperature on the diusion coecient The Stokes-Einstein equation (1) predicts a direct proportionality of D12 with temperature. Furthermore, the dependence of the viscosity on temperature and pressure has an indirect eect on D12 since the viscosity of water decreases with temperature and increases with pressure. A more complex dependence on temperature and pressure is found in the equations of He and Yu, Eq. (6), as well as Fuller and Riazi, Eq. (8) and (9), respectively, including a pressure correction term. No experimental data of diusion coecients in supercritical water is available for which a large range of temperature and pressure have been investigated. Nieto-Draghi et al. 9 found a value for benzene of 63 · 10−9 m2 /s for a temperature of 673 K and a density of 300 kg/m3 , which corresponds to a pressure of 28.8 MPa. Plugatyr et al. 25 measured a diusion coecient for phenol which is of similar size as benzene and found a value for 673 K and 30 MPa of 34.72 · 10−9 m2 /s. From this comparison one can tentatively infer the diusion coecient to decrease with increasing pressure. This is plausible according to the Stokes-Einstein equation since the viscosity of the solvent (water) is in the denominator and it increases with pressure, which leads to a decreasing diusion coecient. 14


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Xiao et al. 37 performed Molecular Dynamics simulations for O 2 and also investigated the inuence of temperature and pressure on the diusion coecient. The calculated results are compared to results of the Stokes-Einstein equation, the Wilke-Chang equation, the He-Yu equation and the Fuller equation to examine whether these equations are able to predict the temperature and pressure dependence of the diusion coecient. Figure 3 and Figure 4 show the predicted pressure and temperature dependence of the diusion coecient. The simulation results by Xiao et al. 37 predict decreasing diusion coecients with increasing pressure which is also qualitatively predicted by the used correlations. The equations of Wilke-Chang and He-Yu overpredict the diusion coecients obtained by Molecular Dynamics simulations. The Stokes-Einstein equation predicts the rst point well but predicts then a decrease of the diusion coecient that is too small compared to the Molecular Dynamics simulations. The Fuller equation predicts a steep decrease of the diusion coecient for pressures below 40 MPa but then also levels out. In the supercritical regime all of the correlations predict a temperature dependence that is too small the temperature dependence starts leveling o at about 693 K. The Molecular Dynamics simulations predict a steeper increase in the diusion coecient above 700 K. 180

diffusion coefficient [10-9 m²/s]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


160

140 120 Xiao et al. (2001)

100

Stokes-Einstein

80

Wilke-Chang

60

He-Yu

40

Fuller et al.

20 0 24

28

32 36 40 pressure [MPa]

44

48

Figure 3: Pressure dependence of the binary diusion coecient of O 2 in water at 703 K, calculated with the correlations by Wilke-Chang (2), He-Yu (6), Stokes-Einstein (1), Fuller and Riazi (8), (9), and with Molecular Dynamics simulations.

15



250 diffusion coefficient [10-9 m²/s]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

200 150

Wilke-Chang He-Yu

100

Stokes-Einstein Xiao et al. (2001) Fuller

50 0 660

680

700 720 740 temperature [K]

760

780

Figure 4: Temperature dependence of the binary diusion coecient of O 2 in water at 30 MPa, calculated with the correlations by Wilke-Chang (2), He-Yu (6), Stokes-Einstein (1), Fuller and Riazi (8), (9), and with Molecular Dynamics simulations.

Prediction of binary diusion coecients for H

2,

CO 2, CO, and CH 4

There are no diusion coecients available in literature for the main reaction products in SCWG processes, which are H 2 , CO, CO2 , and only one value for CH 4 . To give a rst estimation for these values the diusion coecients were calculated with the correlations (1) to (9) for process conditions typical for catalytic SCWG, 2 i.e. for 673 K and 30 MPa. Additionally, the diusion coecients for O 2 and N2 were also calculated at these conditions. The results are shown in Figure 5. For all molecules but H 2 the correlations by StokesEinstein, Wilke-Chang, Scheibel, Reddy-Doraiswamy, Lusis-Ratcli, He-Yu and Kallikragas predict diusion coecients between 50 and 110 · 10−9 m2 /s. The equation by Fuller with the correction by Riazi give considerably lower values in the range of 25 · 10−9 m2 /s. For H2 all correlations give diusion coecients in the range of 80 to 110 · 10−9 m2 /s except for the correlations of He-Yu and Kallikragas. For high temperatures and low molar weights the term (T /M1 )1/2 in the He-Yu equation becomes large and so does the diusion coecient. The equation by Kallikragas is tted to simulation results and also gives a value which is signicantly higher than the other correlations. According to the Stokes-Einstein equation the ratio of the diusion coecients for specic 16


Page 16 of 27

Page 17 of 27

400 Stokes-Einstein, Eq. (1)

350

diffusion coefficient [10-9 m²/s]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Wilke-Chang, Eq. (2) Scheibel, Eq. (3)

300

Reddy-Doraiswamy, Eq. (4) Lusis-Ratcliff, Eq. (5)

250

He-Yu, Eq. (6) Fuller and Riazi, Eq. (8) & (9)

200

Kallikragas, Eq. (10)

150 100 50 0 O2 O2

N2 N2

H2 H2

CO

CO CO2 2

CH4 CH4

Figure 5: Estimated binary diusion coecient for various gases in water at 673 K and 30 MPa. conditions remains constant if solute and solvent remain the same. The binary diusion coecients for H2 and O2 in water at ambient conditions and at innite dilutions are: 8

DH2 = 45 · 10−9 m2 /s and DO2 = 21 · 10−9 m2 /s, respectively, which corresponds to a ratio of approximately 2.1. One would thus expect based on the Stokes-Einstein equation (1) that the diusion coecient for H 2 is also about 2.1 times larger than the one of O 2 at supercritical conditions. Therefore, it seems that the best estimation of the diusion coecient for hydrogen is obtained by the equation of Kallikragas which predicts a value of

250 · 10−9 m2 /s. For all considered molecules (O 2 , N2 , H2 , CO, CO2 , and CH4 ) recommended values of the binary diusion coecients in supercritical water are given for the following conditions: the critical point (647 K, 22.1 MPa), typical conditions during catalytic SCWG (673 K, 30.0 MPa) and conditions at higher temperature (773 K, 30.0 MPa). Figure 6 shows the calculated values for the three conditions. 17



The binary diusion coecients for O 2 , N2 , CO, CO2 , and CH4 at the critical point (647 K, 22.1 MPa) are calculated by using the correlation of Wilke-Chang, Eq. (2), since it performed best for this condition. For conditions well above the critical point (673 K and 773 K, 30.0 MPa) the Stokes-Einstein equation, Eq. (1), seems the most suitable. Therefore, the binary diusion coecients at these conditions for O 2 , N2 , CO, CO2 , and CH4 were calculated with the Stokes-Einstein equation. The correlation by Kallikragas, Eq. (10), was used to calculate the binary diusion coecients for H 2 . The range of validity of the correlation by Kallikragas ends at ρ = 100 kg/m3 . Since the density at 773 K and 30 MPa is 177 kg/m 3 , which is close to the limit of Eq. (10), no bar was drawn for the third condition. However, the given recommendations in Figure 6 are also based on values obtained by Molecular Dynamics simulations which are subject to uncertainties. In Figure 6 the diusion coecients decrease from the critical condition (647 K, 22.1 MPa) to the supercritical condition (673 K, 30.0 MPa). The pressure dependence overcompensates the dependence on temperature. Increasing the temperature at the supercritical condition to 773 K leads to an increase of the binary diusion coecient. 300 diffusion coefficient [10-9 m²/s]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

647 K, 22.1 MPa

250

673 K, 30.0 MPa 773 K, 30.0 MPa

200 150 100 50 0 O2 O2

N2 N2

H2 H 2

CO CO

CO2 CO2

CH4 CH4

Figure 6: Recommended values for the binary diusion coecients of various species in supercritical water for three dierent conditions.

18


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Summary and Conclusion Diusion coecients of various solutes in supercritical water were reviewed. It was found that diusion coecients are mostly in the range of 10 to 100 · 10−9 m2 /s. Various correlations were tested regarding their applicability to predict the measured diffusion coecients in supercritical water. The Stokes-Einstein equation (1) predicts diusion coecients for conditions above the critical point with absolute relative deviations of below 20%. The other correlations, i.e. the Wilke-Chang equation (2), the Scheibel equation (3), the Reddy-Doraiswamy equation (4), the Lusis-Ratcli equation (5) and the He-Yu equation (6) produce inaccurate results for the solutes O 2 , N2 , and phenol above the critical point. These correlations perform similarly or better than the Stokes-Einstein equation only for benzene. Processes with supercritical water, such as supercritical water gasication and supercritical water oxidation, work at temperatures and pressures above the critical point. Therefore, the Stokes-Einstein equation seems to be the most useful equation to estimate the diusion coecients for mass transfer calculations at these conditions. The advantages of the Stokes-Einstein-equation are its simplicity and its low number of input parameters. For conditions near the critical point, the Wilke-Chang equation gives the best results: The absolute relative deviations are below 20%. The Stokes-Einstein equation produces relative deviations larger than 40% for these conditions. Inaccurate input parameters, demonstrated for the molar volume of the solute phenol, can have a great inuence on the calculated diusion coecients. For the Wilke-Chang equation the diusion coecient varies within a range of 70 · 10−9 m2 /s to below 40 · 10−9 m2 /s when the relative deviation for the molar volume, used for the calculation are varied from 50% to +50%. Furthermore, diusion coecients were calculated with all the correlations discussed in this paper for the main reaction products (O 2 , N2 , H2 , CO, CO2 , and CH4 ) in supercritical water processes at the process conditions of 673 K and 30 MPa. 19



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

It was shown that diusion coecients for these molecules are in the range of 75 to

90 · 10−9 m2 /s. However, great discrepancies were found for H 2 . The correlations by He-Yu and Kallikragas gave values which were larger by a factor of about 2 to 3. Since at ambient conditions the diusion coecient of H 2 is also signicantly larger than other molecules like O2 , these higher values for H 2 are considered the most plausible ones. Generally, a signicant extension of the measured data set to include other solutes, pressures, and temperatures is necessary in order to be able to make more generalized recommendations which correlation should be used for which conditions. In summary, based on the analysis in this paper, the equation by Wilke-Chang should be used for conditions near the critical point. For conditions above the critical point, the StokesEinstein equation is recommended, except for H 2 , for which the correlation of Kallikragas produces more plausible results.

Nomenclature symbol

meaning

SI units

a, b , c

coecients in Eq. (9)

ai , b i , c i

coecients in Eq. (10)

D12

diusion coecient of solute 1 solvent 2

m 2 /s

k

Stefan-Boltzmann constant

W/(m 2 K4 )

M

molar mass

kg/mol

p

pressure

Pa

r

molecular radius

m

rdrel

relative deviation

rdrel,vol

relative deviation for the volume

T

temperature

K

x

molar fraction

mol/mol

20


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Greek letters symbol

meaning

SI units

α

constant in Eq. (10)

β

constant in Eq. (2)

ζ

special atomic diusion volume in Eq. (8)

η

dynamic viscosity

kg/(m s)

ρ

density

kg/m3

υ

molar volume at normal boiling point at 1 atmosphere

m 3 /mol

Φ

constant in Eq. (6)

ω

acentric factor

Ω

constant in Eq. (4)

Indices symbol

meaning

1

solute

2

solvent

r

reduced quantity

c

critical quantity

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Table of content graphic supercritical

p

T Figure 7: Table of content graphic.

27


Estimation of Binary Diffusion Coefficients in ... - ACS Publications

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