Chemical Engineering Science 65 (2010) 5092–5104
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Experimental measurement of the effective diffusion and thermodiffusion coefficients for binary gas mixture in porous media H. Davarzani a,b, M. Marcoux a,b,n, P. Costeseque a,b, M. Quintard a,b a ´canique des Fluides de Toulouse), GEMP (Groupe d’Etude des Milieux Poreux) Alle´e Camille Soula, Universite´ de Toulouse; INPT, UPS, IMFT (Institut de Me F-31400 Toulouse, France b CNRS, IMFT; F-31400 Toulouse, France
a r t i c l e in f o
a b s t r a c t
Article history: Received 8 January 2010 Received in revised form 24 May 2010 Accepted 9 June 2010 Available online 16 June 2010
Thermodiffusion or Soret effect, corresponding to a mass flux caused by a temperature gradient applied to fluid mixture, has been taken into account in many porous media applications, particularly in chemical engineering and geophysics. In the literature, the effective macro-scale diffusion coefficients are now well established, while uncertainty remains concerning the relationship between the effective thermodiffusion coefficient and micro-scale parameters (such as pore-scale geometry). Our previous study on theoretical model of effective thermodiffusion coefficient for a pure diffusion regime confirmed that the tortuosity factor acts in the same way on both Fick diffusion coefficient and on thermodiffusion coefficient. In this study, new experimental results obtained with a two bulb apparatus are presented. The diffusion and thermodiffusion of a helium–nitrogen and helium–carbon dioxide system through cylindrical samples filed with glass spheres of different diameter are measured at the atmospheric pressure. Concentrations are determined by analyzing the gas mixture composition in the bulbs with a katharometer device. A transient-state method for coupled evaluation of thermodiffusion and Fick coefficient in two bulbs system is proposed. The obtained results are in good agreement with theoretical results from previous study and emphasize the porosity of the medium influence on both diffusion and thermodiffusion process. The tortuosity of the medium calculated using both effective diffusion and effective thermodiffusion coefficients are the same. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Thermodiffusion Diffusion Soret effect Porous media Two bulb method Volume averaging technique
1. Introduction Transport phenomena in porous media have received considerable attention due to the increasing interest in geothermal processes, petroleum reservoirs, chemical catalytic reactor, waste storage (especially geological or ocean storage of carbon dioxide), etc. Thermodiffusion (or Soret effect) is a heat and mass coupled phenomenon corresponding to the mass flux caused by a temperature gradient. This phenomenon must be taken into account where a multicomponent mixture is subjected to a thermal gradient, including in many porous media applications, in particular in petroleum engineering and geophysics for prediction of the compositional variation in hydrocarbon reservoirs. Related to coupled-transport phenomena, the classical diffusion equation is completed with an additional thermodiffusion
Corresponding author at: Universite´ de Toulouse; INPT, UPS, IMFT (Institut de Me´canique des Fluides de Toulouse), GEMP (Groupe d’Etude des Milieux Poreux) Alle´e Camille Soula, F-31400 Toulouse, France. Tel.: + 33 0 534322876; fax: + 33 0 5 34322899. E-mail address:
[email protected] (M. Marcoux). n
0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.06.007
term. The mass flux, considering a mono-dimensional problem of diffusion, in the x-direction, for a binary system, no subjected to external forces, and in which the pressure, but not the temperature, is uniform, can be written @c1 @T J1 ¼ ÿrb D12 þ DT ð1Þ @x @x where D12 is the ordinary diffusion coefficient and DT the thermodiffusion coefficient. Defining thermodiffusion ratio kT ¼TDT/D12, we can also write (as in Landau and Lifschitz, 1982) @c1 kT @T þ ð2Þ J1 ¼ ÿrb D12 @x T @x Other quantities encountered are the thermodiffusion factor,
aT, (for gases) and the Soret coefficient, ST, defined in literatures by aT ¼ kT =c10 c20 and ST ¼kT/T, respectively.
When kT in Eq. (2) is positive, heaviest species (1) moves toward the colder region, and when it is negative, this species moves toward the warmer region. In some cases, there is a change in sign of the thermodiffusion ratio as the temperature is lowered (see Chapman and Cowling, 1970 and Caldwell, 1973).
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
In a previous study (Davarzani et al., 2010) we developed a theoretical model to predict effective thermodiffusion coefficients from micro-scale parameters (thermodiffusion coefficient, porescale geometry, thermal conductivity ratio and Pe´clet numbers). The results confirm that for a pure diffusion regime, the effective Soret number in porous media is the same as the one in the free fluid. This means that the tortuosity factor acts in the same way on the Fick diffusion coefficient and on the thermodiffusion coefficient. In the present work, the influence of pore-scale geometry on effective thermodiffusion coefficients in gas mixtures have been measured experimentally. In the literature, numerous measurements were made in free medium in 50 and 60 s (see for instance some series of measurements which were done in Grew and Wakeham, 1971; Heath et al., 1941; Humphreys and Gray, 1970; Ibbs and Chapman, 1921; van Itterbeek et al., 1947; Mason et al., 1964, Saxena et al., 1961; Shashkov et al., 1979; Zhdanov et al., 1980). By now direct gas thermodiffusion in porous medium data are not given and there is still uncertainty concerning the question of the relationship between the effective liquid thermodiffusion coefficient and micro-scale parameters (such as pore-scale geometry) (Costeseque et al., 2004; Jamet et al., 1996; Platten, 2006; Platten and Costeseque, 2004). Davarzani et al. (2008) and Davarzani et al. (2010) developed a theoretical model to predict effective thermodiffusion coefficient from micro-scale parameter (thermodiffusion coefficient, pore-scale geometry, thermal conductivity ratio and Pe´clet numbers). The results confirm that for a pure diffusion regime, the effective Soret number in porous media is the same as the one in the free fluid and that it does not depend on the solid to fluid conductivity ratio. On the opposite, in convective regime the effective Soret number decreases. In this case, a change of conductivity ratio will change the effective thermodiffusion coefficient as well as the effective thermal conductivity coefficient. This means that at a pure diffusion regime, the tortuosity factor acts in the same way on Fick diffusion coefficient and on thermodiffusion coefficient. In the present work, the influences of pore-scale geometry on effective thermodiffusion coefficient in gas mixture have been measured experimentally. The use of gaseous mixture has the advantage that the relaxation time will be very smaller compared with a liquid mixture. The main purpose of this paper is to obtain the relationship between the effective diffusion and thermodiffusion coefficients with porosity in porous media. To achieve this goal, we used a specially designed two-bulb apparatus for measuring directly the diffusion and thermodiffusion coefficient for the systems He–N2 and He–CO2 in free medium and non-consolidated porous medium of different porosity. In Section 2, we explain in detail the different parts of the experimental setup. Then, the principle of measurement and calculation for diffusion and thermodiffusion coefficients in twobulb cell will be discussed, respectively. The porosity of the porous samples measured and experimental results for diffusion and thermodiffusion process are presented. Finally, the results are discussed and compared with theoretical results.
2. Experimental setup and working equations In this study we have designed and made-up a new experimental setup that has proven to provide suitable results for the study of diffusion and thermodiffusion in free fluid. It is an allglass two-bulb apparatus, containing two double-spherical layers t (top) and b (bottom) as shown in Fig. 1. The two-bulb technique is the most widely used method for determining the diffusion coefficients (Yabsley and Dunlop, 1976) and thermodiffusion
5093
(Grew and Ibbs, 1952) coefficients in binary and ternary gases as well as liquids, with accurate results. The basic arrangement of a two-bulb cell consists of two chambers of relatively large volume joined by a small-volume diffusion tube. The two-bulb method is a suitable method for this study as we can measure simultaneously both diffusion and thermodiffusion coefficients. In this study, this technique is extended to measure the transport coefficients in porous media. Finally, in this method, thermodiffusion process is not disturbed by natural convection which is negligible in this system. The specific difference between this system and the earliest two-bulb systems is that each bulb contains an interior glass sphere to serve as reservoir bulb and another exterior glass spheres in which, in the space between two glass layers there may be a water circulation to regulate the reservoir temperature. The reservoir bulbs with equal and constant volume Vt ¼Vb ¼ 1000 cm3, are joined by an insulated rigid glass tube of inner diameter d ¼0.795 cm and length 8 cm containing a valve also made especially of 0.795 cm bore, and 5.87 cm long. Therefore, the total length of the tube in which the diffusion processes occur is about ‘ ¼ 13:87 cm. To avoid convection, the apparatus was mounted vertically, with the hotter bulb uppermost. The concentration of binary gas mixtures is determined by analyzing the gas mixture composition in each bulb with a katharometer (Daynes 1933; Jessop, 1966). The method is based on the ability of gases to conduct heat and the property that the thermal conductivity of a gas mixture is a function of the concentration of its components. The thermal conductivity of a gas is inversely related to its molecular weight. Hydrogen has approximately six times the conductivity of nitrogen for example. The thermal conductivity of some gases with corresponding katharometer reading at atmospheric pressure is listed in Table 1. Katharometer which is commonly used as conductivity detector in many experiments and industries, without a chromatographic column, is limited to binary mixtures. It is simple in design and requires minimal electronic support and, as a consequence, is also relatively inexpensive compared with other detectors. Its open cell can form part of the diffusion cell, and so it can indicate continuously the changes in composition without sampling. This is why that in this study, we have used a conductivity detector method with katharometer to analyze the separation process in a two-bulb method. In this study, we have used the analyzer ARELCO-CATARC MP-R model with a sensor operating on the principle of thermal conductivity detection. The electronics high-performance microprocessor of this device allows analyzing the binary gas mixtures with 70.5% repeatability. This type of katharometer works with a circulation of the analyzed and reference gases into the sensors. The first series of the experiment showed that the sampling with circulation cannot be applied in the two-bulb method because gas circulation can perturb the establishment of the temperature gradient in the system. Small changes in the pressure in one bulb may produce forced convection in the system and cause a great error in the concentration evaluation. Therefore, in this study we have eliminated the pump system between the bulbs and katharometer sensors. Instead we have connected the katharometer analyzer sensor directly to the bulbs as shown in Fig. 2. Therefore, the open cell of the katharometer form a part of the diffusion cell, and so it can indicate continuously and without sampling the changes in composition as diffusion and thermodiffusion processes. The other sensor of the katharometer has been sealed permanently with air inside and the readings are the reference readings. For gases, the diffusion coefficient is inversely proportional to the absolute pressure and directly proportional to the absolute temperature to the power 1.75 as given by the Fuller et al. (1969).
5094
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
Vacuum pump
Manometer He
N2
Diffusion zone
Top Bulb
Sensor
Bath temperature controller Valve
Bottom Bulb
Katharometer Manometer
Vacuum pump Fig. 1. Sketch of the two-bulb experimental setup used for the diffusion and thermodiffusion tests.
Table 1 Thermal conductivity and corresponding katharometer reading for some gases at atmospheric pressure and T¼ 300 K. Gas
Air
N2
CO2
He
k (W/m K) (Kaviany, 2009) RK (mV)
0.0267 1122
0.0260 1117
0.0166 976
0.150 2345
An analysis of binary diffusion in the two-bulb diffusion apparatus has been presented by Ney and Armistead (1947). It is assumed that each bulb is at a uniform composition. It is further assumed that the volume of the capillary tube connecting the bulbs is negligible in comparison to the volume of the bulbs themselves. This allows expressing the component material balances for each bulb as follows:
rb Vb
dcib dc ¼ ÿrb Vt it ¼ ÿNi A dt dt
ð3Þ
Pressure and temperature measurements are made with two manometers and thermometers. The temperature of each bulb is kept at a constant value by water circulation from a bath temperature controller. The temperature of the katharometer is controlled with an electrical self-thermostated sensor. In this study, for all diffusion measurements, the temperature of two bulbs system is fixed to 300 K. The gas purities are: He: 100%, N2: 100% and CO2: 100%.
where A is the cross-sectional area of the capillary tube, cit is the mass fraction of component i in the top bulb and cib is the mass fraction of that component in the bottom bulb. The mass flux of species i through the capillary tube Ni is considered to be positive if moving from top bulb to bottom bulb. The density can be computed from the ideal gas law at the average temperature T
2.1. Diffusion in a two-bulb cell
rb ¼
The two-bulb diffusion cell is a simple device that can be used to measure diffusion coefficients in binary gas mixtures. Fig. 1 is a schematic of the setup. Two vessels containing gases with different compositions are connected by a capillary tube. The katharometer sensor itself is connected with the bulb and its volume is negligible compared to the volume of the bulbs. The katharometer cell and the two bulbs were kept at a constant temperature of about 300 1C. The vacuum pumps are used at the beginning of the experiment to eliminate the gas phase initially in the diffusion cell and in the gas flow lines. At the beginning of the experiment (t ¼0), the valve is opened and the gases in the two bulbs can diffuse along the capillary tube.
P RT
ð4Þ
at constant temperature and pressure, the density of an ideal gas is a constant; thus, there is no volume change on mixing and in the closed system the total flux Nt must be zero. The composition in each bulb at any time is related to the composition at equilibrium ci1 by ðVt þ Vb Þci1 ¼ Vt cit þ Vb cib
ð5Þ
0 ðVt þ Vb Þci1 ¼ Vt cit0 þ Vb cib
ð6Þ
The compositions at the start of the experiment are, therefore, related by
0
where c is the mass fraction at time t ¼0. In the analysis of Ney and Armistead it is assumed that, for i¼1 at any moment, the flux N1 is given by its one-dimensional,
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
Heated metal block Sensor connections to Wheatstone bridge
5095
Gas from the bulb to measure (analyzed gas)
Heated metal block Reference gas
Fig. 2. A schematic of katharometer connection to the bulb.
give an effective length ‘eff , given by
steady-state diffusion flux as J1 ¼
rb D12 ‘
ðc1b ÿc1t Þ
ð7Þ
Thus (J1 ¼ N1),
rb Vb
dc1b D12 ¼ ÿrb Aðc1b ÿc1t Þ dt ‘
ð8Þ
To eliminate c1t from Eq. (8) one makes use of the component material balance for both bulbs, Eqs. (5) and (6) dc1b ¼ ÿbD12 ðc1b ÿc11 Þ dt
ð9Þ
where b is a cell constant defined by
b¼
ðVt þ Vb ÞA ‘Vt Vb
ð10Þ
A similar equation for the mass fraction of component 2 in bulb t may also be derived. Eq. (9) is easily integrated, starting from the initial condition o that at t ¼0, c1b ¼ c1b , to give 0 ÿc11 ÞexpðÿbD12 tÞ þ c11 c1b ¼ ðc1b
One can also write this equation as ! c1b ÿc11 ln 0 ¼ ÿbD12 t c1b ÿc11
ð11Þ
ð12Þ
One can conclude from this equation that the variation of the logarithmic dimensionless concentration is a linear function of the time. As a results, when b is known then one value of cb is all that is needed to calculate the diffusivity D12. Alternatively, if an accurate value of D12 is available, Eq. (11) can be used to calibrate a diffusion cell for later use in measuring diffusion coefficients of other systems. In this study, the volume of the two bulbs is equal, Vt ¼Vb. In this case, we can write Eqs. (6) and (10) as ÿ 0 ci1 ¼ cit0 þ cib =2 ð13Þ
b¼
2A ‘V
ð14Þ
‘eff ¼ ‘ þ 0:82d
ð15Þ
where d is the tube diameter. Rayleigh (1945), when investigating the velocity of sound in pipes, showed that one must add 0.41d for thick annulus flange and 0.26d for a thin annulus flange to each end of the tube. Here, r is the tube radius. Wirz (1947) showed that the end corrections for sound in tubes depend on the annulus width, w, and diameter, d. The results fit the correlation (Yabsley and Dunlop, 1976) ÿ0:125d w ¼ 0:60 þ 0:22 exp ð16Þ w where w is the end-correction factor. Analysis of many results on diffusion both in porous media and bulk gas also showed a significant difference between diffusion coefficients measured in different cells (Weller et al., 1974). This difference may arise through a difference in geometry affecting the diffusion (say cell effect) or, in the case of the capillary tube, the end correction factor being incorrect. More recent work indicates that the effect is due to differences in cell geometry (Watts, 1964). The existence of this difference implies that all measurements of bulk gas diffusion by the two-bulb technique may contain systematic errors up to 2% (Weller et al., 1974). Arora et al. (1977) using precise binary diffusion coefficients showed that the end correction formulation is not precise enough when an accuracy of 0.1% in coefficients is required. However, they proposed to calibrate the two-bulb cells with the standard diffusion coefficients. According to this short bibliography, calculated diffusion coefficients in a two-bulb apparatus depend on the cell geometry and end connection tubes. Then, in this study, we will use the standard values of diffusion coefficients to calibrate the two-bulb apparatus for effective tube length. In our work concerning the determination of tortuosity, this error may be small because we have calculated a ratio of the two diffusion coefficients. However, a better understanding of this problem requires doing more experimental or numerical studies.
where V is the bulb volume. 2.3. Thermodiffusion in a two-bulb cell 2.2. Two-bulb apparatus end correction When we determine the diffusion coefficient in a two-bulb system connected with a tube, the concentration gradient does not terminate at the end of the connecting tube, and therefore an end-correction has to be made. This correction was made in the calculation of the cell constants as an end-effect by Ney and Armistead (1947). They adjust the tube length L for end effects to
For calculation of the magnitude of the Soret effect we used the same setup that we have used for diffusion processes. The diameter of the tube is small enough to eliminate convection currents and the volume of the tube is negligible in comparison with the volume of the bulbs. In the initial state, the whole setup is kept at a uniform and constant temperature T0 and the composition of the mixture is
5096
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
uniform everywhere. After closing the valve in the tube, the temperature of the top bulb is increased to TH and the temperature of bottom bulb is lowered to TC, the two bulbs are set at the same pressure. After this intermediate state, the valve is opened. After a short time, a thermal stationary state is reached, in which there is a constant flux of heat from bulb t to bulb b and mass transfer occurs. Measures have been taken such that TC and TH remain constant and, due to the Soret effect, it is observed a difference in mass fraction between the bulbs. Thermodiffusion separation is determined by analyzing the gas mixture composition in the bulbs by katharometric analysis. At steady-state, the separation due to thermodiffusion is balanced by the mixing effect of the ordinary diffusion, there is no net motion of either 1 or 2 species, so that J1 ¼ 0. If we take the tube axis to be in the x-direction, then from Eq. (2) we get @c1 kT @T ¼ÿ @x T @x
TH TC
ð17Þ
ð18Þ
Then the thermodiffusion factor aT is calculated from the following relation:
aT ¼
ÿDc1 ÿ c10 c20 ln TH =TC
ð19Þ
here, c10 and c20 are the initial mass-fractions of the heavier and lighter components, respectively, in the binary gas mixture, and 1 1 Dc1 ¼ c1b ÿc1t . aT values thus obtained refer to an average temperature, T, in the range TC to TH (Grew and Ibbs, 1952; Saxena and Mason, 1959) and these are determined from the formula of Brown (1940) according to which TH TC TH T¼ ln TH ÿTC TC
ð20Þ
which is based on an assumed temperature dependence for aT of the form aT ¼a ÿ bT ÿ 1. The relaxation time tt for this process can be expressed as (Saxena and Mason, 1959)
tt ffi
V‘ TC D12 A TC þ TH
In this section, a transient-state method for thermodiffusion process in a two-bulb apparatus is proposed. In this case, the flux is the sum of Fick diffusion flux and thermodiffusion flux, as D12 D12 kT TH J1 ¼ rb ðc1t ÿc1b Þÿrb ln ð22Þ ‘ ‘ TC at thermal equilibrium and for one-dimensional case. Then the concentration variation in the bottom bulb is given by r D12 rb D12 kT TH dc rb Vb 1b ¼ ÿ b ðc1b ÿc1t Þ þ ln ð23Þ dt ‘ ‘ TC The compositions at the starting of the experiment are related by 0 ¼ cit0 cib
We may ignore the effect of composition on kT, as the variations are small, and integrate this equation on temperature gradient between TC and TH to get the change in concentration of the heavier component at the steady-state in the lower bulb (Shashkov et al., 1979). This gives
Dc1 ¼ ÿkT ln
2.4. A transient-state method for thermodiffusion processes
ð21Þ
where V is the volume of one of the bulbs. The relaxation time is therefore proportional to the length of the connecting tube, and inversely proportional to its cross-sectional area. The approach to the steady-state is approximately exponential, and this was confirmed by following measurements. The variation of pressure is small in each experience. Theory and experiment agree in showing that, at least at pressure below two atmospheres, the separation is independent of the pressure; therefore in this study the thermodiffusion factor is not changed by small variation of pressure. In most gaseous mixture the thermodiffusion factor increases with increase in pressure. The temperature and concentration dependence of the thermodiffusion factor also were found to be affected by pressure (Becker, 1950).
ð24Þ
and, the composition in each bulb at any time is 0 Vt cit þ Vb cib ¼ ðVt þ Vb Þcib
ð25Þ
Then one can eliminate c1b from Eq. (23) using these two component balances dc1b Vt TH 0 þ bD12 c1b ¼ bD12 c1b þ kT ln ð26Þ dt Vt þ Vb TC A similar equation for the mass fraction of component 1 in bulb t may also be derived. The integration of eq. (26), starting 0 from the initial condition (t ¼0), c1b ¼ cib , gives Vt TH 0 ð27Þ c1b ¼ c1b þ kT ln 1ÿeÿD12 bt Vt þVb TC
If the value of D12 is available, then just one value of (c1b, t) is all that is needed to calculate the thermodiffusion factor and then the thermodiffusion coefficient. However, when the experimental time evaluation of the concentration is available, the both D12 and kT (or DT) can be evaluated. This can be done by adjusting D12 and kT until Eq. (27) fits the experimental data. When the volume of the two vessels is equal, Eq. (27) simplifies to 0 c1b ¼ c1b þ
S * 1ÿeÿ2ðt=t Þ 2
ð28Þ
where, S ¼ kT lnðTH =TC Þ and t * ¼ ð‘V=AD12 Þ are a separation rate (or Dc1 in Eq. (18)) and a diffusion relaxation time, respectively.
2.5. Experimental setup for porous media In a porous medium, the effective diffusion coefficient for solute transport is significantly lower than the free diffusion coefficient because of the constricted and tortuous solute flow paths. This effective diffusion coefficient is related to the free diffusion coefficient and tortuosity coefficient. The mono-dimensional solute transport can be write as @c @2 c ¼ D* 2 @t @x
ð29Þ
where Dn is the effective diffusion coefficient. We have used the same apparatus and method explained in the last section to measure the effective coefficients except that, here, one part of the connecting tube (4 cm long, connected to bottom bulb) is filled with a synthetic porous medium made with the spheres of different physical properties. This is presented in Sections 3.3 and 3.4 to measure the effective diffusion and thermodiffusion coefficients in porous media, respectively.
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
5097
3. Results
3.2. Diffusion coefficient
3.1. Katharometer calibration
Usually, five modes of gas transport can be considered in porous media (see also Mason and Malinauskas, 1983). Four of them are related to concentration, temperature and partial pressure gradients (molecular diffusion, thermodiffusion, Knudsen diffusion and surface diffusion), and another one to the total gas pressure gradient (viscous or bulk flow). When the gas molecular mean free path becomes of the same order as the tube dimensions, free-molecule (or Knudsen), diffusion becomes important. Due to the influence of walls, Knudsen diffusion and configurational diffusion implicitly include the effect of the porous medium. In the discussion which follows, no total pressure gradient (no bulk flow) is considered since this is the condition which prevails in the experiments presented in this study. In most of the former studies, surface diffusion was either neglected or considered only as a rapid process since its contribution to the overall transport cannot be assessed precisely. Knudsen diffusion is neglected because the pore size is larger than the length of the free path of the gas molecules. For example, in the atmospheric pressure, the mean free path of the helium molecule at 300 K is about 1.39 10 ÿ 7 m. Thus, in this study, only binary molecular gas diffusion is considered. In this study, it is assumed that the diffusion coefficient of the gas mixture is independent of composition, and the transient temperature rises due to Dufour effects are insignificant. It is also assumed that the concentration gradient is limited to the connecting tube whereas the composition within each bulb remains uniform at all times. In addition, the pressure is assumed to be uniform throughout the cell, so that viscous effects are negligible, and high enough to minimize freemolecular (Knudsen) diffusion. For the setup described in Section 2, the cell constant b is equal to 7.16 10 ÿ 5, therefore we can rewrite Eq. (11) as
To find the relative proportions of the components of a gas mixture, the instrument needs first to be calibrated. This is done by admitting mixtures of known proportions on the open cell and observing the resistance difference between reference values and analyzed readings. The precision of composition measurements depends, of course, on the difference of the thermal conductivities of the two components and this also depends on the difference of the molecular masses. Fig. 3 shows an example of katharometer calibration curve for mixtures of He–CO2, which have been obtained in order to interpolate the changes in concentration as a function of katharometer readings. The katharometer calibration curve obtained in this study gives a very close approximation in shape to the experimental measurement curve of thermal conductivity against concentration reported in the literature (Cheung et al., 1962; Saxena et al., 1966), at temperature about 300 K. The thermal conductivities for gas mixtures at low density can also be estimated by theoretical models like Mason–Saxena approach (Mason and Saxena, 1958) kmix ¼
XN
i¼1
xk Pi i j xj Fij
ð30Þ
Here, the dimensionless quantities Fij are 2 !1=2 32 Mj 1=4 5 1 Mi ÿ1=2 4 mi Fij ¼ pffiffiffi 1 þ 1þ Mj mj Mi 8
ð31Þ
0.138
Estimation
1654.6
0.118
Calibrations
1454.6 1254.6
0.098
1054.6 854.6
0.078
654.6
0.058
454.6 254.6
0.038
54.6 -145.4
0.018 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Katharometer difference reading (mV)
Thermal conductivity of the mixture (W/m.K)
where N is the number of chemical species in the mixture. For each species i, xi is the mole fraction, ki is the thermal conductivity, mi is the viscosity at the system temperature and pressure and Mi is the molecular weight of species i. The properties of N2, CO2 and He required to calculate thermal conductivity of mixture have been listed in Table 2 at 300 K and 1 atm.
Mole fraction of CO2 Fig. 3. Katharometer calibration curve with related estimation of thermal conductivity values for the system He–CO2, T¼ 300 K.
0 ÿc11 Þexpðÿ7:16 10ÿ5 D12 tÞ þ c11 c1b ¼ ðc1b
ð32Þ
The katharometer interval registration data has been set to one minute therefore, there is sufficient data to fit Eq. (32) with experimental data to obtain a more accurate coefficient, compared with a one point calculation. Then, the obtained binary diffusion coefficient is about 0.690 cm2/s for He–N2 system and 0.611 cm2/s for He–CO2 system. In the literature (Wahby and Los, 1987), binary diffusion coefficient for a He–N2 system measured with two-bulb method at conditions of p¼101.325 kPa, and T¼299.19 K, is about 0.7033 cm2/s. This coefficient for a He–CO2 system has been reported as 0.615 cm2/s at 300 K (Dunlop and Bignell, 1995). Using these standard coefficients a new calibrated mean cell constant has been calculated. This constant that will be used for all next experiments is equal to b/1.015. The theoretical estimation of the diffusion coefficients (see Table 10 in Appendix A) from gas kinetic theory also are not different from values obtained in this study which confirms the validity of the measuring method and apparatus (the theoretical formulation is explained in Appendix A).
3.3. Effective diffusion coefficient in porous media Table 2 The properties of CO2, N2 and He required to calculate kmix (T¼ 300 1C, P ¼1 atm) (Poling et al., 2000).
CO2 N2 He
Mi
mi 104 (g/cm s)
ki 107 (cal/cm s K)
44.010 28.016 4.002
1.52 1.76 2.01
433 638 3561
A number of different theoretical and experimental models have been used to quantify gas diffusion processes in porous media. Most experimental models are simply models derived for a free fluid (no porous media) that were simply modified for a porous medium. Attempts have been made to define effective diffusion parameters according to the presence of the porous medium. In literature, the effective diffusion coefficients are now well
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
established, theoretically (Maxwell, 1881; Quintard, 1993; Ryan et al., 1981; Wakao and Smith, 1962; Weissberg, 1963) and experimentally (Currie, 1960; Hoogschagen, 1955; Kim et al., 1987). The comparison of the theoretical and experimental results for the dependence of the effective diffusion coefficient on the medium porosity shows that the results of Quintard (1993) in three dimensional arrays of spheres and the curve identified by Weissberg (1963) are in excellent agreement with the experimental data (Whitaker, 1999). Many experimental studies have been done to determine the effective diffusion coefficient for unconsolidated porous media. The diffusion of hydrogen through cylindrical samples of porous granular materials was measured by Currie (1960). An equation having two shape factor of the form Dn/D ¼ gem has been proposed by these authors which fits with all granular material, m is the particle shape factor. The value of g for glass spheres can be fixed to 0.81 (Carman, 1956). The expected m value for spheres is 1.5. For measuring effective diffusion coefficients we have used the same apparatus and method but here, one part of the connecting tube (4 cm long, connected to the bottom bulb) is filled with the porous medium made of glass spheres. A metal screen was fixed at each end of the tube to border and maintain the unconsolidated porous media (Fig. 4). The mesh size of the screen is larger than the pore size and smaller than spheres diameter. The porosity of each medium has been determined by 3D image analyses of the sample made with an X-ray tomography device (Skyscan 1174 type). A section image of the different samples used in this study is shown in Fig. 5. The various diffusion time evolution through a free medium and porous media made of different glass spheres (or mixture of them) are shown in Figs. 6 and 7 for He–N2 and He–CO2 systems, respectively. These results show clearly that the concentration time variations are very different from free medium and porous medium experiments. In the case of porous medium there is a change associated with the porosity of the medium. The values of the particle diameter, corresponding porosity and calculated diffusion coefficients are shown in Tables 3 and 4. Here, the stared parameters are the effective coefficients and the other ones are the coefficient corresponding to the free fluid. The diffusion coefficients have been obtained by curve fitting of Eq. (32) on the experimental data. We can conclude from these results that there is no significant difference between calculated ratios of Dn/D12 obtained from two different gas systems.
Glass spheres d= 700-1000 μm ε=42.5 %
Glass spheres d= 200-210 μm ε=40.2 %
Glass spheres d= 100-125 μm ε=30.6 %
Mixture of glass spheres d=100-1000 μm ε=28.5 %
Fig. 5. Section images of the tube (inner diameter d ¼ 0.795 cm) filled by different materials obtained by an X-ray tomography device (Skyscan 1174 type).
100
Concentration of N 2 in bottom bulb (% )
5098
Free Fluid
95
Porous medium, ε=42.55
90
Porous medium, ε=30.59
85
Porous medium, ε=28.52
80 75 70 65 60 55 50 0
A
B
C
D
36000
72000
108000
144000
180000
216000
Time (s) Fig. 6. Composition-time kinetics in diffusion experiments for He–N2 system for 0 ¼ 100%). different medium. (TC ¼ 300 K and c1b
3.4. Free fluid and effective thermodiffusion coefficient
Fig. 4. Cylindrical samples filled with glass sphere.
Experimental investigations of thermodiffusion have usually been based on the determination of the difference in composition of two parts of a given gas mixture which are at different temperatures. In this work, after reaching the steady-state in the diffusion process described in Section 3.3, the temperature of top bulb is increased to 350 K. In this stage the valve between the two bulbs is closed. Increasing the temperature in this bulb will increase the pressure then, by opening a tap on the top bulb, the pressure decrease until it reaches an equilibrium value between the two bulbs.
Porous media, ε=42.55
90
Porous media, ε=40.21
85
Porous media, ε=28.52
80 75 70 65 60 55 50 0
72000
144000
216000
5099
may be written as follows:
Free Fluid
95
288000
Time (s) Fig. 7. Composition-time kinetics in diffusion experiments for He–CO2 system for 0 ¼ 100% ). different medium (T¼ 300 K and c1b Table 3 Measured diffusion coefficient for He–N2 and different media. Particle diameter (mm)
Porosity (%)
D12 (cm2/s)
Dn/D12 (dimensionless)
Free fluid 750–1000 100–125 Mixture of spheres
100 42.5 30.6 28.5
0.700 0.438 0.397 0.355
1 0.64 0.57 0.51
Table 4 Measured diffusion coefficient for He–CO2 and different media. Particle diameter (mm)
Porosity (%)
D12 (cm2/s)
Dn/D12 (dimensionless)
Free fluid 750–1000 200–210 Mixture of spheres
100 42.5 40.2 28.5
0.620 0.400 0.375 0.322
1 0.65 0.61 0.52
In a general way, the thermodiffusion coefficient is a complex function of concentration, temperature, pressure, and molecular masses of the components. In this study, we have fixed all these parameters in order to observe only the influence of porosity on the thermodiffusion process. The separation can be found from the change in composition which occurs in one bulb during the experiment, providing that the ratio of the volumes of the two bulbs is known. In the equation expressing the separation, the volume of the connecting tube has been neglected. Then, from Eq. (19) the thermodiffusion factor for He–N2 and He–CO2 binary mixtures is obtained, respectively, as about 0.31 and 0.36. In the literature, this factor, for the temperature range 287–373 K, is reported as about 0.36 for He–N2 binary mixture (Ibbs and Grew, 1931). From experimental results of composition dependence of He–CO2 mixture, done with a swing separator method by Batabyal and Barua (1968) aT increases with increase in concentration of the lighter component. A thermodiffusion factor equal to 0.52 is obtained from the equation proposed in their paper, at T ¼ 341:0 K (Batabyal and Barua, 1968). A study using a two-bulb cell to determine the composition and temperature dependence of the diffusion coefficient and thermodiffusion factor of He–CO2 system has been done by Dunlop and Bignell (1995). They obtain a diffusion coefficient equal to 0.615 cm2/s at 300 K and a thermodiffusion factor of 0.415 at T ¼ 300 K (Dunlop and Bignell, 1995). The theoretical expression for the first approximation of the thermodiffusion factor, according to the Chapman–Enskog theory
* ½a1 ¼ Að6C12 ÿ5Þ
ð33Þ
where A is a function of molecular weights, temperature, relative * is a ratio of concentration of the two components, and C12 collision integrals in the principal temperature dependence given * by the ð6C12 ÿ5Þ factor (Hirschfelder et al., 1964). Calculated values of the thermodiffusion factors for He–CO2 and He–N2 mixtures at T, using this theoretical approach and according to the Lennard–Jones (12:6) potential model gives a thermodiffusion factor for He–N2 mixture about 0.32 and for He–CO2 about 0.41. The detail of formulation and estimation are listed in Appendix B. As we explained in Section 2.4, when the experimental data concerning time evolution of the concentration exist, we can evaluate the both diffusion and thermodiffusion coefficients. Therefore in this study, by a curve fitting procedure on the experimental data, two parameters D12 and DT are adjusted until Eq. (27) fits the experimental curve. In fact, adjusting D12 fits the slope of the experimental data curves and then, DT is related to the final steady-state of the curves. The thermodiffusion kinetics for, respectively, free media and porous media made of different glass spheres (or mixture of them) are shown in Fig. 8 (He–N2 mixture) and Fig. 9 (He–CO2 mixture). Here also, the time history changes with the porous medium porosity. The values for porosity, particle diameter of the porous medium, calculated diffusion coefficients, thermodiffusion coefficients and related thermodiffusion factor are shown in Table 5 for He–N2 mixture and Table 6 for He–CO2 mixture. The diffusion coefficients calculated with this method are larger than the one obtained in diffusion processes for He–N2 system. The theoretical approach (Appendix A) and experimental Concentration of N2 in bottom bulb (%)
100
50.7 50.6 50.5 50.4 Free Fluid
50.3
Porous medium, ε=42.55 50.2
Porous medium, ε=30.59 Porous medium, ε=28.52
50.1 50 0
36000
72000
108000 144000 180000 216000 252000 288000
Time (s) Fig. 8. Composition-time kinetics in thermodiffusion experiments for He–N2 0 binary mixture for different media (DT¼50 K, T ¼ 323:7 K and c1b ¼ 50%).
Concentration of CO 2 in bottom bulb (%)
Concentration of CO 2 in bottom bulb (%)
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
50.7 50.6 50.5 50.4 Free Fluid
50.3
Porous medium, ε=42.55
50.2
Porous medium, ε=40.21
50.1
Porous medium, ε=28.52
50 0
36000
72000
108000
144000
T ime (s) Fig. 9. Composition-time kinetics in thermodiffusion experiments for He–CO2 0 binary mixture for different media .(DT¼50 K, T ¼ 323:7 K and c1b ¼ 50%).
5100
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
Table 5 Measured thermodiffusion and diffusion coefficient for He–N2 and different media. Particle diameter (mm)
Porosity (%)
D12 (cm2/s)
Dn/D12 ( ÿ )
aT (ÿ )
DT (cm2/s K)
DnT/DT (dimensionless)
Free fluid 750–1000 100–125 Mixture of spheres
100 42.5 30.6 28.5
0.755 0.457 0.406 0.369
1 0.605 0.538 0.489
0.310 0.312 0.308 0.304
0.059 0.035 0.031 0.028
1 0.61 0.53 0.48
Table 6 Measured diffusion coefficient and thermodiffusion coefficient for He–CO2 and different media. Porosity (%)
D12 (cm2/s)
Dn/D12 ( ÿ )
aT (ÿ )
DT (cm2/s K)
DnT/DT (dimensionless)
Free fluid 750–1000 200–210 Mixture of spheres
100 42.5 40.2 28.5
0.528 0.304 0.320 0.273
1 0.627 0.567 0.508
0.358 0.362 0.364 0.363
0.047 0.028 0.027 0.024
1 0.61 0.59 0.52
Concentration of N2 in bottom bulb (%)
Particle diameter (mm)
Top bulb
Tube containing porous medium
Bottom bulb
data show that the diffusion coefficient increases with increase in the temperature. From Eq. (A.2), when the ideal-gas law approximation is valid, we can write T 3=2 ÿ
OD T *
61.7 61.6 61.5 Free Fluid
61.4
Porous medium (ε=33.78)
61.3
Porous medium (ε=26.37)
61.2 0
7200
14400
21600 28800 Time (s)
36000
43200
50400
Fig. 11. Composition-time kinetics in thermodiffusion experiments for He–N2 0 binary mixture for different media (DT¼50 K, T ¼ 323:7 K and c1b ¼ 61:25%).
Fig. 10. New experimental thermodiffusion setup without the valve between two bulbs.
D12 p
61.8
ð34Þ
In a second set of thermodiffusion experiments shown in Fig. 10, we eliminate the valve between the two bulbs in order to have shorter relaxation times. In this case, the tube length is equal to 4 cm only (calibrated cell constant ¼2.44 10 ÿ 4 cm ÿ 2) and we filled the system cells with a binary gas mixture (cN0 2 ¼ 61:25%). At the initial state, the whole setup is kept at a uniform and constant temperature about 325 K and the composition of the mixture is uniform everywhere. Then, the temperature of the top bulb is increased to TH ¼350 K and the temperature of the bottom bulb is lowered to TC ¼ 300 K. At the end of this previous process, when the temperature of each bulb remains constant, the pressure of the two bulbs is equal to the one of the beginning of the experience. The thermodiffusion separation in this period is very small because of the forced convection. The katharometer reading data have been recorded with one minute interval. Concentration in bottom bulb has been determined using the katharometer calibration curve. Then, with a curve fitting procedure on the
experimental data, as in the last section, the two coefficients D12 and DT are adjusted until Eq. (27) fits the experimental curve. The adjusted curves for a free medium and different porous media are shown in Fig. 11. The values obtained for porosity, particle diameter of the porous media, calculated diffusion and thermodiffusion coefficients and thermodiffusion factor are listed in Table 7. 3.5. Evaluation of the porous medium tortuosity factor Mathematically, the tortuosity factor, t, defined as the ratio of the length of the tortuous path in a porous media divided by a straight line value. There are several definitions of this factor. The most widely used correlation for gaseous diffusion is the one of Millington for saturated unconsolidated system (Millington, 1959; Millington and Quirk 1961)
t ¼ 1=e1=3
ð35Þ
Tortuosity is also an auxiliary quantity related to the ratio of the effective and free diffusion coefficients. t in many application for homogenous and isotropic environment is also defined as (Nicholson, 2001; Nicholson and Phillips, 1981) rffiffiffiffiffiffiffiffi D12 ð36Þ t¼ D* In this study, we define the tortuosity as a ratio of the effective to free diffusion coefficients as we mentioned also in theoretical
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
5101
Table 7 Measured diffusion coefficient and thermodiffusion coefficient for He–N2 and different media. Particle diameter (mm)
Porosity (%)
D12 (cm2/s)
Dn/D12 ( ÿ )
aT (ÿ )
DT (cm2/s K)
DnT/DT (dimensionless)
Free fluid 315–325 5–50
– 33.8 26.4
0.480 0.257 0.165
1 0.530 0.344
0.256 0.248 0.252
0.029 0.015 0.010
1 0.52 0.34
Table 8 Porous medium tortuosity coefficients. Particle diameter (mm)
Porosity (%)
t ¼ ðD12 =D* Þ (from diffusion
t ¼ ðD12 =D* Þ (from thermodiffusion
t ¼ ðDT =D*T Þ (from thermodiffusion
t
750–1000 200–210 315–325 100–125 Mixture of spheres 5–50
42.5 40.2 33.8 30.6 28.5 26.4
1.57 1.65 – 1.76 1.95 –
1.62 1.76 1.89 1.86 2.00 2.91
1.63 1.71 1.92 1.87 1.99 2.90
1.61 1.71 1.90 1.83 1.98 2.90
experiments)
experiments)
0.6
0.6 Volume averaging, theoretical estimation
Volume averaging, theoretical estimation 0.5
Experimental, He-N2
0.3
0.3
0.2
0.2
0.1
0.1
0.0 0.20
Experimental, He-CO2
Experimental, He-N2
0.4
ε DT* DT
D
0.5
Experimental, He-CO2
0.4
εD *
experiments)
0.0 0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
ε (Porosity)
ε (Porosity) Fig. 12. Comparison of experimental effective diffusion coefficient data with the theoretical one obtained from volume averaging technique for different porosity and a specific unit cell.
Fig. 13. Comparison of experimental effective thermodiffusion coefficient data with theoretical one obtained from volume averaging technique for different porosity and a specific unit cell.
model (Davarzani et al., 2010) D* 1 ¼ D12 t
or
D*T 1 ¼ DT t
ð37Þ
Table 8 presents the tortuosity factors, with uncertainity between 2% and 3%, calculated from values measured in this work (for non-consolidate spheres and the tortuosity definition with Eq. (37)). These factors calculated whether from diffusion or thermodiffusion effective coefficients are not different.
4. Discussion and comparison with theory
for pure diffusion
solid-phase
Fig. 14. Representative unit cell of spatially periodic porous media.
In a previous work, we presented the volume averaging method to obtain the macro-scale equations that describe diffusion and thermodiffusion in a homogeneous porous medium (Davarzani et al., 2010). The results of this model showed that the effective thermodiffusion coefficient at diffusive regime can be estimated with the single tortuosity, results for diffusion case are fully discussed in the literature (Quintard, 1993; Quintard et al., 2006). Here, we rewrite the basic theoretical results for a pure diffusion and binary system as D* D* 1 ¼ T ¼ , D12 DT t
liquid-phase
ð38Þ
Figs. 12 and 13 show, respectively, a comparison of effective diffusion and thermodiffusion coefficients measured in this study
and the theoretical results from the volume averaging technique for different porosity of the medium. We note that, the volume averaging process, have carried out using a model unit cell such as the one shown in Fig. 14. In this system, the effective diffusion and thermodiffusion coefficients for different fractional void space is plotted as the continuous lines. Figs. 12 and 13 show that the experimental, effective coefficient results for the non-consolidated porous media made of spheres are in excellent agreement with volume averaging theoretical estimation. In Fig. 15 the ratio of k*T =kT has been plotted against porosity, where, the effective thermodiffusion ratio, kT, has been defined as
kT* kT
5102
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
k*T ki
Volume averaging, theoretical estimation Experimental, He-N2 Experimental, He-CO2
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
ε (Porosity) Fig. 15. Comparison of the experimental thermodiffusion ratio data with theoretical one obtained from volume averaging technique for different porosity and a specific unit cell.
k*T ¼ TD*T =D* . The experimental results for both mixtures are fitted with the volume averaging theoretical estimation. These results also validate the theoretical results and reinforce the fact that for pure diffusion the Soret number is the same in the free medium and porous media.
5. Conclusion In this study, we used a two-bulb apparatus in order to measure the diffusion and thermodiffusion coefficients in free medium and non-consolidated porous medium having different porosity, separately. The results show that for He–N2 and He–CO2 mixtures, the porosity of the medium has a great influence on the thermodiffusion process. The comparison of the ratio of effective coefficients in the porous medium to the one in the free medium shows that the behavior of tortuosity is the same for the thermodiffusion coefficient and diffusion coefficient. Therefore, the thermodiffusion factor is the same for a free medium and porous media. For nonconsolidated porous media made of the spheres, these results agree with the model obtained by upscaling technique for effective thermodiffusion coefficient proposed in our previous theoretical model. The tortuosity of the medium calculated using both effective diffusion and effective thermodiffusion coefficients are not different to the measurement accuracy.
Nomenclature A A*12 , B*12 , * C12
c10 , c20 cib cit ci1 co D12 d Dn DT D*T Ji kB kmix kT
cross-sectional area of the connecting tube, m2 ratios of collision integrals for calculating the transport coefficients of mixtures for the Lonnard–Jones (6–12) potential Initial mole fraction of the heavier and lighter component mole fraction of component i in the b bulb mole fraction of component i in the bulb mole fraction of component i at equilibrium mole fraction at time t ¼0 diffusion coefficient, m2/s diameter of the connecting tube, m effective diffusion coefficient, m2/s thermodiffusion coefficient, m2/s effective thermodiffusion coefficient, m2/s mass diffusion flux, kg/m2 s Boltzmann constant, 1.38048 J/K thermal conductivity of the gas mixture, W m/K thermodiffusion ratio
‘ ‘eff Mi M N Ni n p RK S S, Q ST TC T0 TH Tn T t tn V Vb Vt xi xj
effective thermodiffusion ratio thermal conductivity of the pure chemical species i, W m/K length of the connecting tube, m effective length of the connecting tube, m molar mass of component I, g/mol particle shape factor the number of chemical species in the mixture mass flux of component i, kg/m2 s number density of molecules pressure, bar katharometer reading, mV separation rate quantities in the expression for aT Soret number, 1/K temperature of the colder bulb, K initial temperature, K temperature of the hotter bulb, K dimensionless temperature averaged temperature, K time, s diffusion relaxation time, s volume of the bulb, m3 volume of the bottom bulb, m3 volume of the top bulb, m3 mole fraction of species i mole fraction of species j
Greek symbols
aT b Dc1
w e12 e mi rb s12 t tt Fij OD O(l,s)n
thermodiffusion factor characteristic constant of the two-bulb diffusion cell defined in Eq. (10), m ÿ 2 change in the concentration of heavier component at the steady-state in the lower bulb end correction factor characteristic Lennard–Jones energy parameter (maximum attractive energy between two molecules), kg m2/s2 fractional void space (porosity) dynamic viscosity of pure species i, g/cm s density, kg/m3 characteristic Lennard–Jones length (collision diameter), A˚ tortuosity thermodiffusion relaxation time, s the interaction parameter for gas-mixture viscosity collision integral for diffusion collision integral
Appendix A. Estimation of the diffusion coefficient with gas kinetic theory The diffusion coefficient D12 for the isothermal diffusion of species 1 through constant-pressure binary mixture of species 1 and 2 is defined by the relation J1 ¼ ÿD12 rc1
ðA:1Þ
where J1 is the flux of species 1 and c1 is the concentration of the diffusing species. Mutual-diffusion, defined by the coefficient D12, can be viewed as diffusion of species 1 at infinite dilution through species 2, or equivalently, diffusion of species 2 at infinite dilution through species 2.
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
Self-diffusion, defined by the coefficient D11, is the diffusion of a substance through itself. There are different theoretical models for computing the mutual and self diffusion coefficient of gases. For non-polar molecules, Lennard–Jones potentials provide a basis for computing diffusion coefficients of binary gas mixtures (Poling et al., 2000). The mutual diffusion coefficient, in units of cm2/s is defined as
D12 ¼ 0:00188T
3=2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M1 þ M2 1 M1 M2 ps212 OD
ðA:2Þ
where T is the gas temperature in unit of Kelvin, M1 and M2 are molecular weights of species 1 and 2, p is the total pressure of the binary mixture in unit of bar, s12 is the Lennard–Jones characteristic length, defined by s12 ¼1/2(s1 + s2), OD is the collision integral for diffusion, is a function of temperature, it depends upon the choice of the intermolecular force law between colliding molecules. OD is tabulated as a function of the dimensionless temperature Tn ¼kBT/e12 for the 12–6 Lennard– pffiffiffiffiffiffiffiffiffi Jones potential, kB is the Boltzman gas constant and e12 ¼ e1 e2 is the maximum attractive energy between two molecules. The accurate relation of the collision integral for diffusion is (Neufeld et al., 1972) OD ¼
1:06036 ðT * Þ0:15610
þ
0:19300 1:03587 1:76474 þ þ expð0:47635T * Þ expð1:52996T * Þ expð3:89411T * Þ
ðA:3Þ
Values of the parameters s and e are known for many substances (Poling et al., 2000). The self-diffusion coefficient of a gas can be obtained from Eq. (A.2), by observing that for a one-gas system: M1 ¼M2 ¼M, e1 ¼ e2 and s1 ¼ s2. Thus, rffiffiffiffiffi 2 1 D11 ¼ 0:00188T 3=2 M ps211 OD
ðA:4Þ
Table 9 shows the necessary data to estimate the diffusion coefficient for the system, He–CO2 and He–N2. The calculation of mixture parameters, dimensionless temperature, collision integral and diffusion coefficient from Eq. (A.2) and for temperatures applied in this study have been listed in Table 10.
Appendix B. Estimation of the thermodiffusion factor with gas kinetic theory From the kinetic theory of gases, the thermodiffusion factor, aT for a binary gas mixture is more complex than one for diffusion coefficient. Three different theoretical expressions for aT are available, depending on the approximation procedures employed: the first approximation and second one of Chapman and Cowling (1953) and the first approximation of Kihara (1949, 1975). The most accurate of these is probably Chapman and Cowling’s second approximation, but this is quite complex. Simple expression of Kihara is more accurate than Chapman and Cowling’s approximations (Saxena, 1956). It therefore seems satisfactory for the present purpose to use Kihara’s approximation written in the form ! S1 x1 ÿS2 x2 * ½aT 1 ¼ 6C12 ÿ5 ðB:1Þ 2 2 Q1 x1 þ Q2 x2 þQ12 x1 x2 The principal contribution to the temperature dependence of * aT comes from the factor ð6C12 ÿ5Þ, which involves only the unlike
(1, 2) molecular interaction. The concentration dependence is given by S1x1 ÿS2x2 term. The main dependence on the masses of the molecules is given by S1 and S2. A positive value of aT means that component 1 tends to move into the cooler region and 2 towards the warmer region. The temperature at which the thermodiffusion factor undergoes a change of sign is referred to as the inversion temperature. These quantities calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # 2,2Þ* Oð11 M1 2M2 s11 2 4M1 M2 A*12 15 M2 ðM2 ÿM1 Þ ÿ ÿ S1 ¼ M2 M1 þ M2 Oð1,1Þ* s12 2 ðM1 þ M2 Þ2 ðM1 þM2 Þ2 12
ðB:2Þ
Q1 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # ð2,2Þ* O11 2 2M2 s11 2 M2 ðM1 þ M2 Þ M1 þ M2 Oð1,1Þ* s12 12 5 6 * 8 ÿ B12 M12 þ 3M22 þ M1 M2 A*12 2 5 5
Mt (g/mol)
e/kB (K)
˚ s (A)
44 28 4
190 99.8 10.2
3.996 3.667 2.576
Table 10 Estimation of diffusion coefficients for binary gas mixtures He–CO2 and He–N2 at
4M1 M2 A*12 M1 ÿM2 2 5 6 * ÿ B12 þ Q12 ¼ 15 M1 þ M2 2 5 ðM1 þ M2 Þ2 " ð2,2Þ* #" ð2,2Þ* # O22 12 * 8ðM1 þ M2 Þ O11 s11 s22 2 B12 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 11ÿ ð1,1Þ* 5 s12 2 5 M2 M1 O12 Oð1,1Þ* 12
with relations for S2, Q2 derived from S1, Q1 by interchange of subscripts. The transport properties for gaseous mixtures can also be expressed in terms of the same collision integral. A*12 , B*12 and * * are function of T12 ¼ kT=e12 defined as C12 A*12 ¼ B*12 ¼
temperatures 300, 350 and T ¼ 323:7 K, pressure 1 bar. He–CO2 T (K) ˚ s12 (A) e12/k (K) Tn (-) OD (-) D12 (cm2/s)
300
6.815 0.793 0.596
* ¼ C12
He–N2 350 3.286 44.02 7.950 0.771 0.772
323.7
7.353 0.782 0.677
300
9.403 0.749 0.715
350 3.121 31.90 10.970 0.731 0.925
ðB:3Þ
ðB:4Þ
Table 9 Molecular weight and Lennard–Jones parameters (Bird et al., 2002).
CO2 N2 He
5103
323.7
10.145 0.740 0.812
Oð2,2Þ* Oð1,1Þ* 5Oð1,2Þ* ÿ4Oð1,3Þ*
Oð1,1Þ*
Oð1,2Þ* Oð1,1Þ*
ðB:5Þ ðB:6Þ ðB:7Þ
The subscripts on the O(l,s)n refer to the three different binary molecular interactions which may occur in a binary gas mixture. By convention, the subscript 1 refers to the heavier gas. To this investigation, Lennard–Jones (12–6) model is applied, which has been the best intermolecular potential used to date for the study of transport phenomena and is expressed by a repulsion term
5104
H. Davarzani et al. / Chemical Engineering Science 65 (2010) 5092–5104
varying as the inverse twelfth power of the distance of separation between the centers of two molecules and an attraction term varying as the sixth power of the separation distance. References Arora, P.S., Shankland, I.R., Bell, T.N., Yabsley, M.A., Dunlop, P.J., 1977. Use of precise binary diffusion coefficients to calibrate two-bulb cells instead of using the standard end correction for the connecting tube. Review of Scientific Instruments 48 (6), 673–674. Batabyal, A.K., Barua, A.K., 1968. Composition dependence of the thermal-diffusion factors in He–CO2 Ne–CO2, and Xe–CO2 mixtures. Journal of Chemical Physics 48, 2557–2560. Becker, E.W., 1950. The thermal de-mixing of gases at high pressures. Die Naturwissenschaften 37 (7), 165–166. Bird, R., Stewart, W., Lightfoot, E., 2002. Transport Phenomena 2nd ed. J. Wiley and Sons, New York. Brown, H., 1940. On the temperature assignments of experimental thermal diffusion coefficients. Physical Review 58, 661–662. Caldwell, D.R., 1973. Thermal and Fickian diffusion of sodium chloride in a solution of oceanic concentration. Deep-Sea Research and Oceanographic Abstracts 20 (11), 1029–1039. Carman, P.C., 1956. Flow of Gases through Porous Media. Academic Press Inc., New York. Chapman, S., Cowling, T.G., 1953. Mathematical Theory of Non-uniform Gases. Cambridge University Press. Chapman, S., Cowling, T.G., 1970. Thermal Combustion and Diffusion in Gases, An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases 3rd ed. Cambridge Mathematical Library. Cheung, H., Bromley, L.A., Wilke, C.R., 1962. Thermal conductivity of gas mixtures. A.I.Ch.E. Journal 8 (2), 221–228. Costeseque, P., Pollak, T., Platten, J.K., Marcoux, M., 2004. Transient-state method for coupled evaluation of Soret and Fick coefficients, and related tortuosity factors, using free and porous packed thermodiffusion cells. European Physical Journal E, Soft Matter 15 (3), 249–253. Currie, J.A., 1960. Gaseous diffusion in porous media. I. A non-steady state method. British Journal of Applied Physics 11 (8), 314–317. Davarzani, H., Chastanet, J., Marcoux, M., Quintard, M., 2008. Theoretical determination of effective thermodiffusion coefficients, application to the description of mass transfer in porous media. In: Lecture Notes of the 8th International Meeting on Thermodiffusion (IMT8), Thermal Nonequilibrium, 3, Forschungszentrum julich GmbH, Bonn, Germany, pp. 181–187. Davarzani, H., Marcoux, M., Quintard, M., 2010. Theoretical predictions of the effective thermodiffusion coefficients in porous media. International Journal of Heat and Mass Transfer 53, 1514–1528. Daynes, H.A., 1933. Gas Analysis by Measurement of Thermal Conductivity. Cambridge University Press, Cambridge, UK, pp. 1–302. Dunlop, P.J., Bignell, C.M., 1995. Diffusion coefficients and thermal diffusion factors for He–CO2, He–N2O and He–COS. Berichte der Bunsen-Gesellschaft 99, 77–79. Fuller, E., Ensley, K., Giddings, J., 1969. Diffusion of halogenated hydrocarbons in helium: the effect of structure on collision cross sections. Journal of Physical Chemistry 73, 3679. Grew, K., Ibbs, T., 1952. Thermal Diffusion in Gases. Cambridge University Press. Grew, K.E., Wakeham, W.A., 1971. A redetermination of the thermal diffusion factor for some inert gas mixtures. I. Journal of Physics B, Atomic and Molecular Physics 4 (11), 1548–1563. Heath, H.R., Ibbs, T.L., Wild, N.E., 1941. The diffusion and thermal diffusion of hydrogen-deuterium, with a note on the thermal diffusion of hydrogenhelium. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 178 (974), 380–389 9749. Hirschfelder, J.O., Curtiss, C., Bird, R.B., 1964. Molecular Theory of Gases and Liquids. Johns Wiley & Sons, Inc., New York. Hoogschagen, J., 1955. Diffusion in porous catalysts and adsorbents. Industrial and Engineering Chemistry 47 (5), 906–912. Humphreys, A.E., Gray, P., 1970. Thermal diffusion as a probe of binary diffusion coefficients at elevated temperatures. I. Techniques, and results for carbon dioxide +nitrogen. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 320 (1542), 397–415 77,835. Ibbs, T.L., Chapman, S., 1921. Some experiments on thermal diffusion. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 99 (700), 385–397 93,924. Ibbs, T.L., Grew, K.E., 1931. Influence of low temperatures on the thermal diffusion effect. Proceedings of the Physical Society 43, 142–156. van Itterbeek, A., van Paemel, O., van Lierde, J., 1947. Measurements on the thermal diffusion in gas mixtures at low temperatures. Physica 13 (4–5), 231–239. Jamet, P., Fargue, D., Costeseque, P., 1996. Determination of the effective transport coefficients for the separation of binary mixtures of organic compounds into packed thermal diffusion columns. Chemical Engineering Science 51 (19), 4463–4475. Jessop, G., 1966. Katharometers. Journal of Scientific Instruments 43 (11), 777–782.
Kaviany, M., 2002. Principles of Heat Transfer. John Wiley & Sons, Inc., New York. Kihara, T., 1949. Imperfect gases. Asakura Bookstore, Tokyo, (English translation by US Air Force Office of Research, Wright-Patterson Air Force Base). Kihara, T., 1975. The Chapman-Enskog and Kihara approximations for isotopic thermal diffusion in gases. Journal of Statistical Physics 13 (2), 137–143. Kim, J.H., Ochoa, J.A., Whitaker, S., 1987. Diffusion in anisotropic porous-media. In: Kim, J.H., Ochoa, J.A., Whitaker, S. (Eds.), Diffusion in Anisotropic PorousMedia. Transport in Porous Media, vol. 2, pp. 327–356. Landau, L.D., Lifschitz, E.M., 1982. Fluid Mechanics 2nd ed. Pergamon Press, Oxford, England. Mason, E., Malinauskas, A., 1983. Gas Transport in Porous Media: the Dusty Gas Model. Elsevier, Amsterdam. Mason, E.A., Islam, M., Weissman, S., 1964. Thermal diffusion and diffusion in hydrogen-krypton mixtures. Physics of Fluids. 7 (7), 1011–1022. Mason, E.A., Saxena, S.C., 1958. Approximate formula for the thermal conductivity of gas mixtures. Physics of Fluids 1, 361–369. Maxwell, J.C., 1881. A Treatise on Electricity and Magnetism 2nd ed. Clarendon Press, Oxford I. Millington, R.J., 1959. Gas diffusion in porous media. Science 130 (3367), 100–102. Millington, R.J., Quirk, J.M., 1961. Permeability of porous solids. Transactions of the Faraday Society 57, 1200–1207. Neufeld, P.D., Janzen, A.R., Aziz, R.A., 1972. Empirical equations to calculate 16 of the transport collision integrals O(l,s)n for the Lennard–Jones (12–6) potential. Journal of Chemical Physics 57 (3), 1100–1102. Ney, E.P., Armistead, F.C., 1947. The self-diffusion coefficient of uranium hexafluoride. Physical Review 71 (1), 14–19. Nicholson, C., 2001. Diffusion and related transport mechanisms in brain tissue. Reports on Progress in Physics 64, 815–884. Nicholson, C., Phillips, J.M., 1981. Ion diffusion modified by tortuosity and volume fraction in the extracellular microenvironment of the rat cerebellum. Journal of Physiology (Cambridge) 321, 225–258. Platten, J.K., 2006. The Soret effect: a review of recent experimental results. Journal of Applied Mechanics 73 (1), 5–15. Platten, J.K., Costeseque, P., 2004. The Soret coefficient in porous media. Journal of Porous Media 7 (4), 329–342. Poling, B.E., Rausnitz, J.M., Connell, J.P., 2000. The Properties of Gases and Liquids 5th ed. McGraw-Hill, New York. Quintard, M., 1993. Diffusion in isotropic and anisotropic porous systems: threedimensional calculations. Transport in Porous Media 11 (2), 187–199. Quintard, M., Bletzaker, L., Chenu, D., Whitaker, S., 2006. Nonlinear, multicomponent, mass transport in porous media. Chemical Engineering Science 61, 2643–2696. Rayleigh, J.W.S., 1945nd. Theory of Sound 2nd ed. Dover, New-York. Ryan, D., Carbonell, R.G., Whitaker, S.A., 1981. Theory of diffusion and reaction in porous media. A.I.Ch.E. Symposium Series 77 (202), 46–62. Saxena, S.C., 1956. On the two schemes of approximating the transport coefficients (Chapman–Cowling and Kihara). Journal of the Physical Society of Japan 11 (4), 367–369. Saxena, S.C., Kelley, J.G., Watson, W.W., 1961. Temperature dependence of the thermal diffusion factor for helium, neon and argon. Physics of Fluids 4 (10), 1216–1225. Saxena, S.C., Mason, E.A., 1959. Thermal diffusion and the approach to the steady state in gases. II. Molecular Physics 2 (4), 379–396. Saxena, S.C., Mathur, S., Gupta, G.P., 1966. The thermal conductivity data of some binary gas mixtures involving nonpolar polyatomic gases. Defence Science Journal 16 (4), 99–112. Shashkov, A.G., Zolotukhina, A.F., Abramenko, T.N., Mathur, B.P., 1979. Thermal diffusion factors for binary gas systems: Ar–N2, Ar–CO2, He–H2, He–N2O, Kr–N2O and He–NH3. Journal of Physics B—Atomic Molecular and Optical Physics 12 (21), 3619–3630. Wahby, A.S.M., Los, J., 1987. Diffusion in Lorentzian and quasi-Lorentzian N2-light noble gas mixtures. Physica, B + C 145 (1), 69–77. Wakao, N., Smith, J.M., 1962. Diffusion in catalyst pellets. Chemical Engineering Science 17 (11), 825–834. Watts, H., 1964. Diffusion of krypton-85 in multicomponent mixtures of krypton with helium, neon, argon and xenon. Transactions of the Faraday Society 60 (10), 1745–1751. Weissberg, H.L., 1963. Effective diffusion coefficient in porous media. Journal of Applied Physics 34 (9), 2636–2639. Weller, K.R., Stenhouse, N.S., Watts, H., 1974. Diffusion of gases in porous solids. I. Theoretical background and experimental method. Canadian Journal of Chemistry 52 (15), 2684–2691. Whitaker, S., 1999. The Method of Volume Averaging. Kluwer Academic Publishers, Dordrecht, The Netherlands. Wirz, P., 1947. Eigenschaften von ein- und mehrfachen zylindrischen akustischen resonatoren. Helvetica Physica Acta 20 (1), 3–26. Yabsley, M.A., Dunlop, P.J., 1976. A study of the two-bulb method for measuring diffusion coefficients of binary gas mixtures. Physica A 85 (1), 160–174. Zhdanov, V.L., Kolenchits, O.A., Mathur, B.P., Saxena, S.C., 1980. Gas-phase transport phenomena at low temperatures: thermal diffusion in an Ar–Kr system. International Journal of Thermophysics 1 (2), 147–157.
