Mass Transfer
Content
STUDENT NUMBER:
200338840
STUDENT NAME:
Osiki Izijesu. O
COURSE NAME:
Advanced Mass Transfer
DEPARTMENT:
Process Systems Engineering
Number 1A-1
TITLE: Determination of Gas diffusion coefficients in saturated porous
media: Helium, He and Methane, CH4 diffusion in Boom clay
BRIEF INTRODUCTION
Boom Clay is presently a potential host rock mainly as a medium of disposal of high level and
long lived radioactive waste in Belgium, and also because production of gas is ineluctable in
geological depository (Yongfeng Deng et al., 2011). Today, gas is produced by a number of
different mechanisms including radiolysis of water and organic materials, microbial debasement
of organic wastes as well as anaerobic corrosion of metals in waste materials (Elke Jacops et al.,
2013). Rodwell et al., 1999 and Yu and Weetjens, 2009 for instance write that, hydrogen gas is
produced from corrosion and radiolysis and conversely methane and carbon dioxide are products
of microbial debasement. Diffusion is the main transport phenomenon in the boom clay since
dissolved gases diffuse from the repository as dissolved species. Importantly, good estimates for
gas diffusion coefficient is necessary in order to estimate and understand the diffusion process
such as gas generation and dissipation through barriers (Elke Jacops et al., 2013).
Literature provides limited information as to the determination of diffusion coefficients of gases
in saturated porous media and available information and data is difficult to analyze or understand
(Elke Jacops et al., 2013). However, there are generally three types of methods that have been
adopted as possible methods to analyze and estimate diffusion coefficients of gases regardless of
individual method’s limitations.
In the first method already used by Bigler et al. (2005) and Gomes-Hernandez (2000), the
diffusion coefficient of gases is determined using gas concentrations from outgassing of the clay
samples stored in a vacuum container and measurement of the concentration of gas released by
the clay samples. In fact, Bigler et al. (2005) estimated the diffusion coefficient of Helium using
a spherical sample of the callovo-oxfordian shale with an uncertainty of 20%. He explains that
calculated uncertainty might have been due to imperfection of the sphere and errors in cutting of
the sample.
The second method determines diffusion coefficient based on the concentration profiles of gases
as natural tracers. Rubel et al. (2002) used this technique and computed the apparent diffusivity
for helium in opalinus clay based on helium’s natural profile obtained at Mount Terri.
Bensenonic et al. (2011) using this technique obtained a helium profile for samples that were
collected from vertical drilled boreholes in the Toarcian shale in France for which the diffusion
coefficients was obtained from outgassing of the water in high vacuum containers. This is an
interesting technique but like the previous method it is only applicable for gases present naturally
in clay and limited to He, Ar and CH 4. Of note is what Mazurek et al. (2011) wrote that, the core
outgassing of noble gases requires advanced equipment and that there is apparent possibility of
leakage at several stages.
The third method for determination of the diffusion coefficient for gases is based on the “in or
through” diffusion technique. This method has been used by Rebour et al. (1997). Although
reported error on the diffusion coefficient of Helium on callovo-oxfordian clay was 20%, this
method has been found to be more versatile as it can be used to compute the diffusion
coefficients for different gases. However, Bigler et al. (2005) argue that interpretation of data
with this method suffered from complications such as anisotropy effects which was not taken
into account and that measured porosity (23%) did not correspond to the porosity value needed
to obtain a good fit (16%). Therefore, a more versatile and simple method that allows for a better
assessment of the diffusivity and suitable for several gases was developed to determine the gas
diffusion coefficient (Elke Jacops et al., 2013).
In this study, the authors developed a method to ascertain the gas diffusion coefficient for
dissolved gases in boom clay and gases, where Helium (He) and Methane (CH 4) served as
reference gases. They write that, their adopted diffusion methodology allows two gases to diffuse
through a clay sample at the same time.
METHODOLOGY
Principle
The experimental work was based on a thorough diffusion test where two dissolved gases were
placed on opposite sides of the Boom clay test core as shown in figure 1 (Shackelford, 1991; Van
Loon et al., 2003). In this method, the porous media is placed between two vessels one of which
contains a known concentration of the diffusant and is the high concentration compartment while
the other known as the low concentration compartment is free of the diffusant. This method used
two diffusants, He and CH4 where the gases are initially concentrated at one side and expectedly
diffuse in different directions afterwards. Importantly, the clay core was sealed in a stainless steel
diffusion cell and connection was via stainless steel plate at both sides to water vessels that were
pressurized with two different gases at the same pressure, shown also in figure 1. Data is
reported in the study only for Helium, He and Methane, CH4.
Experimental Set-up
Since helium diffuses easily through many materials such as polymers used often as seals, Elke
Jacops et al., 2013, emphasize the importance of gas tightness of the entire set up. Thus to avoid
gas leakages, components used are selected based on their gas tightness. For instance, Jacops et
al., 2013 used metal contacts for valves, sensors and couplings as opposed to polymers. Constant
volume was obtained by sealing the diffusion cell properly and the hydraulic press was used to
press the clay core, having a diameter of 80mm and height of 30mm in the diffusion cell. They
also write that perfect sealing between the clay and the diffusion cell was due mainly to the high
plasticity of the clay (Van Geet et al., 2008).
Furthermore, contact between the gas saturated water and the clay was achieved by placing a
porous stainless steel filter plate of diameter 80mm, thickness 2mm and porosity 40% on both
sides of the clay core, where the cover as well as the connections for the water inlet and outlet
were welded onto the body of the container. Gas tightness was ensured by connecting a pressure
sensor, PTX 600 to the two vessels that monitored pressure evolution.
The water with dissolved gas was then circulated over the filters using a magnetically coupled
gear pump, REGLO-Z (ISMTEC, Glattbrugg, Switzerland) calibrated for a flow rate of 3 ± 0.1
ml/min while still ensuring gas tightness and low flow.
Fig 1: Schema of the experimental set-up (vessels and diffusion cell not at scale). Dimensions clay core: diameter 80
mm, length 30 mm. Volume vessels: 1 l, filled with 500 ml water and
500 ml gas (at 10 bar). (Elke Jacops et al., 2013)
Procedure
The clay sample used was taken perpendicular to the bedding plane based on the mineralogical
and physical properties of Boom clay reported by Maes et al (2008). As stated previously, care
was taken to ensure gas tightness prior to the diffusion experiment by pressurizing the set-up
with 10 bar Helium for a 68 day period. Furthermore, gas migration properties had to be
established prior also to the diffusion experiment, by determining the hydraulic conductivity, K
of the clay core. This was done by injecting demineralized water at 0.55MPa, and water flowing
out of the diffusion cell was collected, placed on a precision balance until a stable value was
obtained for five successive intervals. The obtained precision was 8%, with Yu et al. (2013)
reporting that, values for hydraulic conductivity of Boom clay are usually around 1.5 and 8 × 10 12
m/s. Measurement of the hydraulic conductivity was then followed by the preparation of a
solution of oxygen-free water that was 0.014 mol/l NaHCO 3. The water used for preparation of
the solution was stored in an anaerobic compartment and then made to come to equilibrium with
the test gas for a few days to remove N 2. After 500ml of the solution was transferred to each
vessel of the set-up, a gas buffer was placed into the headspace and a sample was then taken to
determine the initial gas composition of both vessels. Once, the diffusion experiment started,
sampling was done regularly in a temperature controlled room (21 ± 2 oC) until about 10 data
points were obtained and to avoid pressure drops sample volume was kept strictly at 6ml. Gas
composition was analyzed with a CP4900 micro GC
Transport Model
Test results obtained from experiment were interpreted with a diffusive model that represents the
transport equation in 1 dimensional geometry and this model is founded on both the first and
second law of Fick for diffusive transport in porous media as represented in equations 1 and 2.
F = - ȠDp
∂C
∂t
∂C
∂X
= Dapp
(1)
∂2 C
2
∂X
(2)
Where F is flux, Dapp is the apparent diffusion coefficient, Dp is the pore diffusion coefficient, Ƞ
denotes porosity; c denotes concentration in the porous medium, t denotes time (s) and x denotes
length (m).
Furthermore, the diffusion coefficients can be correlated by equation 3.
Dapp =
D p Deff
=
R
R
(3)
Of importance, is the decrease of the concentration at the inlet at sampling that could hamper
correct experimental determination of the apparent diffusivity and based on this the model was
implemented numerically. Figure 2 shows the clay core being represented by 30 elements.
Henry’s law (R. Sander, 1999) was applied at one end of figure 2, where the amount of the
dissolving gas is proportional to the partial pressure of the gas in the vessel, while at the other
end the amount of the gas was assumed to be zero. Again, the diffusion accessible porosity set at
0.37 was derived from migration experiments done by Maes et al, (2008) and since assumption
was that there was no adsorption, R is set as 1.At regular time intervals and concentration in
selected elements (grey shades in figure 2), Fick’s law was applied to determine the fluxes at
both faces. Further calculations were performed using COMSOL multi physics version 3.5a,
Earth Science module. Finally, detailed information provided by Glaus et al, (2008) was used to
establish the possible (if any) influence of the filters on the transport process.
Figure 2: Geometry used in the diffusive transport model. Grey cells indicate where the evolution of the dissolved
gas concentrations is recorded. (Elke Jacops et al., 2013)
Results and Discussion
As highlighted in the procedure section, the hydraulic conductivity of the clay core was predetermined as 2.5 × 10
-12
m/s which is a distinctive value for Boom clay (Yu et al., 2013). To
optimize and modify experimental conditions, the sampling volume, frequency and measuring
methodology for the entire experiment, Helium, He and Argon, Ar were used initially even
though both experimental and practical problems resulted in unsatisfactory data. Accordingly, a
second experiment fully described previously and using Helium, He and Methane, CH 4 was
carried out and figures 3 and 4 show graphic representation of results. Furthermore, using the
diffusion model and implementing it in COSMOL, yielded the diffusion coefficients of He and
CH4 as 12.2 × 10 -10 m2/s and 2.42 × 10 -10 m2/s respectively as shown in table 1.
Figure 3: CH4 concentration profile in the opposite vessel, showing the experimental vs. fitted curve. The dotted
lines are calculated results for Dapp (upper line: 3.03 × 10 -10 m2/s; lower line: 1.82 × 10-10 m2/s) and the thick
line is value obtained as best fit, 2.42 × 10 -10 m2/s (Elke Jacops et al., 2013)
From figures 3 and 4, one can observe that not only is the best fit for Dapp plotted but also for
Dapp ± 25%, which indicates the sensitivity of the method as well as the possibility of obtaining
precise measurements for Dapp and this result is in line with work done by Aertsens, 2009.
Again, to estimate diffusion coefficient for other gases, the experimentally obtained diffusion
coefficient is used to compute values for the tortuosity/constrictivity ratio as shown in equation
4.
Do
=RƬ
D app
2
/δ
(4)
Where R is the retardation factor and δ is the constrictivity (Put and Henrion, 1988).
Figure 4: He concentration profile in the opposite vessel, showing the experimental vs. fitted curve. The dotted
lines are calculated results for Dapp (upper line: 15.3 × 10
-10
m2/s; lower line: 9.15 × 10-10 m2/s) and the thick line is
value obtained as best fit, 12.2 × 10 -10 m2/s (Elke Jacops et al., 2013)
The tortuosity, Ƭ is known as the ratio of the effective travelled distance vs the start and end
point distances while constrictivity accounts for the widening and narrowing of clay pore (Collin
and Rasmuson, 1988). Again, since there is no adsorption of gases and pore size distribution of
the boom clay is between 6 and 30 000nm (Hemes et al, 2012), both constrictivity, δ and
retardation, R factor are equal to one.
Table 1: Overview of the fitted diffusion parameters for He and CH4 in Boom Clay
a
Jaehne et al. (1987)
b
Boudreau (1996)
The tortuosity/constrictivity ratio determine for both CH4 and He as shown in table 1, can be
used to predict the Dapp ranges for other gases. This is calculated by rearranging equation 4 and
2
substituting already known Do value and average value of Ƭ /δ
Dapp = (
Table 2:
a
Boudreau (1996)
Do
R
)(
δ
2
Ƭ
)
(5)
as computed in table 2.
Gas diffusion in other clay formations was compared with those of the Boom clay and although
direct comparism cannot be made due to differences in physical and mineralogical properties, it
provides estimates as to whether obtained values are in line with already conducted research
work. For instance, calculated values for Ƭ2/δ in opalinus clay, 3.5 and Oxfordian clay, 3.8
(Gomez-Hernandez, 2000; Bigler et al, 2005)proved to be in line with those determined
experimentally considering factors such as higher densities and lower porosity in these
formations.
In Conclusion, the authors have been able to use a simple method that is both precise and
sensitive in determining the diffusion coefficients of gases in porous media and this method
shows prospects for use in a number of industries and not just for geological disposals.
References
Aertsens, M., 2009. Re-Evaluation of the Experimental Data of the MEGAS Experiment on
Gas Migration Through Boom Clay. SCK•CEN-ER-100. SCK•CEN, Mol, Belgium. (http://
hdl.handle.net/10038/1182).
Bensenouci, F., Michelot, J., Matray, J., Savoye, S., Lavielle, B., Thomas, B., Dick, P., 2011. A
profile of helium-4 concentration in pore–water for assessing the transport phenomena
through an argillaceous formation (Tournemire, France). Phys. Chem. Earth. 36,
1521–1530.
Bigler, T., Ihly, B., Lehmann, B., Waber, H., 2005. Helium Production and Transport in the
Low Permeability Callovo–Oxfordian Shale at the Site Meuse/Haute Marne, France.
Nagra Arbeitsbericht NAB 05-07, Switzerland.
Boudreau, P., 1996. Diagenic Models and Their Interpretation — Modelling Transport and
Reactions in Aquatic Sediments. Springer, Berlin.
Collin, M., Rasmuson, A., 1988. A comparison of gas diffusivity models for unsaturated
porous media. Soil Sci. Soc. Am. J. 52, 1559–1565.
Elke. J, G. Volckaert, N. Maes, E. Weetjens, and J. Govaerts. Determination of gas diffusion
coefficients in saturated porous media: He and CH4 diffusion in Boom Clay. Applied Clay
Science 83-84 (2013) 217-223
Glaus, M., Rossé, R., Van Loon, L., Yaroshchuk, A., 2008. Tracer diffusion in sintered stainless
steel filters: measurement of effective diffusion coefficients and implications for
diffusion studies with compacted clays. Clay Clay Miner. 56, 667–685.
Gomez-Hernandez, J.J., 2000. FM-C Experiment: Part A) Effective Diffusivity and Accessible
Porosity Derived Fromin-situ He-4 Tests. Part B) Prediction of He-3 Concentration
in a Cross-Hole Experiment. Mont Terri Project Technical Note TN 2000-40.
Switzerland.
Grathwohl, P., 1998. Diffusion in Natural Porous Media: Contaminant Transport, Sorption/
Desorption and Dissolution Kinetics. Kluwer Academic Publishers.
Hemes, S., Desbois, G., Urai, J., De Craen, M., Honty, M., 2012. Variability of the Morphology
of the Pore Space in Boom Clay From BIB-SEM, FIB and MIP Investigations on
Representative Samples. 2nd Project Report. SCK•CEN-ER-208. SCK•CEN, Mol,
Belgium (http://hdl.handle.net/10038/7752).
Jaehne, B., Heinz, G., Dietrich, W., 1987. Measurement of the diffusion coefficients of sparingly
soluble gases in water. J. Geophys. Res. 92, 10767–10776.
Kroos, B., Schaefer, R., 1987. Experimental measurements of the diffusion parameters of
light hydrocarbons in water-saturated sedimentary rocks — I. A new experimental
procedure. Org. Geochem. 11, 193–199.
Maes, N., Salah, S., Jacques, D., Aertsens, M., Van Gompel, M., De Cannière, P., Velitchkova,
N., 2008. Retention of Cs in Boom Clay: comparison of data from batch sorption tests
and diffusion experiments on intact clay cores. Phys. Chem. Earth. 33, S149–S155.
Mazurek, M., Alt-Epping, P., Bath, A., Gimmi, T., Waber, N., Buschaert, S., De Cannière, P.,
De Craen, M., Gautschi, A., Savoye, S., Vinsot, A., Wemaere, I., Wouters, L., 2011. Natural
tracer profiles across argillaceous formations. Appl. Geochem. 26, 1035–1064.
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in water-saturated rocks: a new experimental method. J. Contam. Hydrol. 28, 71–93.2
Rodwell, W., Harris, W., Horseman, S., Lalieux, P., Müller, W., Ortiz, L., Pruess, K., 1999. Gas
Migration and Two-Phase Flow Through Engineered And Geological Barriers for a
Deep Repository for Radioactive Waste. EUR19122 EN. Luxembourg. (http://bookshop.
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Rübel, A., Sonntag, C., Jippmann, J., Pearson, F., Gautschi, A., 2002. Solute transport in
formations of very low permeability: profile of stable isotope and dissolved noble
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Organic
Species
of
Potential
Importance
in
(http://www.mpch-mainz.mpg.de/~sander/res/henry.html)
Environmental
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Chemistry
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Number 1A-2
Some other models for estimating Diffusivity include the following;
1. Wilke-Chang model for estimating diffusivity in liquid phases
A number of research work both completed and ongoing have proposed a number of methods for
estimating diffusivity in liquid phase systems. However, the Wilke-Chang model is the most
widely used and diffusion coefficient is given as (Reid R. C. et al, 1977);
Dm =
7.4 X 10−8 T
Ƞ svV
0.6
b, a
. √ M sv α sv
(1)
Where T is the absolute temperature, M is the molecular weight, Ƞ is the viscosity, V b the molar
volume at the normal boiling point, the subscripts a and sv denote the solute and solvent
respectively.
If the aggregation of solute molecules is accounted for, then the Wilke-Chang equation is
modified as follows (Kanji M. and R. Isogai, 2011);
Dm = =
α a V b ,a
¿
¿
¿ 0.6 Ƞ
¿
¿
7.4 X 10−8 T
¿
. √ M sv α sv
(2)
Where αa is the association coefficient of the solute
2. Maxwell-Stefan Diffusivity model for Binary Liquid systems
This model is based on Erying’s absolute reaction rate theory and an extension of the Vignes
model (Glasstone et al, 1941; Vignes. A, 1966). For a non-ideal concentrated binary mixture, the
diffusion coefficient can be expressed as;
D = Did Г (1)
Where, Г is the thermodynamic correction factor, D id is diffusion coefficient for an ideal system
and is equal to the Maxwell-Stefan diffusivity, D12 thus;
Did = D12 (2)
Again, Did is directly proportional to the distance between two successive equilibrium positions,
λ, that is;
Did = λ2k
(3)
k is the rate constant and is computed from the change in Gibbs energy, thus;
k=
T kb
h
−∆ g12
exp(
)
RT
(4)
Δgi is the net activation energy for the diffusion process, and combining equation 2 to 4 gives;
D12 =
λ2 T k b
h
exp(
−∆ g12
)
RT
(4)
For the Vignes model, a linear dependence of Δgi on the composition is assumed (Vignes. A,
1966), thus;
∞
∞
Δg12 = x2 Δ g12 + x1 Δ g21
(5)
And the diffusion coefficient is deduced as;
∞
∞
D12 = ( D 12 )x2 ( D21 )x1
(6)
For non-ideal systems,
V
Δg12 = ϕ2 V 2
∞
Δ g12 +
V
V1
∞
ϕ1 Δ g21
(7)
Where Vi is the partial molar volume of component i, V is the molar volume of the system and ϕ i
is volume fraction of component i.
Replacing ϕi by ϕii the local volume fraction we obtain,
V
Δg12 = ϕ22 V 2
V
V1
∞
Δ g12 +
∞
ϕ11 Δ g21
(8)
We can then obtain a new form of the Vignes model for the Maxwell-Stefan diffusivity as;
∞
∞
D12 = = ( D 12 )Vϕ22/V2 ( D 21 )Vϕ11/V1
(9)
3. Fuller, Schettler, and Giddings Correlation prediction for Binary Gas Diffusivities
Fuller et al, 1966 used 308 experimental values of the diffusivities of various gases to determine
the coefficients a, b, c, d, g, and f equation using a nonlinear least- squares analysis, thus;
DAB =
cT
[
b
[
]
1
1 1
+
M A MB 2
a
g
]
p (∑ V A ) +(∑ V B ) ˄f
(1)
The emperical equation that gives the smallest standard deviation is,
−3
DAB =
10 T
[
1.75
a
[
]
1
1
1
+
˄
M A MB 2
g
]
p (∑ V A ) +(∑ V B ) ˄f
(2)
Where p is the total pressure (atm), Mi is the molecular weight, DAB is the diffusivity (cm2/s), T is
the temperature and Vi is the diffusion volume for component i.
REFERENCES
Fuller E. N, P. D. Schettler, and J. C. Giddings, Ind. Eng. Chem. 58(5), 19 (1966).
Glasstone. S, Laidler K. J. and H. Eyring. The Theory of Rate Processes: the Kinetics of
Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena; McGraw-Hill: New
York, 1941.
Kanji M, and R. Isogai, Estimation of Molecular Diffusivity in Liquid Phase Systems by the
Wilke-Chang equation, Journal of Chromatography A 1218 (2011) 6639 – 6645
Reid R. C, J.M. Prausnitz and T.K. Sherwood, The Properties of Gases and Liquids, McGrawHill, New York, 1977.
Vignes, A. Diffusion in Binary Solutions. Variation of Diffusion Coefficient with Composition.
Ind. Eng. Chem. Fundamentals. 1966, 5, 189-199.
QUESTION 1B
TITLE: On the Modelling of Gas-phase Mass-transfer in Metal sheet
Structured packings
Introduction
The efficiency of mass transfer in gas-liquid reactors is expressed using the volumetric mass
transfer coefficient, KGa. To better understand mas transfer it is imperative that the film
coefficient, KG and the effective interfacial area, a be separated and determined from the
volumetric mass transfer coefficient (Alves S.S et al, 2004). Early works of Chilton and Colburn,
1934 state that, gas-phase mass transfer correlations are often based on fundamental models such
as turbulent flow models, and boundary layer models.
In this article, modelling of gas-phase mass transfer in metal sheet structured packings (M250Y,
M350Y, M500Y, M452Y) was carried out under absorption systems of SO2 chemisorption into
NaOH aqueous solution and CO2 chemisorption into the NaOH aqueous solution. The
determination of the gas-phase mass transfer coefficient for all reviewed packings were
correlated to the dimensionless equation
ShG = 0.409
ℜ0.622
G
ℜ0.0592
L
(1)
Experimental
The experiments were carried out using a column of inner diameter, 0.29m and bed height of
0.84m. For the withdrawal of gas samples by aid of a sampling device (of diameter 30mm), each
packing element was drilled horizontally using an electrical discharge machinery (Valenz et al,
2011). Again, liquid distribution was carried out using a pipe liquid distributor having a drip
point density of 630 dp/m2, while gas entered the column through a large drum. As mentioned
earlier, the aqueous solution of NaOH having a concentration of 1kmol/m 3 served as the
absorption liquid. The batch was replaced when the hydroxyl concentration began to drop,
0.5kmol/m3 for the interfacial area measurement and 0.1kmol/m3 for the volumetric mass transfer
measurements. Furthermore, the temperature of the gas and liquid entering the column was kept
at 20oC and atmospheric pressure. The velocities of the liquid used were 4, 10, 20, 40, 60, 80,
100 m3/(m2 h) and that of the gas superficial velocities were 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 m/s.
Importantly, introduction of the tracer gas into the gas phase was done before its entry into the
column to ensure proper mix and its concentration maintained at the lowest sampling point of at
least 1000 ppm. Continuous measurement of the concentration of CO 2 and SO2 in the gaseous
samples was done using IR analyzer S710, prior to which the gas was dried by passing it through
a cartridge containing crushed CaCl2 (CO2 analysis) or Mg(ClO4)2 (SO2 analysis).
Determination of Volumetric Mass Transfer Coefficient, KGa
To determine the volumetric mass transfer coefficient (K Ga), measurements should be carried out
by experiments where the mass transfer resistance is limited to the gas-phase and where
evaporation of the gaseous solute is used (Danckwerts, 1970; Rejl et al, 2009). In this work,
absorption of SO2 into the NaOH solution was used and the volumetric mass transfer coefficient,
KGa determined with the relation;
KGa =
UG
H
¿
C SO
2
ln C OUT
SO
(2)
2
An independence of volumetric mass transfer coefficient, KGa on hydroxyl concentration in the
NaOH concentration range of 0.1M to 1.0M has been shown by the authors.
Effective area, a determination
Determination included the absorption of diluted CO2 from the air into the lye where the mass
transfer resistance in the gas phase is low and increased reaction of CO 2 with the hydroxyl takes
up more CO2 within the liquid film. This leads to determination of the mass transfer rate by the
reaction transport phenomena in the liquid film correlated to the physical quantities of the
system. Determining the effective interfacial area is by correlation of the CO 2 local mass transfer
rate with its balance along the packing height (Linek et al, 1984) as shown in equation 3.
aR
(K OG a)CO
2
G=0
= He √ DL CO 2 K OH COH , ave
(3)
and substituting value for the volumetric mass transfer as shown in equation 2 gives;
aR
G=0
=
1
He √ DL CO 2 K OH COH , ave
.
UG
H
¿
C SO
2
. ln C OUT
SO
(4)
2
Also, effective area data was correlated to the gas phase mass transfer resistance as shown in
equation 5 (Hoffmann et al, 2007);
a=
aR
1−RG
G=0
(5)
where RG is the relative mass transfer resistance which the authors defined as;
RG =
( K OG a)CO
( K OG a) SO
2
2
.
(
DG ,SO
DG ,CO
2
2
)
2
3
(6)
Mass transfer coefficient determination
The absorption experiments for the volumetric mass transfer coefficient determination and the
effective interfacial area was performed using the aqueous NaOH as the liquid phase and air as
the gas phase. Thus, considering the same liquid and gas flow rate, interfacial area was
considered as equal, as shown in equation 7;
KG =
KGa
a
(7)
Results
Using the power law, experimental results of volumetric mas transfer coefficient and interfacial
area were correlated as show in equation 8 and 9 and table 1 shows the constants for individual
packings; Figures 1a to d also shows a plot of experimental data for which the correlations are
valid.
α
β
KGa = C1 . U G . B
γ
δ
a = C2 . U G . B
(8)
(9)
(Rejl F. J. et al, 2014)
Experimental data collected for the volumetric mass transfer coefficient was compared to those
obtained for Mellpak 250Y and SO2-air-lye system by Klement. M (2002) as shown in figure 1a.
Results further reveal that, the mass transfer coefficient is dependent on the superficial gas phase
velocity, while its dependence on the liquid flow rate is quite weak. Also, since the geometry of
the studied packing were similar, it became imperative to correlate the resulting values of the
mass transfer coefficient as shown in equation 1, for the range of Re G = 240-2500 and ReL = 10300.
Figure 1(a)
M 250Y: volumetric mass-transfer coefficients kGa data used for construction of the correlation Eq. (24).
The lines represent the correlation. Comparison with the data of Klement (2002). (b) M350Y: volumetric masstransfer coefficients kGa data used for construction of the correlation Eq. (24). The lines represent the correlation. (c)
M452Y: volumetric mass-transfer coefficients kGa data used for construction of the correlation Eq. (24). The lines
represent the correlation. (d) M500Y: volumetric mass-transfer coefficients kGa data used for construction of the
correlation Eq. (24). The lines represent the correlation. (Rejl F. J. et al, 2014)
Figure 2: Comparism of experimental and correlated volumetric mass transfer coefficient.
(Rejl F. J. et al, 2014)
Again, mass transfer coefficient for M250Y were correlated in two different ways, effective
phase velocity (UE) or superficial velocity (UG). This further indicated that effective phase
velocity improved fit of the mass transfer coefficient data as seen from the comparative standard
deviation (2.85%) of the correlation, thus;
0.628
KG = 0.0308 . U G
rel.st.dev = 3.16%
(10)
0.703
KG = 0.0209 . U E
rel.st.dev = 2.85%
(11)
Finally, mass transfer data obtained for all packing types under present study was successfully
correlated by the dimensionless form as in equation 1 which shows how the phase velocity and
packing geometry affect the mass transfer (Rejl F. J. et al, 2014).
References
Alves S.S, C. I. Maia, J. M. T. Vasconcelos. Gas-Liquid Mass Transfer Coefficient in Stirred
tanks interpreted through bubble contamination Kinetics. Chemical Engineering and Processing
43 (2004) 823 – 830.
Chilton, T.H., Colburn, A.P., 1934. Mass transfer (absorption) coefficients. Prediction from data
on heat transfer and fluid friction. Ind. Eng. Chem 26, 1183–1187
Danckwerts, P. V., 1970. Gas–Liquid Reactions. McGraw-Hill Book Company, New York.
Hoffmann, A., Mackowiak, J.F., Gorak, A., Haase, M., Loning, J.-M., Runiwski, T.,
Hallenberger, K., 2007. Standardization of mass transfer measurements. Basis for the description
of absorption processes. Chem. Eng. Res. Des. 85 (A1), 40–49.
Klement, M., (Diploma thesis) 2002. Studium transportních charakteristik vyplní Mellapak
(Study of the transport characteristics of the Mellapak packings). ICT, Prague.
Linek, V., Petˇríˇcek, P., Beneˇs, P., Braun, R., 1984. Effective interfacial area and liquid side
mass transfer coefficients in absorption columns packed with hydrophilised and untreated plastic
packing. Chem. Eng. Res.Des. 62, 13–21.
Rejl, F. J . , Linek, V., Moucha, T., Valenz, L., 2009. Methods standardization in the measurement
of mass-transfer characteristics in packed absorption columns. Chem. Eng. Res. Des. 87, 695–
704.
Rejl F. J, L. Valenz, J. Haidl, M. Kordac, and T. Moucha. On the modelling of gas – phase mass
transfer in metal sheet structured packings. Chemical Engineering Research and Design 93
(2015) 194-202.
Valenz, L., Rejl, F. J . , Linek, V., 2011. Effect of gas- and liquid-phase axial mixing on the rate
of mass transfer in a pilot-scale distillation column packed with Mellapak 250Y. Ind. Eng. Chem.
Res. 50 (4), 2262–2271.
Question 1B-2
Other Models to predict the volumetric mass transfer coefficient include the following
1. Model for rising Taylor bubble in circular capillaries developed by Van Baten and
Krishna (2004)
This model considers two contributions to mass transfer, one of which is the caps at either end of
the bubble while the other is the liquid film that seemingly surrounds the bubble (Madhvanand
N. K. et al, 2011). The following correlation was developed by the aforementioned authors for
the overall volumetric mass transfer coefficient, KLa (liquid volume);
KLa = KL, cap acap + KL, filmafilm (1)
Furthermore, the mass transfer coefficients KL, cap and KL, film was derived from the penetration
mass transfer model and is as shown in equations 2 and 3, thus;
√
√
2
KL, cap =
2
KL, film =
2 D m ub
(2)
π2 dt
Dm
(3)
π 2 tc
Then substituting the value of the specific interfacial area of the two caps (a cap) and the film
specific interfacial area (afilm), the volumetric mass transfer coefficient, KLa (liquid volume)
becomes;
KLa =
√
2
2 D m ub
2
π dt
4
Luc
+
√
2
Dm
2
π tc
4 ∅G
dt
(4)
Where Luc is unit cell length, ∅G is gas hold up, ub is bubble rise velocity
2. Dynamic Technique
This technique was proposed by Taguchi and Humphrey, 1966. With this technique, the
volumetric mass transfer coefficient for oxygen transfer in a fermentation process can be
estimated (Atiya, Z. Y, 2012) and is based on oxygen material balance. Thus,
d CL
dt
Where r o
2
= KLa(C* - CL) – r o Cx
(1)
2
is cell respiration rate
Again, if KLa is equal to zero, the slope of a plot of CL versus t will give an estimate of r o Cx.
2
rearranging equation 1 to give a linear relationship we obtain;
CL = C L * -
1 d CL
+r o C x
K L a dt
(
2
)
And a graphical plot will give a slope of
(2)
1
K La
from which the volumetric mass transfer
coefficient can be estimated.
3. Billet and Schultes (1999) gas-phase mass –transfer model
KGa = Cv
[
ag
d p ( ∈−h L )
]
1
2
DG
u G ρG 34
(
)
ag μG
1
Sc G3
(3)
Where Cv is the packing specific constants, a is the area, and the hydraulic dimension of
the packing, dp
References
Atiya Z. Y. Estimation of Volumetric Mass Transfer Coefficient in Bioreactor. Al-Khwarizmi
Engineering Journal, Vol. 8, No.3, (2012) 75-80
Billet, R., Schultes, M., 1999. Prediction of mass transfer columns with dumped and arranged
packings. Updates summary of the calculation method of Billet and Schultes. Trans IChemE 77
(Part A), 498–504.
Madhvanand N. K, A. Renken, and L. Kiwi-Minsker. Gas-liquid and Liquid-Liquid mass transfer
in microstructured reactors. Chemical Engineering Science 66 (2011) 3876 – 3897
Taguchi, H., Humphrey, A.E., 1966. Dynamic measurement of the volumetric oxygen transfer
coefficient in fermentation systems. J. Ferment. Technol. 44, 881–889.
Van Baten, J.M., Krishna, R., 2004. CFD simulations of mass transfer from Taylor bubbles rising
in circular capillaries. Chemical Engineering Science 59,
2535–2545
200338840
STUDENT NAME:
Osiki Izijesu. O
COURSE NAME:
Advanced Mass Transfer
DEPARTMENT:
Process Systems Engineering
Number 1A-1
TITLE: Determination of Gas diffusion coefficients in saturated porous
media: Helium, He and Methane, CH4 diffusion in Boom clay
BRIEF INTRODUCTION
Boom Clay is presently a potential host rock mainly as a medium of disposal of high level and
long lived radioactive waste in Belgium, and also because production of gas is ineluctable in
geological depository (Yongfeng Deng et al., 2011). Today, gas is produced by a number of
different mechanisms including radiolysis of water and organic materials, microbial debasement
of organic wastes as well as anaerobic corrosion of metals in waste materials (Elke Jacops et al.,
2013). Rodwell et al., 1999 and Yu and Weetjens, 2009 for instance write that, hydrogen gas is
produced from corrosion and radiolysis and conversely methane and carbon dioxide are products
of microbial debasement. Diffusion is the main transport phenomenon in the boom clay since
dissolved gases diffuse from the repository as dissolved species. Importantly, good estimates for
gas diffusion coefficient is necessary in order to estimate and understand the diffusion process
such as gas generation and dissipation through barriers (Elke Jacops et al., 2013).
Literature provides limited information as to the determination of diffusion coefficients of gases
in saturated porous media and available information and data is difficult to analyze or understand
(Elke Jacops et al., 2013). However, there are generally three types of methods that have been
adopted as possible methods to analyze and estimate diffusion coefficients of gases regardless of
individual method’s limitations.
In the first method already used by Bigler et al. (2005) and Gomes-Hernandez (2000), the
diffusion coefficient of gases is determined using gas concentrations from outgassing of the clay
samples stored in a vacuum container and measurement of the concentration of gas released by
the clay samples. In fact, Bigler et al. (2005) estimated the diffusion coefficient of Helium using
a spherical sample of the callovo-oxfordian shale with an uncertainty of 20%. He explains that
calculated uncertainty might have been due to imperfection of the sphere and errors in cutting of
the sample.
The second method determines diffusion coefficient based on the concentration profiles of gases
as natural tracers. Rubel et al. (2002) used this technique and computed the apparent diffusivity
for helium in opalinus clay based on helium’s natural profile obtained at Mount Terri.
Bensenonic et al. (2011) using this technique obtained a helium profile for samples that were
collected from vertical drilled boreholes in the Toarcian shale in France for which the diffusion
coefficients was obtained from outgassing of the water in high vacuum containers. This is an
interesting technique but like the previous method it is only applicable for gases present naturally
in clay and limited to He, Ar and CH 4. Of note is what Mazurek et al. (2011) wrote that, the core
outgassing of noble gases requires advanced equipment and that there is apparent possibility of
leakage at several stages.
The third method for determination of the diffusion coefficient for gases is based on the “in or
through” diffusion technique. This method has been used by Rebour et al. (1997). Although
reported error on the diffusion coefficient of Helium on callovo-oxfordian clay was 20%, this
method has been found to be more versatile as it can be used to compute the diffusion
coefficients for different gases. However, Bigler et al. (2005) argue that interpretation of data
with this method suffered from complications such as anisotropy effects which was not taken
into account and that measured porosity (23%) did not correspond to the porosity value needed
to obtain a good fit (16%). Therefore, a more versatile and simple method that allows for a better
assessment of the diffusivity and suitable for several gases was developed to determine the gas
diffusion coefficient (Elke Jacops et al., 2013).
In this study, the authors developed a method to ascertain the gas diffusion coefficient for
dissolved gases in boom clay and gases, where Helium (He) and Methane (CH 4) served as
reference gases. They write that, their adopted diffusion methodology allows two gases to diffuse
through a clay sample at the same time.
METHODOLOGY
Principle
The experimental work was based on a thorough diffusion test where two dissolved gases were
placed on opposite sides of the Boom clay test core as shown in figure 1 (Shackelford, 1991; Van
Loon et al., 2003). In this method, the porous media is placed between two vessels one of which
contains a known concentration of the diffusant and is the high concentration compartment while
the other known as the low concentration compartment is free of the diffusant. This method used
two diffusants, He and CH4 where the gases are initially concentrated at one side and expectedly
diffuse in different directions afterwards. Importantly, the clay core was sealed in a stainless steel
diffusion cell and connection was via stainless steel plate at both sides to water vessels that were
pressurized with two different gases at the same pressure, shown also in figure 1. Data is
reported in the study only for Helium, He and Methane, CH4.
Experimental Set-up
Since helium diffuses easily through many materials such as polymers used often as seals, Elke
Jacops et al., 2013, emphasize the importance of gas tightness of the entire set up. Thus to avoid
gas leakages, components used are selected based on their gas tightness. For instance, Jacops et
al., 2013 used metal contacts for valves, sensors and couplings as opposed to polymers. Constant
volume was obtained by sealing the diffusion cell properly and the hydraulic press was used to
press the clay core, having a diameter of 80mm and height of 30mm in the diffusion cell. They
also write that perfect sealing between the clay and the diffusion cell was due mainly to the high
plasticity of the clay (Van Geet et al., 2008).
Furthermore, contact between the gas saturated water and the clay was achieved by placing a
porous stainless steel filter plate of diameter 80mm, thickness 2mm and porosity 40% on both
sides of the clay core, where the cover as well as the connections for the water inlet and outlet
were welded onto the body of the container. Gas tightness was ensured by connecting a pressure
sensor, PTX 600 to the two vessels that monitored pressure evolution.
The water with dissolved gas was then circulated over the filters using a magnetically coupled
gear pump, REGLO-Z (ISMTEC, Glattbrugg, Switzerland) calibrated for a flow rate of 3 ± 0.1
ml/min while still ensuring gas tightness and low flow.
Fig 1: Schema of the experimental set-up (vessels and diffusion cell not at scale). Dimensions clay core: diameter 80
mm, length 30 mm. Volume vessels: 1 l, filled with 500 ml water and
500 ml gas (at 10 bar). (Elke Jacops et al., 2013)
Procedure
The clay sample used was taken perpendicular to the bedding plane based on the mineralogical
and physical properties of Boom clay reported by Maes et al (2008). As stated previously, care
was taken to ensure gas tightness prior to the diffusion experiment by pressurizing the set-up
with 10 bar Helium for a 68 day period. Furthermore, gas migration properties had to be
established prior also to the diffusion experiment, by determining the hydraulic conductivity, K
of the clay core. This was done by injecting demineralized water at 0.55MPa, and water flowing
out of the diffusion cell was collected, placed on a precision balance until a stable value was
obtained for five successive intervals. The obtained precision was 8%, with Yu et al. (2013)
reporting that, values for hydraulic conductivity of Boom clay are usually around 1.5 and 8 × 10 12
m/s. Measurement of the hydraulic conductivity was then followed by the preparation of a
solution of oxygen-free water that was 0.014 mol/l NaHCO 3. The water used for preparation of
the solution was stored in an anaerobic compartment and then made to come to equilibrium with
the test gas for a few days to remove N 2. After 500ml of the solution was transferred to each
vessel of the set-up, a gas buffer was placed into the headspace and a sample was then taken to
determine the initial gas composition of both vessels. Once, the diffusion experiment started,
sampling was done regularly in a temperature controlled room (21 ± 2 oC) until about 10 data
points were obtained and to avoid pressure drops sample volume was kept strictly at 6ml. Gas
composition was analyzed with a CP4900 micro GC
Transport Model
Test results obtained from experiment were interpreted with a diffusive model that represents the
transport equation in 1 dimensional geometry and this model is founded on both the first and
second law of Fick for diffusive transport in porous media as represented in equations 1 and 2.
F = - ȠDp
∂C
∂t
∂C
∂X
= Dapp
(1)
∂2 C
2
∂X
(2)
Where F is flux, Dapp is the apparent diffusion coefficient, Dp is the pore diffusion coefficient, Ƞ
denotes porosity; c denotes concentration in the porous medium, t denotes time (s) and x denotes
length (m).
Furthermore, the diffusion coefficients can be correlated by equation 3.
Dapp =
D p Deff
=
R
R
(3)
Of importance, is the decrease of the concentration at the inlet at sampling that could hamper
correct experimental determination of the apparent diffusivity and based on this the model was
implemented numerically. Figure 2 shows the clay core being represented by 30 elements.
Henry’s law (R. Sander, 1999) was applied at one end of figure 2, where the amount of the
dissolving gas is proportional to the partial pressure of the gas in the vessel, while at the other
end the amount of the gas was assumed to be zero. Again, the diffusion accessible porosity set at
0.37 was derived from migration experiments done by Maes et al, (2008) and since assumption
was that there was no adsorption, R is set as 1.At regular time intervals and concentration in
selected elements (grey shades in figure 2), Fick’s law was applied to determine the fluxes at
both faces. Further calculations were performed using COMSOL multi physics version 3.5a,
Earth Science module. Finally, detailed information provided by Glaus et al, (2008) was used to
establish the possible (if any) influence of the filters on the transport process.
Figure 2: Geometry used in the diffusive transport model. Grey cells indicate where the evolution of the dissolved
gas concentrations is recorded. (Elke Jacops et al., 2013)
Results and Discussion
As highlighted in the procedure section, the hydraulic conductivity of the clay core was predetermined as 2.5 × 10
-12
m/s which is a distinctive value for Boom clay (Yu et al., 2013). To
optimize and modify experimental conditions, the sampling volume, frequency and measuring
methodology for the entire experiment, Helium, He and Argon, Ar were used initially even
though both experimental and practical problems resulted in unsatisfactory data. Accordingly, a
second experiment fully described previously and using Helium, He and Methane, CH 4 was
carried out and figures 3 and 4 show graphic representation of results. Furthermore, using the
diffusion model and implementing it in COSMOL, yielded the diffusion coefficients of He and
CH4 as 12.2 × 10 -10 m2/s and 2.42 × 10 -10 m2/s respectively as shown in table 1.
Figure 3: CH4 concentration profile in the opposite vessel, showing the experimental vs. fitted curve. The dotted
lines are calculated results for Dapp (upper line: 3.03 × 10 -10 m2/s; lower line: 1.82 × 10-10 m2/s) and the thick
line is value obtained as best fit, 2.42 × 10 -10 m2/s (Elke Jacops et al., 2013)
From figures 3 and 4, one can observe that not only is the best fit for Dapp plotted but also for
Dapp ± 25%, which indicates the sensitivity of the method as well as the possibility of obtaining
precise measurements for Dapp and this result is in line with work done by Aertsens, 2009.
Again, to estimate diffusion coefficient for other gases, the experimentally obtained diffusion
coefficient is used to compute values for the tortuosity/constrictivity ratio as shown in equation
4.
Do
=RƬ
D app
2
/δ
(4)
Where R is the retardation factor and δ is the constrictivity (Put and Henrion, 1988).
Figure 4: He concentration profile in the opposite vessel, showing the experimental vs. fitted curve. The dotted
lines are calculated results for Dapp (upper line: 15.3 × 10
-10
m2/s; lower line: 9.15 × 10-10 m2/s) and the thick line is
value obtained as best fit, 12.2 × 10 -10 m2/s (Elke Jacops et al., 2013)
The tortuosity, Ƭ is known as the ratio of the effective travelled distance vs the start and end
point distances while constrictivity accounts for the widening and narrowing of clay pore (Collin
and Rasmuson, 1988). Again, since there is no adsorption of gases and pore size distribution of
the boom clay is between 6 and 30 000nm (Hemes et al, 2012), both constrictivity, δ and
retardation, R factor are equal to one.
Table 1: Overview of the fitted diffusion parameters for He and CH4 in Boom Clay
a
Jaehne et al. (1987)
b
Boudreau (1996)
The tortuosity/constrictivity ratio determine for both CH4 and He as shown in table 1, can be
used to predict the Dapp ranges for other gases. This is calculated by rearranging equation 4 and
2
substituting already known Do value and average value of Ƭ /δ
Dapp = (
Table 2:
a
Boudreau (1996)
Do
R
)(
δ
2
Ƭ
)
(5)
as computed in table 2.
Gas diffusion in other clay formations was compared with those of the Boom clay and although
direct comparism cannot be made due to differences in physical and mineralogical properties, it
provides estimates as to whether obtained values are in line with already conducted research
work. For instance, calculated values for Ƭ2/δ in opalinus clay, 3.5 and Oxfordian clay, 3.8
(Gomez-Hernandez, 2000; Bigler et al, 2005)proved to be in line with those determined
experimentally considering factors such as higher densities and lower porosity in these
formations.
In Conclusion, the authors have been able to use a simple method that is both precise and
sensitive in determining the diffusion coefficients of gases in porous media and this method
shows prospects for use in a number of industries and not just for geological disposals.
References
Aertsens, M., 2009. Re-Evaluation of the Experimental Data of the MEGAS Experiment on
Gas Migration Through Boom Clay. SCK•CEN-ER-100. SCK•CEN, Mol, Belgium. (http://
hdl.handle.net/10038/1182).
Bensenouci, F., Michelot, J., Matray, J., Savoye, S., Lavielle, B., Thomas, B., Dick, P., 2011. A
profile of helium-4 concentration in pore–water for assessing the transport phenomena
through an argillaceous formation (Tournemire, France). Phys. Chem. Earth. 36,
1521–1530.
Bigler, T., Ihly, B., Lehmann, B., Waber, H., 2005. Helium Production and Transport in the
Low Permeability Callovo–Oxfordian Shale at the Site Meuse/Haute Marne, France.
Nagra Arbeitsbericht NAB 05-07, Switzerland.
Boudreau, P., 1996. Diagenic Models and Their Interpretation — Modelling Transport and
Reactions in Aquatic Sediments. Springer, Berlin.
Collin, M., Rasmuson, A., 1988. A comparison of gas diffusivity models for unsaturated
porous media. Soil Sci. Soc. Am. J. 52, 1559–1565.
Elke. J, G. Volckaert, N. Maes, E. Weetjens, and J. Govaerts. Determination of gas diffusion
coefficients in saturated porous media: He and CH4 diffusion in Boom Clay. Applied Clay
Science 83-84 (2013) 217-223
Glaus, M., Rossé, R., Van Loon, L., Yaroshchuk, A., 2008. Tracer diffusion in sintered stainless
steel filters: measurement of effective diffusion coefficients and implications for
diffusion studies with compacted clays. Clay Clay Miner. 56, 667–685.
Gomez-Hernandez, J.J., 2000. FM-C Experiment: Part A) Effective Diffusivity and Accessible
Porosity Derived Fromin-situ He-4 Tests. Part B) Prediction of He-3 Concentration
in a Cross-Hole Experiment. Mont Terri Project Technical Note TN 2000-40.
Switzerland.
Grathwohl, P., 1998. Diffusion in Natural Porous Media: Contaminant Transport, Sorption/
Desorption and Dissolution Kinetics. Kluwer Academic Publishers.
Hemes, S., Desbois, G., Urai, J., De Craen, M., Honty, M., 2012. Variability of the Morphology
of the Pore Space in Boom Clay From BIB-SEM, FIB and MIP Investigations on
Representative Samples. 2nd Project Report. SCK•CEN-ER-208. SCK•CEN, Mol,
Belgium (http://hdl.handle.net/10038/7752).
Jaehne, B., Heinz, G., Dietrich, W., 1987. Measurement of the diffusion coefficients of sparingly
soluble gases in water. J. Geophys. Res. 92, 10767–10776.
Kroos, B., Schaefer, R., 1987. Experimental measurements of the diffusion parameters of
light hydrocarbons in water-saturated sedimentary rocks — I. A new experimental
procedure. Org. Geochem. 11, 193–199.
Maes, N., Salah, S., Jacques, D., Aertsens, M., Van Gompel, M., De Cannière, P., Velitchkova,
N., 2008. Retention of Cs in Boom Clay: comparison of data from batch sorption tests
and diffusion experiments on intact clay cores. Phys. Chem. Earth. 33, S149–S155.
Mazurek, M., Alt-Epping, P., Bath, A., Gimmi, T., Waber, N., Buschaert, S., De Cannière, P.,
De Craen, M., Gautschi, A., Savoye, S., Vinsot, A., Wemaere, I., Wouters, L., 2011. Natural
tracer profiles across argillaceous formations. Appl. Geochem. 26, 1035–1064.
Put, M., Henrion, P., 1988. An improved method to evaluate radionuclide migration model
parameter from flow-through diffusion tests in reconsolidated clay plugs. Radiochim.
Acta 44 (45), 343–347.
Rebour, V., Billiotte, J., Deveughele, M., Jambon, A., le Guen, C., 1997. Molecular diffusion
in water-saturated rocks: a new experimental method. J. Contam. Hydrol. 28, 71–93.2
Rodwell, W., Harris, W., Horseman, S., Lalieux, P., Müller, W., Ortiz, L., Pruess, K., 1999. Gas
Migration and Two-Phase Flow Through Engineered And Geological Barriers for a
Deep Repository for Radioactive Waste. EUR19122 EN. Luxembourg. (http://bookshop.
europa.eu/en/gas-migration-and-two-phase-flow-through-engineered-and-geologicalbarriers-fora-deep-repository-for-radioactive-waste-pbCGNA19122/).
Rübel, A., Sonntag, C., Jippmann, J., Pearson, F., Gautschi, A., 2002. Solute transport in
formations of very low permeability: profile of stable isotope and dissolved noble
gas contents of pore water in the Opalinus Clay; Mont Terri, Switzerland. Geochim.
Cosmochim. Acta 66, 1311–1321.
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Organic
Species
of
Potential
Importance
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(http://www.mpch-mainz.mpg.de/~sander/res/henry.html)
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Number 1A-2
Some other models for estimating Diffusivity include the following;
1. Wilke-Chang model for estimating diffusivity in liquid phases
A number of research work both completed and ongoing have proposed a number of methods for
estimating diffusivity in liquid phase systems. However, the Wilke-Chang model is the most
widely used and diffusion coefficient is given as (Reid R. C. et al, 1977);
Dm =
7.4 X 10−8 T
Ƞ svV
0.6
b, a
. √ M sv α sv
(1)
Where T is the absolute temperature, M is the molecular weight, Ƞ is the viscosity, V b the molar
volume at the normal boiling point, the subscripts a and sv denote the solute and solvent
respectively.
If the aggregation of solute molecules is accounted for, then the Wilke-Chang equation is
modified as follows (Kanji M. and R. Isogai, 2011);
Dm = =
α a V b ,a
¿
¿
¿ 0.6 Ƞ
¿
¿
7.4 X 10−8 T
¿
. √ M sv α sv
(2)
Where αa is the association coefficient of the solute
2. Maxwell-Stefan Diffusivity model for Binary Liquid systems
This model is based on Erying’s absolute reaction rate theory and an extension of the Vignes
model (Glasstone et al, 1941; Vignes. A, 1966). For a non-ideal concentrated binary mixture, the
diffusion coefficient can be expressed as;
D = Did Г (1)
Where, Г is the thermodynamic correction factor, D id is diffusion coefficient for an ideal system
and is equal to the Maxwell-Stefan diffusivity, D12 thus;
Did = D12 (2)
Again, Did is directly proportional to the distance between two successive equilibrium positions,
λ, that is;
Did = λ2k
(3)
k is the rate constant and is computed from the change in Gibbs energy, thus;
k=
T kb
h
−∆ g12
exp(
)
RT
(4)
Δgi is the net activation energy for the diffusion process, and combining equation 2 to 4 gives;
D12 =
λ2 T k b
h
exp(
−∆ g12
)
RT
(4)
For the Vignes model, a linear dependence of Δgi on the composition is assumed (Vignes. A,
1966), thus;
∞
∞
Δg12 = x2 Δ g12 + x1 Δ g21
(5)
And the diffusion coefficient is deduced as;
∞
∞
D12 = ( D 12 )x2 ( D21 )x1
(6)
For non-ideal systems,
V
Δg12 = ϕ2 V 2
∞
Δ g12 +
V
V1
∞
ϕ1 Δ g21
(7)
Where Vi is the partial molar volume of component i, V is the molar volume of the system and ϕ i
is volume fraction of component i.
Replacing ϕi by ϕii the local volume fraction we obtain,
V
Δg12 = ϕ22 V 2
V
V1
∞
Δ g12 +
∞
ϕ11 Δ g21
(8)
We can then obtain a new form of the Vignes model for the Maxwell-Stefan diffusivity as;
∞
∞
D12 = = ( D 12 )Vϕ22/V2 ( D 21 )Vϕ11/V1
(9)
3. Fuller, Schettler, and Giddings Correlation prediction for Binary Gas Diffusivities
Fuller et al, 1966 used 308 experimental values of the diffusivities of various gases to determine
the coefficients a, b, c, d, g, and f equation using a nonlinear least- squares analysis, thus;
DAB =
cT
[
b
[
]
1
1 1
+
M A MB 2
a
g
]
p (∑ V A ) +(∑ V B ) ˄f
(1)
The emperical equation that gives the smallest standard deviation is,
−3
DAB =
10 T
[
1.75
a
[
]
1
1
1
+
˄
M A MB 2
g
]
p (∑ V A ) +(∑ V B ) ˄f
(2)
Where p is the total pressure (atm), Mi is the molecular weight, DAB is the diffusivity (cm2/s), T is
the temperature and Vi is the diffusion volume for component i.
REFERENCES
Fuller E. N, P. D. Schettler, and J. C. Giddings, Ind. Eng. Chem. 58(5), 19 (1966).
Glasstone. S, Laidler K. J. and H. Eyring. The Theory of Rate Processes: the Kinetics of
Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena; McGraw-Hill: New
York, 1941.
Kanji M, and R. Isogai, Estimation of Molecular Diffusivity in Liquid Phase Systems by the
Wilke-Chang equation, Journal of Chromatography A 1218 (2011) 6639 – 6645
Reid R. C, J.M. Prausnitz and T.K. Sherwood, The Properties of Gases and Liquids, McGrawHill, New York, 1977.
Vignes, A. Diffusion in Binary Solutions. Variation of Diffusion Coefficient with Composition.
Ind. Eng. Chem. Fundamentals. 1966, 5, 189-199.
QUESTION 1B
TITLE: On the Modelling of Gas-phase Mass-transfer in Metal sheet
Structured packings
Introduction
The efficiency of mass transfer in gas-liquid reactors is expressed using the volumetric mass
transfer coefficient, KGa. To better understand mas transfer it is imperative that the film
coefficient, KG and the effective interfacial area, a be separated and determined from the
volumetric mass transfer coefficient (Alves S.S et al, 2004). Early works of Chilton and Colburn,
1934 state that, gas-phase mass transfer correlations are often based on fundamental models such
as turbulent flow models, and boundary layer models.
In this article, modelling of gas-phase mass transfer in metal sheet structured packings (M250Y,
M350Y, M500Y, M452Y) was carried out under absorption systems of SO2 chemisorption into
NaOH aqueous solution and CO2 chemisorption into the NaOH aqueous solution. The
determination of the gas-phase mass transfer coefficient for all reviewed packings were
correlated to the dimensionless equation
ShG = 0.409
ℜ0.622
G
ℜ0.0592
L
(1)
Experimental
The experiments were carried out using a column of inner diameter, 0.29m and bed height of
0.84m. For the withdrawal of gas samples by aid of a sampling device (of diameter 30mm), each
packing element was drilled horizontally using an electrical discharge machinery (Valenz et al,
2011). Again, liquid distribution was carried out using a pipe liquid distributor having a drip
point density of 630 dp/m2, while gas entered the column through a large drum. As mentioned
earlier, the aqueous solution of NaOH having a concentration of 1kmol/m 3 served as the
absorption liquid. The batch was replaced when the hydroxyl concentration began to drop,
0.5kmol/m3 for the interfacial area measurement and 0.1kmol/m3 for the volumetric mass transfer
measurements. Furthermore, the temperature of the gas and liquid entering the column was kept
at 20oC and atmospheric pressure. The velocities of the liquid used were 4, 10, 20, 40, 60, 80,
100 m3/(m2 h) and that of the gas superficial velocities were 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 m/s.
Importantly, introduction of the tracer gas into the gas phase was done before its entry into the
column to ensure proper mix and its concentration maintained at the lowest sampling point of at
least 1000 ppm. Continuous measurement of the concentration of CO 2 and SO2 in the gaseous
samples was done using IR analyzer S710, prior to which the gas was dried by passing it through
a cartridge containing crushed CaCl2 (CO2 analysis) or Mg(ClO4)2 (SO2 analysis).
Determination of Volumetric Mass Transfer Coefficient, KGa
To determine the volumetric mass transfer coefficient (K Ga), measurements should be carried out
by experiments where the mass transfer resistance is limited to the gas-phase and where
evaporation of the gaseous solute is used (Danckwerts, 1970; Rejl et al, 2009). In this work,
absorption of SO2 into the NaOH solution was used and the volumetric mass transfer coefficient,
KGa determined with the relation;
KGa =
UG
H
¿
C SO
2
ln C OUT
SO
(2)
2
An independence of volumetric mass transfer coefficient, KGa on hydroxyl concentration in the
NaOH concentration range of 0.1M to 1.0M has been shown by the authors.
Effective area, a determination
Determination included the absorption of diluted CO2 from the air into the lye where the mass
transfer resistance in the gas phase is low and increased reaction of CO 2 with the hydroxyl takes
up more CO2 within the liquid film. This leads to determination of the mass transfer rate by the
reaction transport phenomena in the liquid film correlated to the physical quantities of the
system. Determining the effective interfacial area is by correlation of the CO 2 local mass transfer
rate with its balance along the packing height (Linek et al, 1984) as shown in equation 3.
aR
(K OG a)CO
2
G=0
= He √ DL CO 2 K OH COH , ave
(3)
and substituting value for the volumetric mass transfer as shown in equation 2 gives;
aR
G=0
=
1
He √ DL CO 2 K OH COH , ave
.
UG
H
¿
C SO
2
. ln C OUT
SO
(4)
2
Also, effective area data was correlated to the gas phase mass transfer resistance as shown in
equation 5 (Hoffmann et al, 2007);
a=
aR
1−RG
G=0
(5)
where RG is the relative mass transfer resistance which the authors defined as;
RG =
( K OG a)CO
( K OG a) SO
2
2
.
(
DG ,SO
DG ,CO
2
2
)
2
3
(6)
Mass transfer coefficient determination
The absorption experiments for the volumetric mass transfer coefficient determination and the
effective interfacial area was performed using the aqueous NaOH as the liquid phase and air as
the gas phase. Thus, considering the same liquid and gas flow rate, interfacial area was
considered as equal, as shown in equation 7;
KG =
KGa
a
(7)
Results
Using the power law, experimental results of volumetric mas transfer coefficient and interfacial
area were correlated as show in equation 8 and 9 and table 1 shows the constants for individual
packings; Figures 1a to d also shows a plot of experimental data for which the correlations are
valid.
α
β
KGa = C1 . U G . B
γ
δ
a = C2 . U G . B
(8)
(9)
(Rejl F. J. et al, 2014)
Experimental data collected for the volumetric mass transfer coefficient was compared to those
obtained for Mellpak 250Y and SO2-air-lye system by Klement. M (2002) as shown in figure 1a.
Results further reveal that, the mass transfer coefficient is dependent on the superficial gas phase
velocity, while its dependence on the liquid flow rate is quite weak. Also, since the geometry of
the studied packing were similar, it became imperative to correlate the resulting values of the
mass transfer coefficient as shown in equation 1, for the range of Re G = 240-2500 and ReL = 10300.
Figure 1(a)
M 250Y: volumetric mass-transfer coefficients kGa data used for construction of the correlation Eq. (24).
The lines represent the correlation. Comparison with the data of Klement (2002). (b) M350Y: volumetric masstransfer coefficients kGa data used for construction of the correlation Eq. (24). The lines represent the correlation. (c)
M452Y: volumetric mass-transfer coefficients kGa data used for construction of the correlation Eq. (24). The lines
represent the correlation. (d) M500Y: volumetric mass-transfer coefficients kGa data used for construction of the
correlation Eq. (24). The lines represent the correlation. (Rejl F. J. et al, 2014)
Figure 2: Comparism of experimental and correlated volumetric mass transfer coefficient.
(Rejl F. J. et al, 2014)
Again, mass transfer coefficient for M250Y were correlated in two different ways, effective
phase velocity (UE) or superficial velocity (UG). This further indicated that effective phase
velocity improved fit of the mass transfer coefficient data as seen from the comparative standard
deviation (2.85%) of the correlation, thus;
0.628
KG = 0.0308 . U G
rel.st.dev = 3.16%
(10)
0.703
KG = 0.0209 . U E
rel.st.dev = 2.85%
(11)
Finally, mass transfer data obtained for all packing types under present study was successfully
correlated by the dimensionless form as in equation 1 which shows how the phase velocity and
packing geometry affect the mass transfer (Rejl F. J. et al, 2014).
References
Alves S.S, C. I. Maia, J. M. T. Vasconcelos. Gas-Liquid Mass Transfer Coefficient in Stirred
tanks interpreted through bubble contamination Kinetics. Chemical Engineering and Processing
43 (2004) 823 – 830.
Chilton, T.H., Colburn, A.P., 1934. Mass transfer (absorption) coefficients. Prediction from data
on heat transfer and fluid friction. Ind. Eng. Chem 26, 1183–1187
Danckwerts, P. V., 1970. Gas–Liquid Reactions. McGraw-Hill Book Company, New York.
Hoffmann, A., Mackowiak, J.F., Gorak, A., Haase, M., Loning, J.-M., Runiwski, T.,
Hallenberger, K., 2007. Standardization of mass transfer measurements. Basis for the description
of absorption processes. Chem. Eng. Res. Des. 85 (A1), 40–49.
Klement, M., (Diploma thesis) 2002. Studium transportních charakteristik vyplní Mellapak
(Study of the transport characteristics of the Mellapak packings). ICT, Prague.
Linek, V., Petˇríˇcek, P., Beneˇs, P., Braun, R., 1984. Effective interfacial area and liquid side
mass transfer coefficients in absorption columns packed with hydrophilised and untreated plastic
packing. Chem. Eng. Res.Des. 62, 13–21.
Rejl, F. J . , Linek, V., Moucha, T., Valenz, L., 2009. Methods standardization in the measurement
of mass-transfer characteristics in packed absorption columns. Chem. Eng. Res. Des. 87, 695–
704.
Rejl F. J, L. Valenz, J. Haidl, M. Kordac, and T. Moucha. On the modelling of gas – phase mass
transfer in metal sheet structured packings. Chemical Engineering Research and Design 93
(2015) 194-202.
Valenz, L., Rejl, F. J . , Linek, V., 2011. Effect of gas- and liquid-phase axial mixing on the rate
of mass transfer in a pilot-scale distillation column packed with Mellapak 250Y. Ind. Eng. Chem.
Res. 50 (4), 2262–2271.
Question 1B-2
Other Models to predict the volumetric mass transfer coefficient include the following
1. Model for rising Taylor bubble in circular capillaries developed by Van Baten and
Krishna (2004)
This model considers two contributions to mass transfer, one of which is the caps at either end of
the bubble while the other is the liquid film that seemingly surrounds the bubble (Madhvanand
N. K. et al, 2011). The following correlation was developed by the aforementioned authors for
the overall volumetric mass transfer coefficient, KLa (liquid volume);
KLa = KL, cap acap + KL, filmafilm (1)
Furthermore, the mass transfer coefficients KL, cap and KL, film was derived from the penetration
mass transfer model and is as shown in equations 2 and 3, thus;
√
√
2
KL, cap =
2
KL, film =
2 D m ub
(2)
π2 dt
Dm
(3)
π 2 tc
Then substituting the value of the specific interfacial area of the two caps (a cap) and the film
specific interfacial area (afilm), the volumetric mass transfer coefficient, KLa (liquid volume)
becomes;
KLa =
√
2
2 D m ub
2
π dt
4
Luc
+
√
2
Dm
2
π tc
4 ∅G
dt
(4)
Where Luc is unit cell length, ∅G is gas hold up, ub is bubble rise velocity
2. Dynamic Technique
This technique was proposed by Taguchi and Humphrey, 1966. With this technique, the
volumetric mass transfer coefficient for oxygen transfer in a fermentation process can be
estimated (Atiya, Z. Y, 2012) and is based on oxygen material balance. Thus,
d CL
dt
Where r o
2
= KLa(C* - CL) – r o Cx
(1)
2
is cell respiration rate
Again, if KLa is equal to zero, the slope of a plot of CL versus t will give an estimate of r o Cx.
2
rearranging equation 1 to give a linear relationship we obtain;
CL = C L * -
1 d CL
+r o C x
K L a dt
(
2
)
And a graphical plot will give a slope of
(2)
1
K La
from which the volumetric mass transfer
coefficient can be estimated.
3. Billet and Schultes (1999) gas-phase mass –transfer model
KGa = Cv
[
ag
d p ( ∈−h L )
]
1
2
DG
u G ρG 34
(
)
ag μG
1
Sc G3
(3)
Where Cv is the packing specific constants, a is the area, and the hydraulic dimension of
the packing, dp
References
Atiya Z. Y. Estimation of Volumetric Mass Transfer Coefficient in Bioreactor. Al-Khwarizmi
Engineering Journal, Vol. 8, No.3, (2012) 75-80
Billet, R., Schultes, M., 1999. Prediction of mass transfer columns with dumped and arranged
packings. Updates summary of the calculation method of Billet and Schultes. Trans IChemE 77
(Part A), 498–504.
Madhvanand N. K, A. Renken, and L. Kiwi-Minsker. Gas-liquid and Liquid-Liquid mass transfer
in microstructured reactors. Chemical Engineering Science 66 (2011) 3876 – 3897
Taguchi, H., Humphrey, A.E., 1966. Dynamic measurement of the volumetric oxygen transfer
coefficient in fermentation systems. J. Ferment. Technol. 44, 881–889.
Van Baten, J.M., Krishna, R., 2004. CFD simulations of mass transfer from Taylor bubbles rising
in circular capillaries. Chemical Engineering Science 59,
2535–2545
