(172d) Evidence for the Need of Probability Density Function Representations of Liquid-Side Mass Transfer Rate Coefficients for Absorption Modeling in Structured Packings

Gas-liquid mass transfer parameters
of structured packings are generally determined from global  absorption or
desorption measurements over some bed height, sometimes with treatment of
entrance and outlet effects. Correlations of these global parameters are then developed
relative to the overall flow conditions and fluid properties and often take the
classical form

                 Eq.
1

where   and  stands either for
the gas phase, G, or the liquid phase, L. These equations are subsequently applied
in models for sizing large absorption or distillation columns using local
average flow conditions and fluid properties. Rejl et al. [1] have recently shown that the dependencies to the flow
conditions in such correlations, while allowing to match global features, such
as HETP, do not accurately reproduce axial concentration profiles.  They
propose a so-called “profile method” to obtain better functions relative to
flow conditions. In this work we present other experimental evidence that
indicates that correlations of mass transfer parameters derived from a global
perspective may be insufficient to properly describe the details of the
processes required for column sizing.

We consider the case of physical absorption or
desorption. Based on the double-film model the global overall gas-side mass
transfer resistance (1/K_Ga ) is mathematically the sum of the
global  gas-side (1/k_Ga ) and liquid –side (m/k_La
)  resistances:

1/K_Ga=1/k_Ga+m/k_La
 or  R_G,ov = R_G+mR_L                     Eq.
2

where m
is the slope of the equilibrium curve.  This equation is valid if all
liquid-gas interfacial areas are subjected to the same mass transfer rates.
However, in many situations such is not the case. Haroun et al. [2] have done CFD calculations of film flow
over a packing structure and indicate that, in coherence with the Higbie
contact time surface renewal model, the liquid side mass transfer coefficient
decreases inversely with the distance or time from the renewal point. Recent
Xray tomographic imaging [3, 4] and CFD simulations [5,6] indicate  the
existence of flow heterogeneities leading to rivulets formation and unwetted
(dry) regions, even with hydrocarbon liquids that have better wetting
characteristics than water.

Now we further consider a case where the global or
average mR_L is secondary to R_G .  If there exists
a local region in the packing where the liquid film velocity is low then it
will likely be characterized by a longer renewal time relative to the mean. This
will lead to a locally higher R_L that will tend to reduce the
overall K_G and hence the mass transfer rate. In contrast, if
there is a region with very low R_L, say along a rivulet, then
there will be no impact on the overall mass transfer rate since the liquid side
resistance is already  globally low. The opposite will hold for a case where
global liquid side resistance is large. Therefore local flow heterogeneities
will, under certain conditions,  alter the true mass transfer rate relative to
one predicted by liquid and gas-side mass transfer parameters determined
independently and on a global basis.

To illustrate this fact, we present experiments
that have been performed with two different chemical systems to measure the
global k_Ga of Mellapak 250.X packing in a 146 mm diameter
column. One system is the fast reactive absorption of SO₂ diluted
with air into  a 2N solution of NaOH.  This system is well known to be only gas
side limited [9]. The second system is the physical absorption of ethanol or
methanol diluted with air into water for which the contribution of liquid side
resistance to the overall resistance is often small. So from a global
perspective, with the reasonable assumption of negligible  liquid side
resistance, both systems should give the same relation for Sh_G as
given in Eq. 1.  Yet the values of Sh_G determined from SO₂
absorption are 50 % higher than from ethanol absorption (Figure 1). Based on
global mass transfer parameters from the literature [7] , under our flow conditions, the contribution of liquid side
resistance to the total does not exceed 10% at the highest gas flowrate. The 50%
difference is well beyond the experimental uncertainties of the global k_L
measurements. For example, it would be necessary to excessively reduce k_L
by 4 to 5 times in order to make the data coherent. In order to help understand 
the different results between the two systems, further experiments were carried
out in a 19 mm ID column over a wire gauze high effective area packing (Sulzer
EX) at very low liquid loads. The liquid side resistance was measured
independently  and found to contribute by at most 5% to the overall resistance.
In this situation the two chemical systems gave the same Sh_G values
within 10%. 

Figure 1.
Comparison of Sh_G values obtained with different systems over
Mellapak 250.X packing at a liquid load of 30 m/h. The values have been
normalized relative to the one obtained with SO₂ absorption at Re^2/3Sc^1/3=100. 

To explain these contrasting results we propose to
consider the nature of the liquid film flow. In the wire gauze packing the
liquid film spreads very uniformly by capillary action and likely stays very
smooth. In the Mellapak 250.X packing the heterogeneities in the film flow are likely
to lead to some form of distribution of local k_L or R_L
values. The existence of k_L distributions has been discussed
in the past by Danckwerts [8] in terms of
a distribution of renewal times and how it impacts the expression for the
global k_L.  However no interaction of this distribution with
the overall resistance to mass transfer was discussed. If the distribution is
described by a probability density function,  f(k_L), then the
fraction of effective area that is subjected to liquid side mass transfer
coefficients of values between k_L ± dk_L/2 is
given by f(k_L)^.dk_L,.  The average
liquid side mass transfer coefficient and overall resistance are then calculated
as

                 Eq.
3

                  Eq.
4

The distribution function should yield = k_L
and, based on the experimental results, /R_G
= 2/3. As an exercise, we found that a log-normal probability density function
on k_L was able to satisfy these conditions whereas simple
normal distribution function could not. The true nature of the distribution
should be the topic of future work.

In conclusion the local liquid flow heterogeneities
that can exist in random and structured packings possibly produce a
distribution of local liquid–side resistances. The use of a global liquid-side
mass transfer coefficients may then lead to erroneous sizing of physical
absorption or distillation columns in situations where both liquid and gas-side
resistances contribute to the overall mass transfer resistance. Under reactive
absorption the k_L distribution will furthermore affect the
local Hatta number (Ha) and lead to a distribution of mass transfer
regimes.  Evaluation of the consequences these distributions should be the
topic of future work.  Last we conclude that physical absorption methods are
not suited for measuring the global k_Ga in random or structured
packing when liquid flow heterogeneities exist, instead, reactive absorption of
SO₂ in NaOH should be used.

[1] F. J. Rejl,
L. Valenz, and V. Linek, “Profile Method” for the Measurement of kLa and kVa in
Distillation Columns. Validation of Rate-Based Distillation Models Using
Concentration Profiles Measured along the Column , Ind. Eng. Chem. Res., 49 (9) (2010)  4383–4398

 [2] Haroun, Y.,
Raynal, L., Legendre, D., Mass transfer and liquid hold-up determination in
structured packing by CFD. Chem. Eng. Sci.,  75 (2012) 342–348.

 [3] S. Schug and 
W. Arlt,  Imaging of Fluid Dynamics in a Structured Packing Using X-ray
Computed

Tomography, Chemical Engineering and Technology
39 (2016) 1561–1569.

[4] A. Janzen,
J.Steube, S.Aferka, E.Y.Kenig, M.Crine, P.Marchot, D.Toye, Investigation of liquid
flow morphology inside a structured packing using X-ray tomography. Chemical
Engineering Science 102 (2013) 451–460

[5] Y. Haroun, L.
Raynal, P. Alix,  Prediction of effective area and liquid hold-up in structured
packings by CFD, Chemical Engineering Research and Design,  92  (2014)
2247–2254.

 [6] Bing Dong,
X.G. Yuan , K.T. Yu, Determination of liquid mass-transfer coefficients for the
absorption of CO₂ in alkaline aqueous solutions in structured
packing using numerical simulations , Chemical Engineering Research and
Design 124 ( 2017 ) 238–251.

[7] C. Wang, D.
Song, F. A. Seibert, G. T. Rochelle,  Dimensionless Models for Predicting the
Effective Area, Liquid-Film, Ind. Eng. Chem. Res., 55 (2016) 5373−5384. .

 [8] Danckwerts,
P.V., 1970. Gas-Liquid Reactions. McGraw-Hill, New York.

[9] L. Hegely, J. Roesler, P. Alix, D. Rouzineau, M. Meyer,  Absorption
methods for the determination of mass transfer parameters of packing internals:
A literature review, AIChE Journal  63 - 9 (2017) 3246–3275.