Direct effect of solvent viscosity on the physical mass transfer for wavy film flow in a packed column
Zhijie Xu1, a), Rajesh Kumar Singh2, Jie Bao2 and Chao Wang1
1Physical and Computational Sciences Directorate, 2Energy and Environment Directorate, Pacific Northwest National Laboratory, Richland, WA 99352, USA
ABSTRACT
The interphase mass transfer plays a critical role in determining the height of packed column used in absorption process. In a recent experiments2, the direct impact of viscosity ( ) on the physical mass
𝐿𝐿 transfer coefficient ( ) was observed to be higher in a packed column as compared𝜇𝜇 to the wetted wall
column. We offer a plausible𝑘𝑘𝐿𝐿 mechanism involving the wavy film and eddy enhanced mass transfer in a
packed column to explain underlying physics via analytical and numerical studies. The analytically
derived mass transfer coefficient matches well with experimental observation in a packed column. The
countercurrent flow simulations in a packed column with both uniform and wavy films also confirm this behavior. The predicted shows steep variation with for a wavy film than a uniform film, further
𝐿𝐿 𝐿𝐿 confirms the proposed theory.𝑘𝑘 A similar relation ( 𝜇𝜇~ . ) for a wavy film is also observed in −0 38 𝐿𝐿 theoretical, experimental and numerical studies. 𝑘𝑘 𝜇𝜇𝐿𝐿
Keywords: Mass transfer, viscosity, wavy film, eddy diffusion, carbon capture
1. Introduction
Solvent absorption by countercurrent flow in a packed column is a promising technology for
3-5 mitigating carbon dioxide (CO2) emissions from power plants . The structured packings provides a large surface area for mass transfer while minimizing pressure drop across the column as compared to
random packings 6. Before deploying this technology to industry, its performance and scalability needs to
be investigated 7. Accordingly, hydraulic characteristics and mass transfer between flue gas and liquid
a) Electronic mail: [email protected]
solvent phases in the packing system need to be better understood for optimal packing design. It is a
challenging problem because of the large number of factors that influence the hydrodynamics and mass transfer behavior, such as solvent properties, surface characteristics, packing design, and liquid loading 8.
In this view, understanding the local hydrodynamics and mass transfer behavior is important as it reflects
the liquid flow pattern within the column that significantly influences the overall mass transfer rate
between phases. In this regard, a number of experimental 9-11, analytical 12 and numerical studies 12-19 were conducted to understand the flow hydrodynamics in the packed column. Most of the numerical studies were focused on the multiphase flow simulations using volume of fluid methods that investigates the liquid film thickness and interfacial areas. Further, the interphase mass transfer is a key factor that governs the overall height of the packed column in absorption process.
A number of experimental and theoretical studies have been conducted to explain the mechanism for mass transfer in structured packings. Various theories, including two-film theory 20, penetration theory 21,
surface renewal theory 22, and boundary layer theory, are applied to explain the mass transfer in a packed
column. Most of the models for physical mass transfer are based on two-film theory, which only considers the molecular diffusion that form each side of the gas-liquid interface 20. On the other hand,
Higbie’s penetration theory 21 considers the residence time of a fluid element at the gas-liquid interface. It
suggests an elementary idea to account for turbulence in mass transfer phenomena. Subsequently, correlation for the mass transfer coefficient is derived in terms of various dimensionless numbers, such as
Reynolds, Schmidt, Froude, or Galileo numbers. Previously, the wetted wall column (WWC) concept was
adopted to predict mass transfer based on the assumption of well-defined geometry and precise surface
area in the structured packings. However, recent investigations have shown complex flow behavior inside
the packed column along with the presence of different flow morphologies. As a result, the mass transfer
prediction significantly deviates from the WWC. The mass transfer between phases is dictated by many
factors. Surface tension was proposed to have negligible effect on mass transfer in the turbulent wavy
film flow 23. On the other hand, viscosity is one of the critical factors affecting the interphase mass transfer in absorption phenomenon. A large value of solvent viscosity can lead to a significant reduction
in the mass transfer coefficient. High solvent viscosity can leads to slow diffusion of gas through the gas-
liquid interface (indirect effect) and modify the local hydrodynamics by generating a more stabilized
liquid film on the packing surface (direct effect). As a result, mass transfer between phases can be
significantly reduced.
The effect of viscosity on the mass transfer coefficient has been investigated in detail over the last
few decades. Rochelle and his coworkers 2, 11, 24-26 have performed extensive experimental studies examining the performance evaluation of structure/random packings. The effects of various factors impacting the mass transfer area 11, 25, 26 and mass transfer coefficients between phases 2, 24 also have been
systematically investigated. Song et al. 2, 24 have experimentally investigated the physical mass transfer for various packings (Structured, Random and Hybrid) utilized in carbon capture process . The effect of viscosity on the overall mass transfer coefficient has been explained in two parts: 1) direct contribution due to hydrodynamics and 2) indirect impact that accounts for solvent diffusivity. An experimental campaign designed to study these factors could be expensive and time consuming. Conversely, computational fluid dynamic (CFD) simulations may be leveraged in the design process to achieve economic and efficient solutions while providing important insights into the physics. Notably, CFD does not replace the need for measurements, but it may significantly reduce the amount of experimentation requirement and complement it. In addition, CFD steadily has been gaining acceptance for studying the flow characteristics in structured packings over time 27-31.
Haroun et al. 32, 33 and Marschall et al. 34 have developed the single fluid formulation for mass transfer
between two phases. It is capable of simulating species transfer across interfaces having arbitrary
morphology using interface capturing methods, such as the Volume of Fluid (VOF) method. Both
approaches utilized Henry’s law with a constant coe cient for thermodynamic equilibrium of chemical
species to capture the jump of the species concentrationffi across the interface. This approach also is
employed 35-38 for various chemical processes involving mass transfer phenomenon. Nieves-Remacha et al. 36 use this approach to simulate the hydrodynamics and mass transfer phenomenon in advanced-flow
reactor technology, an alternative to scale up continuous flow chemistries from micron to millimeter
scales. Hu et al. 39 have conducted a VOF simulation similar to the method suggested by Haroun et al. 33
for mass transfer in a falling wavy film. They characterize the CO2 absorption mass transfer in vorticity in
8, 40 the vicinity of the interface and the local CO2 concentration. In the same context, Singh et al. have conducted VOF simulations for film and rivulet flow over an inclined plate for the shape gas-liquid
interface and flow morphology. Extensive multiphase flow simulations also have been conducted that account for the factors affecting rivulet shape and stability. Note that the shape of the interface
significantly affects the mass transfer across it. Sebastia-Saez et al. 41 have investigated physical mass transfer in a countercurrent flow over an inclined plate. Mass transfer due to physical absorption shows
non-monotonic variation with increased liquid load. The maximum value of mass transfer is observed at the fully wetted film. Recently, Wang et al. 37, 38 have applied a similar approach for physical and reactive mass transfer for CO2 capture in a WWC, where they show the advantage of CFD modeling over the standard two-film theory. In addition, CFD modeling can account for the local flow hydrodynamics, such
33 as wavy interface, and variation in film thickness in CO2 capture. Haroun et al. shows that mass
transfer between phases strongly depends on the packing design. A sheet of triangular channel shows
higher mass transfer than a flat surface. However, their problem setup was limited to concurrent gas-
liquid flow.
In a recent experimental study by Song et al. 2, the direct effect of viscosity on the mass transfer coefficient is shown to be significantly higher than the corresponding one for a WWC. The mass transfer coefficient for a WWC was derived by Pigford 42 and can be expressed as:
a13 = ⋅ 0.333ν − 1 6 1 6 1/2 , (1) kL1.15 ug LL 12 DL L
where uL is solvent velocity at the inlet, ν L is the kinematic viscosity, g is the gravitational
acceleration, DL is the diffusivity of the transport species in the solvent phase, a is the solvent inlet size, and L is the WWC height. This above analytical expression (1) was derived via solving the hydrodynamic
equations for laminar film falling over a vertical plate coupled with mass transfer in accordance with the penetration theory 42, 43. On the other hand, a similar correlation was derived from extensive experimental outcome for the packed columns 2:
0.565−− 2/5 1/6 0.065 1/2 kL=0.12 ⋅ u LLν ga p D L, (2)
where ap is specific area of the packings presented as / . Equation (2) clearly shows a higher 2 3 impact of solvent viscosity on the mass transfer coefficient𝑚𝑚 for 𝑚𝑚a packed column with an exponent of -2/5
as compared to -1/6 for the WWC (Eq. (1)). In addition, a larger dependence of mass transfer on the solvent velocity (uL) also can be identified for the packed column with an exponent of 0.565 compared to
0.33 in Eq. (1). Of note, the flow in a packed column is more complicated compared to WWC because of various factors, including the intermittent nature of the liquid distributor, surface characteristics of the packing material, and turbulent nature of flows due to design and arrangement of packing units. As a result, flow in the packed column significantly deviates from a well-controlled laminar flow present in the
WWC. Subsequently, it may contribute to the enhanced dependency of mass transfer on the viscosity 2.
However, unlike analytical expression for WWC (Eq.(1)) that is corroborated by a rigorous and quantitative theory, the exact mechanism is not fully understood for the packed column, and associated quantitative analysis is still lacking. In this context, an extensive theoretical study with numerical
simulations will provide significant insights and quantitative explanations of the physics behind the correlation in Eq. (2), as well as additional insights into the design optimization of a packed column.
In this paper, an analytical derivation for the dependence of mass transfer on viscosity is presented in
Section 2. Section 3 briefly describes the formulation for VOF simulations followed by a description of the multiphase flow simulation setup and meshing of computational domain. The simulation results and
comparison for the mass transfer coefficient are presented in Section 4 followed by a conclusion. 2. Mathematical Formulation of the Physical Mass Transfer
We consider a wavy film with rolling waves falling over a vertical flat plate due to gravity with the
physical mass transfer of gas species across the gas-liquid interface. Figure 1 shows the schematic of the
problem, where rolling waves can be characterized by an amplitude A, wavelength λ, and wave speed 24TR.
𝑤𝑤 As a result, the thickness profile δ( , ) of the wavy film varies vertically in the falling direction. The𝑢𝑢 x-
axis is directed downward in the streamwise𝑥𝑥 𝑡𝑡 direction (see Figure 1), and the y-axis is perpendicular to the
plate. The governing (continuity and momentum conservation) equations dictating the falling film are
given as:
∂∂uv +=0 , (3) ∂∂xy
∂u ∂ u ∂ u1 ∂ Pµ ∂∂22 uu ++=−+L + +, (4) uv 22g ∂t ∂ x ∂ yρρLL ∂ x ∂∂ xy
∂v ∂ v ∂ v1 ∂ Pµ ∂∂22 vv ++=−+L + uv 22, (5) ∂∂∂t x yρρLL ∂ y ∂∂ xy
where u and v are the vertical and horizontal components of the velocity, ρL is the liquid density, µL is
the dynamic viscosity, g is the gravity and P is the pressure. A parabolic velocity profile in the
streamwise direction for u may be established across the film, which is based on the exact solution of a
steady and fully developed laminar film flow 44:
32qy y2 uxy, = − . (6) ( ) 23 2 wxδδ( ) ( x)
Here, w is the width of film along the direction perpendicular to the x-y plane, and q is the flow rate,
δ (x) q= w u( x, y) dy= w ⋅ u () xδ ( x) , (7) ∫0 av
where uxav () is the average vertical velocity at a location x. Using Eq. (6), the gradient of vertical
velocity u in the x direction can be related to the local film thickness (x),
δ ∂u ∂∂ uδδ3 qy ∂ ==−−(43δ y) . (8) ∂x ∂∂δδ xw2 4 ∂ x
The gradient of lateral velocity v along the y direction at the interface can be computed using the continuity equation (3)
∂∂v uq31 ∂δ =−= , (9) ∂∂δ 2 ∂ yyx=δ ( ) xyx=δ ( ) 2 wx0
where is the average film thickness that can be written as
0 δ 13 3µ 2 Re δ = L 0 2 (10) ρL g
according to the Nusselt theory 45. Here, Re is the film Reynolds number defined as,
ρ qq ua Re =LL = = , (11) µνLLww ν L
where a is the size of the solvent inlet, uL is the solvent velocity at inlet, and ν L is the kinematic
viscosity of the solvent. For a turbulent wavy film flow, transport near a gas-liquid interface is governed
by eddies whose length and velocity scales are characterized by bulk turbulence 23, 46. According to the
mixing length model for eddy diffusion 47, the eddy diffusivity stemming from the local turbulent flow in
the direction normal to the vertical plate can be written as:
∂v 2 DC=⋅∆( y) . (12) Le 1 ∂ y yx=δ ( )
Here, C1 is a proportional constant, y is the penetration depth of gas species into the solvent during a Δ single wave period Tu= λ w (shown in Figure 1). The penetration depth y depends on the wave
Δ frequency f and can be related to the diffusivity of that species in liquid phase ( DL ) and the frequency ( f )
of rolling waves,
∆=y DfL = D Lwλ u. (13) Substitution of Eqs. (9) and (13) into Eq. (12) leads to the expression for eddy diffusivity DLe :
1/3 2 ∂δ ρL g 1/3 DLe = 0.72 Cf1 Re DL . (14) ∂ µ x L
To determine DLe , the thickness gradient ∂∂δ x and the wave frequency f need to be modeled
explicitly. We assume that an arbitrary fluid element at the gas-liquid interface follows a random walk
motion along the y direction (shown in Figure 1) due to the momentum transfer. The velocity of that fluid
element over a single period T of the rolling wave can be written as:
u Vf=ννw = , (15) yLλ L
where, again, ν L is the kinematic viscosity; is the wavelength; u is the wave speed; and fu= w λ is
w the wave frequency. Concurrently, the sameλ fluid element also will experience downward motion along the x direction because of the gravity and vertical velocity gained during the same period T can be written as:
λ g Vgx = = . (16) ufw
The gradient of film thickness along the x direction ( ∂∂δ x ) can be approximated using the wave amplitude A and wavelength λ as:
∂δ 2A ≈ . (17) ∂x λ
A and λ can be related to the distance that the fluid element travels in vertical and lateral directions in a single rolling wave period T. These two quantities should be proportional to the speed in both directions and can be written as:
1 ∂δ A Vy ≈=C2 , (18) 2 ∂xVλ x where C2 is a dimensionless constant on the order of unity. To verify this relation, we use the results obtained from the nonlinear solution of the wavy film falling over a vertical plate by Shkadov 1. The various optimum parameters of the wave regime, such as wavelength, wave propagation speed, and
frequency, can be computed as a function of the Reynolds and Kapitza numbers (Ka), where Ka
incorporates the effect of surface tension:
1 3 1/3 σσρ 4 1 = L = Ka 44 , (19) ρLgg µ L ρν LL
where σ is the surface tension. In Figure 2, we plot the variation of the computed value of A/ with /
𝑦𝑦 at three given Reynolds numbers with the Kapitza number varying between 400 and 4000.λ It can𝑉𝑉 be observed𝑉𝑉𝑥𝑥 that the variation of A/ with / remains unchanged for all Reynolds and Kapitza numbers
𝑦𝑦 𝑥𝑥 studied. Furthermore, it clearly showsλ the𝑉𝑉 linear𝑉𝑉 scaling relation as / = / with = 0.67. The
2 𝑦𝑦 𝑥𝑥 2 numerical solutions (1) are in good agreement with our projection 𝐴𝐴(Eq.𝜆𝜆 (18)𝐶𝐶 )𝑉𝑉 and𝑉𝑉 confirm𝐶𝐶 that the ratio
between wave amplitude and wavelength is proportional to the ratio of two velocity components gained
during a single wave period T.
Next, the ratio A/ can be plotted as a function of Reynolds and Kapitza numbers using the same solution methods 1, λ
A = C Ka−nmRe . (20) λ 3
Here, C is another dimensionless constant. In Figure 3(a), we present the variation of A/ with Reynolds
3 number Re. As expected, A/ increases with an increased value of Re. Conversely, A/ decreasesλ with an
increased value of Ka, whichλ is shown in Figure 3(b) for two different Reynolds numbersλ . Physically, it
can be understood by the fact that the increased value of the Kapitza number leads to a reduction in film
thickness and an increase in wavelength. Subsequently, fluctuations in film thickness also decrease,
causing a lower value of A/ . Both observations also are confirmed in the numerical simulations and will
be explained in a later sectionλ (Figures 6 and 7). Accordingly, A / λ decreases with the increased value of Ka. In addition, A / λ shows nonlinear variation with Re and Ka. To find the scaling relation for the variation of A / λ with Re and Ka, nonlinear regression is conducted, and a scaling relation
0.78− 0.47 A/λ = C3 Re Ka is observed (shown in Figure 4) with C3 ≈1 60 , with 3/4 3/7.
Finally, we can write these relations together as: 𝑚𝑚 ≈ 𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛 ≈
1 ∂δ A Vy ν L 32 −nm ≈=C22 = C f = C 3 Ka Re . (21) 2 ∂xλ Vgx
Substituting Eq. (21) into Eq. (14), the final expression for the eddy diffusivity for wavy film flow with
rolling waves can be presented as:
13 Vy D= 1.44 CC ⋅ Re13 D Le 12V L x 2/3 1/3 (mn+− 1) 3 /3 = 1.44C123⋅ C C Re Ka DL (22) (mn+− 1) 3 /3 = C Re Ka DL
/ / Here, C = 1.44C C C is another empirical constant that aggregates three constants together. To 2 3 1 3 1 derive the expression2 for3 effect of viscosity on the mass transfer coefficient in a wavy film flow with
rolling waves, the molecular diffusivity DL in the analytical expression for WWC (Eq. (1)) can be
replaced by the eddy diffusivity DLe . The difference in mass transfer between wavy film and uniform
film flow primarily is assumed to be primarily due to the difference in diffusivity. It generates the following expression:
1/6 ρ a13 = 1/3 L 1 6 1/2 kuLe 1.15 L g12 DLe µL L − 1/6 (m+1) /6 1/3 n/6 13 1/3 111 6 a 1/2 uaL σ 1/2 =1.15uLL gC D ν 12 ν ρν4 . (23) L LgL LL − (3+m) /6 n/6 + − a σ −+ − = 1.15Cu1/2 (3mn) /6 g( 3) /18 ν (6 3m4n) 18D1/2 LL1/2 L L ρL The original expression for WWC (Eq. (1)) can be fully recovered by substituting the value of the constant C=1, exponents m = -1, and n = 0 in Eq. (23). However, in the case of a packed column, these values can differ because of the turbulent wavy nature of flow. From the solutions of wavy film flow with rolling waves, we find the exact values of exponents as 3/4 3/7 that can be estimated from the scaling relations (Eq. (20)), shown in Figure 4. 𝑚𝑚 ≈ 𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛 ≈
Substituting the value of exponents of m and n into Eq. (22), the eddy diffusivity can be related to the
Reynolds number (Re) and Kapitza number (Ka)
7/12− 1/7 DLe = CRe Ka DL , (24)
and the corresponding mass transfer coefficient ( kLe ) can be expressed as:
−1/14 a5/8 σ = 1/2 5/8 4/21 ν −61 168 1/2 kLe 1.15 CugL 1/2 LLD L ρ −0.07 . (25) a0.625 σ = 1/2 0.625 0.19 ν −0.36 1/2 1.15CuL g 0.5 LLD L ρ
The preceding expression for effective mass transfer coefficient is based on the coupling of hydrodynamic equations and eddy diffusion. It can be quantitatively compared to the correlation derived from extensive
experiments for a packed column (Eq. (2), repeated here for convenience),
0.565 0.17−− 0.065 0.4 0.5 kL=0.12 ⋅ u L ga pLLν D. (26)
It is clear that the mass transfer coefficient for wavy film has significantly higher dependence on both the
solvent flow rate and viscosity compared to the uniform film flow in a WWC (Eq. (1)). This dependence matches well with the correlation (Eq. (26)) developed from an experimental investigation for a packed column 2. The expression presented in Eq. (25) confirms the functional form of the correlation derived
from the experimental observations (Eq. (26)) with a slight difference in the exponent values. The proposed mechanism, i.e., enhanced diffusivity due to wavy film flow in a packed column, reasonably
captures the deviation of physical mass transfer from the WWC. Furthermore, the present expression for
the mass transfer coefficient explores additional contributions from interfacial surface tension. Note that this term is missing in the experimentally derived correlation (Eq. (26)), which highlights the impact of surface tension on the hydrodynamics, thereby influencing the mass transfer phenomena. While it may be
minor for an aqueous amine-based solvent, it can be more significant for a solvent with a higher value of
interfacial surface tension.
Moreover, the mass transfer coefficient can be normalized by /( / ) / , where the term ( / 2 1 3 2 𝐿𝐿 𝐿𝐿 𝐿𝐿 ) / is the viscous length scale. A similar method also has been𝐷𝐷 applied𝜈𝜈 𝑔𝑔 earlier by Yih 46. The𝜈𝜈 1 3 𝑔𝑔 * normalized mass transfer coefficient ()kLe depends on several dimensionless numbers:
2 13 * kLLν 12 (36+m) 12−−n 6 16 kCLe = =1.15 ⋅ Re Sc Ka Ga . (27) DgL
32 Here, Sc=ν LLD is the Schmidt number, and Ga= gL ν L is the Galileo number. In the case of a WWC, the normalized mass transfer coefficient can be obtained by substituting the value as = 1, = 1, and = 0 in Eq. (27): 𝐶𝐶 𝑚𝑚 −
𝑛𝑛 * 13 12− 16 kL() WWC =1.15 ⋅ ReSc Ga . (28)
Therefore, enhancement in the mass transfer for a wavy film can be expressed as:
* k + =Le = 12 (16m) −n 6 E* CRe Ka . (29) kL() WWC
A greater enhancement in the mass transfer can be expected for a large Reynolds number and small
Kapitza number because > 0 and > 0 for a wavy film flow with rolling waves.
3. Numerical Modeling𝑚𝑚 𝑛𝑛
3.1 Mathematical formulation and numerical Scheme
Multiphase flow simulations for mass transfer of a WWC and zigzag column (ZZC) are performed
using the VOF method 48 to verify the analytical derivation of the physical mass transfer in the previous
section. Physical mass transfer across the interface is modeled by a gas dissolution model based on the
Rayleigh–Plesset equation available in Star-CCM+. This method is fully explained in the Star-CCM+ User
Guide 49 with only a brief outline provided here. The governing equations for flow are as follows: ∇⋅u =0 , (30)
∂()ρu T +∇⋅()ρµuu =−∇p + ∇⋅ ∇u +( ∇ u) + ρ g + F , (31) ∂t ( )
where p is pressure and g is the gravitational acceleration. The surface tension force, F, which produces a jump in the normal across the interface, is expressed as a singular body force. The terms and are the phase average density and viscosity, respectively. The surface tension force (F) appears at𝜌𝜌 the gas𝜇𝜇 -liquid interface and is implemented through the continuous surface force (CSF) model 50:
ρκ∇ α F = σ , (32) 1 2 (ρρgl+ ) where is the interfacial tension, is the local curvature of the interface, is the volume fraction of the liquid 𝜎𝜎phase, and can be related𝜅𝜅 to the normal vector that is perpendicular𝛼𝛼 to the interface. The
∇𝛼𝛼 curvature is computed by the divergence of the unit normal nˆ =∇∇αα at the interface,
𝜅𝜅 κ =∇⋅ nˆ (33)
The interface between the phases is captured by solving an additional transport equation (34), where the
value of α is between 0 to 1,
∂α +u ⋅∇α =0 . (34) ∂t
Equation (34) is solved only for the liquid phase, and the volume fraction of the other phase is computed
as (1- ).
α The transient flow simulations are conducted using the implicit scheme in a commercial CFD
code Star-CCM+ 49. In the implicit scheme, specified number time step involves multiple inner iterations to converge the solution. The integration scheme marches inner iterations using pseudo-time steps according to specified value of the Courant number (CO) for stable and converged simulations. The segregated solver was used that solves each of the momentum equations in turn, one for each dimension.
The coupling between the momentum and continuity equations is achieved with a predictor-corrector approach based on the Rhie and Chow interpolation scheme in conjunction with SIMPLE (Semi-Implicit
Method for Pressure-Linked Equations) algorithm 51 for pressure-velocity coupling. The second-order
upwind scheme is used in the spatial discretization of all equations. An implicit solution scheme is used in
conjunction with an algebraic multigrid method (AMG) for accelerating convergence of the solver.
3.2 Problem setup and grid independency test
We have investigated the influence of solvent viscosity on the mass transfer for both wavy film and
uniform film flows in a WWC and a ZZC. The wavy film having rolling waves is introduced via harmonically excited periodic actuations at the solvent inlet. The geometry of WWC is considered from our previous studies 37, 38. The ZZC is selected for its two-dimensional (2D) representation of structured packed columns. A similar approach also has been considered earlier for 2D representation of the structured packed column for investigating the hydrodynamics of countercurrent flow 29, 30. Figure 5
depicts the schematic of the WWC and ZZC column designs and their dimensions. Both columns are placed vertically and gravity acts in a downward direction. Dimension (height and width) of the WWC and ZZC geometries are kept the same to compare the mass transfer coefficient for both cases. The inlet
and outlet of the gas and liquid phases have 1 mm widths. This inlet and outlet size is selected based on a
previous WWC study 38.
The computational flow domain is developed and meshed in StarCCM+. To accurately and efficiently capture the physical mass transfer phenomenon between phases, the domain is discretized using very fine mesh. The mesh density proximate to the left wall where liquid falls appears finer than in
the adjacent region for the surrounding gas. A more refined mesh also is required in the inflexion region
near the zigzag channel apex to capture the flow behavior (see the exploded view of dark region of the mesh domain in Figure 5(c)). A resolution sensitivity study was also conducted to determine a reasonable mesh while maintaining grid independent predictions. Only the grid across the width of the inlet and left wall was varied, as opposed to the whole domain, as mass transfer phenomenon occurs proximate to the
left wall of the flow domain where the inlet was located. The value of physical mass transfer coefficient
was evaluated for the case = 1 , = 1.3 × 10 / , = 0.17 / and = −9 2 𝜇𝜇𝐿𝐿 𝑐𝑐𝑐𝑐 𝐷𝐷𝐿𝐿 𝑚𝑚 𝑠𝑠 𝑢𝑢𝐿𝐿 𝑚𝑚 𝑠𝑠 𝑢𝑢𝐺𝐺 0.049 / at different grid resolutions. The grid resolution is varied from 30 grid points (GP) across
the width𝑚𝑚 of𝑠𝑠 the inlet region to 80 GP. The value of the physical mass transfer coefficient does not
substantially change with resolutions greater than 60 GP (see Table 1). For numerical simulation of
diffusive mass transfer, grid size near the interfacial region needs to be lower than the mass diffusion
length scale ( 4 ). The exposure time (t) was calculated from the interfacial velocity (maximum
𝐿𝐿 velocity) and �height𝐷𝐷 𝑡𝑡 of the column. The grid size near the film region corresponds to 60GP was 16 which is lower than the minimum diffusion length scale (27 ). Therefore, the grid size was assumed𝜇𝜇 𝜇𝜇to be sufficient for the flow simulation. Because of very small grid𝜇𝜇𝜇𝜇 size, flow simulations require a tiny time step ( t~ 10 10 sec) to satisfy the Courant–Friedrichs–Lewy (CFL) condition for stability with −6 −5 a CourantΔ number− = 0.50. As a result, flow simulations become quite expensive computationally.
Table 1: Variation of the Physical mass transfer coefficient ( ) with grid resolutions
𝑳𝑳 Grid points at inlet 𝒌𝒌 width (m/s) 𝒌𝒌𝑳𝑳 30 1.225E-04 40 8.403E-05 50 6.842E-05 60 5.776E-05 70 4.982E-05 80 4.388E-05
Countercurrent flow simulations have been conducted for the multicomponent gas (air and nitrous oxide, N2O) and amine-based solvents ( = 1058 / ) as working fluids. The solvent viscosity is 3 𝐿𝐿 varied over a wide range ( ~ 1 10 𝜌𝜌) to investigate𝑘𝑘𝑘𝑘 𝑚𝑚 its direct impact on the mass transfer between
𝐿𝐿 two phases. The gas phase𝜇𝜇 has −a density𝑐𝑐𝑐𝑐 of ( ) 1 / and viscosity ( ) of 0.015 . The 3 𝜌𝜌𝑔𝑔 𝑘𝑘𝑘𝑘 𝑚𝑚 𝜇𝜇𝑔𝑔 𝑐𝑐𝑐𝑐 diffusivities of N2O in the air and solvent are specified as = 1.67 × 10 / and = −5 2 𝐺𝐺 𝐿𝐿 1.3 × 10 / , respectively. The solvent enters the domain at𝐷𝐷 the top (1) and exits𝑚𝑚 the𝑠𝑠 bottom𝐷𝐷 (3) −9 2 because of 𝑚𝑚gravity.𝑠𝑠 Both uniform (0.17 m/s) and oscillating velocity profiles [0.17(1 + 0.05 2 ) / and = 60 ] are prescribed at the solvent inlet to introduce the flat film flow and wavy𝑠𝑠𝑠𝑠𝑠𝑠 𝜋𝜋𝜋𝜋film𝜋𝜋 flow,𝑚𝑚 𝑠𝑠𝑠𝑠 respectively.𝑠𝑠 𝑓𝑓 On𝐻𝐻𝐻𝐻 the other hand, uniform gas inlet velocity (0.049 m/s) with N2O concertation of 13.31 / is specified at the bottom (see gas inlet (4) in Figure 5 (a and b). The 3 outlet of both phases is 𝑚𝑚𝑚𝑚𝑚𝑚specified𝑚𝑚 as a pressure outlet with zero gauge pressure. The remaining boundaries, including zigzag sheet, are defined as a no-slip wall with a static contact angle ( ) of 40°. The benchmark
results correspond to fixed liquid flow rates. In that case, a flow rate of 𝛾𝛾= 0.17 / and =
𝐿𝐿 𝐺𝐺 0.049 / ensure a stable film flow with a smooth interface for a flat surface𝑢𝑢. The simulat𝑚𝑚 ion𝑠𝑠 cases𝑢𝑢 are summarized𝑚𝑚 𝑠𝑠 in Table 2.
Table 2: Details of the cases in flow simulation
Wetted wall column (WWC) Zigzag column (ZZC) Solvent Flat film flow Wavy film flow Flat film flow Wavy film flow Viscosity with with oscillating flow with with oscillating flow (cP) uniform flow rate rate uniform flow rate rate 1 ~ 10 UWWC OWWC UZZC OZZC
4. Results and Discussion
Analytical and numerical studies have been conducted for the mass transfer via physical
absorption in a countercurrent gas-liquid flow. To explain its impact on the mass transfer coefficient, the
flow simulations have been conducted for a range of solvent viscosity (1–10 cP) for uniform and
harmonically excited oscillating periodic inlet flow. In the simulations, we only consider the direct effect
of solvent viscosity that affects the hydrodynamics. The indirect impact of viscosity on the mass transfer
coefficient via viscosity-dependent diffusivity is not included, which involves the relations between
diffusivity and viscosity of the solvents. The simulated results are presented in terms of dimensionless
numbers: Reynolds, Kapitza, Schmidt, and Galileo numbers. These numbers already are defined in the
previous section. 4.1 Comparison with experiment for hydrodynamics of wavy film flow
The present study involves a complex problem of the physical mass transfer via absorption in wavy film flow in a packed column. Accordingly, preliminary simulations were conducted for validation against experimental studies for wavy film flow19, 52 and physical mass transfer in wetted wall column38.
The CFD simulations are first compared with those from experiment and DNS studies by Denner et al19 to validate the flow prediction for studying wavy film falling over inclined plate. The same method has already been quantitatively validated for studying film flown down an inclined flat plate in our previous studies 8. Specifically, simulations results for wetted area and film thickness were compared against experiment measurements of Hoffmann et al.53 and predictions from Nusselt theory45, respectively8.
Subsequently validation for wavy film flow, the experimental work by Denner et al 19 for wavy film over
a inclined plate was chosen. Similar to the numerical studies by Denner et al19, 2-D flow simulations were conducted in a domain having dimension of 400 × 15 mm2 and 20° inclination angle to the
horizontal (see Figure 6(a)). A harmonically excited periodically varying flow rate (frequency f = 7 Hz
and amplitude A=10% from the mean) is specified at the inlet of the flow domain. Top and outlet of the flow domain was specified as pressure outlet boundary with zero gauge pressure. The bottom, i.e. plate was specified as no slip wall with 10° contact angle. A semi-parabolic velocity profile (35) is specified in the film regime at the inlet whereas remaining portions of inlet was specified as zero velocity.
3yy u =2 −( 1 +A sin(2π ft )) uNN for 0 ≤≤y h 2hhNN (35)
= 0 hN <≤ yH Here, and are the film thickness and film velocity computed from Nusselt theory 45 for uniform film flowℎ𝑁𝑁 as 𝑢𝑢𝑁𝑁
2 1/3 2 3ν Re ghsin β N huNN= and = (36) g sin βν3 The air was chosen as gas phase having density of 1.205 kg/m3 and viscosity of 1.82×10-5 Pa.s.
Aqueous glycerol solutions at three concentrations (A: 45% by weight, B: 65% by weight and C: 45% by weight) were used in the experiments. The physical properties and liquid flow rates are shown in the
Table 3. Consistent with experiments, film velocity and film thickness and wave speed were measured at the 256 mm downstream from the inlet of flow domain.
Table 3: Input parameters and physical properties of the aqueous glycerol solution used in the simulation for wavy film flow over an inclined plate.
Case Ka Re ( / ) ( ) ( / ) ( ) ( / ) 𝝆𝝆 𝝁𝝁 𝝈𝝈 𝒉𝒉𝑵𝑵 𝒖𝒖𝑵𝑵 A1 1214 3 74.9 62.3 14 5.2 0.00261 0.1231 𝐾𝐾𝐾𝐾 𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑠𝑠 B1 1169 18.4 58.7 85 7.5 0.00118 0.0997
B2 1169 18.4 58.7 85 20.6 0.00166 0.1955 C1 1113 6.42 59.7 346 12.4 0.00072 0.0997
C2 1113 6.42 59.7 346 45.1 0.00110 0.2359
C3 1113 6.42 59.7 346 65.1 0.00125 0.3013 C4 1113 6.42 59.7 346 77.0 0.00132 0.3369
The CFD-predicted shape of the wave (f = 0.50) was then qualitatively compared with that of the experiment. As shown in Figures 6 (b-d), the normalized maximum film thickness ( / ) ,
𝑚𝑚𝑚𝑚𝑚𝑚 𝑁𝑁 normalized minimum film thickness( / ) and normalized wave speed ( / ) computedℎ fromℎ the
CFD reasonably matches with the experimentalℎ𝑚𝑚𝑚𝑚𝑚𝑚 ℎ𝑁𝑁 and DNS studies by Denner et𝑐𝑐 al𝑢𝑢𝑁𝑁19. The maximum and minimum film thickness matches excellently with the experiments and DNS studies at low Re value (≤
20). A similar trend for the valuation of these quantiles with Re was observed at all Re and Ka values in
CFD, experimental and DNS studies.
Figure 7 compares the CFD predicted normalized velocity ( / ) inside the film with corresponding ones from experimental and DNS studies by Denner et al 19 𝑢𝑢at 𝑢𝑢three𝑁𝑁 locations of the rolling
wave for two cases namely B1 and C4. The locations: tail (L1), crest (L2) and front (L3) of the rolling
wave are presented in the top of Figure 7. The predicted velocity profiles matches well with the
experimental and DNS studies by Denner et al 19. In particular, the predicted velocity profile excellently
matches with DNS for all cases. A slightly discrepancy in the velocity profile is observed at higher Reynolds number (=77). As shown by Charogiannis et al54, flow field inside the film strongly depends on the local wave shape that quickly vary in space at higher Re value. Subsequently, a small difference in the measuring location can results in a noticeable change in the velocity profile. Subsequently, a small discrepancy in the velocity profile can be seen at Re=77. Overall, present CFD prediction matches well for film height, wave speed and velocity profile in the liquid film with experiments and DNS studies.
4.2 Comparison with experiment for physical mass transfer in wetted wall column
After successful validation of the CFD prediction for wavy film flow, CFD simulations were
further validated for physical mass transfer phenomenon. In our previous study of physical mass transfer
in the wetted wall column 38, 55, VOF prediction for physical mass transfer coefficient were found to compare well with the experimental results. To ensure reliability of the present simulation, the predicted results are quality and quantitatively compared with those obtained from experiments. Wang et al 38, 55
conducted experiments to transport of N2O into MEA solvents on a WWC to generate data for WWC validation. Measurements were performed using a custom-built WWC apparatus consisting of a stainless- steel tube contained within a cylinder. Gas enters the column through three 1/8-inch tubes spaced evenly around the annulus’ perimeter. The gas flows upward, contacting a liquid film falling down the outside of the tube, then exits through a ¼-inch tube at the top. Gases are delivered to the column from cylinders via mass flow controllers (MFC). The shift in concentration while the gas is routed through the column then is used to deduce the rate at which the gas is being absorbed by the column. Gas composition of the stream exiting the WWC was analyzed using a quadrupole mass spectrometer sampling at atmospheric pressure. The mass spectrometer provides an analysis of gas composition at four-second intervals.
Because CO2 and N2O both have an atomic mass of 44, N2O is analyzed on the mass spectrometer at
mass 30, corresponding to a nitric oxide (NO) splitting peak.
Based on the experimental studies, countercurrent flow simulations were conducted with same
parameters in wetted wall column. The input controlled parameters and corresponding value of the
experimentally measured physical mass transfer coefficient are presented in the Table 4. As shown in the Table 4, the physical mass transfer coefficient was evaluated for a wide range of gas and liquid loads,
CO2 loadings and MEA mass fractions in the aqueous solvents. The flow simulations were conducted for a sufficiently long time as to achieve pseudo-steady state for wetted area of the plate and solvent mass flow at the exit. The physical mass transfer coefficient was calculated when flow simulation achieved pseudo-steady state and further compared with corresponding value from the experiment. Figure 8 shows
the comparison between the CFD predicted values of physical mass transfer coefficient with
corresponding experimental values. The data points scatters around the dashed line. Note that the slope of
the dashed line is one correspond to the perfect matching. The difference between the CFD predicted
results and experimental results are within 20%. Overall, the CFD results for physical mass transfer
coefficient reasonably matches with experiments.
Table 4: Input parameters and experimental value of the Physical mass transfer coefficient
for absorption of N2O in MEA in wetted wall column.
Case Temperature MEA CO2 Flow Rate N2O Henry’s Diffusivity kL Loading Conc. Constant Fraction Solvent Gas (×E-09) (×E-05)
1 42 0.25 0.3 450 200 0.325 0.4267 1.3 5.289
2 40 0.1 0.1 509 211 0.228 0.4410 2.08 6.587
3 37 0.1 0.3 586 268 0.43 0.4665 1.87 9.617
4 28 0.2 0.1 309 300 0.483 0.5367 1.14 7.065
5 55 0.2 0.3 413 256 0.299 0.3550 2.15 5.096
6 58 0.2 0.5 358 100 0.435 0.3424 2.13 3.942
7 34 0.3 0.1 431 232 0.203 0.4817 0.97 5.398
8 45 0.3 0.5 350 200 0.15 0.4127 1.01 4.655
9 55 0.3 0.5 350 200 0.15 0.3648 1.31 5.569
10 43 0.4 0.1 451 182 0.289 0.4355 0.89 3.626
12 47 0.4 0.5 536 158 0.165 0.4134 0.64 1.576
13 42 0.25 0.3 450 200 0.325 0.3767 1.00 5.108 After successful validation of CFD prediction for hydrodynamics of wavy film flow over an inclined plate and physical mass transfer in wetted wall column, flow simulation were further conducted to investigate the impact of the viscosity on physical mass transfer.
4.3 Film thickness
The film thickness is a critical factor that impacts the overall efficiency of the column by wet pressure drop across the column. A thin film facilitates the mass transfer between the gas and liquid phases. In addition, it also dedicates the heat transfer between the solvents and wall. In this context, computation of the film thickness is performed first. The effect of viscosity on the film thickness is presented in terms of Kapitza number, a dimensionless number that depends only on the solvent properties. Figure 9(a) shows the variation of film thickness with viscosity for a uniform film flow in
WWC and ZZC (UWWC and UZZC). As expected, film thickness increases with increased viscosity for both cases. The predicted value of the film thickness ( ) matches well with the corresponding value computed from the Nusselt theory 45. A slightly discrepancy𝛿𝛿 in the value of film thickness is observed at
low viscosity ( 2 ). This may be due to the triangular shape of the channel that causes little
instability in the𝜇𝜇 film≤ 𝑐𝑐𝑐𝑐flow. The film velocity accelerates at the tip and retards in the trough region to
satisfy flow continuity. However, the magnitude of the discrepancy in the film thickness is considered to
be negligible. At higher viscosity, both show almost same value. Note that the damping of turbulent
eddies within the liquid film near both the liquid and solid 𝛿𝛿surface is governed mainly by viscosity 46.
Subsequently, smooth and uniform film appears in both cases (UWWC and UZZC).
Next, the film thickness is normalized by the viscous length scale as = /( / ) / . Figure 9(b) ∗ 2 1 3 shows the variation of with Kapitza number for a UWWC. Kapitza number𝛿𝛿 (Ka)𝛿𝛿 𝜈𝜈is𝐿𝐿 a 𝑔𝑔representation of ∗ the ratio of the surface 𝛿𝛿tension force to inertia force. The major advantage of the Kapitza number is that it
depends only the physical properties, therefore, it has fixed value for a given solvent properties. The
Kapitza number was used for explaining the film thickness and mass transfer for a falling film down an
inclined plate. The normalized film thickness increases with an increased Kapitza number. However, absolute value of the film thickness varies the other way, i.e decreases with increased value of Kapitza
number. Note that a higher value of Kapitza number corresponds to a lower value of solvent viscosity. It
also shows a scaling relation ~ / . Earlier, Singh et al. 8 presented similar scaling for film falling ∗ 1 4 over an inclined plate. Film thickness𝛿𝛿 𝐾𝐾𝑎𝑎 also has been reported to increase with solvent viscosity 8, 56, which is consistent with the Nusselt theory 45. As mentioned earlier, film stability is governed by the competing
effects between various forces acting on it. At high viscosity, enhanced viscous forces act against gravity
and make the film more stable 27. As a result, film thickness increases.
As already mentioned, flow simulations have been conducted for the oscillating inlet. Because of the oscillating inlet velocity, a wavy film appears. Wang et al. 38 observe that wavelength ( ) of the wavy
film decreases with increased frequency ( ) for OWWCs. The pronounced impact of the𝜆𝜆 frequency is
visible at = 60 Hz. Accordingly, f = 60𝑓𝑓 Hz is chosen in the flow simulations for all viscosities. The
wavelengths𝑓𝑓 of the wavy film are computed and compared for both WWCs and ZZCs. The computed
value of the wavelength also can be normalized as = /( / ) / . Figure 10 shows the variation ∗ 2 1 3 ∗ with Ka. As expected, the normalized wavelength increases𝜆𝜆 𝜆𝜆 with𝜈𝜈𝐿𝐿 𝑔𝑔 the increased value of Kapitza number𝜆𝜆
for both cases. Note that the lower value of Kapitza number corresponds to higher viscosity. As noted,
viscosity stabilizes the film and acts as damping factors for wave. Furthermore, the normalized
wavelength value is almost the same for both case, except slight discrepancy is observed at a higher ∗ value of Kapitza𝜆𝜆 number.
4.4 Mass transfer coefficient
In this section, we consider the mass transfer from gas to liquid phase due to physical absorption. The
2D simulation using the VOF method of co-current gas-liquid flow in the structure packed column is conducted to compute the interfacial mass transfer 35. The mass transfer is not directly affected by the
recirculation in the liquid side under corrugation cavities. Indeed, the chemical species only penetrate in a
very thin concentration layer. Thus, the transfer process is controlled by the diffusion/advection at the
film interface. The methodology for computing the mass transfer coefficient is the same as in Wang et al. 38 and is
only briefly presented here. The mass transfer coefficient (kL) for physical transport via absorption is calculated using standard formula
kL = jp ∆ , (37)
where J is the mass transfer flux (mol/m2·s) at the gas-liquid interface and is the driving force computed as: ∆𝑃𝑃
PP− ∆=P N22 O,, in N O out . (38) ln (PPN O,, in N O out ) 22
Applying the ideal gas law, can be written in terms of the concentration of the N2O at the gas inlet and outlet: ∆𝑃𝑃
(CN O,, in− C N O out ) RT ∆=P 22 , (39) ln (CCN O,, in N O out ) 22 where is the ideal gas constant (J/K·mol) and is the temperature in the unit of K. To compare the
numerically𝑅𝑅 computed value of the mass transfer 𝑇𝑇coefficient with an analytically calculated one, the unit
of the mass transfer is converted to m/s.
Figure 11 shows the variation of the mass transfer coefficients with viscosity for uniform inlet
velocity (UWWC and UZZC) of = 0.17 / . The UZZC shows higher mass transfer coefficient
(especially at small viscosity) when𝑢𝑢 𝐿𝐿compared𝑚𝑚 to 𝑠𝑠the WWC. This effect is due to the packing geometry
and can be explained by the flow field in the packed column in Figure 12. Note that hydrodynamics plays
a critical role in the mass transfer. In Figure 12(a), the recirculation zone is seen in the gas side for the
UZZC. The recirculation zone enhances the convection. On the other hand, recirculation is absent in case
of wetted wall column (see Figure 12(b)). The stream line is parallel to the gas liquid interface. Note that
the hydrodynamics significantly impact the mass transfer phenomenon. Previous studies by Haroun et al.
32 also uncovered a similar observation. The mass transfer is found to be higher for a triangular channel as
compared to a corresponding one for flat liquid film. Further, effects of the gas flow rate ( ) on the physical mass transfer coefficient were extensively
𝐺𝐺 investigated. Flow simulations were conducted𝑢𝑢 for three solvent viscosities (1cP, 5cP and 10cP) at =
𝐿𝐿 0.17 / , = 1.3 × 10 / and different gas velocities. The range of the gas flow rate𝑢𝑢 was −9 2 considered𝑚𝑚 𝑠𝑠 according𝐷𝐷𝐿𝐿 to the controlled𝑚𝑚 𝑠𝑠 design condition. Figure 13 shows the variation of physical mass transfer coefficient with gas velocity for three values of solvent viscosity. As expected, the value of
physical mass transfer coefficient decreases with increased value of solvent viscosity at all gas flow rates.
On the other hand, the physical mass transfer coefficient slightly increases and then achieves nearly
constant values with increased gas velocity. Overall, gas velocity has marginal impact on the mass
transfer coefficients. A similar observation was drawn by Aroonwilas et al. 57, 58 and Tan et al 59. They
also observed insignificant effect of gas flow rate on the overall mass transfer coefficient using aqueous
2-amino-2-methyl-1-propanol (AMP) solvent. The mass transfer phenomenon is controlled by the liquid phase resistance; subsequently diffusion of solvent molecules in the liquid phase was restricted. As a result, amount of CO2 absorbed by solvent remains nearly constant regardless the change in gas flow rates.
Figure 14 shows the variation of with solvent viscosity for both uniform and wavy film flows in WWC and ZZC. As expected, the mass𝑘𝑘 𝐿𝐿transfer coefficient decreases with increased solvent viscosity in all four cases. Here, wavy film flow shows steep variation with viscosity for both ZZC and WWC. The difference between for wavy and uniform film is observed to be higher (35% at = 1 ) at lower
𝐿𝐿 viscosity. At higher𝑘𝑘 viscosity, the difference in values becomes small (nearly same𝜇𝜇 at 𝑐𝑐𝑐𝑐= 10 ).
Increased value of solvent viscosity damps the initial𝑘𝑘𝐿𝐿 actuation in wavy film flow in the flow field.𝜇𝜇 On𝑐𝑐𝑐𝑐 the other hand, interfacial surface tension and gravitational forces dominate the viscous force at small viscosity. As a result, eddies at the interface occurs and promotes the mass transfer, in addition to higher interfacial area. Effect of the wavy film is found to be more pronounced in WWC than in ZZC. In the
ZZC, the maximum difference of between wavy and uniform film was found to be around 20%. In the
𝑘𝑘𝐿𝐿 ZZC, acceleration of the solvent near the crest and subsequent retardation proximate to trough can
mitigates the inlet flow actuation.
In Figure 15(a), we compare the variation of the mass transfer coefficient with viscosity for expression (25) and numerically compute one at a fixed solvent and gas flow rate and wide range of viscosity ( ~ 1 10 ), where C= 0.115 corresponds to wetted wall column. . The value of C might be different for𝜇𝜇 the− structured𝑐𝑐𝑐𝑐 packed column where flow is generally considered to be more chaotic. Both plots match reasonably well, except the analytically derived expression shows slightly steep variation, i.e., higher exponent value (-0.36). In addition, the scaling for with solvent viscosity shows a scaling
𝐿𝐿 relation ~ . for wavy film in the OWWC (see Figure𝑘𝑘 15(b)). This is nearly the same as the −0 34 exponent𝑘𝑘 𝐿𝐿value𝜈𝜈𝐿𝐿 -0.36 in Eq. (25), as well as the experimental observation (-0.4 in Eq. (26)) by Song et al.
2, 24.
5. Conclusion
Post combustion carbon capture via solvent absorption in a packed column is a promising technology for mitigation of the greenhouse gas emissions. Absorption is carried out through countercurrent gas- liquid flow in a packed column. The mass transfer between the gas and liquid phases plays a critical role in determining the column height. Recent experiments have found the direct impact of viscosity on mass transfer is larger for a packed column ( ~ . ) compared to WWC ( ~ . ). In principle, this −0 40 −0 17 𝐿𝐿 𝐿𝐿 direct influence of viscosity on the mass 𝑘𝑘transfer𝜇𝜇 coefficient may be due to 𝑘𝑘the liquid𝜇𝜇 turbulence enhanced
mass transfer in a packed column. In this context, detailed analytical studies and numerical simulations
have been conducted to explain the underlying physics behind such discrepancy.
Wavy film flow is proposed as a plausible mechanism responsible for the discrepancy and
expressions for eddy-enhanced diffusivity, and the mass transfer coefficient for wavy film falling over a
flat plate is derived based on the hydrodynamic and mass transfer equations. It demonstrates a higher
dependence of mass transfer on viscosity and solvent flow rate in accordance to the experiment. In
addition, to investigate the direct impact of viscosity on the mass transfer coefficient, multiphase flow simulations have been conducted for WWCs and ZZCs. ZZC was chosen for its 2D representation of a structured packed column. CFD prediction for the hydrodynamics of wavy film flow matches well with corresponding experiments and DNS studies by Denner et al19 . Further, predicted physical mass transfer was found to compared well with previous experimental studies for wetted wall column by Wang et al38.
The predicted value of average film thickness ( ) matches well with Nusselt theory. The value of
av av increases with increased value of solvent viscosity.δ For a periodic wavy film, the normalized wavelengthδ
increases with increased Kapitza number i.e. decreased viscosity. The WWC and ZZC show almost
identical wavelength values, except at low viscosity where the WWC showed a slightly higher value.
Notably, the mass transfer coefficient for a wavy film shows larger dependence on the viscosity compared
to a flat uniform film, which is true for both WWC and ZZC. This dependence from simulations
(k ~ ) , = 0.35 reasoably agrees with that found in both analytical ( = 0.36)and experimental −𝑚𝑚 L 𝐿𝐿 ( =𝑣𝑣0.40) studies.𝑚𝑚 𝑚𝑚
Acknowledgments𝑚𝑚
Pacific Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy (DOE)
under Contract No. DE-AC05-76RL01830. This work was funded by the DOE Office of Fossil Energy’s
Carbon Capture Simulation Initiative (CCSI) through the National Energy Technology Laboratory.
Authors also acknowledge the valuable inputs and discussions of Prof. Gerry Rochelle (University of
Texas at Austin).
Disclaimer This manuscript was prepared as an account of work sponsored by an agency of the United States
Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. REFERENCE
1. Shkadov, V. Y., Wave flow regimes of a thin layer of viscous fluid subject to gravity. Fluid Dynamics 1967, 2, (1), 29-34. 2. Song, D.; Seibert, A. F.; Rochelle, G. T., Mass Transfer Parameters for Packings: Effect of Viscosity. Industrial & Engineering Chemistry Research 2018, 57, (2), 718-729. 3. Razi, N.; Bolland, O.; Svendsen, H., Review of design correlations for CO2 absorption into MEA using structured packings. International Journal of Greenhouse Gas Control 2012, 9, (0), 193-219. 4. Rochelle, G. T., Amine Scrubbing for CO2 Capture. Science 2009, 325, (5948), 1652-1654. 5. Spiegel, L.; Meier, W., Distillation Columns with Structured Packings in the Next Decade. Chemical Engineering Research and Design 2003, 81, (1), 39-47. 6. Mackowiak, J., Fluid Dynamics of Packed Columns. Springer-Verlag: Berlin, 2010. 7. Figueroa, J. D.; Fout, T.; Plasynski, S.; McIlvried, H.; Srivastava, R. D., Advances in CO2 capture technology—The U.S. Department of Energy's Carbon Sequestration Program. International Journal of Greenhouse Gas Control 2008, 2, (1), 9-20. 8. Singh, R. K.; Galvin, J. E.; Sun, X., Three-dimensional simulation of rivulet and film flows over an inclined plate: Effects of solvent properties and contact angle. Chemical Engineering Science 2016, 142, 244-257. 9. Tsai, R. E. Mass Transfer Area of Structured Packing. Dissertation The University of Texas at Austin, Austin, TX, 2010. 10. Tsai, R. E.; Schultheiss, P.; Kettner, A.; Lewis, J. C.; Seibert, A. F.; Eldridge, R. B.; Rochelle, G. T., Influence of Surface Tension on Effective Packing Area. Industrial & Engineering Chemistry Research 2008, 47, (4), 1253-1260. 11. Tsai, R. E.; Seibert, A. F.; Eldridge, R. B.; Rochelle, G. T., Influence of viscosity and surface tension on the effective mass transfer area of structured packing. Energy Procedia 2009, 1, (1), 1197-1204. 12. Schmidt, P.; Ó Náraigh, L.; Lucquiaud, M.; Valluri, P., Linear and nonlinear instability in vertical counter-current laminar gas-liquid flows. Physics of Fluids 2016, 28, (4). 13. Singh, R. K.; Galvin, J. E.; Sun, X., Hydrodynamics of the rivulet flow over corrugated sheet used in structured packings. International Journal of Greenhouse Gas Control 2017, 64, 87-98. 14. Singh, R. K.; Galvin, J. E.; Sun, X., Multiphase flow studies for microscale hydrodynamics in the structured packed column. Chemical Engineering Journal 2018, 353, 949-963. 15. Singh, R. K.; Galvin, J. E.; Sun, X.; Sundaresan, S. In Numerical Simulation for Interfacial Forces of Counter-Current Flow over an Inclined Plate, 12th International Conference on Gas-Liquid & Gas-Liquid- Solid Reactor Engineering (GLS12), 2015; AIChE: 2015. 16. Valluri, P.; Matar, O. K.; Hewitt, G. F.; Mendes, M. A., Thin film flow over structured packings at moderate Reynolds numbers. Chemical Engineering Science 2005, 60, (7), 1965-1975. 17. Lavalle, G.; Lucquiaud, M.; Wehrli, M.; Valluri, P., Cross-flow structured packing for the process intensification of post-combustion carbon dioxide capture. Chemical Engineering Science 2018, 178, 284-296. 18. Dietze, G. F., Effect of wall corrugations on scalar transfer to a wavy falling liquid film. Journal of Fluid Mechanics 2019, 859, 1098-1128. 19. Denner, F.; Charogiannis, A.; Pradas, M.; Markides, C. N.; van Wachem, B. G. M.; Kalliadasis, S., Solitary waves on falling liquid films in the inertia-dominated regime. Journal of Fluid Mechanics 2018, 837, 491-519. 20. Whitman, W. G., The two film theory of gas absorption. International Journal of Heat and Mass Transfer 1962, 5, (5), 429-433. 21. Higbie, R., The rate of absorption of a pure gas into a still liquid during short periods of exposure. T Am Inst Chem Eng 1935, 31, 365-389. 22. Danckwerts, P. V., Significance of Liquid-Film Coefficients in Gas Absorption. Industrial & Engineering Chemistry 1951, 43, (6), 1460-1467. 23. Biń, A. K., Mass transfer into a turbulent liquid film. International Journal of Heat and Mass Transfer 1983, 26, (7), 981-991. 24. Song, D.; Seibert, A. F.; Rochelle, G. T., Effect of Liquid Viscosity on the Liquid Phase Mass Transfer Coefficient of Packing. Energy Procedia 2014, 63, 1268-1286. 25. Tsai, R. E.; Seibert, A. F.; Eldridge, R. B.; Rochelle, G. T., A dimensionless model for predicting the mass-transfer area of structured packing. AIChE Journal 2011, 57, (5), 1173-1184. 26. Wang, C.; Perry, M.; Seibert, F.; Rochelle, G., Packing Characterization for Post Combustion CO2 Capture: Mass Transfer Model Development. Energy Procedia 2014, 63, 1727-1744. 27. Ataki, A.; Bart, H. J., Experimental and CFD simulation study for the wetting of a structured packing element with liquids. Chem Eng Technol 2006, 29, (3), 336-347. 28. Fernandes, J.; Lisboa, P. F.; Simões, P. C.; Mota, J. P. B.; Saatdjian, E., Application of CFD in the study of supercritical fluid extraction with structured packing: Wet pressure drop calculations. The Journal of Supercritical Fluids 2009, 50, (1), 61-68. 29. Hosseini, S. H.; Shojaee, S.; Ahmadi, G.; Zivdar, M., Computational fluid dynamics studies of dry and wet pressure drops in structured packings. Journal of Industrial and Engineering Chemistry 2012, 18, (4), 1465-1473. 30. Raynal, L.; Boyer, C.; Ballaguet, J.-P., Liquid Holdup and Pressure Drop Determination in Structured Packing with CFD Simulations. The Canadian Journal of Chemical Engineering 2004, 82, (5), 871-879. 31. Petre, C. F.; Larachi, F.; Iliuta, I.; Grandjean, B. P. A., Pressure drop through structured packings: Breakdown into the contributing mechanisms by CFD modeling. Chemical Engineering Science 2003, 58, (1), 163-177. 32. Haroun, Y.; Legendre, D.; Raynal, L., Direct numerical simulation of reactive absorption in gas– liquid flow on structured packing using interface capturing method. Chemical Engineering Science 2010, 65, (1), 351-356. 33. Haroun, Y.; Legendre, D.; Raynal, L., Volume of fluid method for interfacial reactive mass transfer: Application to stable liquid film. Chemical Engineering Science 2010, 65, (10), 2896-2909. 34. Marschall, H.; Hinterberger, K.; Schüler, C.; Habla, F.; Hinrichsen, O., Numerical simulation of species transfer across fluid interfaces in free-surface flows using OpenFOAM. Chemical Engineering Science 2012, 78, 111-127. 35. Haroun, Y.; Raynal, L.; Legendre, D., Mass transfer and liquid hold-up determination in structured packing by CFD. Chemical Engineering Science 2012, 75, 342-348. 36. Nieves-Remacha, M. J.; Yang, L.; Jensen, K. F., OpenFOAM Computational Fluid Dynamic Simulations of Two-Phase Flow and Mass Transfer in an Advanced-Flow Reactor. Industrial & Engineering Chemistry Research 2015, 54, (26), 6649-6659. 37. Wang, C.; Xu, Z.; Lai, C.; Sun, X., Beyond the standard two-film theory: Computational fluid dynamics simulations for carbon dioxide capture in a wetted wall column. Chemical Engineering Science 2018, 184, 103-110. 38. Wang, C.; Xu, Z.; Lai, C.; Whyatt, G.; Marcy, P.; Sun, X., Hierarchical calibration and validation for modeling bench-scale solvent-based carbon capture. Part 1: Non-reactive physical mass transfer across the wetted wall column. Greenhouse Gases: Science and Technology 2017, 7, (4), 706-720. 39. Hu, J.; Yang, X.; Yu, J.; Dai, G., Numerical simulation of carbon dioxide (CO2) absorption and interfacial mass transfer across vertically wavy falling film. Chemical Engineering Science 2014, 116, 243- 253. 40. Singh, R. K.; Galvin, J. E.; Whyatt, G. A.; Sun, X., Breakup of a liquid rivulet falling over an inclined plate: Identification of a critical Weber number. Physics of Fluids 2017, 29, (5), 052101. 41. Sebastia-Saez, D.; Gu, S.; Ranganathan, P.; Papadikis, K., 3D modeling of hydrodynamics and physical mass transfer characteristics of liquid film flows in structured packing elements. International Journal of Greenhouse Gas Control 2013, 19, (0), 492-502. 42. Pigford, R. L. Counter-Diffusion in a Wetted Wall Column. University of Illinois, 1941. 43. Pacheco, M. A. Mass Transfer, Kinetics and Rate-Based Modeling of Reactive Absorption. The University of Texas at Austin, Austin, TX, 1998. 44. Patnaik, V.; PerezBlanco, H., Roll waves in falling films: An approximate treatment of the velocity field. International Journal of Heat and Fluid Flow 1996, 17, (1), 63-70. 45. Nusselt, W., Die Oberflächenkondesation des Wasserdampfes. Zeitschrift des Vereines Deutscher Ingenieure 1916, 60, (27), 541–546. 46. Yih, S. M., Modeling heat and mass transfer in wavy and turbulent falling liquid films. Wärme - und Stoffübertragung 1987, 21, (6), 373-381. 47. Odier, P.; Chen, J.; Rivera, M. K.; Ecke, R. E., Fluid mixing in stratified gravity currents: the Prandtl mixing length. Physical review letters 2009, 102, (13), 134504. 48. Hirt, C. W.; Nichols, B. D., Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of Computational Physics 1981, 39, (1), 201-225. 49. CD-adapco, STAR-CCM+ 10.04 User Guide. In CD-adapco: USA, 2015. 50. Brackbill, J. U.; Kothe, D. B.; Zemach, C., A continuum method for modeling surface tension. Journal of Computational Physics 1992, 100, (2), 335-354. 51. Patankar, S. V., Numerical heat transfer and fluid flow. Hemisphere Pub. Corp. ; McGraw-Hill: Washington; New York, 1980. 52. Charogiannis, A.; Denner, F.; van Wachem, B. G. M.; Kalliadasis, S.; Scheid, B.; Markides, C. N., Experimental investigations of liquid falling films flowing under an inclined planar substrate. Physical Review Fluids 2018, 3, (11). 53. Hoffmann, A.; Ausner, J.; Repke, J. U.; Wozny, G., Fluid dynamics in multiphase distillation processes in packed towers. Comput Chem Eng 2005, 29, (6), 1433-1437. 54. Charogiannis, A.; Denner, F.; van Wachem, B. G. M.; Kalliadasis, S.; Markides, C. N., Detailed hydrodynamic characterization of harmonically excited falling-film flows: A combined experimental and computational study. Physical Review Fluids 2017, 2, (1). 55. Wang, C.; Xu, Z. J.; Lai, K.; Whyatt, G.; Marcy, P. W.; Sun, X., Hierarchical calibration and validation framework of bench-scale computational fluid dynamics simulations for solvent-based carbon capture. Part 2: Chemical absorption across a wetted wall column. Greenh. Gases 2018, 8, (1), 150-160. 56. Gu, F.; Liu, C. J.; Yuan, X. G.; Yu, G. C., CFD Simulation of Liquid Film Flow on Inclined Plates. Chem Eng Technol 2004, 27, (10), 1099-1104. 57. Aroonwilas, A.; Tontiwachwuthikul, P., Mass Transfer Coefficients and Correlation for CO2 Absorption into 2-Amino-2-methyl-1-propanol (AMP) Using Structured Packing. Industrial & Engineering Chemistry Research 1998, 37, (2), 569-575. 58. Aroonwilas, A.; Veawab, A., Characterization and Comparison of the CO2 Absorption Performance into Single and Blended Alkanolamines in a Packed Column. Industrial & Engineering Chemistry Research 2004, 43, (9), 2228-2237. 59. Tan, L. S.; Shariff, A. M.; Lau, K. K.; Bustam, M. A., Factors affecting CO2 absorption efficiency in packed column: A review. Journal of Industrial and Engineering Chemistry 2012, 18, (6), 1874-1883.
y
Mass
Δy Transfer x
Liquid Gas A
δ0
uw λ
v
u
Figure 1: Schematic description of a wavy film falling over a vertical plate, where u and v are vertical and horizontal velocity, respectively. The falling film with an average thickness δ0 can be described by a rolling wave with an amplitude A, a wavelength λ, and a wave speed uw.
0.02
0.015
λ / A 0.01 Re=20
Re=40 Re=60
0.005
0 0 0.01 0.02 0.03
Vy /Vx Figure 2: Variation of normalized wave amplitude ( / ) with ratio of wave velocity components ( / ) at different Reynolds numbers. It shows the linear scaling ( / ~ / ) and independent of Reynolds number. 𝐴𝐴 𝜆𝜆 𝑉𝑉𝑦𝑦 𝑉𝑉𝑥𝑥 𝐴𝐴 𝜆𝜆 𝑉𝑉𝑦𝑦 𝑥𝑥 𝑉𝑉
0.015 (a)
0.012
0.009 λ / A 0.006
0.003
0 0 20 40 60 80 100 Re 0.015 (b) Re=20 Re=40 0.012
0.009 λ / A 0.006
0.003
0 0 1000 2000 3000 4000 5000 Ka Figure 3: (a) Variation of the normalized wave amplitude ( / ) with Reynolds number at Ka=2850. (b) Variation of ( / ) with Ka at two Reynolds numbers shows that normalized amplitude decreases with increased Kapitza number.𝐴𝐴 𝜆𝜆 𝐴𝐴 𝜆𝜆
0.02
Ka=2850 Re=20 0.015 Re=40 λ / A 0.01
0.005
0 0 0.25 0.5 0.75 1 Re0.78Ka-0.47 Figure 4: Variation of normalized wave amplitude with Reynolds and Kapitza numbers shows the scaling / ~ . . . The exponents of the fitting for Re and Ka reasonably match the previous0 78 studies−0 47by Shkadov 1. 𝐴𝐴 𝜆𝜆 𝑅𝑅𝑒𝑒 𝐾𝐾𝑎𝑎 (1) (2) (1) (2) (a) (b)
(1) Liquid Inlet (2) Gas Outlet
(3) Liquid Outlet (4) Gas Inlet
(3) (4) (3) (4)
(c)
Figure 5: Schematic of flow domain showing the inlet and outlet positions for both phases: (a) wetted wall column and (b) zigzag column as 2D representative of structured packings. The inlet and outlet of both phases have 1 mm width. (c) The dark region at the left in the exploded view of the discretized domain shows very fine mesh to capture the mass transfer phenomenon.
(a) (b)
g
y x
(c) (d)
Figure 6: (a) Schematic of the computational flow domain used for validation shows the direction of rolling waves on an inclined flat plat. The flow dolman has dimension of L×H and 20° inclination angle. Comparison of the present predicted (CFD) normalized (b) maximum film thickness, hmax/hN , (c) minimum film thickness, hmin/hN , (d) wave speed, c/uN , with corresponding ones for experiments (Exp) and DNS simulation by Denner at al18 at f =7 Hz and different Re values.
L3 L1 L2 L4 3/4d d d (a) B1 C4
(b) B1 C4
(c) B1 C4
Figure 7: Comparison of the CFD predicted velocity profile with corresponding ones from Experiments and DNS by Denner at al18 at three locations (a) tail, L1 (b) crest, L2 and (c) front, L3 of the rolling wave for two cases, A1 and C4. The vertical position is normalized by local firm thickness (h(x)). The velocity is normalized by the Nusselt flat film velocity (uN). The position of tail, crest and front are shown at the top.
Figure 8: Comparison of the CFD predicted physical mass transfer coefficient with corresponding experimental value at different cases presented in the Table 3.
1 (a) 0.8
0.6
0.4 )
m UWWC m ( UZZC δ Nusselt Theory [34] 0.2
10-1 100 101 102 µ (cP) 10 (b) 8
6
10
4
8 * δ
* 6 δ
2
4
2 3 4 5 6 7 8 Ka1/4 102 103 104 Ka Figure 9: (a) Plot shows variation of the film thickness ( ) with solvent viscosity ( ) for the wetted wall column. Predicted value of the film thickness matches well with Nusselt theory [34]. 𝐿𝐿 (b) Variation of the normalized film thickness with Kapitza𝛿𝛿 number. Inset of the Figure𝜇𝜇 (b) shows the scaling relation of ~ / . ∗ 1 4 𝛿𝛿 𝐾𝐾𝑎𝑎
103
OWWC OZZC
Gas
* 102 λ
Liquid
1 10 102 103 104 Ka Figure 10: Plot shows variation of the normalized wavelength ( ) of in a periodic wavy film with Kapitza number (Ka) for WWC and ZZC. Inset shows the wavy∗ film (red color) and method to compute wavelength. 𝜆𝜆
Figure 11: Variation of the mass transfer coefficient ( ) with solvent viscosity ( ) wetted wall column and zigzag column in the case of a smooth and uniform film. 𝐿𝐿 𝐿𝐿 The unit of the 𝑘𝑘 𝜇𝜇
(a) (b)
Figure 12: (a) Exploded view in the triangular channel shows velocity streamline and concentration distribution in the flow domain. It also illustrates the recirculation zones on the gas side due to triangular geometry. (b) Velocity streamline and concentration distribution in wetted wall column does not shows the flow recirculation zone and streamline are parallel to the wall.
Figure 13: Effect of the gas flow rates on the physical mass transfer coefficient for uniform film flow in zigzag column at three values of solvent’s viscosity. The physical mass transfer coefficient initially increases marginally and then becomes independent of gas flow rate.
(a)
(b)
Figure 14: Variation of the mass transfer coefficient ( ) with solvent viscosity ( ) for smooth and wavy film flow in: (a) wetted wall column and (b) zigzag column. Prefix U: Uniform and O: 𝑘𝑘𝐿𝐿 𝜇𝜇𝐿𝐿 Oscillatory Inlet.
(a)
0.0001 (b)
8E-05 L k
6E-05
4E-05
2E-05 40 60 80 -0.34 100 120 νL Figure 15: (a) Comparison of the variation of the mass transfer coefficient with solvent viscosity for analytically derived expression (Equation 25) and CFD simulations for OWWC. Both reasonably show the same trend of the variation of mass transfer coefficient with viscosity. (b) Variation of the mass transfer coefficient ( ) with solvent viscosity ( ) shows a scaling ~ . for OWWC. −0 34 𝑘𝑘𝐿𝐿 𝜈𝜈𝐿𝐿 𝑘𝑘𝐿𝐿 𝜈𝜈𝐿𝐿