Wenbin Li,Kuotsung Yu ,Xigang Yuan ,*,Botan Liu

1 Collaborative Innovation Center of Chemical Science and Engineering(Tianjin),Tianjin University,Tianjin 300072,China

2 State Key Laboratory for Chemical Engineering,School of Chemical Engineering and Technology,Tianjin University,Tianjin 300072,China

Keywords:Simulation Adsorption Mass transfer Anisotropic turbulent mass diffusion Packed bed

ABSTRACT Simulations of adsorption process using the Reynolds mass flux model described in Part I of these serial articles are presented.The object of the simulation is the methylene chloride adsorption in a packed column(0.041 m id,packed with spherical activated carbon up to a length of 0.2 m).With the Reynolds mass flux model,breakthrough/regeneration curves,concentration and temperature as well as the velocity distributions can be obtained.The simulated results are compared with the experimental data reported in the literature and satisfactory agreement is found both in breakthrough/regeneration curves and temperature curves.Moreover,the anisotropic turbulent mass diffusion is characterized and discussed.

1.Introduction

Adsorption is an effective technique in gas separation and purification,such as purification of natural gas[1],separation of hydrocarbon mixtures[2],capture of CO2[3]and removal of mercury emissions from coal combustion flue gas[4].Among the applications,packed columns with activated carbon as adsorbent have been widely used due to its wide availability,low cost,high thermal stability,and low sensitivity to moisture[5].

Breakthrough curve as well as regeneration curve measurement is very fundamental in studying the process dynamics since it could provide much valuable information on designing a packed column.Considerable research works[6–10]have been focused on the experimental measurement of breakthrough curve for studying the dynamics and effectiveness of adsorption process.However,this kind of experiment is usually expensive and laborious,and can only be performed in laboratories.Therefore,much attention has been devoted to the numerical model for simulating the adsorption and absorption processes using packed columns.The modeling of adsorption process is complicated,due to the non-ideal flow and concentration distribution.And most of the researches were based on simplified model,such as one-dimensional[11–13]and two dimensional dispersion models[14–19],in which the dispersion coefficient,or the turbulent mass diffusivity,is determined either from empirical correlations or by guessing a constant turbulent Peclet number.Nevertheless,the right choice of Peclet numbers or empirical correlation relies largely on experience.Li et al.[20]applied the?εcmodel[21,22]to determine the turbulent mass diffusivity by solving auxiliary model equations so as to replace the empirical methods and successfully obtained the concentration distributions in an adsorption column.However,the mass diffusivity of the?εcmodel[21,22]is isotropic,but many investigators have found that the turbulent mass diffusion is anisotropic since experimental measurements[23,24]show that axial turbulent mass diffusivity is larger than the radial one.Therefore,the isotropic assumption of the turbulent mass diffusion may produce simulation error.To carry out more precise simulation for adsorption process,the use of the Reynolds mass flux model presented in Part I[25]of this work would be appropriate.

In this second part(Part II),the Reynolds mass flux modelis applied to the simulation of unsteady state methylene chloride adsorption on activated carbon.The simulations on adsorption process are presented,including the breakthrough/regeneration curves,concentration and temperature as well as the velocity distributions.The simulated results are compared with the experimental data in the literature.The characterized anisotropy of the turbulent mass diffusion in the packed column is discussed.

2.Simulation Target and Model Implementations

2.1.Simulated object

The object of the simulation in Part II is the methylene chloride adsorption on activated carbon column reported by Hwang et al.[26].The adsorption column and adsorbent particle properties are given in Table 1.

Table 1 Properties of the adsorption column and the adsorbent particles

The CMT model applied is the Reynolds mass flux model described in Part I[25].The model is numerically solved so as to simulate the unsteady state gas adsorption process.

2.2.Evaluation of various terms

As Tobis and Ziolkowski[27]stated,the transition from laminar to turbulent flow in packed columns occurs at Repequalto 100.In the present study,the value of Repis about 132,which means that our simulation is under the turbulent flow condition.Besides,the following assumptions should be made for the gas adsorption through a packed column:

(1).The gas is incompressible;

(2).The gas flow is steady since the concentration of adsorbate is set to be very low;

(3).The flow is axis-symmetric.

2.2.1.Net mass increase of gas phase and adsorbate

The net gas mass increase termin the continuity equation is equal to net adsorbate mass increasein the turbulent mass transfer equation for the adsorbate as follows:

where ρgis the total gas phase mole concentration;apis surface area per unit volume of packed column;M is the molecular weight of the adsorbate,Y and Y*are the mole fraction of the adsorbate in gas phase and that in equilibrium with solid phase respectively,which can be calculated by the correlation for gas–solid equilibrium isotherm[26]:

where P is the total pressure of gas phase in the column;qsis the saturation mole concentration of adsorbate,b is the Langmuir isotherm constant,and can be calculated by

and q is the mole concentration of adsorbate in solid phase,which can be calculated by

where q0is the initial mole concentration of adsorbate in the solid phase.

KGis the overall mass transfer coefficient,which can be calculated by[28]

where γpis the particle porosity.The intraparticle mass-transfer coefficient kpcan be calculated according to the correlation reported by Gluekauf[29]and the external mass-transfer coefficients kfwere estimated with the correlation of Wakao and Funazkri[30].

2.2.2.Force acted on gas phase

The force acted on gas phasein the turbulent momentum transfer equation,representing the gravity and the resistance of gas flow due to the existence of the solid phase,is given below as[31]

where the local volume fraction of the gas phase β can be calculated from the expression reported by Liu[32].Thus,the local void fraction of packed section is not constant throughout the packed column.

2.2.3.Thermal effects on gas phase and solid phase

The thermal effects on gas phasein the turbulent heat transfer equation for gas phase represents the amount of heat transferred from solid adsorbent to gas phase:

The thermal effects on solid phasein the energy balance equation for solid phase include the heat produced by adsorption and the heat transferred to gas phase:

where ΔH is the heat of adsorption and h is the heat transfer coefficient from solid adsorbent to the gas phase:

where the Nusselt number Nu can be estimated by the following correlation[33]:

2.2.4.Other relevant terms

The molecular diffusivity D was calculated from Chapman–Enskog formula[34].Since the concentration of adsorbate in gas phase is very low,the gas mixture viscosity μ,thermal conductivity kgand density ρ are approximately equal to that of pure carrier gas,which is air for the adsorption step and nitrogen for the desorption step.The viscosity and thermal conductivity of the pure gas are calculated by the method of Lemmon and Jacobsen[35]and the density ρ is obtained by the method of Lee and Kesler,which was reported by Preey and Green[36].

2.3.Boundary conditions

Fig.1 shows the computational domain and boundary arrangement for adsorption and desorption processes,respectively.The boundary conditions for model equations are specified as follows.

Fig.1.Computational domain and boundary arrangement.

2.3.1.Inlet conditions

The inlet boundary conditions for the three sets of model equations are the same as those outlined in Part I[25].

2.3.2.Outlet conditions

The outlet of flow is considered to be close to fully developed,so that zero normal gradients are chosen for all variables except pressure.

2.3.3.Wall conditions

The no-slip condition of flow is applied to the wall.

The heat loss in the adsorption column should be considered.The heat flux from the fluid phase to the inner wall and the heat flux from outer wall to the environment can be calculated by

where hw1and hw2are the heat transfer coefficients from fluid phase to the wall and the heat coefficient from wall to ambient,respectively,T0is the ambient temperature,and Tw1and Tw2are the temperatures of the inner and outer wall.Considering the thin thickness and high solid thermal conductivity of the column wall,it can be assumed that Tw1=Tw2and Qw,1=Qw,2=Qw.Then we have

where the wall heat transfer coefficient hwcan be estimated by[37]

where kfis the thermal conductivity of fluid phase,dpis the packing diameter,Re0is the Reynolds number based on superficial velocity,and Pr is the Prandtl number.

The zero flux condition at wall is applied for the mass transfer equation.

2.4.Numerical procedures

The simulation is done under the condition of unsteady axissymmetrical flow.The model equations are solved numerically by using the commercial software Fluent 6.3.26 with the finite volume method for discretization of momentum equation.The SIMPLEC algorithm[38]is employed for resolving the pressure–velocity coupling problem in the momentum equations.

In present simulation,the grid arrangement of the adsorption column comprised totally 32000 quadrilateral cells.In the radial direction,50 non-uniformly distributed meshes are applied with higher grid resolution in near wall region.In the axial direction,there are 640 uniformly distributed meshes.

3.Results and Discussion

The simulated results,including these for unsteady-state adsorption and regeneration processes,are compared with experimental data reported by Hwang et al.[26].

3.1.Simulation results

Comparisons are presented between the predictions and the experimental measurements on breakthrough/regeneration curves for adsorption and regeneration processes.The simulated velocity,concentration and temperature pro files are presented as well.

3.1.1.Breakthrough curves

The comparison between the simulated breakthrough curves obtained by different models and the experimental measurements is given in Fig.2.As shown in this figure,the simulated curve by using the Reynolds mass flux model demonstrates more close matching to the experimental result than that by the?εcmodel[20,22]especially at t=80–110 min.It indicates that the Reynolds mass flux model is more suitable for the simulation of adsorption process in the packed column.

Fig.2.Comparison of simulated breakthrough curves with experimental measurements.

3.1.2.Gas phase velocity pro files

In Fig.3,no significant variation of the axial velocity at x=0.10 m was observed when r/R increased from 0 to 0.8,which was due to the relative uniform porosity at the column center region.The heterogeneity of porosity near the column wall(see Fig.3(c))resulted in significant variation of velocity ranging from 0.82 to 1.36 m·s?1.This phenomenon has been confirmed by many investigators[39,40].

3.1.3.Axial concentration and temperature pro files

Fig.4 shows the contours of methylene chloride in mole fraction at different time predicted by using the Reynolds mass flux model.The red region in each contour denotes that the adsorbent is almost saturated and in equilibrium with the fluid phase;while large amounts of adsorption are being undertaken within the red–blue transition region.From this figure we can see how the concentration pro files in the column are developed and consequently affect the breakthrough curve.For instance,Yout/Yinis almost equal to zero before t=45 min,and when t=75 min the adsorbate begins to break through the column.

The non-uniform distribution of concentration is clearly seen in Fig.4.The convex concentration fronts have been found by many other researchers using numerical models[15,16,20].As stated by Astrath et al.[41],the uneven concentration pro files are likely to be due to radial heterogeneity of the column packing and the resulted uneven flow.Kwapinski et al.[16]also claimed that in the case of the column-to-particle diameter ratio dcol/dp=11,the flow maldistribution in the vicinity of the wall causes higher concentration in this region than that in the core,and thus resulted in a concave concentration front.While in the case of a higher ratio dcol/dp=25,the concentration could be larger in the core region than near the wall.He explained further that in the case of higher diameter ratio,the effect of flow maldistribution does not dominate,but is overlapped by the reduction of adsorption capacity due to the increase of temperature in the core of the bed.However,as shown in Fig.4,the situation in the present study is even more complicated.At t=5 min,the effect of uneven flow dominates the concentration shape since temperature span in the column is relatively low(see Fig.5),and thus concave concentration front could be found.While,with the adsorption time increasing,the heat effect progressively becomes dominant which leads to lower capacity of the adsorbent in the core than in the near wall region,and thus results in a convex concentration front.The simulated result indicates that the unsteady-state adsorption process is undertaken in an intimate and complex way.The non-uniform concentration pro file is due to the existence of heat effects and non-uniform porosity as well as the hence induced uneven flow.

3.1.4.Regeneration curves

The regeneration of adsorbent by using nitrogen as purge gas after its adsorption in packed column was simulated with the Reynolds mass flux model as well.The simulated results were compared with the reported experimental data[25].

Fig.3.Pro files of gas axial velocity and gas local volume fraction.

Fig.4.Contours of mass fraction of adsorbate at different time.

Fig.6 showed the comparison between the simulated regeneration curves and the experimental measurements.It is seen that the ratio of outlet to pre-adsorption inlet mole fraction of adsorbate Yout/Yin,adswas increased rapidly at the initial stage,reached a maximum value(about 4.20)at 16 min,then decreased to 1.0 at 35 min,and followed by continue dropping to zero.This enrichment phenomenon can be explained by the equilibrium effect caused by temperature changes.It can be found from Fig.7 that the prediction obtained by using the Reynolds mass flux model is more closed to the experimental measurements than that obtained by Li et al.[20]using the?εcmodel[21,22].

The serial contours of mole fraction pro file along the packed column are given in Fig.8,which show the details of the mass transfer behaviors in the packed column.The blue region denotes that the adsorbent is completely regenerated;while desorption is being undertaken in the blue–red transition region.From Fig.9,the concave concentration front is clearly seen which is mainly caused by the heat effect in desorption process.The wall heat loss results in lower temperature in the near wall region,and consequently leads to the decreasing of desorption rate(see Eqs.(1)–(5))at this region.Thus the regeneration of adsorbent in vicinity of the wall is always slower than that in the core.

3.1.5.Temperature curves for regeneration

Fig.5.Contours of temperature at different time.

Fig.6.Simulated and experimental regeneration curves.

Fig.7.Simulated and experimental temperature curves at different packed-column height.

The comparison of the temperature curves at different axial positions(x=0 m,0.1 m,0.2 m,respectively)between the predictions and the experimental measurements is given in Fig.7.It is seen that at the beginning of the regeneration process,deviations exist between the predicted curves and the experimental measurements.It might be due to the uncertainty in calculation of the heat of adsorption.Further explanation may be that the heat of desorption,which is assumed to be equal to the heat of adsorption in magnitude but opposite in sign is overestimated,and thus leading to lower predicted temperature than the measurement.After those times,the regeneration approaching to the end and the heat needed for desorption gradually drop to zero,thus the simulated temperatures are closely checked by the measurements.The temperature curves predicted by the present model are very close to that by the?εcmodel[20,22].Itis noted that the overestimated heat of adsorption or heat of desorption would also produce error in the adsorption process simulation.However,due to lack of experimental data on temperature curve for the adsorption process,comparison of the temperature curve cannot be made for the adsorption process.

The serial contours of temperature along the packed column axis are given in Fig.8.

3.2.Discussion on anisotropic turbulent mass diffusion

The pro files of turbulent mass flux(opposite value of the Reynolds mass flux)predicted by the Reynolds mass flux model in the form ofandare shown in Fig.10 for the adsorption process.It can be seen that although a difference in values exists between the turbulent mass fluxes in two directions,they are practically in the same order of magnitude.

The feature of the Reynolds mass flux model is that the anisotropic turbulent mass diffusion can be characterized.In order to show the anisotropic mass diffusion under turbulent flow condition,the turbulent mass diffusivity could be evaluated by the following relationship based on Fick-like law[42]:

where Dt,xand Dt,rare the axial and radial turbulent mass diffusivities respectively;the concentration gradients in two directions are shown in Fig.11(a)and(b).

For illustration,the turbulent mass diffusivities Dt,xand Dt,rare given in Fig.12.It can be seen that there is no analogy between the diffusivities in the two directions,which implies that the isotropic assumption of the turbulent mass diffusivity in the?εcmodel[20–22]is unreasonable.Moreover,Dt,ris found to be in negative sign in the vicinity of the column center and column wall(see Fig.12(b)).That means the turbulent mass flux in r direction is counter-gradient-transported in the corresponding region.As stated by Eskinazi and Erian[43],in engineering practice the counter-gradient-transport phenomena often occurred.Nevertheless,due to the assumption of Dt,ito be a positive value,the conventional model cannot describe this phenomenon reasonably.

Fig.8.Sequences of contours of gas phase temperatureat different time.

Fig.9.Sequences of concentration contours in mole fraction at different time.

Table 2 gives comparisons between the turbulent mass diffusivity evaluated by using experimental correlations,the?εcmodel[20,22]and the Reynolds mass flux model.The evaluated turbulent mass diffusivities from various experimental correlations show disagreement.As stated by several investigators[51,52],such discrepancies are attributed to the sensitivity of the effective mass diffusivity to experimental errors and the use of different model concepts for estimating the effective mass diffusivity.In Table 2,it is seen that the turbulent axial mass diffusivity is larger than the radial one no matter what experimental correlation is used.And as stated by Yin et al.[53]and Sherwood et al.[54],it is probably attributed to the reason that axial component of flow is larger than the radial one.This anisotropy of the turbulent mass diffusivity is characterized by the Reynolds mass flux model.Nevertheless,isotropic turbulent mass diffusivity is used in the?εcmodel[20,22].Furthermore,both turbulent axial and radial mass diffusivities obtained by the Reynolds mass flux model are lower than that evaluated by experimental correlations and the?εcmodel[20,22].It demonstrates that the determinations of mass diffusivity under turbulent flow condition by experimental correlations and the?εcmodel[20,22]are only rough approximation.

3.3.Study on the influence of grid number and time-step-size on the accuracy of simulation

In order to ensure that the simulated results presented in this paper are independent of mesh density,we meshed the column with different grids of 25×320,50×640 and 100×1280 respectively.And all of the mesh schemes to be used have non-uniformly distributed grids in radial direction with higher grid resolution in near wall region,and uniformly distributed grids in axial direction.

Fig.10.Pro files of turbulent mass flux

Fig.11.Pro files of concentration gradient along radial direction.

Fig.12.Pro files of anisotropic turbulent dispersion coefficient D t,i along the radial direction.

Table 2 Comparisons between the turbulent mass diffusivity evaluated by using experimental correlations,the?εc model[20]and the Reynolds mass flux model

Table 2 Comparisons between the turbulent mass diffusivity evaluated by using experimental correlations,the?εc model[20]and the Reynolds mass flux model

Note:Peclet number based on molecular dispersion coefficient Pe m=|u|d p/D,for the present study Pe m is about202;axial Peclet number for mass dispersion Pex=|u|d p/D e,x and D e,x=D+D t,x;radial Peclet number for mass dispersion Per=|u|d p/D e,r and D e,r=D+D t,r;|u|is the averaged interstitial velocity of gas;averaged porosity of the bed γ∞ =0.42;tortuosity factor for spherical packing τ=2;Reynolds number Re p= ρd p|u|/μ,for the present study Re p is about 132;(av.)means the volume averaged value.

？

As for time step size,it is recommended[38]to be estimated as follows with respect to the stability and precision of the computation.

Following this guideline,the time step size in the present work should be around the order of magnitude of 10?3s.However,it would result in heavy computations since a complete adsorption process needs 150 min,which takes about 20 days of computation-time when mesh density and time step size are respectively 50×640 and 1×10?3s(The simulations are conducted on a workstation(Dell Inc.,Precision T5500,12 cores)).Therefore,larger time step size is used tentatively in the premise of computation stability and precision.It is found that the order of magnitude of 10?2s is appropriate in the present simulation,and then 1×10?2s is chosen as a preset value of the time step size for the grid independence analysis.The simulated breakthrough curves are illustrated in Fig.13 for different mesh densities.The results indicate that outlet adsorbate concentration is increased at higher mesh density,whereas no substantial difference is found with increasing number of grids beyond 50×640.Therefore,the present grid density of 50×640 could provide sufficient simulation precision.

Fig.13.Comparison of simulated breakthrough curves with different mesh densities(time step size of 0.01 s).

To further optimize the time step size,simulations are run for different time step sizes:0.01,0.05 and 0.1 s.And the computation-time consumed are respectively 84,18 and 10 h for a complete adsorption process.It is seen from Fig.14 that the 0.1 s time step size leads to a clear difference in simulated breakthrough curve.While a time step of 0.01 s provides no substantially different results but needs much more computation-time.Therefore,the 0.05 s time step size is used as the based computation setting.

Further on,as mentioned in Part I[25]of these series articles,it is necessary to test whether the use of present grid scale is sufficient for simulating the anisotropic turbulent mass diffusion.Taking Run A0(Yin=2.25 × 10?3,Tin=298 K,P=0.111 MPa,F=33.5 L·min?1)of the adsorption column concerned,the simulated turbulent mass diffusivities with different grid scale are listed in Table 3.

From Table 3,the turbulent mass diffusivities obtained by using different grid scales show significant deviation when the grid scale is larger than 0.461×0.313;while difference is not significant when grid scale to be used is smaller than 0.461×0.313.Thus,it can be concluded that in the present study,the simulation on anisotropic turbulent mass diffusivity becomes independent of grid scale if the grid scale is 0.461×0.313 or smaller.

Fig.14.Comparison of simulated breakthrough curves with different time steps.

Table 3 Simulated anisotropic turbulent mass diffusivity with different grid scale

4.Conclusions

The applications of the Reynolds mass flux model to the simulation of adsorption process undertaken in packed columns have been discussed.Comparison of the Reynolds mass flux model simulations and the experimental data has been performed for both breakthrough and regeneration curves in packed columns.The following conclusions are reached:

(1)Satisfactory agreement is found between the simulated results and the experimental data both for breakthrough and regeneration curves,which confirms the validity of the Reynolds mass flux model.

(2)Anisotropy of the turbulent mass diffusion can be characterized rigorously,which confirms the experimental observation that the turbulent axial mass diffusivity is superior to the radial one for the packed columns.

(3)The predicted breakthrough/regeneration curves for the adsorption process by the Reynolds mass flux model are found to be in better agreement with the experimental data than that by the?εcmodel[20,22].It indicates that the anisotropy of the turbulent mass diffusivity should be taken into account for precise simulation of adsorption process in packed column.

Nomenclature

apsurface area per unit volume of packed column,m?1

b Langmuir isotherm constant for the adsorption process,Pa-1

cpg,cpsspecific heat of gas and absorbent,respectively,J·kg·K?1

D molecular diffusivity,m2·s?1

dpnominal packing diameter,m

g gravity acceleration,m·s?2

ΔH heat of absorption of adsorbate,J·mol?1

h heat transfer coefficient from gas phase to packing,W·m?2·K?1

hwheat transfer coefficient from gas phase to ambient,W·m?2·K?1

hw1heat transfer coefficient from gas phase to column wall,W·m?2·K?1

hw2heat transfer coefficient from column to ambient,W·m?2·K?1

KGoverall mass transfer coefficient

kfexternal mass transfer coefficient for adsorption process,m·s?1

kgthermal conductivity of gas,W·m?1·K?1

kpinternal mass transfer coefficient for adsorption process,m·s?1

ksthermal conductivity of adsorbent particle,W·m?1·K?1

M molecular weight of species,kg·mol?1

Nu Nusselt number(Nu=hRp/kg)

P total pressure of gas phase in the column,Pa

PeiPeclet number based on turbulent mass diffusivity(Pei=|u|d p/D t,i)

PemPeclet number based on molecular dispersion(Pem=|u|dp/D)

Pr Prandtl number(Pr=Cpgμ/kg)

Qw,1the heat flux from gas phase to the inner wall,W·m?2

Qw,2the heat flux from outer wall to the environment,W·m?2

Qwthe heat flux through column wall,W·m?2

q,q0,q*,qsadsorbate,initial adsorbate,equilibrium and saturation adsorbate concentration in solid phase,respectively,mol·kg?1

R inside diameter of the column,m

Rpaverage diameter of the particle,m

RepReynolds number based on particle diameter(Rep=ρd|u|/dp)

r radial distance from the axis of the column,m

SF,isource of interphase momentum transfer,N·m?3

Smnet mass increase of the fluid phase,kg·m?3·s?1

STsthe thermal source term of the static phase,J·m?3·s?1

Tfgas phase temperature,K

Tssolid phase temperature,K

Tw1,Tw2temperature of the inner and outer wall,K

T0ambient temperature,K

t time,min

X total packed height,m

Y,Y* mole fraction and equilibrium mole fraction of adsorbate in the gas phase,respectively

Yininlet mole fraction of adsorbate in the gas phase for the adsorption step

Yin,adspre-adsorption step inlet mole fraction for the desorption step

β volume fraction of the concerned phase

γpparticle porosity

γ∞average column porosity

ρ gas density,kg·m?3

ρgtotal gas phase mole concentration,mol·m?3

ρsbulk density of absorbent,kg·m?3

μ gas molecular viscosity,kg·m?1·s?1

χ local porosity of the random packings

Subscript

e effective

G gas phase

in inlet

out outlet

S solid phase

Acknowledgments

The authors acknowledge the assistance by the staff of the State Key Laboratories for Chemical Engineering(Tianjin University).