Mass transfer from a fluid flowing through a porous media
MMass transfer from a fluid flowing through a porous media
T.G. Myers ∗ , F. Font ∗ September 21, 2020
Abstract
A mathematical model is developed for the process of mass transfer from a fluid flowing througha packed column. Mass loss, whether by absorption or adsorption, may be significant. This is ap-propriate for example when removing contaminants from flue gases. With small mass loss the modelreduces to a simpler form which is appropriate to describe the removal of contaminants/pollutantsfrom liquids. A case study is carried out for the removal of CO2 from a gas mixture passing overactivated carbon. Using the experimental parameter values it is shown, via non-dimensionalisation,that certain terms may be neglected from the governing equations, resulting in a form which maybe solved analytically using a travelling wave substitution. From this all important quantitiesthroughout the column may be described; concentration of gaseous materials, amount of materialavailable for mass transfer, fluid velocity and pressure. Results are verified by comparison withexperimental data for the breakthrough curve (the amount of carbon measured at the column out-let). The advantage of the analytical expression over a purely numerical solution is that it caneasily be used to optimise the process. In the final section we demonstrate how the model may befurther reduced when small amounts of contaminant are removed. The model is shown to exhibitbetter agreement than established models when compared to experimental data for the removal ofamoxicillin and congo red dye from water.
Keywords:
Contaminant removal; Pollutant removal; Adsorption; Absorption; Carbon capture;Packed column; Mathematical model
With oceans overloaded with plastic, the air that we breathe full of noxious substances and evendrinking water laced with legal and illegal drugs it is clear that humanity needs to improve its methodsfor dealing with pollutants. However, cutting down on pollution is not enough, in some cases activeremoval must be carried out. In this paper we will focus on just one aspect of this issue, namelycontaminant removal via column sorption.Column sorption involves forcing a fluid through a confined tube filled with a porous materialcapable of removing certain components of the fluid. As the fluid passes through, the component tobe removed can attach to the surface of the material (adsorption) or enter into the volume (absorption).This may continue until the material becomes saturated, when no more removal occurs, and the fluidpasses through the material unchanged. It is perhaps the most popular practical sorption method[18, 17] and is used for a wide range of processes such as the removal of contaminants includingpharmaceuticals, carbon dioxide, heavy metals, dyes and salts [18, 1, 5, 7, 8, 11].In this paper we will develop a model to describe the flow of a fluid through a porous materialcontained in a cylindrical column. This configuration has been chosen due to its relevance for a ∗ Centre de Recerca Matem`atica, Campus de Bellaterra Edifici C, 08193 Bellaterra, Barcelona, Spain. a r X i v : . [ c ond - m a t . o t h e r] S e p ide number of published experiments and existing extraction equipment. We will focus on a two-component fluid system, with only one component being removed. (The extension to more componentsis straightforward). Once the model is developed we will analyse it within the context of post-combustion carbon capture and compare with experimental results for breakthrough (that is the CO concentration on leaving the column). Subsequently we will show how the model compares withprevious, standard models and test it against experimental data for the removal of contaminants suchas antibiotics and synthetic dyes.There exist a wide literature analysing column sorption. A number of simple models focus solelyon the column outlet and are based on the probability of a gas molecule escaping [19]. These neglectthe evolution of the process through the column and cannot fully explain the physics. However, withcarefully chosen parameter values it is possible to reproduce the breakthrough curves. When flowalong the column is accounted for the simplest model balances advection with mass loss, coupled to arate equation for the mass loss. An early example of this may be found in [4]. Although their full resultis often reduced to a much simplified equation valid only at the outlet. Recent models typically dealwith variables averaged over the cross-section and include advection, diffusion and temperature effects[9, 3]. However, since the sorption process is approximated by a simple kinetic model there is still adegree of fitting to match the breakthrough curve. An issue with many numerical studies is highlightedin [10] where it is shown that a number of errors in the governing equations have propagated throughthe literature. This leads to inaccuracies in the fitting coefficients and therefore incorrect predictionson scaling up.The study of [10] was based on the assumption of negligible mass removal, so that quantitiessuch as the velocity and fluid density are constant and consequently the pressure gradient is linear.In the following we will consider a situation where significant quantities are removed, such that thedensity and velocity may vary along the column. We will follow the style of [10] but do not carryout the averaging which acts to complicate the system. After deriving the governing equations weapply the model to an example of carbon capture, where approximately 15% of a CO /N mixtureis removed by adsorption. Comparison with experimental data for breakthrough shows excellentagreement. Subsequently we compare the full current model, a reduced version of this appropriatefor incompressible flow and previous models against data for the removal of amoxicillin and dye fromsolution.The work contains a number of novel elements for this field. The non-dimensionalisation permits usto identify dominant terms and also negligible ones which then permits the study of a much simplifiedsystem (with a known small error). Except during the very initial period, the simplified systempermits a travelling wave solution and so we find analytical expressions for all important quantities,concentration, available adsorbent, gas velocity and pressure. These solutions have not previously beenpublished. A full numerical solution is therefore not necessary. It is shown that the gas concentrationand available sorbent follow almost identical curves, this means that the experimentally observedbreakthrough curve may be used to determine the form of the kinetic relation. Consider a fluid flowing through a porous medium where mass transfer occurs at the fluid-solid inter-faces. Within the fluid the mass continuity equation may be written in the form ∂ρ∂t + ∇ · j = − S , (1)where S is a sink term representing mass loss and ρ is the density. The flux is composed of advectiveand diffusive components j = − D ∇ ρ + ρu . (2)2he exact shape of the porous media is unknown hence it is impossible to predict the flow, for thisreason it is standard practice to assume plug flow or carry out some radial averaging: the resultingequations are equivalent. Here we will follow the simpler route, assuming plug flow and hence all othervariables also only vary with distance along the column.With the definition u = ( u ( x, t ) , ∂ρ∂t + ∂ ( uρ ) ∂x = D ∂ ρ∂x − (1 − (cid:15) ) ρ q M q ∂q∂t , (3)where (cid:15) is the bed void fraction. The derivation of the mass loss term requires an assumption thatthe solid component accepts material of density ρ q and molar mass M q at a rate ∂q/∂t , where q represents the amount of material transferred. The overbar indicates it is an average. In practicematerial is normally adsorbed or absorbed through the surface or in cracks of the solid. In time thesurfaces become saturated. To avoid dealing with an overly complicated system it is standard to ignorethe precise distribution of the transferred mass and instead define an average throughout the solidmaterial, which is here denoted q . The final term on the right hand side is then a sink representingthe mass lost at all solid-fluid interfaces at a given x , see [10]. If the volume flux at the column inlet is Q ( t ), then the interstitial velocity may be written u ( x, t ) with u (0 , t ) = u ( t ) = Q ( t ) / ( (cid:15)πR ). In theliterature it is also common to work in terms of the superficial velocity, which is simply the interstitialvelocity multiplied by the void fraction, (cid:15)u .The density may be expressed in terms of the molar mass M i and molar concentration c i of thecomponents. For a two component system ρ = M c + M c , while the molar mass of the gas mixture M = ρ/ ( c + c ). There are a number of models to approximate the mass transfer process. These are typically based onassumptions that the rate is proportional to the amount available for transfer and the free sites on thesolid material and lead to different forms of kinetic model. A classical model is presented in [4]. Localequilibrium, linear driving force, pore diffusion, pseudo-first-order, pseudo-second-order and Avramimodels are discussed in [9, 15]. In the present study we will employ the popular linear kinetic relation ∂q∂t = k q ( q ∗ − q ) , (4)where k q is a rate constant and q ∗ the saturation value. The form of q ∗ is also the subject of numerousstudies, see the review of [2].The linear kinetic relation has the unrealistic feature that it has a weak dependence on the com-ponent available for transfer (which can only enter through the definition of ¯ q ∗ ). It must therefore bespecified explicitly that this equation only holds over regions where material is available. There is atendency in published literature to immediately integrate equation (4) to find q = q ∗ (1 − exp( − kt ))[7, 15, 13]. This is incorrect. Firstly, q ∗ typically depends in a complex way on space and time,secondly the constant of integration is dependent on space. In the special case where q ∗ is constant wemay write q = q ∗ (1 − exp( − k ( t − t s ( x ))) where t s ( x ) is the time at which the material to be removedfirst reaches the point x . With a typical form of time-dependent q ∗ the integration cannot be carriedout. If we assume a constant fluid velocity then its value may be easily calculated from the mass or volumeflux and the above equations adequately describe the system. This would be appropriate when a3egligible mass (compared to the total mass) is removed and is typically the case with transfer from aliquid. If mass removal is significant it will affect the velocity and pressure. This often occurs in gasmass transfer processes, which can involve the removal of some 20% of the gas. In this situation weneed more equations to close the system. Since the case of gas flow is more complex henceforth wewill focus on that, the liquid case forms a subset of the model developed below.Firstly, we follow the standard practice of relating the pressure to the concentration via the idealgas law p = R g T c = R g T ( c + c ) . (5)The final governing equation required for the system comes from conservation of momentum. Foruni-directional plug flow the Navier-Stokes equation may be reduced to the form ρ (cid:18) ∂u∂t + u ∂u∂x (cid:19) = − ∂p∂x + 43 µ ∂ u∂x − S u . (6)The term S u represents a reduction in momentum due to mass loss, where S = (1 − (cid:15) ) ρ q M q ∂q/∂t isthe sink from the mass continuity equation. The 4/3 factor is specific to compressible flow: with aconstant density flow this factor becomes unity.The classical relation describing flow in a porous media is Darcy’s law, which was derived for theincompressible flow of a viscous liquid through sand. In the case of a gas flowing through a packedbed Darcy may not be appropriate. To understand Darcy’s law consider conservation of momentumwith no mass sink. Assuming the steady, slow flow of an incompressible fluid (where slow is definedsuch that terms of order u may be neglected) equation (6) reduces to0 = − ∂p∂x + µ ∂ u∂x . (7)The key assumption is that viscous resistance is proportional to velocity u xx ∝ u and then we obtainDarcy’s law − ∂p∂x = µk p u , (8)where k p is termed the permeability. Gases have a very low viscosity so it is possible that for gasflow viscous resistance is small and instead inertial resistance plays a significant role. When inertia isnon-negligible a drag law is usually invoked uu x = C D u d (9)where the length-scale d is chosen as the typical particle diameter and the drag coefficient C D = C D ( (cid:15), Re, d ). In a similar manner to the derivation of Darcy’s law we may use this expression toobtain a relation between pressure gradient and velocity.For the current problem, the momentum equation also contains a sink term proportional to velocity.Including both viscous and inertial resistance as well as the sink we obtain the pressure-velocity relation − ∂p∂x = αρu + ( β + S ) u , (10)where α, β are constant while S is variable. In the absence of the sink term this form of equationis discussed in [14]. Various values for the constants α, β reproduce the Ergun relation (applicableto beds packed with beads or granules), or forms appropriate to beds packed with cylinders, foamsand laminates [12]. With α = S = 0 Darcy’s law is retrieved. Including the sink term we have notbeen able to find this relation in the sorption literature, but it is a natural consequence of momentumconservation with mass loss. Away from the sorption front we expect the sink term to be small and itsneglect is justifiable, however in regions of rapid mass transfer it plays a significant role in the massbalance and it is possible that this also leads to significant momentum loss.4 .3 Summary of governing equations Noting that ρ = M c + M c we may eliminate the gas density from the mass balance ∂∂t ( M c + M c ) + ∂∂x ( u ( M c + M c )) = D ∂ ∂x ( M c + M c ) − (1 − (cid:15) ) ρ q M q ∂q∂t . (11)Conservation of each species then gives ∂c ∂t + ∂∂x ( uc ) = D ∂ c ∂x − (1 − (cid:15) ) ρ q M q M ∂q∂t , (12) ∂c ∂t + ∂∂x ( uc ) = D ∂ c ∂x . (13)If only one species is removed we may set M q = M that is, the molar mass of the component is thesame in the solid and fluid state. The remaining governing equations are ∂q∂t = k q ( q ∗ − q ) (14) p = R g T ( c + c ) , (15) − ∂p∂x = α ( M c + M c ) u + (cid:18) β + (1 − (cid:15) ) ρ q M ∂q∂t (cid:19) u . (16)Equations (12 – 16) provide the system to describe the evolution of the five variables c , c , u, p, q . Thefive equations are required when mass transfer is significant compared to the total mass flow. Whendealing with the transfer of trace amounts of a material, for example drugs in the water supply, thesystem is significantly simpler. First, the velocity is approximately constant u = Q / ( (cid:15)πR ) and theproblem is described by equations (12, 14). If a known pressure drop drives the flow then the velocitymay also be calculated by neglecting the c and q terms in equation (16) (since both terms are smallwhen trace amounts are removed).The assumptions made in deriving the above equations include: the amount of material transferredmay be averaged over the sorbent; q t follows a linear kinetic model (this is discussed later); an idealgas law holds; the flow may be approximated as plug flow. At the inlet, x = 0, there is continuity of mass flux so( ρu ) | x =0 − = (cid:18) ρu − D ∂ρ∂x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x =0 + , (17)where the − , + superscripts indicate just before and just after x = 0. Separating the concentrationsgives u (0 − , t ) c = (cid:18) uc − D ∂c ∂x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x =0 + , u (0 − , t ) c = (cid:18) uc − D ∂c ∂x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x =0 + , (18)where c , c are the concentrations of the inlet gas and u (0 − , t ) = Q / ( πR ), u (0 + , t ) = Q / ( (cid:15)πR ).At the exit we move to a region where no mass transfer occurs, so it is assumed that whatever thedensity on leaving the column it remains the same just outside the exit ∂c ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = L − = ∂c ∂x (cid:12)(cid:12)(cid:12)(cid:12) x = L − = 0 . (19)5he pressure at inlet and outlet are p (0 , t ) = p ( t ) , p ( L ) = p a . (20)The inlet value may vary with time. Initially we assume the solid is fresh and the column is free ofthe component to be removed ρ ( x,
0) = M c ( x,
0) = M p ( x ) RT , q ( x,
0) = 0 , (21)hence c ( x,
0) = 0 , c ( x,
0) = p in ( x ) RT , (22)where the initial pressure p in ( x ) = p (0) − ( p (0) − p a ) x/L . We scale variables in the following mannerˆ p = p − p a ∆ p ˆ c = c c ˆ c = c c ˆ q = qq ∗ ˆ x = x L ˆ t = t ∆ t ˆ u = uu , (23)where L and ∆ t are unknown and q ∗ is the value of q ∗ at t = 0. Since our interest lies with thereaction we choose a time-scale ∆ t = 1 /k q and so ∂ ˆ q∂ ˆ t = (ˆ q ∗ − ˆ q ) . (24)Balancing advection with mass loss gives the length-scale L = u c / ((1 − (cid:15) ) ρ q q ∗ k q ) and then δ ∂ ˆ c ∂ ˆ t + ∂∂ ˆ x (ˆ u ˆ c ) = δ ∂ ˆ c ∂ ˆ x − ∂ ˆ q∂ ˆ t , (25) δ ∂ ˆ c ∂ ˆ t + ∂∂ ˆ x (ˆ u ˆ c ) = δ ∂ ˆ c ∂ ˆ x . (26)In experiments the main gas component is not usually the one being removed, hence we assume c < c and write 1 + δ ˆ p = δ (ˆ c + δ ˆ c ) , (27) − ∂ ˆ p∂ ˆ x = δ (ˆ c + δ ˆ c )ˆ u + (cid:18) δ ∂ ˆ q∂ ˆ t (cid:19) ˆ u , (28)where, assuming the flow to be close to Darcy flow, we have set the pressure scale ∆ p = βu L .The boundary and initial conditions are1 = (cid:18) ˆ u ˆ c − δ ∂ ˆ c ∂ ˆ x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˆ x =0 + , (cid:18) ˆ u ˆ c − δ ∂ ˆ c ∂ ˆ x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ˆ x =0 + , (29) ∂ ˆ c ∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x =ˆ L − = ∂ ˆ c ∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x =ˆ L − = 0 , ˆ p (0 , ˆ t ) = ˆ p (ˆ t ) , ˆ p ( ˆ L ) = 0 , (30)ˆ c (ˆ x,
0) = 0 , δ ˆ c (ˆ x,
0) = 1 + δ ˆ p in , ˆ q (ˆ x,
0) = 0 , (31)6here the initial pressure profile ˆ p in = ˆ p (0)(1 − ˆ x/ ˆ L ). To keep the model general we express thepressure condition at ˆ x = 0 as an unspecified function of time. If the inlet pressure is kept constantat a given value then ˆ p (0 , ˆ t ) = 1, however in a number of experiments the flux is maintained as theconstant (via a flow meter) and this requires a variable inlet pressure. This is then determined as partof the solution process.The five equations (24 – 28), together with the appropriate boundary conditions, are sufficient todetermine the five unknowns ˆ c , ˆ c , ˆ q, ˆ p, ˆ u . They contain eight non-dimensional groupings, δ i , whichindicate the relative importance of the terms δ = L k q u = c (1 − (cid:15) ) ρ q q ∗ , δ = D L u , δ = ∆ pp a , δ = R g T c p a (32) δ = c c , δ = αM c u L ∆ p , δ = M c M c , δ = 1 − (cid:15)β ρ q M q ∗ k q . (33) Standard models for carbon capture in a packed column may be found in the reviews of [9, 3, 14]. Ingeneral these are based on the following assumptions:1. The gas behaves as an ideal gas.2. A plug-flow model is adopted.3. The radial variation of concentration is negligible.4. The bed operates isothermally.5. The CO concentration is low so that the pressure gradient is linear and the velocity constantalong the bed.Assumption 3 follows from 2 (although in mathematical terms the radial variation is the source of themass sink term, see [10]). In experimental studies CO typically comprises 15-20% which, as we willsee later, leads to velocity variation of the same order and non-linear pressure, hence we do not applyassumption 5. We have retained the isothermal and ideal gas assumptions (in [10] it was shown thatthe temperature variation is small during carbon capture).A typical experimental set-up involves a circular cross-section column containing a porous ma-terial, this is then placed inside an oven or furnace to regulate the temperature. Gas is passedthrough the column and the concentration measured at the outlet. Here we consider a specific ex-periment which involves a CO , N mixture passing through a bed of activated carbon, the dataand operating conditions are given in Table 1, see [10, 15]. Mass transfer was by adsorption. Amass flow controller was employed to maintain a constant flow rate Q = 50ml/min, which indicates u = 50 × − / (60 (cid:15)πR ) = 0 . α = 1 . − (cid:15) ) d p (cid:15) ≈ . × β = 150 µ g (1 − (cid:15) ) d p (cid:15) ≈ . × , (34)where d p = 6 . × − m. The length-scale L ≈ . p = βu L ≈ . δ = 0 . δ ∼ . δ ∼ . × − δ = 0 . δ ≈ .
18 (35) δ = 3 . × − δ = 0 . δ = 3 . × − . (36)7ymbol Value DimensionInitial concentration (CO ) c Initial concentration (N ) c Molar mass (CO ) M ) M T p a q ∗ (cid:15) L R d p × − mGas viscosity (15% CO (1.5), 85% N (1.8)) µ g . × − Pa sDensity of adsorbed CO ρ q
325 kg/m Axial diffusion coefficient D . × − m /sInitial volume fraction (CO ) y Q . × − m /sInitial interstitial velocity u ) k q − Solid/gas density ρ s /ρ g Table 1: Values of the thermophysical parameters mainly taken from [15], except k q , ρ q (as discussed later). For a kinetic model we take the version described in [15] since this provides the parameter valuesfor this specific experiment, q ∗ = q m K T p [1 + ( K T p ) n ] /n + q m K T p [1 + ( K T p ) n ] /n , (37)where q m , q m = 0 . , .
57 mol/kg, K T , K T = 8 . × , .
66 atm − , n , n . , .
65 Note, thevariation with temperature discussed in that paper does not apply to the present isothermal study.In non-dimensional form we write q ∗ ˆ q ∗ = q m K T p a (1 + δ ˆ p )[1 + ( K T p n a (1 + δ ˆ p )) n ] /n + q m K T p a (1 + δ ˆ p )[1 + ( K T p n a (1 + δ ˆ p )) n ] /n . (38)Neglecting terms of order δ ∼ − we see that the leading order equation for ˆ q is ∂ ˆ q∂t = 1 − ˆ q (39)where the (constant) adsorbent scaling q ∗ = q m K T p a [1 + ( K T p n a ] /n + q m K T p a [1 + ( K T p n a ] /n . (40)Neglecting terms of order δ , δ ∼ − the concentration equations are ∂∂ ˆ x (ˆ u ˆ c ) = − ∂ ˆ q∂ ˆ t , (41) ∂∂ ˆ x (ˆ u ˆ c ) = 0 . (42)8hese simply state that gas is primarily advected through the column, with the only variation comingthrough the mass loss due to adsorption. In a constant velocity study [10] the leading order c equationwas discussed, where it was pointed out that although diffusion is, in general small, it can play animportant role in the numerical solution. If we have an initial condition where the CO concentrationis zero and then jumps to some non-zero value when first pumped in then the gradient is discontinuousand diffusion is important in smoothing this out. However, this is only important near x = t = 0.For the present we will focus mainly on the outlet, close to first breakthrough and so neglect diffusion(except in Fig. 5 where we compare the approximate solutions with numerics to verify the validity ofthis approach).The leading order pressure relations are1 = δ (ˆ c + δ ˆ c ) , (43) − ∂ ˆ p∂ ˆ x = ˆ u . (44)The form of the non-dimensional gas law, (43), is very informative: the relative change in pressurealong the column is tiny compared to the total pressure, hence the pressure variation along the columnhas a negligible effect on the concentration and we may write one concentration in terms of the other,irrespective of the pressure or velocity, ˆ c = 1 δ − δ ˆ c . (45)The scaling of the momentum balance (44) shows that for this case the Ergun relation is unnecessarilycomplicated: momentum loss is dominated by viscous resistance between the gas and porous material.Hence velocity is related to pressure drop via a simple Darcy law.The necessary leading order boundary and initial conditions are1 = (ˆ u ˆ c ) | ˆ x =0 + u ˆ c ) | ˆ x =0 + , (46) ∂ ˆ c ∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x =ˆ L − = ∂ ˆ c ∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x =ˆ L − = 0 ˆ q (ˆ x,
0) = 0 . (47)Integration of equation (42) shows that ˆ u ˆ c = 1 everywhere. Combining this with equation (45)transforms the mass balance (41) to δ ∂∂ ˆ x (cid:18) ˆ c − δ ˆ c (cid:19) = − ∂ ˆ q∂ ˆ t , (48)where δ = δ δ ≈ .
15. The factor δ / (1 − δ ˆ c ) does not appear in the constant velocity system of[10]. Since δ = 0 .
85 this indicates a change in size of quantities of the order 15% when compared toconstant velocity models. Equation (48) must be solved in conjunction with the adsorbent equation,(39) and then ˆ u, ˆ c , ˆ p may be obtained from the preceding equations.The above equations hold behind the reaction front, ˆ x ≤ ˆ s . Ahead we have ˆ c = ˆ q = 0 and so,from (45), ˆ c = 1 /δ is constant and hence so is ˆ u = δ (i.e. the velocity ahead of the front is 15%lower than the inlet velocity). Equations (39, 48) may be solved numerically to find the behaviour for all time although, due tothe neglect of diffusion, the very early time solution may be inaccurate. For sufficiently large times,such that the initial transient is complete, we do not even need a numerical solution since there9xists a travelling wave solution. To find this we first choose a co-ordinate moving with the c front, η = ˆ x − ˆ s (ˆ t ). Equations (39, 48) then become ∂ ˆ f∂ ˆ η = ˆ s ˆ t ∂ ˆ q∂η , (49) − ˆ s ˆ t ∂ ˆ q∂ ˆ η = (1 − ˆ q ) , (50)where ˆ f = ˆ u ˆ c = δ ˆ c / (1 − δ ˆ c ). A travelling wave solution may be found if ˆ s ˆ t = ˆ v is constant.If this is the case the equation for ˆ f may be integrated immediately. After applying the boundaryconditions ˆ c = ˆ f = ˆ q = 0 at ˆ η = 0 we obtain ˆ f = ˆ v ˆ q . (51)Eliminating ˆ q between (49 – 51) leads to an ODEˆ v ∂ ˆ f∂ ˆ η − ˆ f = − ˆ v (52)with solution ˆ f = ˆ v (cid:104) − e ˆ η/ ˆ v (cid:105) . (53)The constant of integration in this case has been obtained by applying the condition at ˆ η = 0. Thevelocity may be determined by assuming the front is far from the column entrance, ˆ x = 0, henceˆ η = ˆ x − ˆ s is large and negative, formally we apply ˆ η → −∞ , ˆ u ˆ c = ˆ f → v = 1.This verifies the travelling wave assumption that ˆ s ˆ t is constant.The CO concentration may be calculated from (53), with ˆ v = 1,ˆ c = 1 − e ˆ η δ + δ (1 − e ˆ η ) ≈ − e ˆ η − δ e ˆ η . (54)The simplification results from δ + δ = p /p a = 1 + δ ≈
1. Then from (45, 51) and using ˆ u ˆ c = 1, δ ≈ − δ we find ˆ c = 11 − δ e ˆ η , ˆ u = 1 − δ e ˆ η , q = 1 − e ˆ η . (55)The pressure equation (44) may be written as ˆ p η = − ˆ u . For ˆ x ≥ ˆ s , ˆ u = δ is constant and withˆ p = 0 at ˆ η = ˆ L − ˆ s we obtain ˆ p = δ (( ˆ L − ˆ s ) − ˆ η ) for ˆ x ≥ ˆ s . (56)For ˆ x ≤ ˆ s the velocity is specified by (55). The pressure at the column inlet is unknown but equation(56) gives the value at ˆ x = ˆ s (ˆ η = 0), which may be used to determine the constant of integration,leading to ˆ p = δ ( ˆ L − ˆ s ) − (cid:104) ˆ η + δ (1 − e ˆ η ) (cid:105) for ˆ x ≤ ˆ s . (57)To relate the solutions to experiments requires switching between ˆ η and ˆ x, ˆ t . With the definitionˆ s ˆ t = 1 we obtain ˆ s = ˆ t + ˆ s (where ˆ s is unknown) and hence ˆ η = ˆ x − ˆ t − ˆ s . The travelling wave isnot valid at ˆ t = 0 so we cannot apply an initial condition to determine ˆ s and must use informationfrom a later time. In the carbon capture literature the breakthrough curve is generally presented,which shows the CO concentration at the end of the column. If CO is first recorded at the outletat time ˆ t b , then we may use ˆ s = ˆ L − ˆ t b . However, given the uncertainty as to when gas actually firstescapes, a more reliable measure is to note the time t / when c = c / x = ˆ L , ˆ c = 1 / s = ˆ L − ˆ t / + log(2 − δ ) or alternatively we may writeˆ t b = ˆ t / − log(2 − δ ). 10 .2 Dimensional solutions We now summarise, in dimensional form, the solutions starting with the simplest region, ahead ofthe front x ≥ s , where s = L k q t + s = L + L k q ( t − t b ), and t b could be set as the experimentallymeasured breakthrough time or we use t b = t / − log(2 − ( R g T c /p a )) /k q and the length-scale L = u c / ((1 − (cid:15) ) ρ q q ∗ k q ). The velocity of the front ˆ s ˆ t = 1 indicates s t = L / ∆ t = u c / ((1 − (cid:15) ) ρ q q ∗ ).For x ∈ [ s, L ] and t < t b : c = 0 , c = p a R g T , ¯ q = 0 , u = R g T c p a u (58) p = p a + β R g Tp a u c ( L − x ) . (59)For x ∈ [0 , s ] c = c (cid:34) − e ( x − L ) / L e − k q ( t − t b ) − ( R g T c /p a ) e ( x − L ) / ˆ v L e − k q ( t − t b ) (cid:35) , (60) c = c (cid:20) − ( R g T c /p a ) e ( x − L ) / L e − k q ( t − t b ) (cid:21) , (61)¯ q = ¯ q ∗ (cid:16) − e ( x − L ) / L e − k q ( t − t b ) (cid:17) , (62) u = u (cid:16) − ( R g T c /p a ) e ( x − L ) / L e − k q ( t − t b ) (cid:17) , (63) p = p a + βu (cid:20) ( s − x ) − R g Tp a c L (1 − e ( x − L ) / L e − k q ( t − t b ) ) + R g Tp a c ( L − s ) (cid:21) . (64)In experiments it is common to measure the breakthrough curve. The analytical representation isobtained by setting x = L in equation (60) c = c (cid:34) − e − k q ( t − t b ) − ( R g T c /p a ) e − k q ( t − t b ) (cid:35) , (65)where t ≥ t b . In the case studied above the flux is fixed by a flowmeter: the pressure is adjusted to maintain aconstant flow rate. However it is also common practice to fix the pressure at either end and leavethese constant throughout the experiment. For a fixed flux (constant u ) equations (59, 64) determinethe pressure ahead of and behind the front respectively for the specified value of u . If the pressure isfixed such that p (0 , t ) = p , p ( L, t ) = p a then ahead of the front equation (59) holds, while behind p = p − βu (cid:20) x − R g Tp a c L e − k q ( t − t b ) (cid:16) e ( x − L ) / L − e − L/ L (cid:17)(cid:21) x ≤ s . (66)Since the pressure is continuous at the front we may equate these two expressions to determine thevelocity u caused by the prescribed pressure drop u = p − p a β (cid:20) s + R g Tp a (cid:16) c ( L − s ) − c L e − k q ( t − t b ) (cid:16) e ( s − L ) / L − e − L/ L (cid:17)(cid:17)(cid:21) − . (67)The inlet flux Q ( t ) = (cid:15)πR u (0 , t ) is then obviously time-dependent.11 igure 1: Concentrations c ( x, t ) , c ( x, t ), amount adsorbed ¯ q ( x, t ) and velocity u along the column at t = 0 . t b In Figure 1 we show the variation along the column of the concentration of CO , N , the amountof adsorbed material and the velocity of the mixture as predicted by equations (58, 60-63). Theparameter values used are provided in Table 1. The results correspond to a time t = 0 . t b , where t b = 10 . concentration is almost unity at the inlet and decreases smoothly to zero at the front. Thenon-dimensionalisation shows that ˆ q differs from ˆ c by a factor of order 1 / (1 − δ ), where δ ≈ . c must be very similar to ¯ q . This may be observed through the close proximity of the twocurves. This means that if we are able to influence ¯ q then it will have a corresponding effect on the CO concentration and vice-versa. As CO is removed the velocity of the mixture decreases. Conservationof mass requires that the nitrogen concentration must increase (the increase is of the order of theinitial concentration ratio c /c ≈ . is removed. At the outlet we see that c ≈ . c , u ≈ . u .Figure 2 shows the concentrations and velocity at the outlet. Theoretically no CO escapes thecolumn until t = 10 . curve. Both c and ¯ q have their greatest rate of increase at the breakthrough time, the gradient (withrespect to time) then decreases monotonically to zero as the adsorbent is used up. The theoreticalvelocity, u ( L, t ), is around 0.85 u until breakthrough, when it increases to the inlet value. Similarlyfor the nitrogen concentration, which decreases from 1 . c to its inlet value. However, we shouldpoint out that the travelling wave solution is not valid at small times, so the curves may not beaccurate near t = 0. The correspondence between the CO concentration and the experimental dataof [15] (shown as circles) appears very good, showing a similar level of accuracy to previous publishedresults which typically present curves over a large time range (which then makes the curves moredifficult to distinguish). In Figure 3 we show a close-up of the comparison between the predictedCO concentration predictions and experiment. At this scale we observe qualitative differences in theresults. In particular we note that the experimental breakthrough is a more gentle process, with c at12 igure 2: Concentrations c ( L, t ) , c ( L, t ) and velocity u ( L, t ) at outlet, x = L . All divided by their initialvalues. Also shown is the experimentally measured concentration of c ( L, t ) (circles). first slowly increasing to cross the theoretical prediction before increasing in gradient so the two setsof results coincide. We will discuss these differences in more detail in § t = 0 . t b , . t b . The solid line depicts the pressurebehind the front, the dashed line that ahead of it. The dotted line is the linear profile calculated from − p x = βu which indicates p = p a + βu ( L − x ). This is the profile that would be observed if velocityvariation were neglected, it is also the limit of the present theoretical curves for large time (when theadsorbent is full and so the velocity is constant). The variation of p (0 , t ) verifies our earlier statementthat a flow meter will have to vary the pressure to maintain a fixed flow rate.To convince the numerically minded reader of the accuracy of the travelling wave solution, inFigure 5 we compare numerical and travelling wave results. The curves correspond to those presentedin Figure 1. The dashed lines represent the numerical solution, the solid lines the travelling wave.Full details of the system solved numerically and the scheme are provided in Appendix A. In Figure5a) we compare the CO and N concentrations, the CO concentration obviously corresponds to thecurves reaching zero around 0.18m. In the first runs of the code it turned out that there was a smalldiscrepancy: the numerical solution moved slightly faster than the travelling wave, by around 4%.The main difference between the two methods comes from the retention of δ , δ in the numericalscheme. The values δ = 0 . , δ = 0 .
036 suggest errors of the order 3.6%, so this discrepancy isentirely in line with the approximations made. The diffusion parameter δ controls the spread of thefront while δ = L k q /u = c / ((1 − (cid:15) ) ρ q q ∗ relates to the speed. The parameter for which we have theleast information is ρ q consequently we increased this by 4% to ρ q = 338kg/m . With this adjustmentwe achieve the excellent agreement between numerics and the analytical solution. Figure 5b) showsthe variation of velocity (top curves) and available adsorbate (bottom curves). Again the agreement isexcellent. Consequently we may state that the travelling wave provides solutions within the accuracyof neglected terms, where here the largest was 0.036. If even higher accuracy is required then thenumerical solution may be used to fine tune the density of adsorbate.13 igure 3: Blow-up of breakthrough curve, showing the predicted c ( L, t ) and experimental data (circles).Figure 4: Pressure variation along the column for t = 0 . t b (red), t = 0 . t b (blue). The dashed section, nearthe end of the column, shows the pressure ahead of the front. The standard linear profile (dotted line) is alsoshown. igure 5: Comparison of numerical (dashed) and travelling wave (solid) results for the same conditions as thoseof Figure 1: (a) shows c ( x, . t b ) , c ( x, . t b ), (b) shows q ( x, . t b ) , u ( x, . t b ). There exist a variety of breakthrough models designed to model different sorption processes. Typicallythey are based on the probability of a component escaping and the amount of material available formass transfer. For example in [19] the assumptions on the probability of escape lead to a standardlogistic equation for the concentration c t = kc ( c − c ). In this section we discuss a numberof mathematically equivalent breakthrough models and also derive the form of the present modelappropriate for describing incompressible fluid flow. The models are then compared with experimentaldata for the adsorption of amoxicillin and a dye from water. It is shown that in these two examplesthe form of the previous models is incapable of capturing the whole breakthrough curve, whereas theforms (compressible and incompressible) presented in this paper both provide a good approximation. An early, classic model to describe the concentration and amount absorbed was developed by Bohartand Adams [4]. They wrote down a constant velocity model where the time derivative and diffusionterms are neglected from the conservation equation for c and the absorption rate depends on theamount already absorbed and the available absorbate ∂c∂x = − k BA v (¯ q ∗ − ¯ q ) c , ∂ ¯ q∂t = k BA (¯ q ∗ − ¯ q ) c , (68)where ¯ q ∗ is constant. They provide the solution c = c − exp( − k BA c t ) + exp( k BA (¯ q ∗ x/v − c t )) , (69)¯ q = ¯ q − exp( k BA (¯ q ∗ x/v ) + exp( k BA ( c t − ¯ q ∗ x/v )) (70)see [4, eq. (21,22)]. The breakthrough curve is obtained by setting x = L . This is a much abusedresult and is often misquoted, as discussed in [6]. In fact even in [6] a ‘simplified’ version is studiedwhich results from neglecting the first exponential in the denominator in (69) c ≈ c k BA (¯ q ∗ x/v − c t )) . (71)15his may be justified by assuming a sufficiently large time such that exp( − k BA c t ) (cid:28)
1. Equation(71) is often referred to as the Thomas model [1, 8, 16]. If we divide the top and bottom by theexponential term and again assume that an exponential term is small, exp( − k BA (¯ q ∗ x/v − c t )) (cid:28) c ≈ c exp( − k BA (¯ q ∗ x/v − c t )) (72)see [1, 8, 16] for example. It is equivalent to the Wolborska model [11, 16].In arriving at (71) we assumed exp( − k BA c t ) (cid:28)
1, usually this is justifiable after substituting forthe parameter values and considering a sufficiently large time. In arriving at (72) we assumed thata second exponential is small, leaving an expression where the concentration is proportional to thisneglected exponential, hence it is only valid for small concentrations. Equation (72) is, rather harshly,usually termed the Bohart-Adams equation instead of their more widely applicable result (69). Sinceit only holds for small c it is often stated that their model is only valid near the start of breakthrough.We now focus on the breakthrough curve and write equation (71) at x = L in a slightly moregeneral form c = c (cid:20)
11 + A exp( − A t ) (cid:21) , (73)where for the Bohart-Adams model A = exp( k BA ¯ q ∗ L/v ) , A = k BA c . This form also covers theYoon-Nelson, Thomas and Bed Depth Service Time models where the parameters A , A have slightlydifferent interpretations in each case (see Table 3 in the review paper [11]). However, since each involvesome fitting to experimental data they are mathematically equivalent. Similarly we may write thepresent model, equation (65), as c = c (cid:20) − A exp( − A t )1 − A exp( − A t ) (cid:21) , (74)where A = exp( k q t b ) , A = k q , A = ( R g T c /p a ) A . The model derived in § ∂ ˆ c ∂x = − ∂ ˆ q∂ ˆ t , ∂ ˆ q∂ ˆ t = 1 − ˆ q . (75)The non-dimensional velocity ˆ u = 1 and the scale u = Q / ( (cid:15)πR ). The travelling wave analysis thenleads to the incompressible form of equation (65) c = c (1 − A exp( − A t )) , (76)where A = exp( k q t b ) , A = k q . In Figures 6a), b) we show comparisons of the current model against data for the removal of dye [16]and amoxicillin [7]. In each case we determine the time when the concentration is half the inlet value, t / , from the experimental data (for the dye t / = 28 ×
60 s, for the amoxicillin t / = 13 . ×
60 s)16 igure 6: Experimental data (circles) for the adsorption of a) congo red dye in water by soil, [16, Fig. 4], b)amoxicillin in water by activated carbon, [7, Fig. 9]. Lines are least-squares fit to equation (74) (solid), equation(73) (dashed), equation (76) (dotted). and use this to eliminate an unknown from each model. For equation (73) we find A = exp( A t / ),for equation (74) A = 2 A − exp( A t / ) and for equation (76) A = exp( A t / ) /
2. Then weapply a least-squares fit to determine the remaining unknowns. The experimental points in Figure6a) relate to the removal of congo red dye from solution after being passed through soil, see [16,Fig. 4, H = 5 cm]. The solid line comes from the current model, equation (74) with A = 0 . A = 2 . × − s − , the dashed line corresponds to equation (73) with A = 6 . × − s − , the dottedline (76) with A = 3 . × − s − . The experimental points in Figure 6b) relate to the removal ofamoxicillin from water using activated carbon, see [7, Fig. 9]. Again the solid line represents thecurrent model, equation (74), now with A = 2 . A = 0 . − , the dashed line equation (73)with A = 0 . − , the dotted line (76) with A = 0 . − . In both graphs the best fit is providedby the current compressible flow model. Of the one parameter models the best fit is the present modelfor incompressible flow. In the above we carried out a least-squares fit to determine the system unknowns. Other researchersuse different methods, such as linear regression. Whatever the method it is clear that the form ofequation (73), which describes at least four different previous models, is not capable of producing abetter fit to the data used in Figure 6 than either of the two current models. However, this is not tosay that the present model solves all problems.Consider the form of the adsorption equation ¯ q t = k q (¯ q ∗ − ¯ q ). The equilibrium adsorption dependson the total pressure which varies throughout the column, as shown in Figure 4. However, it is usuallythe breakthrough curve which is measured, this occurs at the column outlet where the pressure isapproximately constant throughout the process. Since ¯ q increases monotonically from zero to ¯ q ∗ therate ¯ q t is greatest at first breakthrough. From the non-dimensionalisation it is clear that c differs from¯ q by a factor of around (1 − δ ) where δ is small. Consequently the concentration variation withtime must have a similar form to that of ¯ q (this is apparent from Figure 1) and the highest gradientin concentration therefore also occurs at first breakthrough. This means that, although the time offirst breakthrough will depend on system parameters such as the flow-rate, column length, initialconcentration, void fraction etc, it is the form of the mass transfer model that primarily determinesthe shape of the breakthrough curve . 17he two examples shown in Figure 6 involve experimental data where the gradient ∂c /∂t isgreatest at first breakthrough, so the present linear driving force model results in an excellent fit. Thedata presented in Figure 3 indicates ∂c /∂t ≈ q t = kc (¯ q ∗ − ¯ q ) . (77)This form has a zero adsorption rate when there is zero concentration or no transfer sites available,and a maximum close to the middle of the process. So perhaps the best system to describe CO transfer in a column will involve the current set of equations but with a mass transfer model such asequation (77). Whilst the transfer of pollutants from a liquid solution best follows the present model. In this paper we have developed a mathematical model to describe isothermal mass transfer froma fluid flowing through a porous medium contained within a cylindrical tube. The model was keptrelatively general, to permit the inclusion of adsorption and absorption processes and also the removalof relatively large quantities of material, such that the velocity and pressure vary nonlinearly alongthe column.Since the model permits the removal of a significant amount of the fluid it is suitable to gaspollutant studies. As the amount removed becomes smaller then the model may be reduced to onemore appropriate to the removal of contaminants in aqueous solutions. Hence we first validated itagainst experimental data for CO removal by adsorption. Subsequently we considered contaminantremoval from aqueous solutions.Perhaps the key limitation of this work is the assumption of an isothermal reaction. In a numberof studies on carbon capture it has been shown to be a small effect, similarly, with removal of tracequantities from a liquid it is clearly small however there are, no doubt, situations where this willnot be an appropriate assumption. The travelling wave solution cannot capture the very early timebehaviour. This may not be viewed as a problem since practical sorption equipment usually runs forvery long periods and the start-up is of little interest. The model reduction was based on the size of thenon-dimensional parameters, their relative size may change for different materials and experimentalset-ups. Consequently they should be checked whenever a new process is investigated.It was shown that sufficiently far from the inlet a travelling wave solution holds, thus there is noneed for a full numerical solution. To verify this we compared the travelling wave solution against thenumerics. Results showed that the difference, below 4%, was exactly in line with the terms neglectedin the analytical approximation. The numerical solution therefore turned out to be most useful forfine-tuning the value of the density of material transferred to the column or the saturation value,otherwise the travelling wave appears sufficiently accurate.The travelling wave solution was compared against experimental data for the removal of amoxicillinand dye from water. The agreement was excellent, it also outperformed standard previous models.A key result of the analysis is that the concentration of the material to be removed closely fol-lows the amount of sorbate available. In the CO example the difference was less than 10%. Theexperimentally observed breakthrough curve may then be used to guide the form of the mass transfermodel. For example, in the cases of amoxicillin and dye removal the breakthrough curve had its steep-est gradient at first breakthrough which then slowly decreased to zero: this suggests a kinetic relationof the form q t ∝ q ∗ − q . In the case of CO removal published data often shows a small gradientat first breakthrough, suggesting q t ∝ c ( q ∗ − q ) or q ( q ∗ − q ). Since the pressure near the outlet isapproximately ambient the value of q ∗ is constant (with respect to pressure), which can simplify the18reakthrough calculation (there may still be some temperature variation, which was not considered inthis paper).The analytical solutions provided in our analysis, the gas concentrations, gas velocity, amount ofsorbate and pressure as well as the front velocity, have been shown to accurately describe the evolutionin a cylindrical column. This means that we now have explicit expressions to describe the role playedby the operating parameters which may then be used to improve or optimise the process. Acknowledgements
F. Font acknowledges that the research leading to these results has received funding from la CaixaFoundation. T. G. Myers acknowledges financial support from the Ministerio de Ciencia e Innovacin,Spain Grant No. MTM2017-82317-P.
A Numerical solution
Numerically solving the full model (24)-(31) can be relatively complicated, mainly due to the nonlin-earities in the momentum equation (28). However, a reduced version of the model can be formulatedby neglecting terms of order 10 − by setting δ = δ = δ = 0 in (24)-(31). The neglect of these termsindicates errors of the order 0.1% when compared to a solution of the full system.The governing equations of the reduced model are δ ∂ ˆ c ∂ ˆ t + ∂∂ ˆ x (ˆ u ˆ c ) = δ ∂ ˆ c ∂ ˆ x − ∂ ˆ q∂ ˆ t , (78) δ ∂ ˆ c ∂ ˆ t + ∂∂ ˆ x (ˆ u ˆ c ) = δ ∂ ˆ c ∂ ˆ x , (79) ∂ ˆ q∂ ˆ t = (1 − ˆ q ) , (80)1 = δ (ˆ c + δ ˆ c ) , (81) − ∂ ˆ p∂ ˆ x = ˆ u , (82)with the boundary conditions1 = ˆ u | ˆ x =0 ˆ c | ˆ x =0 − δ ∂ ˆ c ∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x =0 , ∂ ˆ c ∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x = l = 0 , (83)1 = ˆ u | ˆ x =0 ˆ c | ˆ x =0 − δ ∂ ˆ c ∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x =0 , ∂ ˆ c ∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x = l = 0 , (84) ∂ ˆ q∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x =0 = ∂ ˆ q∂ ˆ x (cid:12)(cid:12)(cid:12)(cid:12) ˆ x = l = 0 , (85)and the initial conditions given by (31). Note that (78)-(85) include terms of order 10 − previouslyneglected in the derivation of the travelling wave solution. Hence, the numerical solution of (78)-(85)can be used to validate the accuracy of the travelling wave solution.Expression (81) leads to ˆ c = 1 δ − δ ˆ c . (86)Substituting (86) in (79) provides a relation between ˆ u and ˆ c , which can be combined with (78) togive ∂ ˆ u∂ ˆ x = − δ δ ∂ ˆ q∂ ˆ t . (87)19his is consistent with the travelling wave solution (55). In a similar fashion, by substituting (86) in(84), and using (83), we obtain the following boundary condition for the gas velocityˆ u | ˆ x =0 = (1 + δ ) δ = 1 , (88)(after noting that (1 + δ ) δ = 1 + δ ≈ u = 1 − (cid:90) ˆ x δ δ ∂ ˆ q∂ ˆ t d ˆ x . (89)We note that adsorption can only occur in the region where ˆ c is present. In the travelling wavesolution this corresponds to the growing region x < s ( t ). For the numerical solution, we take a differentapproach and define the function H (ˆ c ) = (cid:40) c >
00 otherwise (90)which we will use to enable/disable equation (80) if a particular region within the column contains ˆ c or not (see [10]), thereby avoiding the difficulty of dealing with a moving boundary.Using (90) as a multiplying factor in (80), and substituting (80) in (89), the equations of the modelreduce to δ ∂ ˆ c ∂ ˆ t + ∂∂ ˆ x (ˆ u ˆ c ) = δ ∂ ˆ c ∂ ˆ x − ∂ ˆ q∂ ˆ t (91) ∂ ˆ q∂ ˆ t = (1 − ˆ q ) H (ˆ c ) , (92)ˆ u = ˆ u | ˆ x =0 − (cid:90) ˆ x δ δ (1 − ˆ q ) H (ˆ c ) d ˆ x (93)which are subject to the boundary conditions (83), (85) and (88). The concentration ˆ c is obtainedvia (86) and the pressure ˆ p can be constructed by numerically integrating (82) a posteriori.The set of equations (91)-(93) are solved using second-order central finite differences in space andexplicit Euler in time. The boundary conditions are discretised using one-sided second-order finitedifferences. The nonlinear advection term in (91) is dealt with by using an upwind scheme with the u profile from the previous time step. The scheme was implemented in Matlab, using the function trapz for the numerical integration of (93) (at each node and time step). The choice of ∆ t and ∆ˆ x is madeensuring that the stability criteria ∆ˆ t δ / (∆ˆ x δ ) ≤ . u ) ∆ˆ t/ (∆ˆ x δ ) ≤ t = 0 . × − and ∆ˆ x = 0 . References [1] M.J. Ahmed and B.H. Hameed. Removal of emerging pharmaceutical contaminants by adsorptionin a fixed bed column: A review.
Ecotoxicology and Environmental Safety 149 (2018) 257266 ,149:257–266, 2018.[2] N. Ayawei, A.N. Ebelegi, and D. Wankasi. Modelling and interpretation of adsorption isotherms.
Journal of Chemistry , 2017.[3] R. Ben-Mansour, M.A. Habib, O.E. Bamidele, M. Basha, N.A.A. Qasem, A. Peedikakkal,T. Laoui, and M. Ali. Carbon capture by physical adsorption: Materials, experimental inves-tigations and numerical modeling and simulations - A review.
Applied Energy , 161:225 – 255,2016. 204] G. S. Bohart and E. Q. Adams. Some aspects of the behaviour of charcoal with respect to chlorine.
J. Am. Chem. Soc. , 42(3):523–544, 1920.[5] Z. Z. Chowdhury, S. M. Zain, A. K. Rashid, R.F. Rafique, and K. Khalid. Breakthrough curveanalysis for column dynamics sorption of mn(ii) ions from wastewater by using mangostanagarcinia peel-based granular-activated carbon.
Journal of Chemistry , 2013.[6] K.H. Chu. Fixed bed sorption: Setting the record straight on the BohartAdams and Thomasmodels.
Journal of Hazardous Materials , 177:1006–1012, 2010.[7] Marcela Andrea Espina de Franco, Cassandra Bonfante de Carvalho, Mariana Marques Bonetto,Rafael de Pelegrini Soares, and Liliana Amaral Feris. Removal of amoxicillin from water byadsorption onto activated carbon in batch process and fixed bed column: Kinetics, isotherms,experimental design and breakthrough curves modelling.
Journal of Cleaner Production , 161:947–956, 2017.[8] R Han, Y. Wang, X. Zhao, Y. Wang, F. Xie, J. Cheng, and M. Tang. Adsorption of methylene blueby phoenix tree leaf powder in a fixed-bed column: experiments and prediction of breakthroughcurves.
Desalination , 245:284–297, 2009.[9] Shuangjun Li, Shuai Deng, Li Zhao, Ruikai Zhao, Meng Lin, Yanping Du, and Yahui Lian.Mathematical modeling and numerical investigation of carbon capture by adsorption: Literaturereview and case study.
Applied Energy , 221:437 – 449, 2018.[10] Tim G. Myers, Francesc Font, and Matt G. Hennessy. Mathematical modelling of carbon capturein a packed column by adsorption.
Applied Energy , 278:115565, 2020.[11] H. Patel. Fixed-bed column adsorption study: a comprehensive review.
Applied Water Science ,9(45), 2016.[12] Fateme Rezaei and Paul Webley. Optimum structured adsorbents for gas separation processes.
Chemical Engineering Science , 64(24):5182 – 5191, 2009.[13] Ariful Islam Sarker, Adisorn Aroonwilas, and Amornvadee Veawab. Equilibrium and kineticbehaviour of CO2 adsorption onto zeolites, carbon molecular sieve and activated carbons.
EnergyProcedia , 114:2450 – 2459, 2017. 13th International Conference on Greenhouse Gas ControlTechnologies, GHGT-13, 14-18 November 2016, Lausanne, Switzerland.[14] Mohammad Saleh Shafeeyan, Wan Mohd Ashri Wan Daud, and Ahmad Shamiri. A review ofmathematical modeling of fixed-bed columns for carbon dioxide adsorption.
Chemical EngineeringResearch and Design , 92(5):961 – 988, 2014.[15] Mohammad Saleh Shafeeyan, Wan Mohd Ashri Wan Daud, Ahmad Shamiri, and Nasrin Aghamo-hammadi. Modeling of carbon dioxide adsorption onto ammonia-modified activated carbon: Ki-netic analysis and breakthrough behavior.
Energy & Fuels , 29(10):6565–6577, 2015.[16] Camelia Smaranda, Maria-Cristina Popescu, Dumitru Bulgariu, Teodor Malut, and MariaGavrilescua. Adsorption of organic pollutants onto a romanian soil: Column dynamics andtransport.
Proc. Safety and Environmental Protection , 108:108–120, 2017.[17] L.S. Tan, A.M. Shariff, K.K. Lau, and M.A. Bustam. Factors affecting co2 absorption efficiencyin packed column: A review.
J. Ind. and Engng Chem. , 18:18741883, 2012.[18] Zhe Xu, Jian-Quo Cai, and Bing-Cai Pan. Mathematically modeling fixed-bed adsorption inaqueous systems.
J Zhejiang Univ-Sci A (Appl Phys and Eng) , 14(3):155–176, 2013.2119] Y.H. Yoon and J.H. Nelson. Application of gas adsorption kinetics i. a theoretical model forrespirator cartridge service life.