Mathematical modelling of carbon capture in a packed column by adsorption
MMathematical modelling of carbon capture in a packed column byadsorption
T.G. Myers ∗ , F. Font ∗ , M. G. Hennessy † September 15, 2020
Abstract
A mathematical model of the process of carbon capture in a packed column by adsorptionis developed and analysed. First a detailed study is made of the governing equations. Due tothe complexity of the internal geometry it is standard practice to average these equations. Herethe averaging process is revisited. This shows that there exists a number of errors and someconfusion in the standard systems studied in the literature. These errors affect the parameterestimation, with consequences when the experimental set-up is modified or scaled-up. Assuming,as a first approximation, an isothermal model the gas concentration equation is solved numerically.Excellent agreement with data from a pressure swing adsorption experiment is demonstrated. Anew analytical solution (valid away from the inlet) is obtained. This provides explicit relations forquantities such as the amount of adsorbed gas, time of first breakthrough, total process time andwidth and speed of the reaction zone, showing how these depend on the operating conditions andmaterial parameters. The relations show clearly how to optimise the carbon capture process. Bycomparison with experimental data the analytical solution may also be used to calculate unknownsystem parameters.
Keywords:
Carbon capture; Pressure swing adsorption; Mathematical model; Adsorption
The issue of excessive amounts of carbon in the atmosphere, which is still being added to at an alarmingrate, and the resultant effect on the climate is well-documented. Consequently mankind must look toa range of solutions including a reduction in current production, removing existing carbon from theenvironment and safe storage or reuse (in a sensible way). The process of carbon capture falls withinthese measures and in this paper we will develop and examine a mathematical model for carbon captureby adsorption. The governing equations developed, although similar to previous works, contain thecorrect terms and coefficients. Further, we present a novel analytical solution which predicts how thegas concentration and amount of adsorbate vary along the column as functions of the experimentalconditions.The particular process of interest for this study involves a gas being forced through a columnpacked with an adsorbent material. This is a standard method and the literature abounds withexperimental, numerical and theoretical papers. Due to the complexity of the gas flow around theporous media mathematical models for the process typically involve cross-sectional averaged equationsfor the heat in the gas and the solid as well as an advection-diffusion equation for the gas concentration.The equations are linked through a term describing the adsorption rate. In the advection-diffusion ∗ Centre de Recerca Matem`atica, Campus de Bellaterra Edifici C, 08193 Bellaterra, Barcelona, Spain. † Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom a r X i v : . [ c ond - m a t . o t h e r] S e p quation this appears as a mass sink, while in the solid heat equation it appears as a source due tothe exothermic reaction. Various forms of this type of system may be found in the review papers[4, 13, 24], the modelling of fixed bed experiments [7] and the high feed rate experiments of [20].Isothermal models are investigated in the experimental and numerical studies of [1, 25], the numericalstudy of [19] and the studies on adsorption equilibrium and breakthrough [26, 31]. Similar modelsoccur in the literature to describe absorption and liquid (as opposed to gas) flow [30, 17].In many publications the averaged governing equations are immediately written down, referringthe reader to the book of Ruthven [22]. In the following we will start at an earlier point, using thestandard heat and advection-diffusion equations and then carrying out the averaging process. Throughthis method we are able to identify a number of errors and inconsistencies which have propagatedthrough the literature. We discuss these in detail later and also explain why despite the errors theystill permit good agreement with experiment.In the following section we will present the system of equations describing the cross-sectionalaveraged temperature and concentration in a circular column. In §
4, by non-dimensionalising theequations, we are able to identify dominant terms and also the scales describing the general featuresof the reaction. We then compare the model results with data from the pressure swing adsorptionexperiments of [25]. The final section concerns the derivation of an exact analytical solution, valid awayfrom the inlet. This solution has important consequences concerning our understanding of the physicalprocess. If all system parameters are known this solution permits the prediction of quantities such asthe first breakthrough time, width of the reaction zone and adsorbed mass. If certain parameters areunknown comparison with breakthrough data permits their determination: in this study using only afew data points we calculate the density of adsorbate and the adsorption rate.
The typical experimental set-up involves a circular cross-section column containing an adsorbing ma-terial which is placed inside an oven or furnace to regulate the temperature. Gas is passed throughthe column and the concentration measured at the outlet. A schematic of the experimental set-up ispresented in Fig. 1. In the following we will take data and parameter values from [25], which involvesa CO , N mixture passing through a bed of activated carbon. A full description of the experimentmay be found in [25]. Similar experiments are described in detail in the reviews of [4, 13, 15] and theexperimental study [7], for example. The only reason to choose the data of [25] is that they clearlystate all experimental conditions necessary to reproduce their results. The model should work equallywell with any other similar experiment.Here we summarise the assumptions that will be made during the model derivation:1. The amount of adsorbate is much less than existing solid, or fills up gaps in the porous media,so the bed void fraction, (cid:15) , remains constant.2. Heat released by adsorption is directly transferred to the solid.3. The adsorption saturation is constant during the process (this will be discussed later).4. The problem geometry is axisymmetric. Since the domain under study consists of a packed porous media it is complex and varies along thecolumn. To treat the governing equations in any sensible way they must be averaged hence it isstandard to apply a cross-sectional average. Not all cross-sections will have the same fraction of solid2 able 1: Nomenclature
Symbol DimensionSpecific heat capacity at constant pressure c J/ (kg K)Diffusion coefficient of component i D i m /sHeat transfer coefficient h W/ (m K)Concentration of component i C i mol/m Thermal conductivity λ W/(m K)Pressure p PaAmount of adsorbed solid q mol/kgInner column radius R mUniversal gas constant R g J/ (mol K)Gas temperature T KWall thickness t w mVelocity u = ( u ( x, r ) , w ( x, r )) m/sGas molar fraction y -Adsorption heat ∆ H J/molBed void fraction (cid:15) -Dynamic viscosity µ Pa sComposite temperature φ KSolid temperature θ KDensity ρ kg/m Subscripts
Ambient a -Gas g -Solid s -Adsorbed q -Wall w -3 as in Gas outCarbon adsorptionCross-sectionCO2 N2 Activated carbon material xr x = 0 x = L Figure 1: Illustration of the carbon capture by adsorption process. A gas containing CO (typically CO and N in standard experimental configurations) passes through a cylindrical column bed of adsorbent, which adsorbsCO molecules onto its surface or within its pores. to gas so strictly we are thinking in terms of an ensemble average, which would be a typical averageover a number of cross-sections.The average gas temperature is defined as (cid:15)πR (cid:101) T = 2 π (cid:90) R T r dr , ⇒ (cid:101) T = 2 (cid:15)R (cid:90) R T r dr , (1)where T is the temperature in the gas and (cid:15) the void fraction. In the same manner we average thesolid temperature, θ , concentration of gas, C , and the amount adsorbed, q , (cid:101) θ = 2(1 − (cid:15) ) R (cid:90) R θr dr , (cid:101) C = 2 (cid:15)R (cid:90) R Cr dr , (cid:101) q = 2(1 − (cid:15) ) R (cid:90) R qr dr . (2)Other parameters are defined in the Nomenclature table. In the T integral the integrand is only non-zero over an area (cid:15)πR of the cross-section. Similarly, the gas occupies the same region. The solid (andhence material adsorbed onto the solid) occupies an area (1 − (cid:15) ) πR . The omission of the radial angleindicates that we restrict our attention to an axisymmetric cross-sectional average. A more rigorousway to deal with the averaging through the column would be to follow an averaging process such asthat described in [9]. This requires introducing an indicator function χ , where χ = 1 corresponds tothe gas phase and χ = 0 the solid. We could then integrate the governing equations, multiplied by χ , over a given volume, leading to volume averaged quantities. In regions where χ switches valuesa law for the interchange between phases must be applied. In the Supplementary Material we applya specific form of this technique, that is we restrict our attention to an axisymmetric cross-sectionalaverage. The choice of axisymmetry avoids the need for a more complex mathematical analysis andcoincides with the equations studied in the carbon capture literature.4he averaging process leads to the following equations describing the heat flow,( ρc ) g (cid:32) ∂ (cid:101) T∂t + ∂ ( u (cid:101) T ) ∂x (cid:33) = λ g ∂ (cid:101) T∂x + 2 (cid:101) h gs (cid:15)R (cid:16)(cid:101) θ − (cid:101) T (cid:17) + 2 h wg (cid:15)R (cid:16) T w − (cid:101) T (cid:17) − p(cid:15) ∂u∂x , (3)( ρc ) s ∂ (cid:101) θ∂t = λ s ∂ (cid:101) θ∂x + 2 (cid:101) h gs (1 − (cid:15) ) R (cid:16) (cid:101) T − (cid:101) θ (cid:17) + 2 h ws (1 − (cid:15) ) R (cid:16) T w − (cid:101) θ (cid:17) + (cid:88) i ∆ H i ρ qi ∂ (cid:101) q i ∂t . (4)The parameter c is the specific heat capacity measured at constant pressure , this will be discussed later.The solid heat equation contains a source term due to the heat released as the gas is adsorbed.An important point to note here is that the rate of heat generation is proportional to ρ q ∆ H , where ρ q is the density of the adsorbed material (adsorbate), this is typically much lower than that of theadsorbing material (adsorbent) [2]. In fact the whole source term is an approximation, the correctform would link the heat flow to an extra energy balance describing the creation of new solid (a Stefancondition). This may be seen in studies of ablation, see [16] for example. To avoid solving the fullmoving boundary problem at each interface, further complicating the solution it is standard to addthe heat generated to the solid temperature. Heat exchange with the gas is then accounted for by aconvective boundary condition.When the gas and solid are taken to have different temperatures it is termed the ‘non-equilibriumthermal model’ [13]. If we set (cid:101) T = (cid:101) θ = (cid:101) φ then the ‘equilibrium thermal model’ is found. A singleequation for (cid:101) φ may be obtained by adding (cid:15) times equation (3) to (1 − (cid:15) ) times equation (4)( ρc ) p ∂ (cid:101) φ∂t + (cid:15) ( ρc ) g ∂ ( u (cid:101) φ ) ∂x = k p ∂ (cid:101) φ∂x + 2( h wg + h ws ) R ( T w − (cid:101) φ ) + (1 − (cid:15) ) (cid:88) i ∆ H i ρ qi ∂ (cid:101) q i ∂t − p ∂u∂x , (5)where the subscript p denotes the porous medium ( z p = (cid:15)z g + (1 − (cid:15) ) z s ).The average concentration, (cid:101) C i ( x, t ), is described by ∂ (cid:101) C i ∂t + ∂ ( u (cid:101) C i ) ∂x = D i ∂ (cid:101) C i ∂x − − (cid:15)(cid:15) ρ qi ∂ (cid:101) q i ∂t . (6)The coefficients D i represent an ’effective axial dispersion coefficient’ which lumps all mechanismscontributing to axial mixing (e.g. molecular diffusion or turbulent mixing as flow passes round particlesand recombines) [22, P. 208, 209]. Since it is dominated by the flow if there is more than one componentto the gas it may be assumed to have the same value for each component. Note, this equation onlyholds where C i >
0, i.e. for x ∈ [0 , s i ( t )] where x = s i ( t ) denotes the boundary beyond which theconcentration is zero. Since u ( x ) represents the velocity at which energy is being advected by the gasit must be interpreted as the interstitial velocity (rather than the superficial velocity). If the total gasflux at the column entrance is Q then u (0) = Q / ( (cid:15)πR ). At the inlet the gas has concentration C i of component i , just inside the column it moves with velocity u and is adsorbed by the porous media. The velocity just outside the column will be the same as thatjust inside, and occupying the same area (any gas above a solid section will have zero velocity and sonot contribute to the gas flux). Hence we apply a Dankwert condition to the average concentration uC i = (cid:32) u (cid:101) C i − D i ∂ (cid:101) C i ∂x (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 . (7)5t the outlet x = L a similar balance holds, however it is standard to assume that (cid:101) C i ( L − , t ) ≈ C i ( L + , t )and so ∂ (cid:101) C i ( L − , t ) ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = L = 0 . (8)However, we note that before breakthrough not all gas components reach the outlet. In this case weimpose ∂ (cid:101) C i ( s i , t ) ∂x = (cid:101) C i ( s i , t ) = 0 , (9)where x = s i ( t ) is the point where the i -th gas concentration reaches zero. The condition (8) has beenreplaced by two conditions. The additional condition is necessary to determine the new unknown s ( t ).For the temperature we may carry out a similar balance, in this case for the energy flux u ( ρc ) g T a = (cid:32) u ( ρc ) g (cid:101) T − λ g ∂ (cid:101) T∂x (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 . (10)As above, at the outlet we impose ∂ (cid:101) T∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 = 0 . (11)Similar conditions hold in the solid. In experiments where the bed has been saturated by N the initialcondition for N is approximated with an ideal gas law (cid:101) C i ( x,
0) = pR g (cid:101) T , (cid:101) q i ( x,
0) = ˜ q ∗ i . (12)The CO initial condition for N saturated and unsaturated scenarios should be (cid:101) C i ( x,
0) = (cid:101) q i ( x,
0) = 0 . (13) There exist a variety of formulae to describe the adsorption rate, but possibly the most common is alinear kinetic relation, ∂ (cid:101) q i ∂t = k i (˜ q ∗ i − (cid:101) q i ) , (14)where (cid:101) q i is the averaged concentration within the adsorbing particle. This form is termed a LinearDriving Force model in the literature. Other forms such as the second order, Avrami and porediffusion relations are discussed in [13, 25]. The premise of the LDF is that the rate of change of aspecies is directly proportional to the difference between its saturation concentration and the meanconcentration within the particle. The model is very simple and contains only two parameters, k i is arate constant and ˜ q ∗ i is the maximum possible value for the average concentration. This latter valuediffers from values measured at a surface due to the averaging process ˜ q ∗ i = 2˜ q ∗ surf / ((1 − (cid:15) ) R ). Anumber of isotherm models exist for its estimations such as Langmuir [13], Langmuir-Freundlich [29]and Toth [25]. Since we take experimental data from [25], we will use their formula which takes theform ˜ q ∗ i = q c + q p , which incorporates chemical and physical adsorption mechanisms. Both components6ave the same form, which is identical to the Toth isotherm, but with different exponents. Details areprovided in A.An important point to note is that throughout the derivation we have written ∂ (cid:101) q i /∂t rather than d (cid:101) q i /dt as in all previous works. We use the partial derivative since (cid:101) q i also depends on position. Thisbecomes clear if we consider two points within the domain, x , x where x > x . The column at x has had more time to adsorb the gas and so (cid:101) q i ( x , t ) > (cid:101) q i ( x , t ). If we take the specific example with x = 0 and x = s , then at any time t > (cid:101) q i (0 , t ) > (cid:101) q i ( s, t ) = 0: (cid:101) q i is therefore dependent onposition. The x dependence is also apparent from the numerical results presented later.The average fluid velocity in a packed bed may be expressed by the Ergun equation − ∂p∂x ≈ ∆ pL = 150 µ g d p (1 − (cid:15) ) (cid:15) u + 1 . ρ g d p (1 − (cid:15) ) (cid:15) u . (15)This relation is often quoted in terms of the superficial velocity v s = (cid:15)u and so has slightly differentfactors. The terms on the right hand side represent pressure loss due to viscous and kinetic termsrespectively. The Ergun equation can deal with both turbulent and laminar flows. If the second termof the right hand side is neglected then the Carman-Kozeny equation for laminar flow is retrieved.For pressures up to a few atmospheres an ideal gas law may be employed (cid:101) C = pR g T , (16)and then (cid:101) C i = y i (cid:101) C where (cid:101) C = (cid:80) i (cid:101) C i and y i is the volume fraction. The system presented above, consisting of the governing equations (3, 4, 6) and the boundary condi-tions of § ρ q ∂ (cid:101) q/∂t . The density ρ q is that of the adsorbatewhich is generally significantly less than that of the adsorbent ρ q < ρ s [2]. The energy releaseassociated with this process results in a source term proportional to the mass sink, ∆ Hρ q ∂ (cid:101) q/∂t . Itis common practice to represent the sink by ρ s d (cid:101) q/dt . Given that ρ s > ρ q employing ρ s could predicta significantly faster reduction in the gas concentration than in practice and a higher temperatureincrease. In [4, 13, 24, 7, 25, 26, 8, 27] the solid particle density, ρ s is used in both the concentrationand heat equations. In [5, 18, 21] the bulk density is employed. However, in [22], the oft quoted sourceof the governing equations, the variable considered is in fact the product of density and adsorbate, soavoiding the issue. In § ρ q ≈
325 kg/m , whichis approximately one sixth of the particle density ρ s = 1818 kg/m .The standard heat equation quoted in studies such as [4, 13, 24, 7, 29, 8] neglects heat conductionin the solid. In fact the standard derivation of this heat equation stems from a basic energy balancefor a single isolated particle, see [22], which is then assumed to hold throughout the column. Due tothe simplicity of the balance, heat transfer between the solid and wall is neglected. For the gas, heatconduction and transfer at the wall is always included. In [11] values for the thermal conductivityof various forms and porosities of activated carbon are given, typically λ s is of the order 0.4 W/mK. For gases the thermal conductivity is typically an order of magnitude lower, λ g ∼ . c is often simply referred to as the heat capacity, rather than the valuemeasured at constant pressure or constant volume. For a solid the two values are virtually identical:the difference depends on the thermal expansion. For a compressible gas the difference is large. Forpressure swing adsorption experiments the pressure is approximately constant at any given pointduring the adsorption phase, hence the value at constant pressure should be employed. In certainworks both forms, constant volume and constant pressure, are employed, one multiplying the timederivative the other the advection term, see [7, 8, 3] for example, this is incorrect.The pressure-work term proportional to ∂u/∂x is neglected in most studies, this is consistent withthe assumption that mass loss is small compared to the total flow of mass. In the results section we willalso apply this approximation. However, if this term is included in the model then to be consistentthe velocity must also be retained inside the advection derivative, i.e. in the terms ( uC ) x , ( uT ) x .For example in [18, Supplementary Material] the gas energy equation is based on constant u forthe advection term but varying u for the pressure work. In many examples around 20% of thegas is removed, calling the assumption on small mass loss into question. However, it is clearly anunderstandable first approximation for preliminary studies, provided the equations are written in aconsistent manner.A final issue is that a number of authors write down equations involving radial flow, heat exchangeat the boundaries (both at the wall and between the gas and solid) and mass loss from the gasconcentration, see [13, Eqs. (19)-(20), (25)-(27)], [15, Eq. (1)]. The inclusion of the heat exchangeterms in the governing equations (as opposed to in the boundary conditions) and the mass sink termin the concentration equation is a direct result of the cross-sectional averaging, which eliminatesradial variation: the presence of both radial derivatives and boundary exchange terms is thereforeincompatible and should not be used.A number of these issues will be discussed in more detail later. Non-dimensionalisation often allows a system of equations to be simplified, particularly by identifyingdominant and negligible terms. It can also determine the controlling parameters as well as time andlength-scales for the process. To correctly carry out the non-dimensionalisation we must focus on aspecific example, with realistic parameter values. Hence from now on we will focus on a two componentsystem composed of CO and N , following the experimental work of [25]. Specifically we will use datafrom the experiments of CO adsorption on material ‘OXO-GAC’ at 303.15K, appropriate parametervalues are listed in Table 2. From now on we will use C to denote the CO concentration. Forsimplicity we will also assume u to be constant, i.e. the rate of mass removal is small compared tothe total mass flow. The equilibrium saturation q ∗ is calculated from [25, eq. (7)], which accounts forboth physical and chemical adsorption mechanisms, details are provided in A. The interstitial velocityis u = Q/ ( (cid:15)πR ). The apparent density, ρ a = 800 kg/m , quoted in [25] indicates a particle density ρ s ≈ ρ a / (1 − (cid:15) ) ≈ . This density is used in their mass sink term. In our calculations weemploy ρ q = 325 kg/m . Instead of their estimate k = 0 . − , we use k = 0 . − . Thesevalues will be discussed later. Most authors calculate the axial diffusion through a formula involvingthe molecular diffusivity, Schmidt number and Reynolds number, see [7, 25] for example. In [7] thisleads to values ranging between 10 − to 10 − . Here we simply use a value taken from [6, Table 16.2-2]and adjust due to the porous nature of the solid, D = 1 . × − /(cid:15) , this will also be discussed later.We scale concentration with the initial value of CO , the amount of adsorbent will then be scaledwith ˜ q ∗ (which for the present study is assumed to be constant). Length and time-scales will be left8ymbol Value DimensionInitial concentration (CO ) C Adsorption saturation ˜ q ∗ ε ρ q
325 kg/m Axial diffusion coefficient D . × − m /sVolume fraction (CO ) y L p Q . × − m /sBed radius R q ∗ calculation) T u ) k − Specific heat capacity of solid/gas c s /c g ρ s /ρ g Thermal conductivity λ s /λ g H . × J/kg
Table 2: Values of the thermophysical parameters mainly taken from [25], except k , ρ q , solid thermal propertiesare taken from [11, 28, 12]. Gas thermal properties are those of air at 300K and 1 atmosphere. unspecified for the moment (cid:98) C = (cid:101) CC , ˆ x = x L , (cid:98) t = t ∆ t , (cid:98) q = (cid:101) q ˜ q ∗ . (17)The CO concentration is then governed by C ∆ t ∂ (cid:98) C∂ (cid:98) t + uC L ∂ (cid:98) C∂ (cid:98) x = DC L ∂ (cid:98) C∂ (cid:98) x − − (cid:15)(cid:15) ρ q ˜ q ∗ ∆ t ∂ (cid:98) q∂ (cid:98) t , (18) ∂ (cid:98) q∂ (cid:98) t = k ∆ t (1 − (cid:98) q ) . (19)Much information may be gained from these equations. For example, the second equation indicatesthat CO adsorption occurs on a time-scale ∆ t = 1 /k = 1 / . ≈
73 s. So significant changes occurover time-scales of approximately 1 minute. Since ρ q ˜ q ∗ (cid:29) C it is clear that the time derivative ofthe concentration is negligible compared to the sink term and therefore may, in general, be neglected.Except for when the interstitial velocity is extremely low, u ∼ D/ L , advection dominates over diffusionhence we expect the advection term to balance with the sink term, this necessitates the choice oflength-scale uC L = 1 − (cid:15)(cid:15) ρ q ˜ q ∗ ∆ t ⇒ L = (cid:15)uC (1 − (cid:15) ) k ρ q ˜ q ∗ . (20)Taking the values of Table 2 gives L ≈ adsorption onsilica gels). We can now see that diffusion becomes important for an interstitial velocity of the order D/ L ∼ . u (cid:29) D/ L .9sing the length-scale L and time-scale ∆ t = 1 /k we obtain δ ∂ (cid:98) C∂ (cid:98) t + ∂ (cid:98) C∂ (cid:98) x = P e − ∂ (cid:98) C∂ (cid:98) x − ∂ (cid:98) q∂ (cid:98) t , (21) ∂ (cid:98) q∂ (cid:98) t = (1 − (cid:98) q ) , (22)where P e − = D/ ( L u ) is the inverse Peclet number and δ = L /u ∆ t = k L /u . Here we find P e − ≈ .
066 and δ = 0 . P e − indicates that, in this case, diffusion plays a minorrole in the process (as discussed earlier). Consequently a detailed calculation of the value of D isunnecessary. Neglecting this term would lead to errors of the order 7%, however, retaining it may beuseful for a numerical scheme if sharp concentration gradients occur such as at small times. The term δ is even smaller, its neglect will lead to errors of the order 1%.Under the standard assumption (cid:98) q = (cid:98) q ( t ) the integration of (22) is simple and leads to (cid:98) q = 1 − e − (cid:98) t ,as quoted in [25, 23]. This satisfies the initial condition (cid:98) q ( (cid:98) t = 0) = 0. However, as discussed earlierthere is an obvious (cid:98) x dependence in the adsorption term. The issue is easily resolved. If we denotethe time for the gas to reach a given point (cid:98) x as (cid:98) t = (cid:98) t s ( (cid:98) x ) then the correct initial condition at thatpoint is (cid:98) q ( (cid:98) x, (cid:98) t s ( (cid:98) x )) = 0. Consequently the correct solution to the adsorption equation is (cid:98) q ( (cid:98) x, (cid:98) t ) = 1 − e − ( (cid:98) t − (cid:98) t s ) , (23)which holds over the domain where (cid:98) C >
0, i.e. (cid:98) x ≤ (cid:98) s ( (cid:98) t ). The time (cid:98) t s could for example be foundby matching to experiment (or the numerical solution), i.e. if the time of first breakthrough, (cid:98) t fb , isrecorded then (cid:98) t s = (cid:98) t fb at (cid:98) x = (cid:98) L . The evaluation of (cid:98) t s is discussed in § concentration equation holds over (cid:98) x ∈ [0 , (cid:98) s ( (cid:98) t )] and is to be solvedsubject to 1 = (cid:32) (cid:98) C − P e − ∂ (cid:98) C∂ (cid:98) x (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) x =0 , (24) ∂ (cid:98) C∂ (cid:98) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) x = (cid:98) s = (cid:98) C ( (cid:98) s, (cid:98) t ) = 0 . (25)To investigate the temperature we must define pressure and temperature scales (cid:98) T = (cid:101) T − T a ∆ T , (cid:98) θ = (cid:101) θ − T a ∆ T , (cid:98) p = p − p a ∆ p , (26)where T a , p a are the ambient temperature and pressure. The scale ∆ T is, as yet, unspecified while∆ p represents the pressure drop along the column. Since the rise in temperature is clearly due tothe source term in the solid heat equation we choose ∆ T = ∆ Hρ q ˜ q ∗ / (( ρc ) s ) ≈ T /T (cid:28) §
6. With this choice of ∆ T the solidtemperature is described by ∂ (cid:98) θ∂ (cid:98) t = δ ∂ (cid:98) θ∂ (cid:98) x + F (cid:101) h gs (cid:16) (cid:98) T − (cid:98) θ (cid:17) + F h ws (cid:16) (cid:98) T w − (cid:98) θ (cid:17) + ∂ (cid:98) q∂ (cid:98) t , (27)where δ = λ s ∆ t L ( ρc ) s ≈ . , F = 2∆ t (1 − (cid:15) ) R ( ρc ) s ≈ . . δ ∂ (cid:98) T∂ (cid:98) t + ∂ (cid:98) T∂ (cid:98) x = P e − g ∂ (cid:98) T∂ (cid:98) x + F (cid:101) h gs (cid:16)(cid:98) θ − (cid:98) T (cid:17) + F h wg (cid:16) (cid:98) T w − (cid:98) T (cid:17) , (28)where P e − g = λ g u L ( ρc ) g ≈ . , F = 2 L (cid:15)Ru ( ρc ) g ≈ . . (29)For further details see the Supplementary Information.From the non-dimensional form of the heat equations we may obtain a good understanding ofthe physical process. The solid heat equation shows that the solid temperature increases in time dueto the heat generated by adsorption, this is offset by heat loss due to transfer to the walls and gas.The heat transfer coefficient between a slow moving gas and solid is of the order 10 W/m K, hence F (cid:101) h gs ≈ F h ws ≈ . δ ≈ .
04 we see that conduction through the solid has a minor effect.Since δ (cid:28) P e − g ≈ δ , hencediffusion in the solid is just as important as in the gas, so indicating that its neglect in standard modelsis inconsistent: if diffusion is neglected in the solid, for consistency, it should also be neglected in thegas . The coefficient F is an order of magnitude larger than the solid counterpart F which indicatesthat the gas temperature primarily varies due to advection and exchange with the solid and walls.The large difference between F and F shows that the solid plays an important role in giving heat tothe gas, while the gas has a smaller effect on the solid temperature, this is a consequence of the factthat the volumetric heat capacity of the gas is significantly lower than that of the solid, ( ρc ) g (cid:28) ( ρc ) s . For the present study we will investigate CO adsorption under an isothermal assumption. Con-sequently our numerical scheme only requires equations to describe the CO concentration and theamount adsorbed. The validity of this approximation will be discussed in the results section.At time (cid:98) t = 0 the concentration of gas to be adsorbed is zero everywhere inside the column, (cid:98) C ( (cid:98) x,
0) = 0. At the boundary (cid:98) x = 0, according to equation (24), (cid:98) C (0 , (cid:98) t ) is close to unity (since P e − (cid:28) In the Supplementary Information we demonstrate how, by changing to a short time-scale we are ableto determine approximate solutions for both the concentration and amount of adsorbate: (cid:98) q = (cid:98) t − (cid:98) t e , (cid:98) s = 1 , (30) (cid:98) C = 1 − P e − + P e − e − P e (1 − (cid:98) x ) − (cid:98) x . (31)These hold for small times such that (cid:98) t ≥ (cid:98) t e , where δ (cid:28) (cid:98) t e (cid:28)
1. We make the standard choice (cid:98) t e = √ δ which lies in the required range (in the present study this correspond to times of the order9s). Interestingly, we see that the amount of adsorbate is approximately independent of x at small11imes. This independence is lost as time progresses. In practice we start the numerical scheme at time (cid:98) t = (cid:98) t e = √ δ , with the profiles defined by (30, 31).The conclusion that to leading order, for sufficiently small time (cid:98) s = 1 is a constant may appearstrange (and that (cid:98) C is independent of time). It represents the distance that the incoming gas travelsbefore being completely used up. Since for small times the adsorbent is relatively fresh the gas canonly travel a fixed distance before being completely adsorbed. This will continue until the amountof adsorbate becomes significant and so the adsorbent is less efficient at removing gas. The achievednon-dimensional value (cid:98) s = 1 indicates that the length-scale is well-chosen and that it can also representthe distant over which the concentrate travels over fresh adsorbent before being completely used up.An even shorter time-scale is possible, where the gas first enters the column and has not yet reached (cid:98) s = 1. For numerical purposes (30, 31) are sufficient. t > t e For most of the process the problem is a free boundary problem and adsorption only occurs in thegrowing region (cid:98) x ∈ [0 , (cid:98) s ( (cid:98) t )] or, in other words, in the region where (cid:98) C ( (cid:98) x, (cid:98) t ) is strictly larger than 0.To overcome the numerical difficulty of solving our equations in a growing domain, we rewrite theproblem in the form δ ∂ (cid:98) C∂ (cid:98) t + ∂ (cid:98) C∂ (cid:98) x = P e − ∂ (cid:98) C∂ (cid:98) x − ∂ (cid:98) q∂ (cid:98) t , (32) ∂ (cid:98) q∂ (cid:98) t = (1 − (cid:98) q ) H ( (cid:98) C ) , (33)where H ( (cid:98) C ) represents the Heaviside function H ( (cid:98) C ) = (cid:40) (cid:98) C > , . (34)The Heaviside function ensures that (cid:98) q ( (cid:98) x, (cid:98) t ) only increases in the regions where CO is present. Note,whilst we pointed out earlier that diffusion (also conduction) is small we retain it here firstly to allowus to verify this statement and secondly since it can help smooth the rapid change in concentrationdue to the initial condition. Although the use of a small time solution removes the actual jumpdiscontinuity.Because we have effectively removed the moving boundary (cid:98) x = s ( (cid:98) t ), we can modify the boundarycondition (25) to read ∂ (cid:98) C∂ (cid:98) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) x = (cid:98) L = 0 . (35)We then use standard second-order central finite differences in space and explicit Euler in time. Wealso employ one-sided second-order finite differences to discretise the derivatives in the boundaryconditions, thereby ensuring that the solution is overall second-order in space. The numerical schemeis implemented in Matlab. We choose a time step ∆ (cid:98) t and a mesh size ∆ (cid:98) x that allow the stabilitycondition P e − ∆ (cid:98) t/δ ∆ (cid:98) x < . (cid:98) t and ∆ (cid:98) x to ensurethat the solution converges. The simulations shown in the present study correspond to ∆ (cid:98) x = 0 .
02 and∆ (cid:98) t = 2 · − . 12 igure 2: Variation of CO concentration with time at the inlet (solid), middle (dash) and outlet (dash-dot) ofthe column.Figure 3: Variation of (cid:101) q with time at the inlet (solid line), middle (dashed line) and outlet (dash-dotted line)of the column. The circles represent the analytical expression for (cid:101) q of equation (36). For ease of interpretation we present all results in dimensional form. The concentration is plotted as (cid:101) C/ (cid:101) C where (cid:101) C = 6 .
03 mol/m , the adsorbate is presented as (cid:101) q/ ˜ q ∗ with ˜ q ∗ = 1 .
57 mol/kg.In Figure 2 we present the variation of CO concentration with time at the inlet, middle of thecolumn and outlet. Parameter values are all given in Table 2. Referring to the solid line we see that atthe inlet most of the CO passes straight through and after the first few minutes the gas passes throughunchanged. The dashed line shows that CO reaches the centre of the column after approximately t = 4 .
83 minutes, at t = 7 .
73 minutes it is at 90% of the inlet value. The same pattern is repeated atthe end of the column,the dash-dot curve, the CO reaches the end after t = 10 .
77 minutes and is at90% of the inlet value at 13.57 minutes.Figure 3 shows the variation of (cid:101) q at the inlet (solid), middle (dashed) of the column and outlet(dash-dot). At the inlet (cid:101) q starts at zero and after 2.95 minutes reaches 90% of the equilibrium value.This is repeated at the centre and exit of the column, gas first reaches the centre at 4 .
89 minutes andthe end at 10 .
92 minutes and is at 90% at 7.72 and 13.72 minutes respectively. The behaviour at the13 igure 4: The variation of concentration with x at times t = 12 . , . , , , ,
825 s. centre and exit is very similar to that exhibited by the concentration. In each case the time betweenfirst reaching a point and being at 90% of the original value is approximately 2.95 minutes, suggestingthat the front moves with a constant speed. The circles plotted on the figure, show the analyticalsolution (see eq. (23)) (cid:101) q = ˜ q ∗ (1 − e − k ( t − t s ) ) . (36)Here we have chosen the value t s = 4 . ×
60 seconds, so that we may compare the analytical andnumerical solutions at the centre of the column. Obviously this is virtually indistinguishable from thedashed line, thus verifying the numerical solution.Figure 4 shows the concentration along the channel at times t = 12.8, 45.6, 155, 374, 648, 825 s.At very early times, t = 12 . , . t = 374 ,
648 s,we see the concentration wave takes on a self-similar form (the drop from 90% of C to zero occursover a length 4.7 cm on both curves). This suggests that beyond the initial transient the concentrationhas a travelling wave form. The final time shown, t = 825 s is when the concentration at the outletis 90% of C , everywhere else it is above this value and we may consider the process to be effectivelycomplete.Figure 5 shows the adsorption curves corresponding to the concentration of the previous figure. In § (cid:101) q is independent of x and grows with time, (cid:101) q/ ˜ q ∗ ≈ k ( t − t s ). Thisis quite clear from the results for times t =12.8, 45.6s: the curve is approximately flat, followed by asharp drop to zero. Once (cid:101) q/ ˜ q ∗ ≈ (cid:101) q wave then starts to move in a self-similar form(dropping from 90% of ˜ q ∗ to zero over the same region as the concentration).A feature apparent from Figures 4, 5 is that beyond the initial transient the concentration andadsorption curves are hard to distinguish. We will explain this in the following section.A quantity shown in virtually all experimental papers is the gas concentration at the outlet, thebreakthrough curve, which corresponds to the final curve of Figure 2. In Figure 6 we compare thenumerical prediction with experimental data taken from [25]. In general the agreement is excellent.The experimental data indicates a slightly earlier breakthrough with a brief slow rise in CO concen-tration, followed by a sharper rise. This discrepancy may be attributed to the kinetic relation for ˜ q .Any model where (cid:101) q t ∝ ˜ q ∗ − (cid:101) q (where ˜ q ∗ is constant) will have the highest gradient in the adsorptioncurve where the gas first meets fresh adsorbent, that is, where (cid:101) q = 0. As observed from the numericalresults beyond the initial transient the concentration behaviour is almost identical to the adsorbate,so the highest concentration gradient will also be at the moving front.14 igure 5: The variation of (cid:101) q with x at times t = 12 . , . , , , ,
825 s.Figure 6: Comparison of CO concentration at breakthrough from the present isothermal model prediction(solid line) and experimental values taken from [25] (circles), for the adsorption of CO on activated carbon. igure 7: Percentage variation in the equilibrium saturation for a temperature rise of 7 K and 2 K (solid anddashed lines, respectively). In § D and how instead of using this we favoured asimple estimate from Bird et al . [6]. The non-dimensionalisation shows that gas diffusion is describedby the inverse Peclet number, which we determined to be small. To verify its small contribution wecarried out simulations changing D by factors 0.1 and 10. This had an effect on the concentrationnear the inlet (since P e − ∝ D also enters the boundary condition there), however it diminishesrapidly as the gas moves down the pipe. For the case of 0 . D there was no noticeable change tothe breakthrough curve, with 10 D the time for first breakthrough increased by less than 1% but theform remained the same. This backs up our assertion that it is not necessary to carry out detailedcalculations to determine D .In the non-dimensionalisation process we demonstrated a temperature rise of the order 7 K. Thiswill have a small effect on the material properties, however, the equilibrium adsorption has a strongtemperature dependence, see A. For this reason we now examine the effect of temperature on thevalue of ˜ q ∗ . In Figure 7 we show the percentage change 100(˜ q ∗ ( T ) − ˜ q ∗ ( T + 7)) / ˜ q ∗ ( T ) and 100(˜ q ∗ ( T ) − ˜ q ∗ ( T + 2)) / ˜ q ∗ ( T ). The solid line corresponds to the higher temperature rise and has a maximum justbelow 23%. The maximum of the dashed line, which corresponds to the 2 K rise, is around 7%. Forthe current problem we work at 303 K, which will result in an approximately 22% maximum differencein ˜ q ∗ when compared to an isothermal model. Since the heat is quickly dissipated we would expectthis to decrease rapidly. A second source of error is the mass loss, here of the order 15%, which impliesa reduction in velocity ahead of the front of the order 15%. Since this is outside the reaction zonethe velocity there does not affect the calculations. Behind the front the mass loss will vary between0 and 15% over a length-scale of around 2cm. An average of 7.5% mass loss occurring over 10% ofthe column suggests an order of magnitude for the error incurred of around 0.75%. Similarly thetemperature variation should lead to small errors, consequently we do not expect either effect to besignificant in the analysis but they are certainly worth investigating in future studies. The numerical results show that once the initial transient is passed the concentration and adsorptionmove with a self-similar form, this observation coupled with the constant speed of the front suggeststhat the reaction may be described by a travelling wave. In this section we will derive analyticalexpressions for the shape of the concentration and adsorption curves. These analytical solutions16re extremely important to our physical understanding of the process. They provide an accuratedescription of the wave forms, showing clearly the factors affecting the reaction. At the outlet thesolution characterises the breakthrough curve. Comparison with experimental results then permits usto estimate quantities such as the adsorption rate constant k , the speed and width of the reactionzone and density of adsorbate.If we consider only the CO equation and define a new variable (cid:98) η = (cid:98) x − (cid:98) s ( (cid:98) t ), which has its originat the reaction front, then the analysis provided in the Supplementary Material leads to the followingexpressions for the concentration and adsorption (cid:101) C ( x, t ) ≈ C (cid:16) − αe (( x − s ) / (cid:98) v L− k t ) (cid:17) , (37)˜ q ( x, t ) = ˜ q ∗ (cid:16) − e (( x − s ) / ˆ v L− k t ) (cid:17) , (38)where (cid:98) v = 1 / (1 + δ ) is constant and α = (cid:98) v P e (cid:98) v P e − (cid:18) (cid:98) v P e + 1( (cid:98) v P e ) · · · (cid:19) . (39)The first part of the equality is the exact expression for α , the second shows how it may be approx-imated given that ˆ v P e ≈ (cid:29)
1. If all terms involving (cid:98) v P e are neglected in the series then theconcentration expression is identical to the adsorption (with errors of the order 7%). Retaining thefirst order term will give errors of the order 0.04%. So the small difference between the form of theconcentration curves and adsorption, discussed in §
6, can now be seen as an effect of diffusion in thegas.The position of the front of the concentration wave is s ( t ) = (cid:98) v L k t + s = (cid:15)uC (1 − (cid:15) ) ρ q ˜ q ∗ + (cid:15)C t + s , (40)where s is a constant to be determined. Since measurements are primarily taken at the outlet we may use the above expressions to characterisebreakthrough. If we denote Λ = exp((( L − s ) / ˆ v L ) then the concentration and amount of adsorbateat the outlet are described by (cid:101) C ( L, t ) ≈ C (cid:16) − α Λ e − k t (cid:17) , ˜ q ( L, t ) = ˜ q ∗ (cid:16) − Λ e − k t (cid:17) . (41)Note, in both the adsorption and absorption literature there exist a number of models to predict theshape of the breakthrough curve based on assumptions regarding the rate of decline of CO at theexit, typically that it is proportional to the amount of CO remaining in the gas. These lead to aclassic logistic type equation which integrates to an exponential relation between the concentrationand time. For example the Thomas and Yoon-Nelson models take identical forms (cid:101) C ( L, t ) = C Ae − k t , (42)with different definitions for the constants, see [14, 10]. Assuming the exponential term to be small,the first order Taylor series leads to the form of equation (41a). Below we will see that Λ ∼ A is largehence these earlier models can provide a good approximation to the later stages of breakthrough (when Ae − k t (cid:28) s .We may use the breakthrough curve to determine the values of k , Λ without worrying about s . Consider two points on the experimental breakthrough curve (here we choose them at eitherside of the central region). From Figure 6 we find (cid:101) C ( L, t ) = 0 . C when t = 60 × .
16 s and (cid:101) C ( L, t ) = 0 . C at t = 60 × .
17 s. This allows us to write down (cid:101) C ( L, t ) C = 0 .
164 = 1 − α Λ e − k t , (cid:101) C ( L, t ) C = 0 .
838 = 1 − α Λ e − k t . (43)Eliminating Λ between the two equations leads to k = 1 t − t ln (cid:32) C − (cid:101) C ( L, t ) C − (cid:101) C ( L, t ) (cid:33) ≈ . s − . (44)This process was repeated using a number of data points within the breakthrough curve, to determinean average k = 0 . − . This value is within the typical range reported in [25] and has been usedin our numerical calculations, see Table 2. Taking this value of k the unknown Λ may be found,Λ = (cid:32) − (cid:101) C ( L, t ) C (cid:33) e k t = (cid:32) − (cid:101) C ( L, t ) C (cid:33) e k t ≈ . (45)The concentration and amount of adsorbate at the outlet may therefore be determined without knowl-edge of s .In § (cid:101) q ( x, t ) = ˜ q ∗ (cid:16) − e − k ( t − t s ( L )) (cid:17) . (46)Comparison with equation (41)) shows that the travelling wave reproduces the exact solution.The unknown s may be determined in a number of ways, for example we could look at the timeof first breakthrough predicted by the numerical solution, in this case t fb ≈ .
75 minutes. Then,according to (40), we obtain s = L − (cid:98) v L k t fb ≈ . s ≈ L − L ln Λ ≈ . s = 0 . t fb can be written in terms of the system parameters by setting s = L in equation (40) t fb = L − s ˆ v L k = (1 − (cid:15) ) ρ q ˜ q ∗ + (cid:15)C (cid:15)uC ( L − s ) ≈ (1 − (cid:15) ) ρ q ˜ q ∗ (cid:15)uC ( L − s ) . (47)The term (1 − (cid:15) ) ρ q ˜ q ∗ is related to the mass of carbon adsorbed during the experiment, whereas theterm (cid:15)C represents the mass of gaseous CO in the column. This second term is about 1% of thefirst and so may be neglected. For standard operating conditions, s (cid:28) L , so we may also neglect s .Equation (47) then makes clear all the factors which affect the breakthrough time and, importantly,it is independent of k . 18 igure 8: Evolution of the point, ˜ s ( t ), where the concentration of CO reaches zero. If we define the process time as being when the outlet concentration is 90% of its inlet value thenby setting (cid:101)
C/C = 0 . t = t fb + 1 k log(10 α ) , (48)where we have made use of the fact log Λ = ( L − s ) / ( (cid:98) v L ) = k t fb to reduce this expression. The finalterm represents the time taken for the width of the wave to pass through the outlet.Previously we have discussed the choice of the adsorbate density. If we obtain a breakthrough timefrom the experimental data then we may use this to calculate ρ q = (cid:15)C (1 − (cid:15) )˜ q ∗ (cid:20) ut fb L − s − (cid:21) ≈ (cid:15)C (1 − (cid:15) )˜ q ∗ (cid:18) ut fb L − s (cid:19) . (49)The density of adsorbate is discussed in [2], between 1 and 5 bar it varies between approximately 200and 400 kg/m . The data of [25] shows that breakthrough occurs somewhere between 10.17 and 10.49minutes. Taking the average t fb = 60 × .
33 s equation (49) indicates ρ q = 311 . . Given thatthere are a number of approximations involved in this work as well as experimental uncertainty (suchas not knowing the exact time of first breakthrough) we take this value as an initial guess. In Figure6 we adjust it by 4.5%, ρ q = 325 kg/m , to obtain the good agreement with the breakthrough curve.This is the only fitting of our model.Of course there are other ways to calculate the density of the adsorbate. For example, the mass ofadsorbed carbon can easily be measured from the experiment by comparing the starting mass of thecolumn with that at the end. The mass flux of CO entering the column is M CO C u kg/m s (where M CO = 44 . × − kg/mol is the molar mass of CO ) this occurs over an area (cid:15)πR m . Hence themass of CO in the column at any time may be expressed as M ex = (cid:40) M CO u(cid:15)πR C t t ≤ t fb M CO u(cid:15)πR (cid:16) C t fb + (cid:82) tt fb C − (cid:101) C ( L, t ) dt (cid:17) t ≥ t fb . (50)The final integral is easily evaluated to give M ex = M CO u(cid:15)πR C (cid:20) t fb + α Λ k (exp( − k t fb ) − exp( − k t )) (cid:21) t ≥ t fb , (51)19here t fb is given by equation (47). These formulae demonstrate that before first breakthrough theincrease in mass is linear in time, afterwards there is an exponential decrease in the rate that mass isadded. In the limit of large time, t → ∞ , this expression reduces to M lim → M CO u(cid:15)πR C (cid:20) t fb + αk (cid:21) . (52)This is the maximum mass that may be extracted during the experiment.To determine the density we could simply stop an experiment at first breakthrough, determine theadsorbed mass and then from equations (47), (50) we obtain ρ q = 1(1 − (cid:15) )˜ q ∗ (cid:18) M fb M CO πR ( L − s ) − (cid:15)C (cid:19) . (53)The final term in the brackets represents the mass of the gas in the column: this is approximately 1%of the total mass, hence ρ q ≈ M fb M CO πR (1 − (cid:15) )˜ q ∗ ( L − s ) o r M fb ≈ ρ q M CO πR (1 − (cid:15) )˜ q ∗ ( L − s ) . (54)The first expression gives a simple formula for the density of adsorbate in terms of the system param-eters and the mass of adsorbed CO at first breakthrough. The second expression shows how the massat first breakthrough depends on the syetem parameters. So, to increase the adsorbed mass requireseither an increase in column dimensions, adsorbate density or equilibrium concentration or a decreasein (cid:15) . The interstitial velocity and adsorption rate constant play no role in the mass adsorbed at firstbreakthrough . However the interstitial velocity does affect the time for first breakthrough, as evidencedby equation (47). If the measurement is made some time after first breakthrough then equation (51)does show a dependence on u and k , however even in the limit t → ∞ this contribution only accountsfor 10% of the adsorbed mass, so the dependence on these two parameters is weak.The approximate expression for (cid:101) C , equation (41), is compared with the experimental data forbreakthrough and also the numerical solution in Figure 9 (using ρ q = 325). The full expression forthe travelling wave (the first part of equation (34) in the supplementary information) is not plottedbecause it is indistinguishable from the approximate version. Clearly the approximate expression forthe travelling wave provides excellent agreement with the experimental data. So we may use it tocharacterise breakthrough.To find the width of the reaction zone consider the equation for (cid:101) C ( x, t ), (37a). If we define thereaction zone as having (cid:101) C ( x, t ) ∈ [0 , . C then the co-ordinates where the extreme values are achievedat a given time t are x | (cid:101) C =0 = s + (cid:98) v L (cid:20) k t + log (cid:18) (cid:98) v P e − (cid:98) v P e (cid:19)(cid:21) , (55) x | (cid:101) C =0 . C = s + (cid:98) v L (cid:20) k t + log (cid:18) . (cid:98) v P e − (cid:98) v P e (cid:19)(cid:21) . (56)The width of the reaction zone is then W = x | (cid:101) C =0 − x | (cid:101) C =0 . C = − (cid:98) v L ln 0 . ≈ (cid:15)uC (1 − (cid:15) ) k ρ q ˜ q ∗ ln 10 . (57)To verify this we take the values provided in Table 2 and obtain W ≈ . L ≈ . . L ≈ . igure 9: Comparison of the travelling wave solution for the gas concentration (dashed line) against the numer-ical solution (solid line) and experiment (circles). width of the reaction zone is provided in [31]. If we define this using the times for first breakthroughand when (cid:101) C ( x, t ) /C = 0 . W Z = 2 L t . − t fb t . + t fb ≈ . . (58)This equates to an 8% increase from the numerical width. So this formula is slightly less accuratethan the present one, equation (57). Equation (57) has the added advantage of clearly showing thefactors affecting the reaction zone. The primary purpose of developing a mathematical model of a process is to help understand and soimprove or optimise that process. Consequently, it is essential to start from the correct model. In thispaper we have revisited the averaging process to derive an accurate system of equations describingthe adsorption of a gas in a circular cross section packed column. In doing so we were able to identifya number of common errors in carbon capture which are prevalent the literature, such as:1. the use of an incorrect adsorbate density (which then affects the mass sink and heat generationterms);2. inconsistent neglect and retention of certain terms, such as heat conduction in the gas but notthe solid, or heat transfer at the walls;3. simultaneous inclusion of varying and constant velocity;4. radial flow in previously radially averaged equations;5. the incorrect integration of the adsorbate equation.Despite these errors many of the published works show excellent agreement with experimental data,often better than presented here. This may be explained through the choice of parameter values, forexample, if an incorrect adsorbate density is used it may be compensated for by adjusting the reactionrate. In general the models involve advection-diffusion equations, the experiment involves advection21nd diffusion so it is relatively straightforward to achieve good agreement using well chosen parametervalues. However, the parameter values obtained will be incorrect and so lead to inaccurate predictionswhen the experiment is scaled up or altered in some way.Once the correct averaged model was derived we proceeded to non-dimensionalise the system. Thisshowed which terms were important and which negligible. For example, it is clear that diffusion ofconcentrate in the gas is negligible, hence a detailed calculation of its value is not necessary. Similarlyconduction in the gas and solid are both small and have little effect on heat flow.Numerical results were presented for an isothermal system. Good agreement was shown withexperimental data. A brief analysis indicated that neglecting temperature variation, specifically onthe adsorption saturation may lead to errors of the same order as incurred by the neglect of mass loss,so future work should account for both of these effects.The travelling wave solution removes the need for a numerical solution and provides a simple wayto characterise the breakthrough curve. It also clearly demonstrates how the physical process dependson the operating conditions. Specifically through this analysis we obtained exact expressions for1. the time of first breakthrough;2. the process time;3. the width of the reaction zone;4. the (time-dependent) mass of adsorbed CO in the column.Perhaps the key parameter in all of this analysis is the adsorbed mass. The analytical solutionshows that the main ways to increase this are by increasing the adsorbate density, the column dimen-sions (both length and radius) and the adsorption saturation or by decreasing the void fraction. It isindependent of the adsorption rate, gas velocity and diffusion.The analysis in this paper has therefore led to an improved set of equations describing the ad-sorption of gas in a packed column. The novel analytical solution provides the relation betweenexperimental operating conditions and outputs, particularly the amount of adsorbed gas. The methodis applicable to standard operating conditions for carbon capture in a column, as studied in numerousother papers. Possible sources of error come from the neglect of temperature variation and mass loss.Whilst common practice to neglect these effects, temperature variation mainly affects the saturationconcentration, while mass loss affects the fluid velocity. Heat is dissipated quickly (in comparison tothe process time-scale) so this error will be limited by the specific temperature-saturation concentra-tion relation and the heat diffusion rate. In both cases we expect their neglect to lead to errors ofonly a few percent. Consequently we do not expect either effect to be significant in the analysis butthey are certainly worth investigating in future studies.Although the theory provides a clear description of the process, in practice it may not be sensibleto alter certain parameters. For example the adsorption saturation increases with pressure and de-creases with temperature, but there are operating costs associated with this. Decreasing void fractionincreases the surface area for adsorption but also requires a higher pressure drop across the column tomaintain the flow rate. A balance must therefore be found between the optimal theoretical solutionand operating considerations. A Isotherm equation
In [25] the adsorption saturation is assumed to consist of a chemical and physical component˜ q ∗ = q c + q p (59)22 m (mol/kg) K (1/atm) n α T η T ∆ H (kJ/mol)Physical adsorption 3.57 0.66 0.65 1.05 12.45 22.23Chemical adsorption 0.69 8.14 × Table 3: Isotherm parameters from [25, Table 2]. where each component takes the form q = q m K T P (1 + ( K T P ) n ) /n . (60)Certain parameters show a pressure and temperature dependence K T = K exp (cid:20) ∆ HRT (cid:18) T − TT (cid:19)(cid:21) q m = q m exp (cid:20) η T (cid:18) T − TT (cid:19)(cid:21) (61) n = n + α T (cid:18) T − T T (cid:19) . (62)The values for the various constants are shown in Table 3. In all of our calculations the referencetemperature T was set to 303 K, which corresponds to the base temperature in the considered exper-iment. Acknowledgements
T. G. Myers acknowledges financial support from the Ministerio de Ciencia e Innovacin, Spain GrantNo. MTM2017-82317-P. F. Font acknowledges that the research leading to these results has receivedfunding from la Caixa Foundation and MICINN through the Juan de la Cierva programme, GrantNo. IJC2018-038463-I.
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