Modeling of fixed-bed columns for gas physical adsorption

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in Chemical Engineering Journal, Vol. 378, on December 2019, available at: https://doi.org/10.1016/j.cej.2019.121985 © 2019 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND 1 Modeling of fixed-bed columns for gas physical adsorption Francisco Javier Gutiérrez Ortiz a, Manuel Barragán Rodríguez a, Ralph T. Yang b a. University of Seville, Escuela Técnica Superior de Ingeniería, Department of Chemical and Environmental Engineering, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain b. University of Michigan, College of Engineering, Department of Chemical Engineering, 3074 H.H. Dow, 2300 Hayward Street, Ann Arbor, MI 48109-2136, USA Corresponding author e-mail: “Francisco Javier Gutiérrez Ortiz” [email protected] Abstract Adsorption processes may be classified as purification or bulk separation, depending on the feed concentration of the compounds to be adsorbed (adsorbates). The adsorption dynamics of a fixed-bed is crucial for a well-designed adsorption process. This work focuses on physical adsorption and provides a modeling to predict the breakthrough curves using a minimum set of experimental data. The modeling does not require one to obtain key parameters from the experiment unlike other modeling found in the literature. In addition, boundary conditions, thermal effects, correlations to be used, and the homogeneous/heterogeneous modelings are discussed for 1-D modeling, after verifying that it provides the same output as 2-D modeling does. Moreover, it is demonstrated that pseudo-homogeneous modeling is realistic, so a more complex heterogeneous modeling is not necessary. The modeling has been tested against five sets of experimental data: three cases of bulk separation and two cases for purification. The simulation was carried out by Comsol Multiphysics software, and a good match between the experimental data and the simulation output was achieved, which demonstrates the applicability of the modeling, so it may be used with confidence. Purification can be modeled as an isothermal process, and no energy balance equation is needed. However, for bulk separation, noticeable thermal effects may take place due to the relatively high adsorbed gas that 2 consequently releases a higher amount of heat; at the same time, the superficial fluid velocity is reduced due to the decrease in the flow-rate, and the gas properties change, affecting the breakthrough curves. Keywords Breakthrough curve, fixed-bed column, adsorption, gas separation, modeling, simulation 3 1. Introduction In a real adsorption fixed-bed, the outlet concentration and temperature profiles do not replicate the input because of the dispersion and mass transfer resistance that needs to be considered in addition to adsorption equilibrium. The latter one involves the time delay kinetics due to the hold-up in the column, while the former one is related to hydrodynamics, especially regarding the extent of mixing of the fluid elements occurring due to turbulence as well as molecular diffusion and conduction. Usually, axial dispersion is considered, but sometimes radial dispersion might be also important, depending on the extent of adsorption and the heat released in the medium and exchanged with the surroundings. In relation to the former, adsorption processes may be classified as purification or bulk separation, depending on the feed concentration of the components to be adsorbed. So, the radial effects should not be disregarded a priori for bulk separation processes, while the radial dispersion is almost negligible for purification processes. Likewise, the separation behavior changes in an adsorption bed from isothermal to adiabatic systems, and of course for non-adiabatic and non-isothermal fixed-beds. In addition, the overall dynamics is affected by the presence of other solutes that may be absorbable, i.e., multicomponent systems. For a well-designed system, a modeling to predict breakthrough curves from basic kinetic and equilibrium data is very important, thus reducing the number of experimental tests, which may be very laborious and time-consuming. Thus, a comprehensive adsorption modeling for both purification and bulk separation processes would be very useful for design engineers. Process simulation is a tool more and more used for process design because it reduces the cost relative to tests carried out in the lab and pilot plant before scaling up. Although many phenomena may occur simultaneously in an adsorption process, there is a challenge to achieve a modeling and simulation methodology that allows a reasonable prediction of the system behavior even if a number of simplifications are used. 4 In a previous paper, a specific adsorption model for hydrogen sulfide removal from landfill biogas on activated carbon from treated sewage-sludge was developed, as a purification casestudy [1]. Simulation results were compared with experimental data obtained in an experimental fixed bed, showing good agreement between them. In the present work, the model is extended to be applied to both purification and bulk separation adsorption processes. A discussion about the use of one-dimensional (1-D) and two-dimensional (2-D) pseudo-homogeneous and heterogeneous modelings, as well as thermal effects caused by the heat of adsorption, and the right selection of boundary conditions and correlations are carried out. Thus, by solving a set of partial differential equations (PDE) using Comsol Multiphysics software, the breakthrough curves, temperature and concentration profiles for gas and solid can be obtained. Comsol is focused on the numerical solution of systems of partial differential equations (PDE) by the Finite Element Method. This software has a series of modules that can be coupled to model different processes involving different branches of physics and chemistry. However, after testing all of them, the module termed Coefficient Form PDE, which is a mathematical module without any specific physical function that allows the user to enter a set of PDE, is the most appropriate to code the modeling in Comsol in a dimensional form because it requires less computing resources; this is another novelty with respect to the previous model. When the equations vary with time (transitory regime), which is a characteristic of any adsorption process, the solution method is divided into four steps: (1st) Spatial discretization by the Finite Element Method. After this step, the time derivatives are the only ones present in the equations. (2nd) Time-iterated integration using only the BDF (Backward Differentiation Formulas) Method, whereby the differential equations are solved at each time interval with data coming 5 from the previous time interval. After this step, a system of nonlinear algebraic equations remains to be solved. (3rd) Iterative linearization of the system of nonlinear equations by the Damped Newton Method, which uses a damping factor to ensure convergence. The smaller the factor, the slower the algorithm is, but the convergence is better addressed. The result of applying this method to nonlinear equations is a system of linear equations. (4th) Direct or iterative solution with algorithms to solve the matrix equations of the system of linear equations, which gives the final simulation outputs. 2. Aims and scope The main aim of the paper is to provide a predictive modeling of fixed-bed adsorber dynamics both for purification and for bulk separation processes based on physical adsorption applied to gas separation process with favorable Type I isotherms, including multicomponent systems. The main novelty is the applicability of the modeling, which needs only a few number of parameters and does not require information from experimental breakthrough data to obtain preceding mass transfer or axial dispersion coefficients as usually done by other researchers. The paper describes and discusses the equations and boundary conditions, and how to deal with them in order to reach a suitable numerical solution, using correlations and a minimum number of experimental data, mainly related to the equilibrium and sorbent characteristics. In order to avoid lengthening the document excessively, the paper is focused on physical adsorption, so chemisorption is out of the scope of the study, which will be developed in a future communication. As commented on below, two-dimensional (2-D) axially dispersed plug flow modeling will be further developed in a future work because no noticeable difference between 1-D and 2-D modeling was verified. 6 The model predictions are compared with the results obtained from five experimental studies published in the literature regarding both purification and bulk separation processes in order to test the predictive power of the proposed model using minimum data, and to demonstrate its applicability. 3. Mathematical modeling The mathematical modeling used to describe the dynamics of adsorption considers a nonlinear adsorption isotherm with a plug flow contact model. It also includes transport phenomena due to dispersion and mass transfer resistance regarding the adsorption rate on the particles, as well as because of thermal effects relative to the heat exchange throughout the bed. The overall mass transfer rate is represented by the Linear Driving Force (LDF) approximation, which considers the adsorbent particles as a homogeneous phase in which the diffusion occurs with a constant effective diffusivity. For physical adsorption, other surface phenomena are assumed to be fast enough, so the adsorption rate is proportional to the deviation of the equilibrium through an overall mass transfer coefficient. According to the LDF model, the overall mass transfer coefficient (KG) determines the rate at which the adsorption occurs, so a high value of KG increases the rate of adsorption, due to the decrease in the mass transfer resistance; the system performance improves, and the slope of the breakthrough curve rises. However, the fact that many simulated breakthrough curves are steeper than the real ones may be because of the overall mass transfer coefficient is underestimated, so axial dispersion must be included to achieve a simulation to be closer to reality. The dispersion involves a back mixing for the adsorbate that enlarges the width of the mass transfer zone in the bed. Thermal effects may be also important on the adsorption performance. In the next sub-section, a set of equations and boundary conditions are described and discussed; variables, parameters and properties symbols are detailed in the Nomenclature section. 7 3.1. Assumptions In a previous study [1], isothermal conditions, negligible radial dispersion and constant fluid velocity were assumed. In the present paper, these assumptions are revisited, but the sorbent particles are assumed to be spherical and homogeneous in size and density. Isothermal conditions might be valid for a purification process, because the adsorbate amount on the solid surface is small and the heat released in the adsorption is low, so the changes in temperature may be neglected. However, this assumption is no longer valid for bulk separation processes a priori, due to the higher adsorbate concentration in the feed. Therefore, a non-isothermal process or an adiabatic system (if a good thermal isolation is used) must be considered. If the temperature changes, the equilibrium isotherm and the breakthrough curve will change. The axial flow velocity will also change over time due to the loss of adsorbate in a bulk separation process, but not for purification processes, due to the very low feed concentration of adsorbate. On the other hand, there may be a radial gradient of velocity or void fraction, so both concentrations and temperature might change radially. Moreover, a high thermal gradient due to cooling or heating through the adsorber wall can lead to a radial concentration distribution. Thus, a 2-D heterogeneous axially dispersed plug flow model should be considered. Generally, it is accepted that if the diameter of the adsorber-to-particle sorbent diameter is greater than 10, the radial concentration gradient is negligible. Furthermore, in physical adsorption processes, the heat source is not high enough to involve radial variations. A number of simulations have allowed verifying this for two case-studies, and therefore, no radial variations in velocity, concentration and temperature are considered. Thus, 2-D modeling is to be only taken into account in chemical adsorption, in another paper, as above mentioned. The pressure drop along the bed may be calculated by Eq. (1): 8 𝜌𝜕𝑢 𝜕𝑡𝜌𝑢𝜕𝑢 𝜕𝑧𝜕𝑃 𝜕𝑧𝜇𝜕𝑢 𝜕𝑧𝜑𝑢 (1) where 𝜑 represents an inertia term given by Ergun correlation (Eq. 2): 𝜑150𝜇󰇛1𝜀󰇜 𝜀𝑑 1.75𝜌󰇛1𝜀󰇜 𝜀𝑑 (2) For purification processes, the pressure drop is negligible because the fluid velocity does not change. For bulk separation, since these systems usually operate at a constant fixed-bed pressure by using a pressure controller and a control valve set downstream from the bed, Eq. (1) is not strictly necessary to obtain the breakthrough curves, and a constant pressure value can be considered. 3.2. Balance equations Mass and energy balance equations along with a kinetic rate expression for the adsorption are the set of equations to be considered in the modeling. Since the radial gradient of concentration may be neglected (1-D modeling), the differential mass balance for the i-th adsorbate (for a multicomponent system) in the fixed-bed is Eq. (3) 𝜕𝐶 𝜕𝑡𝜕 𝜕𝑧󰇛𝑢𝐶󰇜𝐷,𝜕𝐶 𝜕𝑧𝜌1𝜀 𝜀𝜕𝑞 𝜕𝑡 (3) where 𝜕𝑞 𝜕𝑡𝐾,󰇛𝑞∗𝑞󰇜 (4) The equilibrium adsorbed amount 𝑞∗ in Eq. (4) can be calculated from the equilibrium isotherm, which is associated with the extended Langmuir isotherm (Eq. 5) as a representative isotherm for Type I isotherms: 𝑞∗𝑞,𝐾,𝐶 1∑𝐾,𝐶 (5) 15 To solve Eq. (4), the General Form PDE can be selected with 𝑢󰇟𝑞𝑞𝑞󰇠 (Eq. 12): 𝑒𝑑𝑢 𝑑𝑡𝑑𝑑𝑢 𝑑𝑡 𝛻 𝑓 (12) 𝑒000 000 000  𝑑100 010 001  000 𝑓 󰇯𝐾,󰇛𝑞∗𝑞󰇜00 0𝐾 ,󰇛𝑞∗𝑞󰇜0 00𝐾 ,󰇛𝑞∗𝑞󰇜󰇰 4. Results and discussion Experimental data have been taken from references [19-25] to validate the proposed modeling, which is based on a minimum set of experimental data and the use of a reduced catalogue of correlations. Most references are based on activated carbon (AC) due to its extremely large, accessible internal surface, and large pore volume, especially, micropore volume, which allows multiple applications. In addition, AC does not require prior stringent moisture removal, and the heat of adsorption, or bond strength, on activated carbon is generally lower than that on other sorbents, so lower energy is needed for the sorbent regeneration [26]. In the proposed modeling, simulated using Comsol, parameters are calculated by equations and correlations, but few experimental data are needed, which is one of the key points of this paper. These data are the following: Adsorbent data: ρs: density of the adsorbent particles εp: porosity of the adsorbent particles dp: diameter of the adsorbent particles dpore: diameter of the pores of the adsorbent Bed data: ε: porosity of the bed D: bed diameter 16 L: length of the bed, although this only influence the time in which the breakthrough point is reached Operation data: P: operating pressure T0: fluid temperature at the bed inlet yi,0: molar fractions of the feed compounds at the bed inlet F0: fluid flow-rate at the bed inlet. Isotherm data (with Langmuir isotherm): KL: equilibrium constant of the Langmuir isotherm. qsat: saturation charge of the adsorbent. ΔHads: enthalpy of adsorption. Any isotherm can be adapted to a Langmuir isotherm. Likewise, physicochemical properties of the compounds can be obtained from open database. Most of these parameters will be normally known, since the adsorbent is characterized, the sizing of the column is available, and the operating conditions must be set; isotherms should have been determined but if not, some experiments should be carried out before applying the modeling and simulation to predict the behavior of the fixed-bed. In addition, some relationships among the 14 parameters allow the reduction of this number. Thus, except for very low column diameters, bed porosity ranges from 0.3 to 0.5. Likewise, from the column diameter and the flow-rate, the gas superficial velocity is obtained, which should be kept constant when scaling-up, so dimensionless Reynolds number is constant as present in almost all the correlations used. Similarly, the pore characteristics influence the effective diffusion coefficient, but usual values of micropore diameters (about 1-2 nm) and particle porosities (around 0.2-0.3) might be used if these parameters were unknown, as they are not too relevant. 17 4.1. Modeling validation Two case-studies of purification and three cases of bulk separation have been simulated by Comsol in order to verify the goodness of the modeling proposed in this paper. Along with these cases, the reference from which the experimental data were taken is cited. Tables S1-S5 show the data withdrawn from experimental papers to implement the code in Comsol (see Supplementary Material) for the five case-studies included in this work. All the case-studies have been performed using both heterogeneous and pseudohomogeneous modeling. In all the cases, the results are exactly the same. The reason for such a behavior is in the heat transfer from sorbent to fluid, which is very high. In fact, as the heat transfer coefficient ranges from 50 to 100 Wꞏm-2ꞏK-1, and particle diameter is about 1-2 mm, the heat transfer resistance is very low, and the heat released during the adsorption simultaneously increases the temperature in both the solid and the gas. In addition, most of references used in the case-studies include a section of modeling apart from the experimental data, whose results are very close to those obtained in the modeling presented in this work, with minor differences. In fact, the current output is oftentimes even better than that used by others, and even sometimes no differences can be found between the previous and current outputs. Figures S1 and S2 (Supplementary Material) show a comparison between the modeling results previously provided and the output of the current modeling for the case-studies 2 (purification) and 3 (bulk separation) selected to illustrate the discrepancies. These are not substantial, and among the most important results regarding the modeling now presented is that no fitting parameter is used in the current modeling. 4.1.1. Purification case-studies No article providing experimental data of temperature for a purification adsorption has been found, probably because the thermal effects observed have been negligible. From all the simulation outputs, the increase in temperature predicted by the model is always less than 1 K. 18 The increase in temperature due to the release of adsorption heat in purification processes is insignificant because the amount of adsorbate adsorbed on the sorbent is very small. Therefore, purification processes can be modeled as isothermal processes, and no energy balance equation is needed. In addition, simulation was performed using both Danckwerts and Dirichlet boundary conditions at the bed inlet, verifying that the model output is the same. - Case-study 1. Hydrogen sulfide removal [19] Experiments were carried out in a fixed-bed cylindrical column (30-mm ID, 100 mm height) and a thermally treated sewage–sludge was used as an adsorbent. A gas composed of a mixture of CH4 (60 vol.%), H2S (variable, but less than 2000 ppm) and CO2 (balance) was entered the bed. A detailed description of the experiment and results can be found elsewhere [19]. The H 2S adsorption isotherms and breakthrough curves were obtained at H 2S concentrations of 1980, 1065, 570 and 162 ppm. Figure 1 illustrates the comparison between the results from the experiment (points) and the model (solid lines), at different H2S inlet concentrations. The breakthrough curves are steeper when the H2S feed concentration is higher. The simulation outputs are the same as those previously reported [1] but using less input parameters and direct equations without converting them into dimensionless equations, thus optimizing the simulation performance in Comsol. Note that the overall mass transfer and the axial dispersion coefficient were not fitted from experiments. It is observed that the predictions match well with the experimental data, although the final slight curvature exhibited in the experimental breakthrough curves deviates from the prediction; the small slope in the final adsorption stages means that adsorption rate becomes very small as the adsorbent is close to saturation. It might be because of the overall mass transfer coefficient decreases over time, which cannot be represented by the constant value used in simulation. This effect of KG on the shape of the breakthrough curve is corrected in the case studies relative to bulk separation, where the value of KG is continuously updated. The other 19 calculated parameter, the axial dispersion coefficient, also influences the adsorption performance and is updated in bulk separation cases as well. - Case-study 2. Thiophene removal [20] Sorption experiments were performed in a laboratory setup for the removal of heterocyclic sulfur compounds by a commercial activated carbon. Breakthrough curves were experimentally obtained at 100, 150 and 200 °C for C4H4S inlet concentrations of 8.5, 15 and 30 ppm, and flow-rates of 30, 42.5 and 55 NmLꞏmin-1, at temperature 100 and 150 °C. The thiophene was mixed in Ar as the carrier while the adsorption of Ar was assumed negligible relative to thiophene. Figure 2 shows the experimental results (points) as well as the curves simulated (solid lines). The agreement between the modeling output and the experimental results is very good in most tests. Like in the case-study 1, a higher concentration of the adsorbate at the inlet causes the breakthrough time to be reached earlier, since the amount of sorbent is the same but is spent earlier because a greater amount of adsorbate is adsorbed. No thermal effects were detected in any of the purification case-studies. 4.1.2. Bulk separation case-studies The gas adsorbed in a purification process is, usually, less than 1% of the total gas entering the column, so the effect on the fluid flow-rate and any thermal effect are negligible. To observe an appreciable increase in the temperature, the concentration of the adsorbate should be about or greater than 10 wt.% of the total feed that corresponds to bulk separation processes, in which the amount adsorbed may be of the same magnitude order as that non-adsorbed. In this case, the effect on the fluid flow-rate can become significant because the superficial fluid velocity is reduced as the flow-rate decreases, and the gas density, the fluid specific heat and other 20 properties change due to the variation in the mass fraction of compounds. In addition, noticeable thermal effects may be expected due to the relatively high amount of adsorbed gas that consequently releases a higher amount of heat. These thermal effects also affect the velocity and properties aforementioned. - Case study 3. Methane and carbon monoxide adsorption [21, 22] The simulated system is the multicomponent adsorption of the effluent coming from a steam methane reforming to purify the hydrogen obtained. The gas is composed of H2, CH4 and CO, and all of them are adsorbed, although hydrogen is barely adsorbed. A ternary mixture (H2/CH4/CO of 60.4/28.1/11.5 vol%) was fed into the fixed-bed containing a commercial activated carbon. The pressure was kept constant in the fixed-bed by a back-pressure regulator. The adsorption bed was initially saturated with H2 at the adsorption pressure. Figure 3 shows the breakthrough curves of the gas mixture, which match reasonably well with the experimental data (points). The evolution of H2 is determined by the evolution of CO and CH4. CO is less adsorbed than CH4 and starts to increase near 500 s, with a slope that superposes on the experimental data. The maximum value is somewhat lower than that experimental, and a small deviation is exhibited between 550 and 650 s; during this period, a plateau of the molar fraction of the CO and H2 can be seen. After that, the agreement is pretty good. When the sorbent becomes saturated of CH4, the corresponding breakthrough curve appears showing a slight deviation regarding the experimental data, probably because of the overall mass transfer coefficient is not high enough or the axial dispersion coefficient should be somewhat lower, although the values required for that are not very different from those provided by correlations. Figure 4 shows the trend of temperature with time at 0.1L, 0.3L, 0.5L, 0.75L and L from the fixed-bed entry, all together to illustrate how the temperature profile changes over time depending on the section of fixed-bed. Figure 5 compares the profiles obtained by simulation 21 (solid lines) with experimental data (points) at four different sections. It can be seen that those match well with each other from the initial time up to reach the maximum or peak value, corresponding to the maxima of 𝜕𝑞/𝜕𝑡 for CO and CH4 that take place at different moments; after that, the simulated temperatures decrease because the bed (note that solid and gas temperatures are equal) is eventually cooled by the heat exchange with the environment and by the feed gas entering the fixed-bed at a lower temperature. However, the cooling is faster than that obtained experimentally. There may be some reasons for this: (1) the values of the heat transfer coefficients (especially, ℎ) are computed from correlations, but a better fitting can be obtained if those values are manually adjusted; (2) the effective axial thermal conductivity (𝑘,) has also a significant effect. Thus, for example, if ℎ is reduced and/or 𝑘, is increased, the tail beyond the maxima has a lower slope, i.e., temperatures diminish slowly over time. However, a too low value of ℎ implies an approximation to an adiabatic system, and hence, higher temperatures, depending also on the value of ℎ. Similarly, a too high value of 𝑘, leads to a lower maximum for temperature. Moreover, in these cases, the maxima might increase over the length, contrary to the evolution shown in Figure 5. Nevertheless, the first stage of the evolution of temperatures due to the active adsorption is the key to know when the maxima are reached and the rate for it. The slower rate of cooling after that is less important in these predictions. - Case study 4. Carbon dioxide adsorption [23] The behavior of an adsorbent based on commercial granular activated carbon (AC) and activated carbon modified by ammonia derivatives (N-doped AC) adsorbents for carbon dioxide capture was tested. The modified adsorbent enhances the performance, because of the additional chemical adsorption taking place. In this work, no chemisorption is considered, so only pure (unmodified) sorbent (AC) is used to validate the proposed modeling. Two experiments were carried out for this latter sorbent at two inlet volumetric flow-rates of 50 22 mL/min and 100 mL/min, and temperatures of 30, 45 and 60 °C. The inlet composition of the feed gas is 15/85 CO2/N2 vol% at 1 bar. From the isotherm proposed in the study, only the term relative to physical adsorption is used, which was fitted following the Tóth isotherm. In the modeling, in addition to this latter, Langmuir isotherm is applied by adjusting the isotherm to 30 ºC and varying the parameters with temperature by the van’t Hoff Equation. Since Reynolds number is about 1, the correlations to be used are that by Rastegar-Gu for the axial dispersion coefficient and the Dwivedi-Upadhyay correlation to obtain the external mass transfer coefficient. In Figure 6, the breakthrough curves are drawn in different colors at 50 and 100 mL/min at the three mentioned temperatures. In addition, the increase in temperature is lower than 1 ºC, and no thermal effects are exhibited. As expected, the breakthrough curves move towards the right, i.e., to higher times, at lower flow-rate and temperature values, showing a good agreement between the model and the experiment in all the tests. - Case study 5. Carbon dioxide and nitrogen separation [24, 25] The study deals with CO2 capture by adsorption in post-combustion. The stream of gas is composed of 80% N2 and 20% CO2. The adsorbent used was activated carbon. The breakthrough curves were obtained at different temperatures (25, 50, 100 and 150 °C), at 1.0 bar. Figure 7 shows the experimental breakthrough curves obtained by the researchers (points), as well as the simulation output of the proposed modeling (solid lines). As the temperature increases, the simulations match equally well with the experimental results; indeed, the agreement is very good for all the tests. No experimental curves for temperature were presented in the original study. However, Figure 8 shows the evolution of temperature in the four experiments performed under adiabatic conditions at the exit of the bed, which does not exceed 6 K at 373 and 423 K, so similar results 23 should have to be obtained by simulating under isothermal conditions. However, at 301 K and 323 K, the increase in temperature is 22 and 15 K, respectively, so there is a thermal effect on the fixed-bed performance. Note that the effect described in the case-study 3, relative to low values of ℎ, is clearly shown here; in fact, some simulations have been performed using a non-adiabatic system, and the breakthrough curves widen over time, because the heat released to environment leads to temperature profiles with lower peaks, which take place sooner. 4.2. Thermal effects on breakthrough curves When the temperature changes, the equilibrium constant of the adsorption also varies, so that the breakthrough curves will change. The modification of the equilibrium constant can be approximated by integrating the van't Hoff equation (Eq. 6) by assuming that the adsorption heat does not change with temperature; in this way, the new equilibrium constant can be obtained from that obtained experimentally at a reference temperature (𝑇): 𝐾,𝐾,exp∆𝐻 𝑅𝑇󰇧1𝑇 𝑇󰇨 As adsorption is an exothermic process, the equilibrium constant decreases when the gas temperature increases. Therefore, the performance of the process enhances at a lower temperature, as can be seen in some of the previous Figures. In addition, the axial dispersion coefficient increases with temperature, because it is directly proportional to the diffusivity that is proportional to the 1.75th power of the temperature. If the axial dispersion coefficient rises, the slope of the breakthrough curve will decrease, and the breakthrough time will be reached sooner. It is because a high axial dispersion coefficient enlarges the width of mass transfer zone that, hence, arrives earlier at the end of the bed and a non-zero exit concentration is detected earlier. The temperature and concentration profiles have been shown for five case-studies, although those relative to bulk separation are more significant than those for purification 24 processes. The heat released in the adsorption is directly related to the breakthrough curve, since it is proportional to the adsorption rate. The heat generated over time reaches a maximum at any bed section, which corresponds to the maximum slope of the breakthrough curve. From that point, the slope changes from being increasing to decreasing (Figures 5 and 8), because when the slope begins to rise from zero (at a certain bed zone), heat of adsorption starts to be released and reaches a maximum value corresponding to maximum value of 𝜕𝑞/𝜕𝑡. Beyond this point, the slope begins to decrease because of the heat released by adsorption decreases to zero and the cooling through the environment. In this way, the released heat leads to an increase in temperature of this part of the bed; likewise, heat is transmitted from the bed (solid) to the gas. As the product of the heat transfer coefficient and the interface area is very high, the heat transfer is fast, and the temperatures of both phases are similar, so the curves appear overlapped; indeed, the simulation output of heterogeneous and pseudo-homogeneous modeling are the same. The gas heats up the solid disposed in the sections of sorbent downstream from the mass transfer zone, which has not participated yet in the adsorption process and, therefore, it is still at room/initial temperature. Temperature peaks will be achieved just in the end of adsorption process itself. The shape of temperature profiles depends on the axial thermal conductivity. Furthermore, if no axial thermal conductivity was considered, the maximum temperature achieved would be higher, because the width of mass transfer zone would decrease, and the adsorption would take place in a narrower band of fixed-bed where the heat is released (Figure 9). Similarly, a higher inlet concentration would lead to a higher temperature, since the amount adsorbed per unit time at any fixedbed section would be greater, so the heat accumulated in that area would be also higher as more heat is released. The change of concentration in the radial axis (2-D model) is not perceptible because the increase in the temperature that causes the radial dispersion is not enough to show significant Gas thermal conductivity (kg) This parameter does not have a high influence on the breakthrough curves, so it can be approximated to the conductivity of the main compound. If such a compound does not exist, Wilke approach to the Chapman- Enskog model, valid for gas mixtures [16], can be also used: 𝑘 𝑦𝑘 ∑𝑦𝜑     , where 𝜑󰇯1󰇧𝜇 𝜇󰇨 󰇧𝑀 𝑀󰇨 󰇰󰇯4 √ 2󰇩1󰇧𝑀 𝑀󰇨󰇪 󰇰 Effective axial thermal conductivities (ke) - Yagi, Kunii and Wakao correlation [17] modified by Krupiczka [18], disregarding radiation effects because of temperatures are relatively low: 𝑘,𝑘󰇧𝑘𝑘 ⁄.. 0.75𝑅𝑒𝑃𝑟󰇨 𝑘,𝑘 In this way, for fluid phase two terms are considered for effective thermal conductivity: one due to molecula r conduction in the fluid (quiescent bed) and another due to fluid convection, while for the solid phase there is only one term related to molecular conduction in the solid. Figure 1. Breakthrough curves for the case-study 1 (H2S removal) at four H2S feed concentrations at 1 atm and room temperature (source of experimental data: [19]) C (ppm) 20000150001000050000 400 800 1200 1600 Time (s) C 0 : 1065 ppm C 0 : 570 ppm C 0 : 162 ppm C 0 : 1980 ppm Figure 2. Breakthrough curves for the case-study 2 (thiophene removal) at 1.5 bar: (a) at 100, 150 and 200 °C (15 ppm C4H4S, 30 NmL min−1); (b) for flow-rates of 30, 42.5 and 55 NmL min−1 (15 ppm C4H4S, 150 °C); (c) for 8.5, 15 and 30 ppm C4H4S (100 °C, 30 NmL min−1) (source of experimental data: [20]) 0140 0.0 12010080604020 0.2 0.4 0.6 0.8 1.0 1.2 200 ºC 150 ºC 100 ºC C/C0 Time (min) 040 0.0 3010 0.2 0.4 0.6 0.8 1.0 1.2 20 55 NL/min 42.5 NL/min 30 NL/min C/C 0 Time (min) 0.0 Time (min) 0.2 0.4 0.6 0.8 1.0 C/C0 1.2 1801401006020 40 80 120 1600 8.5 ppm 15 ppm 30 ppm (a) (b) (c) Figure 3. Breakthrough curves for the case-study 3 (CH4 and CO adsorption) at 9.8 bar and 289.1 K (source of experimental data: [21, 22]) Figure 4. Simulated temperature evolution over time for the case-study 3 (CH4 and CO adsorption) at different locations (9.8 bar, 289.1 K) 0.2 1.0 0.8 0.6 0.4 Molar fraction 0 200 400 600 800 1000 1200 1400 1600 Time (s) 0.0 H 2 model CH 4 exper. CO exper. 289 291 293 295 297 299 301 303 305 307 309 311 313 315 0 200 400 600 800 1000 1200 1400 1600 Temperature (ºC) Time (s) T_0.1L T_0.3L T_0.5L T_0.75L T_L Figure 5. Temperature evolution for the case-study 3 (CH4 and CO adsorption) at three locations: (a) 10 cm, ((b) 30 cm, (c) 50 cm and (d) 75 cm, at 9.8 bar and 289.1 K (source of experimental data: [21, 22]) 0 290 295 300 305 310 315 400200 600 800 1000 1200 1400 1600 Temperature (K) Ti me (s) 0 290 295 300 305 310 315 400200 600 800 1000 1200 1400 1600 Temperature (K) Ti me (s) 0 290 295 300 305 310 315 400200 600 800 1000 1200 1400 1600 Temperature (K) Ti me (s) 0 290 295 300 305 310 315 400200 600 800 1000 1200 1400 1600 Temperature (K) Ti me (s) (a) (b) (c) (d) Figure 6. Breakthrough curves for the case-study 4 (CO2 adsorption) at 1.0 atm and (a) 50 mLꞏmin-1 and (b) 100 mLꞏmin-1 (source of experimental data: [23]) 016 C t /Co 0.2 0.4 0.6 0.8 1.0 0.0 Time (min) 1412108642  T 30 ºC  T 45 ºC  T 60 ºC 016 C t /Co 0.2 0.4 0.6 0.8 1.0 0.0 Time (min) 1412108642  T 30 ºC  T 45 ºC  T 60 ºC (a) (b) Figure 7. Breakthrough curves for the case-study 5 (CO2 and N2 adsorption at (a) 301 K, (b) 323 K, (c) 373 K, (d) 423 K, at 1.02 bar and 30 mL min-1 (source of experimental data: [24, 25]) 1.2 025020015010050 C/Co 0.2 0.4 0.6 0.8 1.4 1.0 0.0 Time (s) N 2 CO 2 1.2 025020015010050 C/Co 0.2 0.4 0.6 0.8 1.4 1.0 0.0 Time (s) N 2 CO 2 1.2 010080604020 C/Co 0.2 0.4 0.6 0.8 1.4 1.0 0.0 Time (s) N 2 CO 2 1.2 010080604020 C/Co 0.2 0.4 0.6 0.8 1.4 1.0 0.0 Time (s ) N 2 CO 2 (b) (c) (d) (a) Figure 8. Temperature evolution for the case-study 5 (CO2 and N2 adsorption) at the outlet (z=L), 1.02 bar and 30 mL min-1, and at: (a) 301 K, (b) 323 K, (c) 373 K, (d) 423 K (source of experimental data: [24, 25]) 300 302 304 306 308 310 312 314 316 318 320 322 324 0 20 40 60 80 100 120 140 160 180 200 220 240 260 Temperature (K) Time (s) 322 324 326 328 330 332 334 336 338 0 20 40 60 80 100 120 140 160 180 200 220 240 260 Temperature (K) Time (s) 372 373 374 375 376 377 378 379 380 0 20 40 60 80 100 120 140 160 180 200 220 240 260 Temperature (K) Time (s) 422 423 424 425 426 427 0 20 40 60 80 100 120 140 160 180 200 220 240 260 Temperature (K) Time (s) (a) (b) (c) (d) Figure 9. Simulated temperature evolution over time for the case-study 3 (CH4 and CO adsorption) at different locations under adiabatic conditions: (a) 𝒌𝒇,𝒆 (b) 𝒌𝒇,𝒆𝟓 (c) 𝒌𝒇,𝒆𝟎 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 0 200 400 600 800 1000 1200 1400 1600 Temperature (ºC) Time (s) T_0.1L T_0.3L T_0.5L T_0.75L T_L 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 0 200 400 600 800 1000 1200 1400 1600 Temperature (ºC) Time (s) T_0.1L T_0.3L T_0.5L T_0.75L T_L 289 291 293 295 297 299 301 303 305 307 309 311 313 315 317 319 321 0 200 400 600 800 1000 1200 1400 1600 Temperature (ºC) Time (s) T_0.1L T_0.3L T_0.5L T_0.75L T_L (a) (b) (c)