Simulation of temperature fields in a narrow tubular adsorber by thermal lattice Boltzmann methods

Chemical Engineering Science 63 (2008) 4269 -- 4279

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Simulation of temperature fields in a narrow tubular adsorber by thermal lattice Boltzmann methods Nishith Verma a,∗ , Dieter Mewes b a Department b Institute

of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India of Process Technology, Leibniz University of Hannover, Hannover 30167, Germany

A R T I C L E

I N F O

Article history: Received 7 December 2007 Received in revised form 21 May 2008 Accepted 27 May 2008 Available online 14 July 2008 Keywords: Adsorption Lattice Boltzmann modelling Mathematical modelling Packed bed Porous Heat Transfer

A B S T R A C T

Significant temperature gradients may exist in packed bed adsorbers due to the exothermic heat of adsorption. In this study we investigate temperature fields in a narrow tubular packed bed adsorber having tube to particle size ratio < 10. Theoretical calculations are carried out using lattice Boltzmann methods (LBM) to simulate three-dimensional concentration and temperature profiles in macro- as well as micro-pores of the adsorption bed. Model simulation results show non-uniform temperature gradients across the tube's cross-section. Zones of significantly high temperature are observed within macro-voids. Temperature gradient is found to be primarily dependent on the amount of heat released, internal BET surface area, and hydrodynamic conditions prevailing in the adsorber. Agreement between the model results and the experimental data obtained with the aid of the tomography technique for a tubular adsorber is observed to be reasonable. The study is important from the point of view of a realistic design of packed bed adsorbers. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Adsorption is extensively used in many separation and purification applications. Considering complexities of velocity profiles due to varying bed porosities and irregular packing arrangements, design of packed bed adsorbers based on rigorous mathematical analysis is often not feasible. The status of the present theoretical analysis is that most of the mathematical models developed to predict concentration profiles and breakthrough curves in the adsorption bed make 1D approximations by averaging gas velocity and concentration profiles across the tube's cross-section and using empirical quantities such as dispersion coefficient and average bed porosity (Yang, 1997). While these engineering approximations may be considered reasonable in the case of a relatively larger tube-to-particle-size adsorber, the same is not true for a narrow size adsorber. A number of studies have indeed shown that for the latter type of adsorber (small d/dp ratio) where concentration gradients are significant, approximations assuming point size particles lead to over-simplification of the problem and result in the numerical values that are not consistent with the experimental observations (Vortmeyer and Michael, 1985; Tobis and Vortmeyer, 1998). On a similar note, the assumption of isothermal conditions prevailing in adsorber beds may not always be justified. In principal,

∗

Corresponding author. Tel.: +91 512 2597704; fax: +91 512 2590104. E-mail address: [email protected] (N. Verma).

0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.05.040

true isothermality never exists during adsorption. In fact, the gas temperature in the thermally insulated packed bed increases during adsorption, reaches a maximum level before decreasing to the level of the inlet gas temperature. At this instance the adsorbents are saturated with the gas at the concentration level corresponding to the inlet condition. The maximum gas temperature in the bed is influenced by a number of parameters including gas flowrate, heat of adsorption, and the internal BET surface area of the adsorbent particles. In narrow tubular adsorbers, the presence of localized hot zones is common even under moderate operating conditions, due to the regions of stagnant flow in the bed voids. This study is intended to ascertain temperature profiles due to the exothermic heat of adsorption in the narrow tubular packed bed. Full 3D (axial, radial, and circumferential directions) simulation of velocity, concentration, and temperature fields in the tube is carried out using lattice Boltzmann methods (LBM). Similar to the concentration, temperature breakthrough curves are also theoretically calculated. One of the special features of LBM is its ability to handle complex and irregular geometries in terms of applying boundary conditions. The programming code is relatively simpler to write. Drawing analogy between momentum and mass conservations, the programming code written to simulate hydrodynamics can be altered with minimal changes to simulate concentration profiles. Considering that the “particles” are permitted to stream only in straight lines, the LBM implementation of mapping of the circular cross-sections of spherical objects on the rectangular grids is also simple. Here, it is appropriate to point out that the successful application of LBM in simulating

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hydrodynamics in a number of complex scenarios including flow in porous media is well recognized, with numerous reviews reported in open literature (Chen and Doolen, 1998; Nourgaliev et al., 2003). With the advent of fast computational platform in recent times, the LBM studies related to mass transfer, for example, adsorption, catalytic reaction, diffusion in porous media, etc., have also been aptly addressed (Tessier et al., 1997; Filippova and Hanel, 2000; Zeiser et al., 2001; Kandhai et al., 2002; Freund et al., 2003; Sullivan et al., 2005). Yet, most of the calculations have been done assuming that isothermal conditions exist in the packed bed. The development of thermal LBM is relatively newer and its application has started emerging in the past few years only.

2. Macroscopic mass and thermal energy conservation equations Fig. 1 is a schematic of the packed bed adsorber containing the spherical adsorbent particles of relatively larger size. The particles are porous and packing arrangement is assumed to be structured, for example, body centered cubic, tridiagonal packing, etc. It is assumed that the adsorbent particles are fresh (devoid of any gaseous solute) to begin with. At certain instance the packed bed is challenged with an inert carrier gas laden with the calibrated amount of the solute at low concentration levels (usually in parts per million levels). This scenario may correspond to the control of emission of undesirable gaseous species present at trace levels, in the context of environmental applications, or drying of a carrier gas containing moisture, in purification applications. The temperature of the wall of the adsorber may be set at a predetermined value different from that of the inlet gas. Alternatively, the wall may be thermally insulated (adiabatic condition) implying zero heat flux at the wall. The former situation is commonly cited as Gratz and the latter as Gratz–Nusselt problems in literature (Dean, 2000). In this study these problems have been used as benchmark cases for the validation of the thermal lattice models. To this end, we are interested in solving for temperature fields associated with those of concentration and flow during dynamic adsorption and desorption of the solute on the adsorbent surfaces. Temperature effects due to the exothermic heat of adsorption are incorporated in the theoretical analysis. Considering both inter- and intra-phase transport, the conservation equation for the solute in the

macro-pores (voids between the particles) of the adsorber may be mathematically written as

jt Cg + ji Cg ui = Di ∇ 2 Cg + where i = x, r, 

(1 − )



Dpore ∇Cp |r=Rp × ap (1)

Eq. (1) describes the unsteady-state 3D concentration distribution of the solute in the bed voids incorporating the convective as well as dispersive effects. The last term in Eq. (1) represents the source of diffusion flux from or into the spherical particles due to adsorption/desorption within the micro-pores of the particles. In Eq. (1), Dpore is the intra-particle diffusivity due to Knudsen effect. Similar conservation equation without convection may be written for the intra-phase transport:

jt Cp = Dpore ∇ 2 Cp − ajt Cs

(2)

where the last term in Eq. (2) represents the rate of change in the surface concentrations of the adsorbate and may be related to the change in the gas phase concentration via the slope of the equilibrium isotherm as follows:

jt Cs = jt Cp (dC s /dC p )

(3)

In the above equation, the last term in parenthesis is identified as the slope of the equilibrium isotherm, which may be allowed to vary with temperature and the solute's concentration. The macroscopic conservation balance for thermal energy in the packed bed may be written as

jt T + ji Tui = ∇ 2 T − a Hjt Cs

(4)

Eq. (4) includes transport of thermal energy due to the combined effects of convection and conduction in the macro-pores of the adsorption bed. The last term in Eq. (4) represents the rate of increase in the gas temperature due to the exothermic heat of adsorption and is dependent upon the rate of change in the surface concentration of the adsorbed species through Eq. (3). Eq. (4) in its present form assumes that there is no temperature gradient within the pores of the adsorbent particles. Such assumption may be considered to be valid considering that the thermal conductivity of most of the common

Fig. 1. Schematic of tubular packed bed (top) and d3q19 lattice molecule (bottom).

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adsorbent materials like zeolites and activated carbons is relatively larger. Nonetheless, thermal lattice models were also developed in the present analysis to ascertain the magnitude of temperature gradients inside the porous particles, similar in approach adopted to simulate concentration profiles due to intra-phase transport. There is a significant price, however, to pay for the computational time if solving the mass and thermal conservation equations for both micro and macro voids. The analogous thermal energy balance equations for the macro- and micro-pores of the adsorbent particles may be written as (1 − ) jt T + ji Tui = i ∇ 2 T + pore ∇Tr=Rp × ap  where i = x, r, 

(5)

Eq. (5) describes the unsteady-state 3D temperature profiles in the bed voids incorporating the convective as well as conductive effects, with the last term representing the source of thermal flux from the spherical particles due to the exothermic heat of adsorption during adsorption/desorption inside the pores. In Eq. (5), pore is the intraparticle diffusivity due to Knudsen effect. The conservation equation for the intra-phase heat transport is written as

jt T = pore ∇ 2 T − a Hjt Cs

(6)

Finally, the momentum balance equation (Navier–Stokes) and the equation of state for the ideal gas complement the macroscopic conservation equations (1)–(6) to completely describe flow, concentration, and temperature fields in the adsorption bed packed with porous particles. Appropriate initial and boundary conditions were used at the inlet, outlet and the wall of the tube, and at the surfaces of the adsorbent particles. 3. Thermal LBM In this section we describe the mesoscopic conservation equations for the transport of thermal energy. A comprehensive discussion of the lattice Boltzmann equations for hydrodynamic and concentration fields in a packed bed adsorber may be obtained in our previous studies (Agarwal et al., 2005; Manjhi et al., 2006a,b; Verma et al., 2007). Various lattice models reported in literature for simulating temperature profiles in a fluid flow may be broadly classified into two categories: (a) multi-speed approach (Alexander et al., 1993; Shan and Doolen, 1996; McNamara et al., 1995; Chen et al., 1997) and (2) passive-scalar approach (Eggels and Somers, 1995; Shan, 1997; Inamuro et al., 2002; Yuan and Schaefer, 2006a,b; Kao and Yang, 2007). The former approach is apparently an extension of the lattice models for isothermal conditions in which only the density distribution function is used, however, with additional speeds to obtain the temperature evolution. In addition, the equilibrium distribution must include the higher-order velocity term. In the latter approach, an auxiliary lattice Boltzmann equation is used assuming that the temperature field is passively advected. This scheme is relatively simpler to implement, numerically stable and also permits use of varying Prandtl numbers over a wider range. Considering that viscous heat dissipation and compression work done due to pressure may be neglected without loss of accuracy in a packed bed adsorber, the thermal LBM based on the passive-scalar approach essentially suits the conditions for the present modeling in this study and is adopted here and described below. Drawing analogy with the transport of a species, the lattice Boltzmann equation for the macroscopic thermal energy balance equation (5) may be written as hi (r + ei , t + 1) − hi (r, t) =

−1

h

(hi − h0i ) + (T)

(7)

Eq. (7) contains distribution functions and the local equilibrium distribution function, with the thermal relaxation constant h related

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to thermal diffusivity in the similar fashion as mass diffusivity and kinematic viscosity are related to the corresponding relaxation constants of the LBGK transport equations for the mass and momentum transport of a solute in the carrier gas:   2 h − 1  = e2 t 6     2d − 1 2m − 1 t and  = e2 t (8) D = e2 6 6 Eq. (7) is analogous to the equation written for the density distribution function, gi (r, t) for determining difference in the concentrations of the carrier gas and the solute, whereas the sum of the concentrations of the two species is obtained from the other distribution function, fi (r, t): gi (r + ei , t + 1) − gi (r, t) = fi (r + ei , t + 1) − fi (r, t) =

−1

d

−1

m

(gi − gi0 )

(fi − fi0 )

(9)

(10)

Chapman's Enskog expansions on Eqs. (9) and (10) are performed to recover the macroscopic advective-diffusion and Navier–Stokes equations for incompressible fluid, respectively. The similar mathematical manipulation on Eq. (7) may also be carried out to recover thermal energy balance equation (5). The exothermic heat generation rate is incorporated via (T) in Eq. (7) by following the similar approach as adopted for incorporating gravity as an external body force in the momentum conservation equation. With lattice thermal, mass, and momentum diffusivities related to the respective relaxation constants as shown in Eq. (8), Prandtl and Schmidt numbers may be set independently in the simulation. Finally, the development of local equilibrium distribution function h0i is also analogous to that of the equilibrium distribution functions, fi0 and gi0 corresponding to  (sum density) and  (difference density) of a binary miscible fluid as per the diffusion model proposed by Swift et al. (1996). h0i (u) = A0 + C0 u. u,

i=0

= h01i (0)[A + Bei . u + C(ei . u)2 + D(u. u)], i = 1, . . . , 9 = h02i (0)[A + Bei . u + C(ei . u)2 + D(u. u)], i = 10, . . . , 18

(11)

The values of the coefficients used in Eq. (11) for a d3q19 cubic lattice are reported in the Appendix. In the case of porous adsorbent particles, the intra-particle thermal diffusivity is set at values different from those in the bulk region (voids between the particles) by segregating two domains of computations, one for the inside and the other for the outside, and using two different hs related to macro and micro as per Eq. (8). The numerical scheme for solving temperature profiles is identically the same as that used for solving concentration profiles in the adsorber (both macro and micro) and described in our previous study (Verma et al., 2007). The details of the scheme adopted for mapping a spherical object (adsorbent particles) on the rectangular grids and the refinement of the grids near the vicinity of the spherical particle may also be obtained in the aforesaid study. 4. Lattice thermal boundary conditions and model validation There are two types of boundary conditions: one is the constant wall temperature and the other is the constant heat flux. The flux may be set at zero for a thermally insulated wall, i.e. adiabatic condition. In the case of known wall temperature, the scheme originally proposed by Yuan and Schaefer (2006a) for 2D geometry is adopted and extended for the present case (3D). First, distribution functions after streaming for the known links are determined. For example,

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Fig. 2. Model validation for the constant wall flux case at varying wall flux (Pe = 56).

referring to the 3D lattice molecule of Fig. 1 (bottom), the distribution functions which are known at the bottom plane of the tube are: h0, h1, h3, h4, h5, h6, h9, h10, h11, h12, h13, h14, h16, and h17. The unknown functions are h2, h7, h8, h15, and h18. An intermediate quantity, say T  , is calculated from the known distribution functions: T  =Twall −(h0+h1+h3+h4+h5+h6+h9+h10+h11+h12+h13+h14+ h16 + h17). The unknown distribution functions are temporarily set at their equilibrium values given by Eq. (11) with temperature set at  T  . Thus, realizing that hi (i = 1, . . . , 18) = Twall , the intermediate temperature is accordingly corrected as Tcorrect = T  × Twall /(h02 + h07 + h08 + h015 + h018 ). Finally, the unknown distribution functions are calculated as: h2 = h02 /Twall × Tcorrect ; h7 = h07 /Twall × Tcorrect , etc. In the case of wall flux boundary condition, the second-order finite difference scheme based on discretization of three points is carried out:

jT = (4Ti,2 − Ti,3 − 3Ti,1 )/(2 y) jYi,1 Thus, knowing flux at the wall, the wall temperature Ti,1 is calculated. The after-procedure is the same as that adopted for the constant wall temperature case. The implementation of the boundary conditions on the surfaces of the spheres follows exactly the same approach as used in the case of (mass) diffusion in the porous spherical particles by applying continuity in the temperature and heat flux across the surface of the sphere T|Rp+ = T|Rp− , and −k∇T|Rp + = −kpore ∇T|Rp −

(12)

Validation of the thermal lattice model was carried out by comparing the predicted results to the analytical solutions for two benchmark cases: constant wall temperature and constant wall heat flux. The analytical solutions for both cases are reported in literature. For brevity we produce here the validation results for the latter case only. Fig. 2 compares the model results for the radial temperature profiles in the tube to the analytical solutions. The results are shown for a fixed Peclet number (=56) and varying (dimensionless) wall flux at three different locations in the tube (Z = L/4, L/2, and L). The simulation was carried out on 40 × 40 × 320 grids in a tube of L/d ratio = 8, with Prandtl number set at 0.7 and the thermal relaxation constant

h set at 0.6. Thermal conductivity, k, was calculated from equation, k = 13 h x (Yuan and Schaefer, 2006a).

As observed in Fig. 2, the model simulation results are in reasonably good agreement with the analytical solutions obtained for the temperature profiles at the exit of the tube, for all two cases. Difference between the simulation and theory, however, increased for the locations away from the exit (i.e., towards the entrance) of the tube. Further simulation results in this study showed that difference also increased for relatively smaller Peclet numbers (< 10). The thermal model results are consistent with the theory. Referring to Appendix given at the end of this paper, it may be pointed out that the limiting form of the analytical solution to the steady-state temperature profiles in tubular flow under forced convection with prescribed wall flux is valid for (1) large Peclet numbers, in which case the axial conduction term may be neglected in comparison to the convective term, and (2) for the location far downstream from the beginning of the heated section, where the shape of the radial temperature profiles may be assumed not to undergo further change with increasing distance. We also validated the present thermal lattice model by comparing the predicted concentration profiles under the extreme condition of H set at zero, to those predicted from the mass LBM developed in the previous study (Manjhi et al., 2006a) for isothermal conditions. The two results were shown to converge within the numerical errors.

5. Temperature fields in packed bed As pointed out earlier in the text on theoretical analysis, thermal LBM is the extension of the mass LBM, considering analogy between thermal and mass transport and neglecting viscous heating and compression effects. In our recent study we have extensively discussed unsymmetricity observed in concentration profiles in macro and micro voids of the adsorber bed, due to the non-uniformity in the radial velocity profiles (Manjhi et al., 2006b). In the aforesaid study, model parametric calculations were carried out for different packing arrangements, voids between the particles, d/dp ratios, and flowrates to ascertain the influence of these variables on the uniformity of concentration fields in the narrow tubular adsorber packed

N. Verma, D. Mewes / Chemical Engineering Science 63 (2008) 4269 -- 4279

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Fig. 3. Velocity vectors in adsorber of four-sphere packing arrangement with one sphere placed alternately. Plots are shown for different x–y planes (from top left z = 0, z = R/4, z = R/3, z = R/2 in clockwise direction).

Fig. 4. Temperature contours across the tube's cross-section (top: X = L/2, bottom: X = L/2 + R, T = 5000 s).

with porous adsorbent particles. For brevity, we do not reproduce here those results related to the mass transport and confine our discussion to the temperature effects only. However, for a meaningful description of the results related to the thermal model we discuss the representative results for the velocity profiles.

Fig. 3 describes the simulation results of the steady-state velocity vectors obtained for the flow (Re, p = 20) in a tubular adsorber (L = 200 mm, I.D.=50 mm). The packing arrangement corresponded to the four equal size spheres (d/dp = 5) placed circumferentially at equal distances on a vertical plane, with one sphere placed at the center of the tube's cross-section alternately between two such vertical planes. The particles were assumed not to contact the walls of the vertical tubular adsorber. However, the particles centrally located between the two horizontal planes containing four particles each were assumed to touch with the neighboring particles. The plots are shown for different vertical planes cut across the horizontal tube at different locations. There are two important observations that may be made from the simulation results. First, velocity profiles across the cross-section of the packed section are non-uniform for all four plots shown in Fig. 3, with magnitude of the velocity larger near the wall than at the center of the tube. Non-uniformity in the velocity profiles is attributed due to variation in the radial porosity (larger voids near the wall than towards the center) in the tube packed with relatively larger size particles. These characteristics, typical of flow in a narrow tubular adsorber, have also been experimentally observed and reported in literature. The second important observation is the regions of nearly stagnant gas flows around the particles in the bed voids. The two observations were consistently observed in all the simulation results obtained for different packing arrangements. Similar to the effects of non-uniformity in the flow fields on the mal-distribution of the solute in macro- and micro-voids, non-symmetricity in the temperature profiles was also observed and is discussed next. We first present the simulation results obtained for temperature contours in the macro voids. The simulation conditions are the same as chosen for Fig. 3. To begin with, the LBM calculations for temperature and concentration were carried out only for the outside regions (voids between the particles). This way the effects of intra-particle diffusion (or conduction) were delineated. This situation corresponds to the mass transport in the adsorber bed packed with the non-porous adsorbent particles having small thermal conductivity (k < 1 W/m K). In other words, adsorption was assumed to

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Fig. 5. Transient temperature distribution in the adsorber: temperature in voids is maximum at t = 4000 s.

occur at the external surface of the particles, and the heat released at the surface was assumed to dissipate to the bulk gas flow. It is important to note that most of the adsorbent particles such as zeolites and activated carbons have relatively larger thermal conductivity (usually in the order of 1–10 W/m K) to the extent that thermal Biot number (defined as the ratio of intra-phase resistance to interphase resistance) is usually much smaller than 1. Yang (1997) has done analysis to conclude that in most of the adsorber the major resistance for mass transfer is within the pellet, whereas the opposite is true for heat transfer. On the similar note, Aris (1995) estimated that the mass Biot number for a packed bed is in the range between 5 and 500, whereas the heat Biot number ranges from 0.05 to 5. In the present work, the model analysis indeed confirmed that while mass diffusion inside the porous zeolite particles must be considered in the analysis for a realistic design of an adsorption packed column, the intra-phase thermal resistance may be assumed to be negligible. Consequently, while concentration gradients inside the zeolite particles were calculated to be often significant, temperature within the particle was nearly uniform. Fig. 4 describes temperature contours (T/Tinlet ) across the tube's cross-section at two adjacent vertical planes, the first containing four spheres, and the second containing one sphere placed concentrically. The results are shown for t = 5000 s. It was assumed that the gas temperature and concentration are initially uniform in the tube and the adsorbent particles are fresh (devoid of any solute). The tube's walls are thermally insulated. As observed in the figure, the temperature in the bed increases by nearly 40% of the inlet gas temperature due to release of the exothermic heat of adsorption at the external surface of the zeolite particles. In addition, the gas temperature towards the center of the tube is larger than that near the

tube's walls. At both locations (refer top and bottom plots) similar patterns in temperature distribution may be observed. Heat is released at the adsorbing surface of the spheres at a rate proportion to the rate of adsorption/desorption of the solute and is removed or swept away from the surface by convection. As observed in Fig. 3, there are zones of nearly stagnant gas in the bed voids around the center of the tube and the gas velocity is much larger near the wall than at the center. Thus, the relatively larger gas temperature observed near the center of the adsorber is attributed due to the non-uniformity in the gas velocity. During a typical adsorption/desorption cycle equilibrium never exists in the adsorber, except at the beginning of the cycle when the adsorbents are fresh and towards the end of the cycle when the adsorbents are saturated. The net rate of adsorption is initially large. As the bed is gradually saturated with the adsorbate, the rate asymptotically decreases. As a consequence, the heat released due to adsorption/desorption goes through a maximum value. The resulting temperature profiles at each location in the adsorber also go through the similar transience. Fig. 5 describes the unsteady-state temperature profiles at a fixed location in the adsorber. The conditions chosen for the simulation are the same as those chosen for the results shown in Fig. 4. As observed, temperature of the gas in the macro-voids increases above the inlet gas value and reaches maximum value at t = 4000 s, before gradually approaching the inlet temperature level. At around t = 10, 000 s, the maximum temperature observed is 1. 05Tinlet near the center of the four spheres. Further model parametric study showed that depending upon the gas flowrate and the exothermic heat of reaction assumed in the simulation, there were temporal zones of “hot spots” in the adsorber bed, especially between the two adjacent planes

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containing the four spheres and one sphere where the temperature was significantly higher than the gas inlet value. As shown in Fig. 5, the gas temperature at the center of the adsorber at t = 4000 s is approximately 1.5 times the inlet gas temperature.

Fig. 6. Temperature profiles within and outside adsorbent particles in large thermal conducting zeolite particles.

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In the previous text we have pointed out that most of the common adsorbent particles, including zeolites (considered in the present study) are good thermal conductors. Therefore, it intuitively follows that while mass transfer in an adsorber packed with porous particles like zeolites may be controlled by Knudsen diffusion (intraparticle diffusion), heat transfer is predominantly controlled by the flow conditions in the adsorber bed. The model simulation results in this study corroborated these effects. The simulation carried out for varying thermal conductivities showed that temperature profiles within the particles were nearly uniform, in contrast to the concentration profiles, which were observed to be significant over moderate range of intra-particle diffusivity assumed in the simulation. In the case of porous adsorbents, heat is released within the pores of the particles due to adsorption/desorption at the pore walls. Thus, there are three rates to be considered in understanding the thermal effects: (1) rate of diffusion of the solute within the pores, (2) rate of adsorption/desorption (or the proportionate rate of release of heat), and (3) rate at which heat is transported by conduction from the internal surface to the outside in the bed voids. Since the last rate is usually much faster in comparison to the other two rates, the temperature profiles within bed voids qualitatively follow the same pattern (temporal hot zones towards the center of the tube due to relatively smaller velocity) as in the case of non-porous particles, except that increase in the temperature is higher in the case of the porous particles than in the non-porous particles due to the relatively larger (BET) area of adsorption/desorption available in the former. Fig. 6 describes the representative model simulation results for the temperature contours obtained in both macro- and micro-voids under the identical conditions chosen in the simulation for the results of Figs. 4 and 5. As observed, maximum gas temperature in the bed increases to 1.7 times the Tinlet at 8000 s after change over to the influent gas. During the entire transience, temperature within the particles remained nearly uniform. Before moving on to the description of the temperature breakthrough curves obtained for the adsorber bed, we summarize in Table 1 the simulation conditions, including the physical properties of the adsorbents, and the corresponding model parameters used for the results shown in Figs. 4–6. Table 1 also includes the experimental conditions used in the study of Salem (2006), which will be discussed later in this paper. Fig. 7 describes the model simulated temperature breakthrough curves at the end of the packed section of the tube. The curves correspond to the center of the tube (r =0). Simulation was carried out for

Table 1 Simulation/experimental conditions and model parameters S. no.

Simulation/experimental conditions

This study ( Figs. 4–6)

Salem (2006)

1 2 3 4 5 6 7 8 8 9 10

Length of the tube (mm) Inside diameter of the tube (mm) Particle size (mm) Gas superficial velocity (m/s) Gas inlet temperature (K) Gas inlet concentration (% or ppm) Tube wall thermal condition Density of zeolite particles (kg/m3 ) External area of adsorption (m2 ) Specific internal area (BET) of adsorption (m2 /gm) Thermal conductivity of zeolite particles (W/m K)

200 50 10 0.02 303 1 or 10,000 Adiabatic 1220 3. 14 × 10−4 55 8.0

200 50 4.5 0.04–0.1 323 1 or 10,000 323 K 1220 6. 36 × 10−5 55 8.0

LBGK momentum relaxation constant, m LBGK mass relaxation constant, d LBGK thermal relaxation constant, h Number of grids Incompressible gas density, 0 (lattice units) Particle Reynolds number Peclet number (mass/thermal)

0.8 0.9 0.85 800 × 20 × 20 1.0 20 17

1.1 0.8 0.85 800 × 20 × 20 1.0 25 19

Model parameters 11 12 13 14 15 16 17

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Fig. 7. Temperature breakthrough in a packed bed adsorber (L = 0. 16 m, dt = 0. 04 m,  = 0. 000033 m2 /s, Pr = 0. 7).

Fig. 8. Concentration breakthrough curves at the exit of the packed bed adsorber with and without heat of adsorption.

varying exothermic heats of adsorption and assuming tube's walls to be thermally insulated. As observed, gas temperature increases during adsorption and reaches a maximum value before gradually

Fig. 9. Temperature breakthrough curves in the packed and empty sections of the adsorber.

decreasing to the inlet gas temperature level. The higher the heat of adsorption, the larger the peak in the temperature. The plot in the inset of Fig. 7 describes the corresponding concentration

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Fig. 10. Comparison between model predictions and experimental data for temperature breakthrough at the center of the tube in a tubular adsorber (L = 200 mm, ID = 40 mm).

breakthrough curves. Comparing the temperature and concentration breakthrough curves, peak in the temperature may be observed to occur approximately at the instance of the maximum change in the concentration level (point of inflexion marked in the plot), which is consistent with the physics. Referring to the previous section on the model analysis, heat was assumed to be passively advected and the solution to the thermal energy balance equation was obtained simultaneously with that to the species balance equation. Thus, the effect of temperature on concentration profiles was realized through the dependence of mass diffusivity on temperature. Since diffusivity has weak dependence on temperature (T 1.5 ), difference between the concentration breakthrough curves should be insignificant. As observed in the figure, some differences do arise, which is attributed due to the influence of stagnant gas in the macro-voids, where the solute transport is predominantly determined by diffusion rather than convection. Fig. 8 describes the simulation results for the concentration breakthrough curves in the empty section of the tube (after the packed section). The simulation results for the fluid flow show that velocity profiles in the empty section gradually settle to parabolic in shape, unlike in the packed section where the velocity near the wall is larger than towards the central core of the tube, where the velocity is approximately uniform. Thus, breakthrough in the concentration appears earlier at the center than near the wall. In the figure, the results obtained for the special case of isothermal condition (i.e. assuming zero heat of adsorption) are also compared to those for the case of adiabatic condition, with the heat of adsorption included. As observed, there is only marginal difference between the two curves corresponding to the center of the tube, where the velocity is maximum. However, near the wall where the velocity is relatively smaller, breakthrough response is faster in the case of adiabatic than that of isothermal condition due to increase in the mass diffusivity with increase in the temperature. The effects of difference between the velocity profiles in the packed and the empty sections of the tube on the corresponding temperature breakthrough curves were also ascertained. As shown in Fig. 9, the temperature at the centerline (r = 0) is larger than that at the wall (r = R) in the region immediately adjacent to the end of the packed section, unlike in the empty section where the opposite effects are observed in temperature profiles

due to the velocity profiles gradually settling to radially parabolic. Further increase in the temperature is due to the release of heat of adsorption at the tube's wall. We finally present the experimental data related to the temperature measurements taken in the adsorber using the tomography technique. In recent times, the non-intrusive techniques such as optical tomography have made it possible to investigate the spatial variation of concentrations and temperature in a packed adsorber or chemical reactor (Yuen et al., 2003; Salem et al., 2005, 2006). More recently, Salem (2006) has experimentally investigated the influence of the wall effects on the concentration and temperature profiles in a tubular adsorber packed with the zeolite particles. The measurements were made at the outlet cross-section of the adsorber (L × d = 200 mm × 50 mm) with the help of the near-infrared tomography. The details of the experimental set-up and the tomographic technique employed in the measurement may be obtained from the aforesaid study. The spherical zeolite particles (d/dp ∼ 10) packed at randomly were challenged with the calibrated moisture in nitrogen at the inlet of the bed. The experimental results pertaining to the concentration measurements have been discussed in our previous studies. Fig. 10 describes the temperature breakthrough data obtained at the outlet cross-section of the tube. As observed, temperature increases to 1.05 times the gas inlet temperature, corresponding to the gas velocity at 0.04 m/s before asymptotically decreasing to the inlet temperature. At large gas flowrate, increase in the temperature is expectedly relatively smaller. In both cases, LBM thermal model predictions are observed to be in good agreement with the data. The inset of the figure shows the packing arrangement assumed in the model, which best matched the experimental conditions. As mentioned earlier in the text, Table 1 lists the primary experimental conditions, including the physical properties of zeolites, used in the study of Salem (2006) and the corresponding LBM parameters used in the simulation. 6. Conclusions In the narrow tubular adsorber having small tube to particle size ratio, increase in the gas temperature due to the exothermic heat of adsorption may be significant. Using lattice Boltzmann techniques,

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model simulation was carried out to show that there may be regions in the bed voids, where the flow conditions may be nearly stagnant, and consequently the gas temperature may be significantly larger than the inlet gas temperature. Significant variation in the temperature in the radial direction may also exist. The concentration breakthrough curves were found to be affected by temperature variation. For a tubular adsorber having small d/dp ratio and operated under non-isothermal conditions, it is essential that full 3D simulation of velocity, concentration, and temperature profiles be carried out in order to get a realistic picture of various operating conditions affecting those profiles.

Appendix (A) The numerical values of Lagrangian coefficients of Eq. (11) were evaluated as follows: A0 = T(1 − 2Te), h02i (0) = T/36,

C0 = − 12 T, A = 3Te,

B = 3,

C = 92 ,

D = − 32

(B) The analytical solution to the problem of temperature profiles for the steady-state forced convection in a tube with the constant wall flux is given as

= −4 − 2 + 14 4 +

Notation

h01i (0) = T/18,

7 24

where a ap CG Cs Cp d dp D f g h H k L q r R Re, p Rp t T Te u x

surface area per unit volume of the pore of the fiber, 1/m external surface area per unit volume of the fiber, 1/m gas phase concentration in the bed, mol/m3 surface concentration of adsorbed species, mol/m2 gas phase concentration of adsorbed species inside the pores, mol/m3 diameter of the tube, m particle diameter, m dispersion/diffusion coefficient, m2 /s particles density distribution function difference-density distribution function particles thermal distribution function heat of adsorption, J/kg thermal conductivity of the particle , W/m K length of the tube, m directions of lattice-particle speed grid locations radius of the tube, m particles Reynolds number radius of the particle, m time, s temperature, K lattice model parameter fluid velocity, m/s x-direction

Greek letters



∈

bed porosity circumferential direction

  m d h

fluid viscosity, Pa s fluid kinematic viscosity, m2 /s fluid density, kg/m3 relaxation time for momentum transport relaxation time for mass diffusion relaxation time for thermal diffusion

Subscripts i p

lattice index (0–18), x, r,  of cylindrical coordinates particle

Superscript 0

equilibrium

Acknowledgement N. Verma acknowledges the Alexander von Humboldt Research Fellowship (IV0INI/1114920) to conduct the present study at the Leibniz University of Hannover (Germany) during 2006-07.

=

T − Tinlet , qwall R/k

=

zk

Cp vz,max

R2

,

=

r R

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