Simulation of Gas Exothermic Chemical Reaction in Porous Media Reactor with Lattice Boltzmann Method

Transcription

1 Journal of Thermal Scence Vol.22, No.1 (2013) DOI: /s Artcle ID: (2013) Smulaton of Gas Exothermc Chemcal Reacton n Porous Meda Reactor wth Lattce Boltzmann Method Xn Fang 1,2, L Xun-Feng 1, Xu Mn 1, Hua Xu-Lan 1, Ca Jun 1 and Guo Zh-Xong 3 1. Insttute of Engneerng Thermophyscs, Chnese Academy of Scences, Bejng , Chna 2. Graduate School of the Chnese Academy of Scences, Bejng , Chna 3. Department of Mechancal and Aerospace Engneerng, Rutgers, the State Unversty of New Jersey, 98 Brett Road, Pscataway, NJ 08854, USA Scence Press and Insttute of Engneerng Thermophyscs, CAS and Sprnger-Verlag Berln Hedelberg 2013 Exothermc reactor s the man part n a chemcal heat pump. It nvolves complex mult-component exothermal chemcal reacton n catalyst-flled porous meda. The lattce Boltzmann method (LBM) s developed to smulate the characterstcs of flud flow, heat and mass transfer couplng chemcal reacton n the exothermc reactor of the sopropanol/acetone/hydrogen chemcal heat pump system. Fractal theory s used to structure a porous medum model n the reactor. The smulaton results show that LBM s sutable for the smulaton and the converson has an optmal value wth dfferent nlet veloctes. Keywords: Chemcal heat pump, Hydrogenaton reacton, LBM. Introducton The hydrogenaton exothermc reacton of acetone, whch occurs at 473K, s an mportant step n the sopropanol/acetone/hydrogen chemcal heat pump system [1]. The reacton equaton s shown as the followng: ( CH3) 2CO( g) + H2( g) ( CH3) 2CHOH ( g) (1) Δ H = 55.0 kj / mol The catalyst-flled reactor can be treated as a porous meda, n whch the heat and mass transfer as well as the exothermal reacton exst. Though the nvestgatons on the flud flow and heat transfer n porous meda have been a hot topc and made great progress [2], few studes have been carred out to nvestgate the characterstcs ntegrated wth the chemcal reacton. Furthermore, most of the nvestgatons for chemcal reacton neglected the effect of heat transfer. And the chemcal reacton and heat transfer were studed separately. Therefore, the heat and mass transfer n the reactor are stll poorly understood and ntractable. Although the conventonal CFD methods can be appled to the problems of complex geometry boundary condtons, the LBM has been proposed n recent years for smulatng the flud flow and presentng the detaled analyss of heat transfer problem n porous meda feld [3-5]. A strkng capablty of LBM s that t can handle very complcated geometry and boundary condtons [6]. Based on the porous meda reactve flows, Huber et al. developed a Lattce Boltzmann model ncludng the effect of mass dffuson and advecton-dffuson and appled the advecton-dffuson model to the problem of flud flow n a porous meda [7]. Kao et al. conducted an nvestgaton on the flud flow and heat transfer n a mcroarray structure [8]. A smple mathematcal model was proposed to evaluate the chemcal reactve effcency just Receved: September 2012 HUAI Xulan: Professor Ths work was supported by the Natonal Natural Scence Foundaton of Chna (Grant No ) and the Natonal Basc Research Program of Chna (Grant No. 2011CB710705)

2 Xn Fang et al. Smulaton of Gas Exothermc Chemcal Reacton n Porous Meda Reactor wth Lattce Boltzmann Method 43 Nomenclature Abbrevaton X Dstrbuton functon CHP Chemcal Heat Pump Greek symbols LBM Lattce Boltzmann Method α Thermal dffusvty coeffcent [m 2 /s] Symbols ε Porosty c Lattce speed [m/s] λ Stochometrc coeffcent c p Specfc heat capacty [J/(kg K)] ν Knetc vscosty [m 2 /s] D Mass dffusvty coeffcent [m 2 /s] ρ Densty [kg/m 3 ] e Lattce partcle speed τ Relaxaton tme k e Reacton equlbrum constant [1/Pa 2 ] ω Weght coeffcent K r Reacton coeffcent [mol/(m 2 s Pa 2 )] Superscrpts L 0 Characterstc length [m] eq Equlbrum M Mole weght Source term Q Reacton heat [J/mol] Chemcal component R react Reacton rate [mol/(m 2 s)] Subscrpts Δt Tme step ace Acetone T Temperature [K] hydro Hydrogen T 0 Characterstc temperature [K] I Dscrete velocty drecton U Macroscopc velocty[m/s] p Isopropanol U 0 Characterstc velocty [m/s] mx Mxture x Poston n space Chemcal component based on the steady-state temperature feld. Actually, the chemcal reacton n the reactor was not computed by LBM. Thus, there was lack of detaled observaton and understandng of basc physcal-chemstry processes n the reactor. In ths study, followng the method n [9], LBM s extended to nvestgate the characterstcs of flud flow and heat and mass transfer accompaned by the chemcal reacton. The classcal regular Serpnsk carpet fractal structure s desgned to smulate the porous meda, n whch the 2-D fractal porous structures wth porosty of 0.79 (order 2) and porosty of (order 3) are shown n Fgure 1. Fg.1 The fractal structure of Serpnsk carpet (a. porosty of 0.79; b. porosty of 0.702) Numercal method The followng assumptons are adopted for smulatng the flow, temperature and concentraton felds wthn the reactor usng the LBM: (1) The body forces are not appled to the flow feld and the flow s ncompressble. (2) The radoactve heat transfer s gnored. (3) The flow, temperature and concentraton felds are ndependent and can be solved separately. (4) The thermophyscal propertes of the reactants and product are constant. The D2Q9 lattce BGK models are used for the flow, temperature and concentraton smulaton. The common expresson of the model s shown n Eq.(2). x denotes the poston n space. Δt s the tme step. X represents the dstrbuton functons of flow f, temperature T and concentraton Y. s the speces of components. τ X s the sngle relaxaton tme. X eq s the correspondng equlbrum dstrbuton functon. e denotes the dscrete velocty, whch s expressed as Eq.(3). 1 X( x+ eδ t, t+δ t) = X(,) x t [ X(,) x t τ X (2) eq X ( x, t)] + S = 0 8 ( ) ( ) c ( ) ( ) c ( ) 0,0 = 0 e = cos θ,snθ θ = α 1 π 2 = 1 4 2cos θ,snθ θ = α 5π 2+ π 4 = 5 8 (3) The source term s S X =0 for the flow, X S X ωqt = for

3 44 J. Therm. Sc., Vol.22, No.1, 2013 the temperature and S X ωq Y = for the concentraton, whch are avalable only on the reacton surface. ω s the weght coeffcent wth ω 0 =4/9, ω 1 4 =1/9, ω 5 8 =1/36. The Q T and Q Y n the source term ncludng the effects of the chemcal reacton and can be expressed as: T Q = QL R /( ρ c T U ) (4) Y 0 react mx p o 0 0 react mx 0 Q = λ M L R /( ρ U ) (5) where Q s the reacton heat. c p s the specfc heat capacty of the mxture. ρ mx s the densty of mxture. M s the molecular weght of components speces. λ are the stochometrc coeffcents, λ ace = 1, λ hydro = 1, λ p =1. U 0 s the characterstc velocty. L 0 s the characterstc length. T 0 s the characterstc temperature. eq The equlbrum dstrbuton functon X s of the form as : eq ω [1 1.5( e u) / 4.5( e u) / X = P + c + c u / c ] where P s the macroscopc value correspondng to ρ for flud, T for temperature and Y for concentraton. c s the lattce speed and take 1 here. u s the macroscopc velocty. The macroscopc quanttes can be obtaned from the dstrbuton functon: ρ = f u= f e / ρ T = T Y = Y The relaxaton tme τ X can be obtaned from the knematc vscosty of ν of the mxture flud for the flow smulaton, from the thermal dffuson coeffcent of α for the temperature calculaton, and from the mass dffuson coeffcents of D of every component for the concentraton solvng. 2 2 ν = c ( τ 1/2) Δ t/3, α = c ( τ 1/2) Δt/3 D = c ( τ 1/2) Δt/3 ν 2 The value of ν s m 2 /s, α s m 2 /s, D p s m 2 /s, D ace s m 2 /s, D hydro s m 2 /s. The reacton rate R react n formulas (4) and (5) s determned referrng as the followng: R = K p p [1 p /( k p p )] (7) react r ace hydro p e ace α hydro where K r s the reacton coeffcent, mol/(m 2 s Pa 2 ). K e s the reacton equlbrum constant, Pa 1. p ace, p hydro and p p are the partal pressure of acetone, hydrogen and sopropanol, respectvely. The half way scheme bounce-back boundary s appled for all no-slp sold boundares for the flow smulaton [10]. The velocty boundary s appled at the nlet, whch can be deduced from the expresson of the macroscopc (6) quanttes. The developed boundary condton s mposed at the outlet. The Drchlet boundary for the temperature boundary (nlet) and the Neumann boundary for the heat flux boundares (top and bottom wall) are appled for the thermal smulaton [11]. The zero dffusve flux boundares for the channel walls and the constant concentraton boundary for the nlet are used to calculate the components concentraton referrng to the thermal boundary. The half way scheme bounce-back boundary s also set for the nteror flud-sold nterfaces for the temperature and concentraton smulaton. The developed boundary condton s used at the outlet for both of them [12]. The unform grd s used for the numercal smulatons. The convergence crteron s set as follows: u(, j, t + δt) u(, j, t) + v(, j, t + δt) v(, j, t) Err _f = u (, jt, + δt) + v (, jt, + δt) Err _g =, j, j Err _ θ =, j T(, j, t + δ t) T(, j, t) T(, j, t + δ t) Y (, j, t + δ t) Y (, j, t) Y (, j, t + δ t) 8 Error = max Err _ f, Err _ g, Err _ θ < 10 Before applyng the present lattce Boltzmann method to solve the complex confguraton, a few benchmarks are carred out for valdaton of the LBM. Wth regard to hydrodynamcs, the smlar flow past a square wth the same boundares s smulated. Fgure 2a descrbes the Fg. 2 (a) Velocty vector of flow past a square wth Re=10; (b) Comparson of smulaton results by LBM and analytcal soluton for the Poseulle flow

4 Xn Fang et al. Smulaton of Gas Exothermc Chemcal Reacton n Porous Meda Reactor wth Lattce Boltzmann Method 45 numercal profle for ths scenaro. Ths phenomenon s n lne wth the flow pattern of small Re Number flow around the bottom of column. Besdes, the well-known Poseulle flow s also carred out for the model valdaton. Fgure 2b descrbes the predcted results by the LBM compared wth the analytcal results. The calculated result fts well wth the exact soluton. That shows that the LBM for the flud flow s relable. Yet, the expermental data or the analytcal results of the hydrogenaton reacton of acetone cannot be got at present. So the valdatons of the LBM for the temperature and concentraton are not carred out. But the expresson and the soluton of those models are the same as the model for the flud flow. Results and dscussons In ths study, the flow and heat transfer couplng chemcal reacton wth dfferent nlet veloctes and porostes are smulated. The smulaton doman s performed usng a lattce wth nodes. Fgure 3 depcts the representatve smulaton results for the x-drecton velocty, temperature and components concentraton n the doman wth the porosty of 0.79 and the nlet velocty of It s found that the maxmum of the concentraton gradent s near the sold matrx because of the surface reacton. The concentraton of the hydrogen decreases and the concentraton of the sopropanol ncreases gradually along the flow drecton. The temperature also ncreases because of the exothermc reacton. The concentraton of the sopropanol centralzes after the central matrx, whch shows that the dffuson of components s slow n that regon. The temperature also centralzes n the same locaton. The dstrbutons of the temperature and velocty along the x-drecton at Y/L=1/2 wth porosty of 0.79 are shown n Fg.4. The change of temperature s larger before the central matrx than after that locaton. The ncrease of the nlet velocty has lttle effect on the x-drecton velocty before or after the central matrx, but causes the maxmum temperature to move from the front of the central matrx to the back. Those prove that the natural dffuson s predomnant at the lower velocty regon. So the temperature or the concentraton of sopropanol s hgher n that locaton. The effects of the process varables ncludng the mole rato(r), nlet velocty (U), and porosty (ε) on the temperature feld and converson are studed. The dstrbutons of the converson wth the nlet velocty at dfferent porostes are shown n Fgure 5. The ratos of the hy- Fg.3 Contour plots for (a) velocty, (b)hydrogen concentraton, (c) sopropanol concentraton, (d) temperature

5 46 J. Therm. Sc., Vol.22, No.1, 2013 drogen and acetone are 1 and 2, respectvely. The nlet velocty vares from 0.01 to 0.1. It s seen evdently from the profle that the maxmum value of the converson s obtaned when the mole rato of hydrogen/acetone s hgher at the same porosty n the research range. The excess of the hydrogen causes the converson ncreasng. the nlet velocty enhances the transport of the product. Thus, the converson rses as the ncrease of the nlet velocty. However, the tme of the reactants stayng on the surface decreases when the nlet velocty becomes hgher, whch nduces the converson fallng when the nlet velocty s hgher. The nlet velocty correspondng to the peak converson becomes dfferent when the porosty s dfferent at the same mole rato. The converson rses as the decrease of the porosty at hgher velocty because of the larger reacton surface. Nevertheless, the converson decreases as the decrease of the porosty at lower velocty. Though the reacton area s larger at lower porosty, the dstance of the reacton surface becomes shorter. That counterworks the natural dffuse of the product of sopropanol, whch s predomnant at lower velocty. So the converson decreases at the lower velocty and porosty. Fg.4 Dstrbutons of (a) temperature and (b) velocty at Y/L=1/2 Fg.6 Varatons of temperature rate at the outlet wth nlet velocty under dfferent porostes Fgure 6 descrbes the varaton of T out /T n at dfferent mole ratos, veloctes and porostes. The changes of the temperature are smlar to the converson wth the nlet velocty. The heat produced by the reacton s correspondng to the converson of the reacton. Conclusons Fg.5 Varatons of converson rate at the outlet wth nlet velocty under dfferent porostes Besdes, the converson has a peak wth the rse of the nlet velocty. When the velocty s lower, the dffuson nduced by the concentraton dfference s predomnant before the peak pont of the converson, whch causes the product concentratng on the reacton surface and restrans the reversble chemcal reacton. The ncrease of The lattce Boltzmann method s ntroduced to study the exothermc reacton of acetone hydrogenaton over the catalyst, whch s an mportant process n a chemcal heat pump. The fractal theory s used to construct the porous meda catalyst. Some well-known cases are conducted to verfy the lattce Boltzmann code. The smulaton results show that the LBM can be developed to smulate the mult-component flow, heat and mass transfer couplng wth the chemcal reacton. In summary, the nlet velocty has a great effect on the temperature and concentraton felds. The locaton of the hghest temperature and concentraton of sopropanol changes from the front to the back of the central matrx

6 Xn Fang et al. Smulaton of Gas Exothermc Chemcal Reacton n Porous Meda Reactor wth Lattce Boltzmann Method 47 as the velocty rsng. The temperature and converson have a peak as the rse of the velocty. Furthermore, the excess of the hydrogen can mprove the converson. However, the converson s lower wth lower porosty at lower velocty, and s hgher wth lower porosty at hgher velocty. The LBM s sutable for studyng the flow, heat and mass transfer couplng the chemcal reacton n the porous meda reactor; and t revealed the complex mechansm of the mult-feld couplng n exothermal reactor. Acknowledgment Ths work was supported by the Natonal Natural Scence Foundaton of Chna (Grant No ) and the Natonal Basc Research Program of Chna (Grant No. 2011CB710705). References [1] Ajah, A.N., Mesbah, A., Grevnk, J., Herder, P.M., Falcao, P.W., and Wennekes, S., 2008, "On the Robustness, Effectveness and Relablty of Chemcal and Mechancal Heat Pumps for Low-Temperature Heat Source Dstrct Heatng: A Comparatve Smulaton-Based Analyss and Evaluaton" Energy, 33(6), pp [2] Basagaoglu, H., Meakn, P., Succ, S., and Welhan, J., 2006, "Boundary Effects on the Onset of Nonlnear Flow n Porous Domans", Europhyscs Letters, 73(6), pp [3] Succ, S., Fot, E. and Hguera, F., 1989, "Three-Dmensonal Flows n Complex Geometres wth the Lattce Boltzmann Method", Europhyscs Letters 10(5), pp [4] Spad, M.A.A. and Phelan, F.R., 1997, "Lattce Boltzmann Methods for Modelng Mcroscale Flow n Fbrous Porous Meda", Physcs of Fluds, 9(9), pp [5] Alazm, B., and Vafa, K., 2001, "Analyss of Flud Flow and Heat Transfer Interfacal Condtons between a Porous Medum and a Flud Layer", Internatonal Journal of Heat and Mass Transfer, 44(9), pp [6] Chen, S., Dawson, S.P., Doolen, G.D., Janecky, D.R., and Lawnczak, A., 1995, "Lattce Methods and Ther Applcatons to Reactng Systems", Computers & Chemcal Engneerng, 19(6-7), pp [7] Huber, C., Chopard, B. and Manga, M., 2010, "A Lattce Boltzmann Model for Coupled Dffuson", Journal of Computatonal Physcs, 229(20), pp [8] Kao, P.H., Ren, T.F. and Yang, R.J., 2007, "An Investgaton nto Fxed-Bed Mcroreactors Usng Lattce Boltzmann Method Smulatons", Internatonal Journal of Heat and Mass Transfer, 50(21-22), pp [9] Yu, H., Luo, L. and Grmaj, S.S., 2002, "Scalar Mxng and Chemcal Reacton Smulatons Usng Lattce Boltzmann Method", Internatonal Journal of Computatonal Engneerng Scence 01(3), pp [10] Succ, S., 2001, The Lattce Boltzmann Equaton for Flud Dynamcs and Beyond, Clarendon Press, Oxford. [11] Peng, Y. and Schaefer, L., 2006, "A Thermal Lattce Boltzmann Two-Phase Flow Model and Its Applcaton to Heat Transfer Problems - Part 1. Theoretcal Foundaton", Journal of Fluds Engneerng-Transactons of the ASME, 128(1), pp [12] Sukop, M.C. and Thorne, D.T., 2006, Lattce Boltzmann Modelng: An Introducton for Geoscentsts and Engneers, Sprnger, Berln.