Reaction kinetics in chemical engineering

Transcription

1 Bugarian Chemica Communications Voume 5 Specia Issue K (pp ) Reaction inetics in chemica engineering Christo Boyadiev Institute of Chemica Engineering Bugarian Academy of Sciences 1113 Sofia Bugaria Submitted Juy 31 18; Revised October 7 18 In the paper is presented a theoretica anaysis of the roe of the reaction inetics in chemica engineering for the soution of the main probems in the chemica industry (biotechnoogy heat energy) i.e. the optima design of new devices and the optima contro of active processes. The thermodynamic and hydrodynamic approximations for the modeing of the industria process rates are presented and anayzed. The reation between the Onsager s inearity coefficient and mass transfer coefficient is presented. Keywords: reaction inetics optima design optima contro thermodynamic approximation hydrodynamic approximation. INTRODUCTION The main probems in the chemica industry (biotechnoogy heat energy) are the optima design of new devices and the optima contro of active processes i.e. minimization of the investment and operating costs. These probems are soved by chemica engineering with modeing and simuation methods [1]. Mathematica modes of processes in the chemica industry contain equations invoving variabes and parameters. Depending on the probem soved variabes can become parameters and vice versa. They are input mode variabes (X) output mode variabes (Y) and construction parameters (A): F X Y A (1) where F is a vector containing a the equations in the mode X(Y) are the vectors of the input (output) variabes that contain the fow rates and temperatures of the input (output) phase fows and the concentrations of their components A is the vector of the constructive parameters which contains the constructive parameters of the apparatuses. The soutions to the optima design probems of new apparatuses use agorithms where input mode variabes (X) and output mode variabes (Y) are set (as parameters) and optima construction parameters must be obtained (as variabes): A F1 X Y. () The probems of optima contro of current processes use agorithms where the output mode variabes (Y) and constructive parameters (A) are set (as parameters) and the optima input mode Address for correspondence: E-mai: chr.boyadiev@gmai.com variabes A are searched: X F Y A. (3) In the case of renovation (optima reconstruction) part of the input mode variabes and construction parameters are set. Optima design and contro in the chemica industry is uniquey reated to process rates so a mathematica descriptions of processes are ined to agorithms to determine these rates. THE PROCESSES RATES IN THE CHEMICAL ENGINEERING The processes in the chemica industry (biotechnoogy heat energy) are the resut of the deviation of the systems from their thermodynamic equiibrium []. One system is not in a thermodynamic equiibrium when the concentrations of the components (substances) and the temperature at the individua points in the phase voumes are different. These differences are the resut of reactions i.e. of processes that create or consume substance and (or) heat. The presented anaysis shows that processes in the chemica industry are resut of reactions that occur in the phase voume (homogeneous) or on the boundary between two phases (heterogeneous). Homogeneous reactions are generay chemica whie heterogeneous reactions are chemica cataytic physica and chemica adsorption interphase mass transfer in gas-iquid and iquidiquid systems (on the interphase surface the substance disappears from one phase and occurs in the other phase). The rates of these processes are determined by the reaction inetics which ies at the basis of modeing and simuation in chemica engineering and soving the basic probems in the chemica industry (biotechnoogy heat energy). 18 Bugarian Academy of Sciences Union of Chemists in Bugaria

2 MODELING AND SIMULATION The basics of modeing and simuation in chemica engineering as part of human nowedge and science are reated to the combination of intuition and ogic that has different forms in individua sciences [3]. In the mathematics the intuition is the axiom (unconditiona statements that cannot be proven) whie the ogic is the theorem (the ogica consequences of the axiom) but ogic prevais over intuition. In the natura sciences (physics chemistry bioogy) the "axioms" (principes postuates aws) are not aways unconditiona but ogic prevais over intuition too. The processes in chemica engineering tae pace in the industria apparatuses where gas iquid and soid phases move together or aone. They are described by variabes which are extensive or intensive. In the case of merging of two identica systems the extensive variabes are doubed but the intensive variabes are retained. The processes in the chemica engineering are the resut of a deviation from the thermodynamic equiibrium between two-phase voumes or the voume and phase boundaries of one phase and represent the pursuit of the systems to achieve the thermodynamic equiibrium. They are irreversibe processes and their inetics use mathematica structures derived from Onsager s principe of inearity []. According to him the average vaues of the derivatives at the time of the extensive variabes depend ineary on the mean deviations of the conugated intensive variabes from their equiibrium states. The principe is vaid near equiibrium and the proportionaity factors are the inetic constants. When the process is done away from equiibrium (high intensity processes) inetic constants become inetic compexes depending on the corresponding intensive variabes. MECHANISM OF INFLUENCE OF REACTION KINETICS In the chemica industry (biotechnoogy heat energy) processes tae pace in moving phases (gas iquid soid). Reactions (reaction processes) ead to different concentrations (and temperatures) in the phase voumes and the phase boundaries. As a resut hydrodynamic processes diffusion mass transfer and heat conduction are oined to the reaction processes. Under these conditions there are various forms of mass transfer (heat transfer) that are convective (as a resut of phase movements) and diffusion (as a resut of concentration (temperature) gradients in the phases). Convective mass transfer (heat transfer) can be aminar or turbuent (as a resut of argescae turbuent pusations).diffusion mass transfer (heat transfer) can be moecuar or turbuent (as a resut of sma-scae turbuent pusations). Mathematica modes of industria processes aim at determining the concentration of substances (fow temperatures) in the phases. Mathematica modes represent a materia (therma) baance in an eementary (sma) phase voume that is equivaent to a mathematica point. Components in this baance are convective mass transfer (heat transfer) diffusion mass transfer (heat transfer) and homogeneous reactions (heat effect of reactions). Heterogeneous reactions tae part in the boundary conditions of the equations in mass transfer (heat transfer) modes. On this basis modes of cassica theory of mass transfer were created. THEORY OF MASS TRANSFER The modern theory of mass transfer is based on the theory of the diffusion boundary ayer [4]. This approach repaces (physicay ustified) eiptic partia differentia equations with paraboic partia differentia equations which faciitates their mathematica soution and offers a mathematica description of physica processes with free (not predetermined) ends. The theory of the diffusion boundary ayer deveops in the case of drops and bubbes [5] fim currents [6] non-inear mass transfer and hydrodynamic stabiity [7 8]. The modeing of chemica engineering processes has two eves of detai - thermodynamic and hydrodynamic approximation. Thermodynamic approximation The processes in chemica engineering are the resut of a deviation from the thermodynamic equiibrium between two-phase voumes or the voume and phase boundaries of one phase and represent the pursuit of systems to achieve thermodynamic equiibrium. They are irreversibe processes and their inetics use mathematica structures derived from Onsager's principe of inearity. According to him the average vaues of the derivatives at the time of the extensive variabes 113

3 depend ineary on the mean deviations of the conugated intensive variabes from their equiibrium states. The principe is vaid cose to equiibrium and the Onsager's inearity coefficients are inetic constants. When the process is done away from equiibrium (high intensity processes) inetic constants become inetic compexes depending on the corresponding intensive variabes. The thermodynamic approximation modes cover the entire voume of the phase or part of it. Hydrodynamic approximations The hydrodynamic eve uses the approximations of the mechanics of continua where the mathematica point is equivaent to an eementary physica voume which is sufficienty sma with respect to the apparatus voume but at the same time sufficienty arge with respect to the intermoecuar voumes in the medium. In this eve the moecues are not visibe as is done in the next eve of detai of Botzmann. The modes of the hydrodynamic approximations are possibe to be created on the basis of the mass (heat) transfer theory whose modes are created by the modes of the hydrodynamics diffusion therma diffusion and reaction inetics using the ogica structures of three main axioms reated with the impuse mass and heat transfer: 1. The postuate of Stoes for the inear reationship between the stress and deformation rate which is the basis of the Newtonian fuid dynamics modes;. The first aw of Fic for the inear reationship between the mass fow and the concentration gradient which is the basis of the inear theory of the mass transfer; 3. The first aw of Fourier for the inear reationship between the heat fux and the temperature gradient which is the basis of the inear theories of the heat transfer. These are the aws of the impuse mass and energy transfer. In Botzmann's inetic theory of the idea gas these are three "theorems" that derive from the axiom of the "eastic shoc" (in a shoc between two moecues the direction and the veocity of the movement change but the sum of their inetic energies is retained i.e. there is no oss of inetic energy) and the rate coefficients are theoreticay determined by the average veocity and the average free run of the moecues. Rate of thermodynamic processes In the chemica engineering the Onsager s inearity principe is used to determine the mass transfer rate in one phase or between two phases where the mass of the transferred substance is an extensive quantity but its concentration is an intensive quantity. The mass m [g-mo] of a substance dissoved in the phase voume and its derivative at the time (its rate of change over time) [g-mo.s -1 ] depends ineary on the mean deviation from the thermodynamic equiibrium c c [g-mo.m -4 ] where c [g-mo.m -3 ] is the concentration of the dissoved substance in a point in the phase c is its equiibrium concentration on the phase boundary [m] is the distance between this point and the phase boundary i.e. J dm dt c c J. (4) is the mass fow through the surface [m ] where [m 4.s -1 ] is the Onsager s inearity The mass fow per unit surface J = J /s [gmo.m -.s -1 ] is obtained directy J c c (5) where [m.s -1 ] is the mass transfer rate s This is a fundamenta resut representing the reationship between the thermodynamic inetics (the Onsager s inearity principe) and the mass transfer inetics where the mass transfer coefficient is proportiona to the Onsager s inearity In a two-phase system (gas-iquid iquid-iquid) the concentrations in phase voumes are ci i 1 and at the phase boundary there is aways a thermodynamic equiibrium of the dissoved substance in the two phases c c where s 1 c i i 1 are the equiibrium concentrations of the dissoved substance. This is the aw of Henry and is the number of Henry (in iquid-iquid systems this is the distribution coefficient). In this case the reaching rate of the thermodynamic equiibrium (Onsager s inearity principe) is c1 c1 c c J 1. (6) 1 The mass fow in this case is 114

4 J c c c c (7) where i i i1 [m.s -1 ] are the mass transfer s i rate coefficients. Simiary the veocity of the interphase mass transfer can be expressed: c1 J K1 c1 c K c (8) where [m.s -1 ] are the interphase mass transfer rate coefficients: K i1 i K K. (9) An anaogous resut can aso be obtained in the case of heat conduction in the presence of temperature differences. The Onsager's principe of inearity is the thermodynamic approximation of the mathematica description of the inetics of the compex irreversibe processes but it does not show the way to achieve the equiibrium (the mechanism of the process) an as a resut the veocity coefficients are unnown. Obviousy this "thermodynamic eve" dc dx The condition I J maes it possibe to determine the diffusion mass transfer coefficient D. This is a fundamenta resut that ins the thermodynamic and hydrodynamic approximation in determining the read of the industria processes i.e. the reationship between Onsager's inearity principe and Fic's first aw where Fic's first aw is a consequence of Onsager's principe of inearity in diffusion processes. An anaogous connection can aso be obtained with the first Fourier aw. does not aow for a rea quantitative description of the reaction inetics of irreversibe processes in chemica engineering and the use of the next eve of description detai the so-caed "hydrodynamic eve" is required. Reationship between thermodynamic and hydrodynamic approximation The presence of concentration differences in phase voumes (as a resut of the reactions) eads to moecuar diffusion and the diffusion mass transfer fow [g-mo.m -.s -1 ] is determined by the first aw of Fic: I I Dgrad c. (1) where [m.s -1 ] is the moecuar diffusion In the case of one-dimensiona diffusion D I D dc dx where c is the soution of the equation: dc ; x c c ; x c c. dx As a resut is obtained c c c c c c c c c c x I D I D. (11) (1) (13) This approach wi be iustrated in the cases of mass transfer in fowing iquid fims and one-phase and two-phase diffusion boundary ayers. Liquid fim fows Let's considers the absorption of sighty soube gas in a aminar iquid fim fow on a vertica fat surface y. The equation of convectiondiffusion has the form [1]. g c c c h y y D ; x x y c x c c; x c c ; y ; y h c c. y (14) There is a thermodynamic equiibrium at the surface y h of the fim and c is the equiibrium concentration. The soid surface y h is impermeabe to the diffusion substance whose concentration at the inet is c < c (absorption). A fim with ength wi be considered. The thicness of the diffusion boundary ayer is ess than the thicness of the iquid fim h which aows the diffusion boundary ayer approximation to be used. As a consequence of this approach the foowing generaized variabes can be introduced: x X y h Y c c ( c c ) C (15) where h and h. The introduction of the generaized variabes eads to 115

5 116 C D C C 1 Y h X uav X Y (16) where gh D D D uav Fo 1 Pe 1. (17) uav h uavh uav In these equations is the fim average veocity diffusion boundary ayer approximation ( 1 - Fourier number) - Pecet number. In 1 Pe ) i.e. these conditions the probem must be soved in 1 C C FoY ; X C ; Y C 1; Y C. (18) X Y Fo u av The mass transfer rate in a fim of ength is the average vaue of the oca mass fux through the fim surface ( ). It can aso be represented by the mass transfer coefficient i.e. D c J dx ( c c ). y (19) yh The resuting expression aows the determination of the Onsager s inearity coefficient after soving the convection-diffusion equation (18). The introduction of the generaized variabes eads to: Pe y h As a resut the average vaue of the mass fux through the fim surface is: 1 u J x J xdx c c av c c s D (3) from where the reation between of the Onsager s inearity coefficient and the mass transfer coefficient is obtained: C Sh Pe dx. D Y () Y In (18) is a sma parameter and the perturbation method must be used. As a resut is obtained: 6Pe Fo 19Fo Sh 1 (1) 6 1 where is the Sherwood number and represent the dimensioness form of the mass transfer The thicness of the diffusion boundary ayer varies aong the fim ength and as a resut the Onsager's inearity principe has the form: Dx av Sh c c uav J x x J x c c. () x u s Dx s D (4). uav One-phase diffusion boundary ayers u u u u v c c c u v u v D ; x y y x y x y y Fo The interphase mass transfer in the gas (iquid) soid systems is reaized at a fixed phase boundary. The fow on a smooth semi-infinite pate of a potentia fow with a constant veocity wi be considered. The substance of the soid phase proceeds with a concentration and on the soid surface y = its equiibrium concentration is c. Depending on the sign of the concentration difference (c - c ) there is a process of deposition (crystaization) or dissoution. In this case the veocity and the concentration satisfy the equations of the aminar boundary ayer and the diffusion boundary ayer [1]: x u u c c; y u v c c ; y u u c c. The mass transfer rate in a diffusion boundary D c ayer of ength is the average vaue of the oca J mass fux through the soid surface ( y ). It can y dx ( c c ). (6) y aso be represented by the mass transfer coefficient The soution of (5) permits to be obtained the i.e. Sherwood number: u 1 c (5)

6 5 3 Sh Pe Sc where () Re (7) D 3 1 ; The thicness of the diffusion boundary ayer varies aong the soid surface and as a As a resut the average vaue of the mass fux through the soid surface is: from where the reation between of the Onsager s inearity coefficient and the mass transfer coefficient is obtained: Two-phase diffusion boundary ayers The interphase mass transfer into the gas-iquid and iquid-iquid systems is reaized at a moving () is a soution of:eqs.(8): 1 () ) () ) () 1 ( ) ( ). (8) resut the Onsager's inearity principe has the form: c c u J x x J x c c. (9) x u s Dx Dx 1 u J x J xdx c c c c (3) s D s D. (31) u interphase boundary. In the approximations of the boundary ayer theory the processes are described by the foowing set of Equation [1]: u u u u v c c c u v u. v D x y y x y x y y u1 u x u u c c ; y u1 u 1 y y c1 c c1 c D1 D v 1; y y y u1 u1 c1 c1 ; y u u c c. The index for the first phase 1 is for where K 1 are the interphase mass transfer gas or iquid and for the second phase coefficient and 1 are mass transfer is for iquid. At the phase boundary there is coefficient in the phases. thermodynamic equiibrium and is the The oca mass fuxes are Henry s number (in iquid-iquid systems is the c I D 1 (34) distribution coefficient). y y The average rate of the interphase mass and the Sherwood numbers are: transfer through a phase boundary with a ength 1 K c is simiary determined: Sh dx 1. D c1 c 1 y y J K1 c1 c I1dx 1( c1 c1 ) (35) From (33) and (35) is possibe to be obtained: c1 1 K c Idx ( c c) c1 c (33) K1 1 ; K1 1; K ( 1) ; K. (36) At the end for the Sherwood numbers is u obtained: Sh Pe () Pe D 1 (3) (37) 117

7 where () 1 is the soution of the equations set: 1 ; 1 1 () ( ) ( ) 1; 1 1() 1 () () 5 1() 1 () () () () 1. (38) The thicness of the diffusion boundary ayers changes aong the interphase surface and as a resut the Onsager's inearity principe has the form: c c D x u J x x J x c c 1. (39) x u s D x As a resut the average vaues of the mass fuxes through the interphase surfaces are: 1 u J x J x dx c c c c 1 s D (4) from where the reation between of the Onsager s inearity coefficients and the mass transfer coefficients are obtained: s D 1. (41) u CONCLUSION The main probems in the chemica industry (biotechnoogy heat energy) are the optima design and optima contro of the industria processes using the aws of the reaction inetics. Industria processes are the resut of reactions i.e. creation or disappearance of a substance and (or) heat as a resut of chemica and (or) physica processes and their rate is determined by the reaction inetics. The reactions deviate the systems from the thermodynamic equiibrium and as a resut processes arise who are trying to restore that equiibrium. The rate of these processes can be determined by Onsager's "inearity principe" where the rate of the process depends ineary on the deviation from the thermodynamic equiibrium. The Onsager s inearity coefficient can be determined after soving the hydrodynamics mass transfer and heat transfer equations where it is proportiona to the mass transfer (heat transfer) The dimensioness form of the mass transfer coefficient is the Sherwood number. REFERENCES 1. Chr. Boyadiev Theoretica Chemica Engineering. Modeing and simuation Springer-Verag Berin Heideberg 1.. J. Keizer Statistica Thermodynamics of Nonequiibrium Processes Springer-Verag New Yor Chr. Boyadiev Open Access Library Journa 6 1 (14). 4. L.D. Landau E.M. Lifshitz Fuid Mechanics Pergamon Oxford V.G. Levich Physicochemica Hydrodynamics Prentice Ha New Yor Chr. Boyadiev V. Beschov Mass Transfer in Liquid Fim Fows Pub. House Bug. Acad. Sci. Sofia V.S. Kryov Chr. Boyadiev Non-inear Mass Transfer (in Russian) Edition of the Institute of Thermophysics Siberian Branch of the Russian Academy of Sciences Novosibirs Chr.B. Boyadiev V.N. Baba Non-Linear Mass Transfer and Hydrodynamic Stabiity Esevier New Yor. 118

8 РЕАКЦИОННАТА КИНЕТИКА В ХИМИЧНОТО ИНЖЕНЕРСТВО Христо Бояджиев Институт по инженерна химия Българска академия на науките 1113 София Постъпила на 31 юли 18 г.; Коригирана на 7 октомври 18 г. (Резюме) В работата е представен теоретичен анализ на ролята на реакционната кинетика в химичното инженерство за решаване на основните проблеми в химичната промишленост (биотехнологиите топлоенергетиката) т.е. оптимално проектиране на ново оборудване и оптимално управление на действащи процеси. Представен е анализ на термодинамичното и хидродинамичното приближение при моделирането на скоростта на промишлените процеси. Представена е връзката между коефициента в принципа на линейността на Онзагер и коефициента на масопренасяне. Ключови думи: реакционна кинетика оптимално проектиране оптимално управление термодинамично приближение хидродинамично приближение 119