FUNDAMENTALS OF CHEMISTRY Vol. II - Microkinetics Versus Macrokinetics: The Role of Transport Phenomena in Determining Reaction Rates - Rota R.

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1 MICROKINETIC VERU MACROKINETIC: THE ROLE OF TRANPORT PHENOMENA IN DETERMINING REACTION RATE Rota R Poltecnco d Mlano, Italy Keywords: reacton rate, chemcal knetcs, transport phenomena, mcroknetcs, macroknetcs, multphase reactons, heterogeneous reactons, flud flud homogeneous reactons, effectve reacton rate, effectve actvaton energy, effectve reacton order, nterphase dffuson, ntraphase dffuson, nterphase effectveness, ntraphase effectveness, consecutve reactons, parallel reactons, smultaneous reactons, yeld, selectvty, characterstc tme, Damköhler number, Hatta number, Thele modulus, Bot number. Contents 1. ummary of Basc Concepts 1.1. Rates of Reacton and Producton 1.2. Mass-Transfer Rates 1.3. Mcroknetcs and Macroknetcs 1.4. A mple Example of Mcroknetcs versus Macroknetcs: Frst-Order Heterogeneous Reactons 2. Flud old Interphase Dffuson and Reactons 2.1. Frst-Order ngle Irreversble Reacton 2.2. Postve-Order ngle Irreversble Reactons 2.3. Negatve-Order ngle Irreversble Reactons 2.4. Frst-Order Irreversble Consecutve Reactons 2.5. multaneous Irreversble Reactons of nth Order 2.6. Frst-Order Irreversble Parallel Reactons 3. Intraphase Dffuson and Reactons 3.1. Intraphase Dffuson and Reactons n Partcles and Drops 3.2. Interphase and Intraphase Dffuson and Reactons n Partcles and Drops 4. Flud Flud Homogeneous Reactons 4.1. Irreversble Pseudo-Frst-Order Reactons Fast Reactons low Reactons Glossary Bblography Bographcal ketch ummary After a short account of the man concepts related both to chemcal reacton rate and transport phenomena, a smple example s used to ntroduce the concepts of mcroknetcs and macroknetcs. tuatons where mass transport and reactons proceed n seres are then dscussed, wth partcular reference to flud sold nterphase dffuson and reactons. In ths context, the concepts of nterphase effectveness and of the

2 Damköhler number are ntroduced. Moreover, the effects of mass-transfer rate on both yeld and selectvty are dscussed. tuatons where dffuson and reactons proceed smultaneously wthn a phase are then presented for both partcles and drops, ntroducng both the Thele modulus and the concept of ntraphase effectveness. multaneous external and nternal mass transfer processes n partcles and drops are dscussed through the Bot number, as well as n terms of overall effectveness. Fnally, flud flud homogeneous reactons are dscussed as examples of smultaneous dffuson processes and reactons nsde the body of a reactng phase. In ths context, the Hatta number s also ntroduced. 1. ummary of Basc Concepts Ths secton summarzes some basc concepts, concernng both reacton rates and transport phenomena, that wll be used n the followng materal. A more detaled dscusson of reacton rates s gven elsewhere (see Rate of Chemcal Reactons: Ther Measurement and Mathematcal Expressons), whle fundamentals of transport phenomena can be found n the materals lsted n the relevant bblography Rates of Reacton and Producton When a chemcal reacton takes place n a system, t s usually possble to characterze the change of the moles of each compound usng a stochometrc equaton. Ths balanced reacton accounts for the rearrangement of atoms n varous molecules, allowng for the numbers of each atomc speces to be conserved. Let us consder, for example, a smple reacton lke: CO + 2H2 = CH3OH (1) uch a reacton can be convenently wrtten n a dfferent way: 0= CO 2H2 + CH3OH (2) that s, n the general form: Nc = 1 0 = ν A (3) In ths relaton the sum s taken over all compounds A and defnes the stochometrc coeffcent, ν for each speces. Ths s the number assocated wth each compound, whch s assumed to be postve for products, negatve for reactants, and zero for nert compounds. Gven the followng vector of speces nvolved n the reacton n Eq. (2): [ CO H CH OH] T A = (4) 2 3 the assocated stochometrc vector s ν = [ ] T (5)

3 Ths vector provdes a lnk between the change n the moles of each speces due to the chemcal reacton, through the general relaton: n = n + ν λ (6) 0 Ths expresson, whch s known as the law of defnte proportons, defnes the extent of reacton, λ, usng the same dmensons as n. Consderng a closed system (that s, one that does not exchange mass wth the surroundngs: see Thermodynamc ystems and tate Functons) where the number of moles of each compound can only change as a result of a chemcal reacton, n s the number of moles of speces A at a gven tme, 0 whle n s the number of ts moles ntally present n the system. In vector form, Eq. (6) becomes: o n= n + νλ (7) The extent of reacton changes over tme, and s a natural reacton co-ordnate. Consequently, we can defne the rate of reacton as: dλ r ' = (8) dt Ths equaton defnes the rate of reacton as an extensve property of the system, snce λ tself has been defned as an extensve property. However, a more convenent defnton nvolves an ntensve property: 1 dλ r = (9) X dt X s a measure of the extenson of the system where the reacton proceeds. For nstance, for a homogeneous reacton X s the volume of the system, V, leadng to the tradtonal defnton of the reacton rate n mol m 3 s 1 : 1 dλ r = (10) V dt However, for multphase systems the reactons take place only when reactants can contact one another, such as on the partcle surface for flud sold reactons. In these stuatons, the nterfacal surface area avalable for reacton, A, s a better measure of the extent of the reacton system, and the reacton rate assumes the dmensons of mol m 2 s 1. Moreover, the nterfacal area s often ll-defned, snce t nvolves both the external partcle surface and the nternal pore surface. In ths case, an alternatve measure of the reacton system extenson s the weght of the sold, W, leadng to a reacton rate n mol kg 1 s 1. Obvously, all these defntons can be nterchanged readly usng constant parameter values characterstc of the reacton system.

4 The reacton rate s a property of the reacton, whle the producton rate, defned as the number of moles of a gven speces produced per unt tme and volume (or surface area, or sold weght), s a property of each speces: R 1 dn = (11) X dt Reacton rate and producton rate can be related to each other through the defnton of the extent of reacton Eq. (6) as follows: 1 dn 1 dλ R = = ν = ν r (12) X dt X dt Ths relaton provdes a mean of computng the reacton rate through measurng the change n the number of moles of a gven speces over tme: 1 dn r = ν X dt (13) For a homogeneous system wth constant volume X = V. Therefore, ths relaton becomes 1 dc r = (14) ν dt showng that the reacton rate s proportonal to the tme dervatve of the molar concentraton of each speces (C = n /V) through ts stochometrc coeffcent. For a reactant wth stochometrc coeffcent equal to one, Eq. (14) can be further smplfed to gve dc r = (15) dt whle for a product wth stochometrc coeffcent equal to one: dc r = (16) dt It s worthwhle stressng agan that these relatons provde the reacton rate when the change n moles or n molar concentraton s caused only by chemcal reactons. However, when dealng wth multphase systems, at least one of the speces nvolved has to move before reactng towards the nterfacal surface where the reactants meet one another; the reacton then proceeds. The rate of ths transport phenomenon can nfluence the overall producton rate. Two lmtng stuatons are often encountered n practce, namely homogeneous and heterogeneous reactons. In the frst stuaton, reactons proceed only n one phase and

5 nvolve several reactants, at least one of whch s orgnally present n a dfferent phase. Ths means that ths speces must be transferred nto the phase where reactons proceed before converson of reactants can begn. Ths s the typcal stuaton n gas lqud reactons: the gas-phase reactant must dffuse from the gas phase nto the lqud one where the reactons proceed before startng the converson of the reactants. Heterogeneous reactons usually take place at the nterface between two phases, most commonly at a catalytc surface (see Mechansm of Elementary Reactons n Gaseous and Lqud Phase; Mechansm of Heterogeneous Reactons). In ths case, reactants have to be transported to the nterface where the reactons proceed. Reacton rate depends on the state varables of the system, such as temperature and composton. (Apart from n some pecular cases, a dependence on pressure s usually mplct n the dependence on composton, whch s expressed as molar concentraton, mol m 3 : for more detals see Rate of Chemcal Reactons: Ther Measurement and Mathematcal Expressons.) In the case of an rreversble reacton, t s usually assumed that dependence on temperature and concentraton can be factorzed as: r = k( T) f( C) (17) When the dependence on concentraton s represented by a power law, ths s: r = k( T) C C C... (18) α β γ The exponents α, β, γ are termed the order of reacton wth respect to each speces, whle ther sum s called total order of reacton. The temperature dependence usually assumes an Arrhenus-lke form, that s: E kt ( ) = ko exp RgT R g s the perfect gas constant; E s the actvaton energy, whch can be computed from several measurements of reacton rate values at dfferent temperature, as follows: (19) dln( r) dln( k) E = Rg = Rg (20) d(1/ T) d(1/ T) The aforementoned concepts can easly be generalzed to a system where NR dfferent reactons proceed through consecutve, smultaneous, and parallel steps. Each reacton can be wrtten n a compact form smlar to Eq. (3) as: Nc 0 = ν j A j = 1... NR (21) = 1 where for each speces A a stochometrc coeffcent, ν j, s defned wth respect to each reacton j. Obvously, the stochometrc vector Eq. (5) becomes a stochometrc matrx and the lnk between the change of moles of each speces due to the chemcal reacton (Eq. (6) s:

6 0 NR j= 1 j j n = n + ν λ (22) In ths case, we defne for each ndvdual reacton ts extent of reacton, λ j. The extent of each reacton changes over tme, and allows defnton of the rate of each reacton as: r j 1 dλ j = (23) X dt In ths case X s also a measure of the extent of the system where the reactons take place. Reacton rates and producton rate can be related to one another, as n the case of a sngle reacton, as follows: NR R = ν r (24) j= 1 j j When several reactons proceed n a system, usually not all of them lead to the desred product. To nvestgate the success of a reacton network n producng the desred compound, t s useful to defne the yeld (Y) and selectvty () as follows: rate of producton of the desred product Y = (25) rate of converson of the most mportant reactant rate of producton of the desred product = (26) rateof producton of the undesred product The rate of converson s equal to mnus the rate of producton as defned prevously. Usually, both yeld and selectvty should be maxmzed Mass-Transfer Rates Mass transfer between phases s usually assocated, n at least one phase, wth dffuson. Ths s the movement, under the nfluence of a drvng force (usually a concentraton gradent or, more precsely, a gradent of chemcal potental), of a speces through a mxture. The presence of a concentraton gradent tends to move a speces n a drecton capable of reducng the concentraton gradent: that s, to make the concentraton unform. The rate of mass transfer s represented by the molar flux relatve to the nterface (assumed as a plane at rest): n other words, the number of moles of a speces crossng the nterface per unt surface area and tme, J, n mol m 2 s 1. The molar flux can be related to the concentraton gradent through the followng relaton, whch s known as the Fck frst law of dffuson: dc J D dx = (27) j

7 where a mxture of components and j s consdered. D j s the dffusvty of component n the mxture, whle x s the coordnate perpendcular to the surface across whch dffuson s occurrng. However, mass transfer through nterfaces s usually descrbed usng a mass-transfer coeffcent, K, n m s 1. The mass-transfer coeffcent s defned as the rate of mass transfer per unt surface area and concentraton dfference, leadng to the followng relaton for the molar flux: (, ) J = K C C (28) where C, s the concentraton of speces at the nterface. Assumng that the concentraton gradent s confned n a thn stagnant layer close to the nterface, the mass-transfer coeffcent can be consdered as the molecular dffusvty dvded by the effectve layer thckness, δ. Ths s the basc concept of the so-called flm model, as llustrated n Fgure 1 for the mass transfer of a speces from the gas phase to the lqud one. Fgure 1. chematzaton of the two-flm model Wth regard to ( C C) ln( C/ C) << 1, the mass-transfer coeffcent n the absence of chemcal reactons does not depend on the bulk concentraton of the dffusng speces. Its value depends both on the local flud dynamcs (for nstance, n the case of dffuson towards sold partcles t depends on bulk flud velocty and partcle sze), as well as on the physcal parameters of the flud, and t s usually measured expermentally. Emprcal dmensonless relatons are readly avalable n the techncal lterature, and allow the computaton of mass-transfer coeffcents n many dfferent stuatons. These relatons are usually expressed n terms of the herwood number, h = K dp/d, or tanton number, t = εk/u: for example

8 tc 2/3 udp = 0.81 ν 0.5 (29) where c=ν/d s the chmdt number, d P the partcle dameter, ε the volumetrc fracton of the flud phase, u the superfcal flud velocty, and ν the knematc vscosty. Ths relaton, whch s vald for mass transfer to partcles surface as far as 5<u d P /ν<50, clearly shows K s weak dependence on temperature. When dealng wth heterogeneous reactons, and when the sold s mpervous (that s, non-porous), one needs only to consder dffuson n the flud phase, and a sngle flm close to the sold surface s nvolved. Reactants dffuse from the flud towards the sold surface where the reacton proceeds, whle products have to dffuse back from the surface to the flud bulk. When dealng wth homogeneous reactons, however, two flud phases are nvolved, and one must consder a flm layer on both sdes of the nterface. It s clear from ts defnton that we must use a dfferent mass-transfer coeffcent for each sde of the nterface. Assumng steady-state condtons for the two flms, we may deduce a relaton between the two mass-transfer coeffcents (see also Fgure 1): (, ) (, ) J = K C C = K C C (30) lq lq lq gas gas gas It s assumed that, n almost all practcal stuatons, the speces are n equlbrum at the nterface. Consequently, the nterface concentratons are related each other by a thermodynamc equlbrum relaton (see Equlbrum n Multphase Reactng ystems): C = H C (31) α β,, Here α and β refer to the two phases, and the proportonalty constant H can depend on speces concentratons, temperature, and pressure. When speces concentraton dependence s dsregarded, the relaton above s known as the Henry law for supercrtcal gas lqud equlbrum, the Raoult law for subcrtcal vapor lqud equlbrum, and the Nernst law for lqud lqud equlbrum. In ths case, the nterface concentraton can be elmnated easly from Eq. (30), gvng an expresson for the molar flux as a functon of the bulk concentratons alone: 1 1 H gas lq = + lq gas ( ) J H C C K K (32) In the gas phase there s a smple proportonalty between partal pressure and concentraton: consequently, both can be used n the relatons above as long as the correct values for the coeffcents H and K are used Mcroknetcs and Macroknetcs

9 In heterogeneous systems, the two steps (mass transfer and chemcal reacton) usually proceed n seres. As a consequence, f the rate of dffuson from the bulk of the flud phases towards the nterface (a physcal phenomenon) s much hgher than the rate of the chemcal reacton (a chemcal phenomenon) the producton rate s entrely controlled by the chemcal knetcs, and the reacton rate computed through Eq. (13) provdes a measure of the true chemcal knetcs: n other words, of the chemcal phenomena alone. Ths rate s usually called mcroknetc. On the other hand, physcal phenomena also nfluence the producton rate when the dffuson velocty s not much hgher than the chemcal reacton rate, and Eq. (13) does not provde a measure of the rate of the chemcal phenomena alone. Ths rate s usually called macroknetc, and accounts for both chemcal and physcal phenomena. In the followng, for the sake of smplcty, we wll focus only on sothermal condtons. However, t s mportant to stress that the presence of thermal gradents (due, for nstance, to exothermc or endothermc reactons) can change system behavor sgnfcantly. The reader nterested n these pecular condtons s referred to materals lsted n the Bblography A mple Example of Mcroknetcs versus Macroknetcs: Frst-Order Heterogeneous Reactons When the overall transformaton requres several steps n seres, the rate of all the steps must be the same n steady-state condtons: that s roverall = r1 = r2 =... = rn (33) These rates depend on the concentraton values at the nterface, whch are usually unknown. When all the rates are frst-order wth respect to the speces concentraton, t s possble to deduce a smple expresson for the overall reacton rate; when nonlnear dependences are nvolved n the rate expressons, the overall rate becomes much more complcated. As an example, let us consder a smple rreversble reacton among a speces A 1 n the flud phase and a sold compound A 2 : A + A = A (34) gas sol gas The two steps n seres (dffuson of speces A 1 from the bulk of the flud phase to the gas sol sold surface and reacton between A 1 and A 2 ) can be shown as n Fgure 2, where C s the concentraton of speces A 1 n the bulk of the flud phase and C ts concentraton on the sold surface.

10 Fgure 2. ketch of a smple gas sold reacton: macroknetc condtons The rate of mass transfer of speces A 1 from the bulk to the surface can be computed as (unts mol m 2 s 1 ): physcal ( ) r = J = K C C (35) whle that of the frst-order chemcal reacton, also n ths case n mol m 2 s 1, s: r = r = kc (36) chemcal At steady state, the two rates must be the same, thus leadng to the followng relaton for the producton rate of speces A 1 : ( ) R = r = r = kc = K C C (37) chemcal physcal The surface concentraton can then be expressed as: C K = C K + k (38) In addton, the overall rate becomes: 1 R = kc = C = keffc 1 K + 1 k (39) Eq. (39) shows that when all the rates n seres are lnear wth respect to a sngle reactant concentraton the expresson of the overall rate s also lnear, and nvolves an effectve rate constant. In ths case, the two resstances (1/K and 1/k) can smply be

11 added, and Eq. (39) s a macroknetc rate equaton accountng for both physcal and chemcal rates. However, when the mass-transfer rate s much hgher than the chemcal reacton rate (that s, K>>k) the prevous relaton can be smplfed to gve: R kc (40) Ths means that the surface concentraton n ths case s almost equal to the bulk concentraton (see Eq. (38): ths s to be expected snce mass transfer s much faster than a chemcal reacton, thus allowng for prompt replacement of the speces A 1 consumed by reacton at the sold surface. Ths condton, whch s a mcroknetc one, s shown n Fgure 3. Fgure 3. ketch of a smple gas sold reacton: mcroknetc condtons Moreover, snce n ths case keff k, the measured temperature dependence of keff follows the usual Arrhenus law. However, when the chemcal reacton rate s much faster than the mass-transfer rate (that s, k>>k), Eq. (39) gves: ( 0) R KC = K C (41) whch shows that n ths case the surface concentraton of A 1 s close to zero (see also Eq. (38), snce the reactant s suddenly consumed as t reaches the sold surface. It should be noted that n ths case keff K and the measured values of keff are almost unaffected by temperature. 2. Flud old Interphase Dffuson and Reactons Ths secton dscusses the stuaton nvolvng heterogeneous reactons among reactants present n dfferent phases, as n the prevous example. In partcular, flud sold reactons wll be consdered where at least one reactant has to dffuse from the flud phase towards the surface of a nonporous sold, whch can convert the flud reactant to dfferent products: A + A = A + (42) flud sol flud Here the frst reactant (A 1 ) dffuses from the bulk of the flud phase, and only reacts

12 wth the second reactant (A 2 ) on the sold surface. In order to show the nfluence of mass-transfer lmtatons on the overall reacton rate, let us consder sothermal steady-state condtons where the physcal mass-transfer rate of the reactng compound must equal ts converson rate: Ja = R( C ) = r( C ) (43) In the relaton above, we have consdered the converson rate n mol m 3 s 1 ; consequently, we have multpled the molar flux, J, by the nterfacal surface area per unt volume, a, n m 2 m 3. Moreover, snce ths equaton refers only to the dffusng speces A 1 the subscrpt 1 has been dsregarded. As ndcated n Eq. (43), the producton rate depends on the surface concentraton of the dffusng speces, C. If we further assume that a smple power law of order n apples for the mcroknetcs of the surface reacton, ths relaton can be recast n the followng form: n Ja = Ka( C C ) = R = r = kc (44) It s clear that the maxmum reacton rate can be obtaned n the absence of any masstransfer lmtaton. In ths case, the surface concentraton of the reactant, C, approaches ts bulk value, C, and the macroknetc rate equaton equals the mcroknetc one. However, the presence of relevant mass-transfer lmtatons reduces the surface concentraton wth respect to the bulk one, and a reducton n the surface reacton rate results. In ths case, the macroknetc rate equaton dffers from the mcroknetc one TO ACCE ALL THE 38 PAGE OF THI CHAPTER, Vst: Bblography Brd R.B., tewart W.E., and Lghtfoot E.N. (2002).. Transport Phenomena. pp895, New York: Wley. [A forty-year-old book whch remans the man reference source for transport phenomena fundamentals.] Boudart M. (1968). Knetcs of Chemcal Processes. pp246, Englewood Clffs, NJ: Prentce-Hall. [Despte ts age, ths offers a clear exposton of the basc concepts of chemcal knetcs.] Butt J.B. (1980). Reacton Knetcs and Reactor Desgn. pp431, Englewood Clffs, NJ: Prentce-Hall. [A

13 clearly-wrtten book on knetcs wth an mportant chapter on macroknetc phenomena.] Carberry J.J. (2001). Chemcal and Catalytc Reacton Engneerng. pp642, New York: Dover Publcatons (McGraw-Hll Chemcal Engneerng eres: Internatonal tudent Edton).[One of the best books consderng flud sold reactons n the presence of mass-transfer lmtatons.] Fogler H.. (1999). Elements of Chemcal Reacton Engneerng. pp967, Englewood Clffs, NJ: Prentce- Hall. [A ddactc book, useful as a basc dscusson of macroknetc phenomena n chemcal reactors.] Levenspel O. (1999). Chemcal Reacton Engneerng. pp668, New York: Wley. [Offers a clear exposton of several macroknetc phenomena that are relevant to analyss of chemcal reactors.] Westerterp K.R., Van waaj W.P.M., and Beenackers A.A.C.M. (1984). Chemcal Reactor Desgn and Operaton. pp767, New York: Wley. [A very practcal book, wth many sectons consderng the nfluence of mass-transfer on reactor desgn.] Bographcal ketch Renato Rota was born on June and s marred wth two sons. He took hs degree n Chemcal Engneerng n 1986 at the Poltecnco of Mlano, Italy. From 1986 to 1988 he worked at the chemcal engneerng company namprogett.p.a. He obtaned a poston for a three-year Corso d Perfezonamento (.e., a Ph.D.) at the cuola Normale uperore d Psa, Italy, n He was a vstng student at the Purdue Unversty, IN, UA, n 1988/1989. From 1990 to 1998 he worked at the Poltecnco d Mlano as a researcher, dong research and teachng n several courses (e.g., Appled Physcal Chemstry; Thermodynamcs; Appled Chemcal Knetcs; and Fundamentals of Technology). nce 1998 he has been a professor of Chemcal Engneerng, Thermodynamcs, and afety and Relablty n the Process Industres at the Poltecnco d Mlano. In 1986 he won the Natonal Award ENIChem Pano govan, reserved for degree theses. and n 1996 the VIII Natonal Award Federchmca per un futuro ntellgente, reserved for chemcal researchers and professors. Hs scentfc work s summarzed n about 150 publcatons.