GENERAL EQUATIONS OF PHYSICO-CHEMICAL

GENERAL EQUATIONS OF PHYSICO-CHEMICAL PROCESSES Causes and conons for the evoluton of a system... 1 Integral formulaton of balance equatons... 2 Dfferental formulaton of balance equatons... 3 Boundary conons... 5 Consttutve relatons... 5 Causes and conons for the evoluton of a system Tme evoluton (or advance or progress) of a materal system s dctated by two possble causes: a) Internal, due to ntal conons of non-equlbrum. b) External, due to oundary conons of non-equlbrum. Tme evoluton s constraned by some restrctons (laws of Nature), best expressed for an solated system (be t consdered a contnuum or an ensemble of partcles) as: 1. In every real evoluton of an solated system, there s a state functon called entropy, S=ΣSk(Uk,Vk,nk), measurng the nternal dstrbuton of the conservatve functons, that ncreases wth tme, ds/>0 (=0 only n the state of equlbrum). As a consequence of equlbrum, the senstvty of that functon to the ndependent varables cannot be dfferent from one place to the other and, n absence of external felds, temperature, pressure, and the chemcal potental, µ, of each chemcal speces, must be unform wthn the system. The fundamental equaton that relates all consttutonal varables at equlbrum s: C 1 p µ ds = du + dv dn (0) T T T = 1 where temperature T measures the 'escapng force' of thermal energy, pressure p measures the 'escapng force' of mechancal energy, and the chemcal potental m measures the 'escapng force' of chemcal energy. As a consequence of the tendency to equlbrum, t can be demonstrated that some varables and materal propertes are bounded (e.g. at equlbrum T>0, p>0, cp>0, cv>0, cp>cv, κ>0, and durng evoluton k>0, µ>0, D>0), and, from the bunch of propertes and functons used (ρ, p, T, u, h, s, etc.), only a few can be ndependently changed (e.g. only two, for smple compressble system), the others becomng functons of these varables. 2. In every real evoluton of an solated system, one can fnd conservatve components, n=σnk (molecules n physcal evolutons, atoms n chemcal evolutons, more elementary partcles n nuclear evolutons), such that dn/=0 for every speces. Sometmes, ths law s replaced by mass conservaton,.e. dm/=0, wth m=mn and M beng the molar mass of speces. Mass s so lttle varant wth atomc nteractons that t can be taken as a constant; n realty, t s always m= E/c 2, but m<<m n most nteractons. General equatons of physco-chemcal processes 1

3. In every real evoluton of an solated system, there s a state functon called (lnear) momentum, p= mr, whch does not change wth tme. k k 4. In every real evoluton of an solated system, there s a state functon called angular momentum, L = r m r, whch does not change wth tme. Ths law becomes redundant f the prevous one k k k s appled to nfntesmal parts of the system, as done n a contnuum. 5. In every real evoluton of an solated system, there s a state functon called energy, E, measurng all nternal motons and nteractons n a certan ntegral way, ndependent of the orgn of tme, whch does not change wth tme. 6. In every real evoluton of an solated system, the total electrc charge does not change wth tme. But we dsregard electrc and magnetc effects altogether. These unversal laws for an solated system can be grouped as: Conservaton laws: there are a few magntudes (accordng to Noether's theorem of correspondence between contnuous symmetres and conservaton laws), let us name them as Φ, whch cannot change wth tme n solated systems,.e. dφ/=0. Equlbrum laws: after suffcent tme, all magntudes of an solated system dstrbute n just one manner, called equlbrum dstrbuton, such that the entropy functon gets a maxmum value, whch for a sngle-phase system n absence of external forces mples that all magntudes (mass, speces, energy, speed) dstrbute unformly,.e. dφ dx = 0. Knetc laws: n the evoluton, the rate of change of a magntude s proportonal to the ds-equlbrum amount; e.g. the flux of thermal energy s proportonal to the temperature gradent. Consttutve laws, relatng the equlbrum and knetc propertes of matter (e.g. densty, thermal capacty, conductvty, vscosty...) to the macroscopc state of a system. For non-solated systems, these unversal laws are expressed n the form of balance equatons, ascrbng every nput to the followng common-language budgetary terms: accumulaton, producton (generaton-consumpton), or flux (nput-output), such that accumulaton producton + net flux n, ether n ntegral form (amounts) or n dfferental form (rates of change). Conservatve magntudes are those wthout nternal producton term (all of them may be conservatve, except for entropy and ts derved functons). Integral formulaton of balance equatons For a control mass,.e. a closed system, mpermeable to matter but permeable to energy: Magntude accumulaton producton (dffusve) flux dm mass (total) = 0 +0 (1) speces mass momentum dm d( mv ) = m, gen +0 (2) = mg fda (3) General equatons of physco-chemcal processes 2

energy entropy d( me) d( ms) = Egen q nda + f vda (4) = Sgen q n da T (5) In (2), a mass-producton term has beng ncluded, m, gen, namely the mass producton of speces nsde the system by chemcal reactons (t can be the source or snk, cancellng n the whole, and t s usually related to other varables by means of Arrhenus' law, see below). In (3), a momentum-producton term has beng ncluded, mg, due to the effect of an external volumetrc force feld (represented as a constant gravty feld), and f s the external force per unt area at the fronter. In (4), an energy-producton term has beng ncluded, E gen, as a convenent term to separately account for changes n nternal energy of non-thermal orgn, lke mechancal dsspaton by vscosty, electrcal dsspaton by Joule effect, chemcal dsspaton, phase changes, mxng enthalpy, etc.), and q s the energy flux n the fronter due to a temperature dfference (the heat transfer by molecular dffuson (and electrons flow n metals), and t s usually related to other varables by means of Fourer's law, see below). In (5), the entropy-generaton term, S gen, s an unknown n real processes. It s assume to be zero n the thermodynamc lmt of deal non-dsspatng processes, and can be computed n terms of the transport coeffcents and the detaled nternal evoluton. It can be emprcally computed wth Eq. (5) f all other varables are measured. Besdes, for prelmnary desgn purposes, ths entropy producton term can be approxmated f data from smlar processes allow a generc behavour n terms of sentropc effcences (emprcal ratos between real and deal processes). To pass from ths control-mass formulaton to one for open systems, frst the model of a contnuum system (feld varables) s ntroduced, and them use s made of two mathematcal theorems: the Reynolds Transport Theorem to pass from a closed system entraned by the flow to a permeable system, d CM d t ( t ) d V ( v v A ) na Φ = Φ + Φ d V, and the Gauss-Ostrogradsk Theorem to transform surface A ntegrals at the boundary nto volume ntegrals n the doman, Ψ na d = ΨdV A. All ntegrals now V beng volume ntegrals, they are appled to a generc elemental volume and one gets what the typcal Euleran descrpton of the system, where every functon, e.g. T( xt,), represents the value of temperature at poston x and any tme t, although n some cases t may be convenent to use a Lagrangan descrpton of the system, Txx ( (,),) t t, where varables refer to movng materal partcles (movng wth the flow). 0 Dfferental formulaton of balance equatons For a unt control volume, at every pont n the spatal doman boundng the system: Magntude accumulaton producton dffusve flux convectve flux ρ mass = 0 +0 ( ρv) (6) General equatons of physco-chemcal processes 3

speces momentum energy entropy ρ ( ρv) e s = w j ( ρ v ) (7) = ρg + = τ ( ρvv) (8) = e gen q ( + τ v ) ( ev) (9) = s q gen T ( sv) (10) where ρ s the mass-densty of speces (mass of speces per unt volume of mxture), although equvalent equatons can be formulated n terms of the mass fracton, y m/m, the mol fracton, x n/n, or the concentraton, c n/v, of the speces, all of them related by (M s the molar mass of speces ): m ρ M y = = x = Mc m ρ xm (11) Other terms ntroduced n (6-10) are the stress tensor τ (such that the force over an elementary area wth external normal n s: f = τ n = p n+ τ ' n (12) wth p beng pressure and τ ' the vscous part of τ. Besdes, the denstes for accumulated energy, e, for generated energy, e gen, for accumulated entropy, s, and for generated entropy, s gen, have been ntroduced. We nsst that n realty there s no energy generaton (energy s conservatve) and e gen refers to varaton n the nternal energy not accounted for n e, where we only account for thermal energy. Usually, for a smple system (sngle component), one takes as ndependent space-tme varables the local velocty, local pressure and local temperature, v ( x, t), p( x, t), T( x, t) (and composton for compound systems), and all the other functons are related to the former by consttutve laws (equatons of state, ρ(t,p), h(t,p), and transport equatons, τ = τ( vt,, p), q = qvt (,, p), v = v ( vt,, p), and w= w( vt,, p) ). d The energy balance s usually developed n terms of temperature (nstead of energy), what yelds, n absence of phase changes and reactons: d Magntude accumulaton producton dffusve flux convectve flux T Dp energy ρc p = τ ': v + αt q ρc pv T t Dt (13) that reads: the accumulaton of thermal-enthalpy equals the thermal-enthalpy producton due to frcton and work nput, plus the net heat conducton, plus the convected thermal-enthalpy (n absence of phase changes, speces dffuson, and chemcal reactons). General equatons of physco-chemcal processes 4

Boundary conons Of the utmost mportance for the evoluton of a system are the boundary conons, snce they usually are the man dfference between one problem and another (the nternal equatons been always the same). In many cases, however, the boundary conons are not drectly prescrbed, but must match those of a neghbour system. If [φ]=φnner φouter represents the jump n the value of a generc functon φ across the nterface, the boundary conons n the smplest case of a geometrcal boundary wthout other physcal propertes (no possblty of accumulaton and producton of any magntude at the nterface), are: Magntude local equlbrum no-flux conon mass 0 = [ µ ] 0 = [ v ] (14) speces 0 = [ µ ] 0 = ρd ( y + ct S ) (15) = 0 = τ n (16) 0 = k T n (17) momentum 0 [ v] energy 0 [ T ] = [ ] (where µ stands for the chemcal potental), although there are cases where one has to consder a net flux of mass (e.g. ablatve processes), a net flux of speces (e.g. accumulaton/depleton of tenso-actve speces at the nterface), a net flux of momentum (e.g. a normal and tangental forces due to capllarty), a net flux of energy (e.g. energy deposton by sold frcton at the nterface, radaton absorpton, nterface stretchng, etc.), and a net flux of entropy assocated to the above processes or due to the modellsaton (e.g. when one assumes unform temperatures, but dstnct at both sdes, an entropy producton term, ( k T n) / T, must be added). Consttutve relatons To complete the feld equatons one has to make explct (ether from emprsm or from knetc theory) the ntal conons, the boundary conons, and the consttutve relatons at equlbrum, whch are: Thermal) equaton of state: ρ=ρ(p,t,y). As an example, the deal gas equaton of state: ρ = p RT u 1 y M (18) Energetc equaton of state: h=h(p,t,y). As an example, the deal gas enthalpy: ( ) p (19) h= y h + c T T Chemcal equatons of state for each speces except one: µ=µ(p,t,y). As an example, for an deal gas mxture: p M µ = µ ( T, p ) + RT u ln + RT u ln y p M (20) plus the consttutve relatons for evoluton (transport equatons, that relate all dffusve fluxes to the forces that orgnate them, and chemcal knetcs): General equatons of physco-chemcal processes 5

Mass dffuson (generalsed Fck's law wth Soret effect): j = D + ct ( ρ / ρ) S Momentum dffuson (Stokes' law): c h F I HG K Jb g = = = τ = p I+ µ v + v T 2 ( ) µ µ v v I 3 (21) (22) Energy dffuson (generalsed Fourer's law wth speces dffuson). q = k T + ρ v h (23) d Chemcal knetcs (Arrhenus' law): w ' ν " ' F ρ = M ( ν ν ) H G Ba exp M I KJ F HG Ea RT I K J (24) where µ s the chemcal potental for speces, D s the coeffcent of mass dffuson due to concentraton gradents, cs s the Soret coeffcent of mass dffuson due to thermal gradents, µ and µv are the dynamc vscosty coeffcents (shear and volumetrc), v d s the dffuson velocty such that ρvd = D ρ, w s the mass of speces produced by unt of mass of the mxture and tme due to chemcal reactons, ν'' and ν' the stochometrc coeffcents for the forward and backward reacton consdered (multples w must be consdered for multple reactons), and Ba and Ea two emprcal Arrhenus coeffcents. The knetc theory of gases provde a smple (although sometmes not very accurate) formulaton of all the transport coeffcents and equatons of state n terms of pressure, temperature and composton, but n practce one usually resorts to tabulated expermental data. Back to Index General equatons of physco-chemcal processes 6