J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1865-1874 Irreversible Thermodynamic Coupling between Heat and Matter Fluxes across a Gas/Liquid Interface -Scott C. Doney Advanced Study Program, National Center for Atmospheric Researcht P.O. Box 3000, Boulder, CO 80307, USA The governing equation for irreversible thermodynamic coupling phenomena, the dissipation function, is derived for a generalized gas/liquid interface following two routes and is shown to differ significantly from previous results (L. F. Phillips, Geophys. Res. Lett, 1991, 18, 1221; J. Chern. SOC.,faraday Trans., 1991, 87, 2187). The magnitude and direction of the coupled heat-mass effect is then computed for an idealized interface based on results from the kinetic theory of evaporation and condensation.The characteristic parameter for the coupling behaviour is the heat of transfer (93, the ratio of the conductive heat to mass flux arising from the cross-terms in the irreversible thermodynamic formulation. 9: for interfacial transport is shown to have a value of -0.43RT for both liquid-vapour and liquid-gas mixture systems. The coupled or cross-effect mass flux forced by the temperature field occurs from the cold fluid to the warm, counter to the conductive heat flux. This work suggests that the coupled mass fluxes across a gas/liquid interface are generally weaker than the direct, pressure-driven mass fluxes and are much smaller, by approximately a factor of 20, and in the opposite direc- tion from earlier model predictions.1. Introduction The transport of mass between gas and liquid phases is a complex problem involving both the physical chemistry of the gaspiquid interface as well as the fluid dynamic properties of the two fluids, in particular the presence and nature of the boundary layers adjacent to the interface. In addition to being of interest as a strictly theoretical question, the mass flux between gases and liquids, termed either evaporation/ condensation or gas exchange depending on whether the species is a bulk or trace constituent in the liquid, has numer- ous applications in fields as diverse as chemical engineering,' aerosol science,, and 0ceanography.~9~The fundamental thermodynamic and kinetic processes influencing gas exchange rates have been discussed for some time,'s5 but many important qpestions still remain.6 Recently, Phillips',' has re-examined the topic of gas exchange in terms of Onsager's irreversible thermodynamic This work7,', is focused primarily on under- standing the flux of CO, between the atmosphere and ocean, but is relevant to other fields as well.The res~lts~,*,~~ suggest that the coupling of heat and mass fluxes at the gas/liquid interface arising due to irreversible thermodynamic pheno- mena is an important factor in determining the magnitude and, in some cases, even the direction of the gas flux at a gas/liquid interface. In this paper, a development is presented of irreversible thermodynamics as it applies to the flow of trace gases and solvent across a gaspiquid phase boundary.The entropy pro- duction rate and dissipation function that govern the irre- versible heat-mass coupling phenomena are derived in Section 2 following two different routes. The two approaches are shown to yield an equivalent result, which differs signifi- cantly from that of previous work.' The proper equations describing the heat and mass coupling are presented in Section 3. The direction and approximate magnitude of the irreversible coupling effect are then computed for a simple gas/liquid interface based on results from the kinetic theory of evaporation and condensation (Section 4). For most systems, the irreversible effects are shown to be smaller and in t The National Center for Atmospheric Research is sponsored by the National Science Foundation.the opposite direction from those predicted by previous mod- elling studies.',' Fluid boundary layers near the interface are not considered in these calculations, which focus strictly on the phase interface itself; the bulk fluids outside of the inter- face region are assumed to mix at an infinite rate. The role of irreversible phenomena such as thermal diffusion in the inter- facial boundary layers is explored in a companion paperI3 with particular emphasis on the atmosphere-ocean system. Irreversible thermodynamic processes are distinguished from idealized, reversible processes by the production of internal entr~py,~,' that is '*14 dSi = 0; reversible dS, > 0; irreversible (1) Unlike in a reversible system, irreversible processes occur at a finite rate and involve finite gradients in temperature T, chemical potential ,u, and electric potential.The field of irre- versible or non-equilibrium thermodynamics developed as an extension of classical thermodynamics, which deals strictly with closed systems at equilibrium, and is particularly useful for studying open systems involving the transport of heat, matter and electricity.' Two key properties for characterizing irreversible processes are the production rate of internal entropy o = dSJdt and the related dissipation function To. The dissipation function quantifies the amount of dissipated energy that could have been utilized for useful work in a reversible process.In general, the dissipation function is equal to the sum of a set of fluxes Ji times forces Xi Ta= 1JiXi I The fluxes Jiinclude such things as heat, mass and charge while the forces result from differences or gradients in tem- perature, chemical and electrical potentials. A basic tenet of irreversible thermodynamics is that for slowly evolving systems close to equilibrium the fluxes Ji can be written as a linear combination of the forces" Xi Ji= c LiiXj (3) i where Lij are phenomenological coefficients linking .Ii and Xi. [Eqn. (3) may also be written in the reciprocal form, Xi = xiBijLj, where Bij are the corresponding jump 1866 c~efficients.'~] The choice of appropriate flux-force pairs (Ji and Xi) to use in the linear flux eqn.(3) is guided by the dissipation function; not any choice of flux-force pairs will produce a consistent set of flux equations.'.'' The phenom- enological coefficients are independent of the forces Xi, but may depend on temperature, pressure or system composi- tion. ''Two constraints apply to the phenomenological coef- ficients: the first, known as Onsager's relationships, states that the matrix of phenomenological coefficients is sym- metric,' i.e. Lij = Lji;the second results'' from the condition that a > 0 leading to Lii2 0 and LiiL, 2 Le .The criteria for whether a system is close enough to equilibrium to apply eqn. (3) can be difficult to address when chemical reactions are involved, but the assumption of linearity generally holds for most transport problems (e.g.diffusion, heat conduction).'* '' The exchange of heat and mass across a gas/liquid phase boundary can be written in terms of generalized irreversible thermodynamic flux-force pairs, eqn.(3). The conductive heat flux J,, sometimes referred to as the sensible or measurable heat flux, and mass flux Ji are often parametized using linear, phenomenological relationships such as J, = -C,pc,AT (4) (5) where AT and APi are the temperature and partial pressure differences between the two phases, p and c, are the density and specific heat for one of the phases, and Ki is the Henry's law coefficient relating the equilibrium liquid concentration of species i to the gas partial pressure. C, and Ci are empiri- cally derived transfer coefficients or transfer velocities and are dependent on the specific configuration of the system.For many systems, C, and Cican be specified as a function of the turbulence level in either the gas or liquid phases. The empirical flux relationships for J, and Ji [eqn. (4) and (5)] can be partitioned into a thermodynamic forcing term and a kinetic term, much like the generalized flux-force form, eqn. (3), for a single flux-force pair. For the conductive heat and gas flux equations [eqn. (4) and (5)], the thermodynamic Xj and kinetic Lij terms are the interfacial temperature and partial pressure differences and the transfer coefficients, respectively. In contrast to many irreversible thermodynamic problems, the kinetic or phenomenological coefficients L, linking the fluxes and forces for gas-liquid exchange may be controlled by both molecular and turbulent transport pro- cesses.Eqn. (4) and (5) are developed for systems involving a single forcing term. Irreversible thermodynamic theory, however, suggests that when there are multiple force terms, for example, coexisting temperature and partial pressure dif- ferences, additional cross phenomena may arise due to coup- ling between the different flux-force pair^.^-'' That is, one would expect mass fluxes driven solely by the temperature difference across the interface and a conductive heat flux due to partial pressure differences. The remainder of the paper examines the importance of these cross-effect terms right at the gas/liquid interface; a more general discussion including thermal and concentration boundary layers away from the interface region is presented elsewhere.' 2.The Dissipation Function for the Gasbiquid Interface 2.1 Traditional Derivation We are interested in the fluxes of heat and matter (solvent and/or trace gas) between a liquid and gas phase under J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 steady-state conditions. For environmental and chemical engineering applications, the gas phase is often a mixture of a bulk carrier gas, trace gas species and vapour from the solvent. Below, the generalized dissipation function is derived for heat and mass phase transfer, applicable for both bulk solvents and trace species. For simplicity, the gas phase is assumed to behave as an ideal gas but this is not necessary for the derivation.First, consider an idealized gas-liquid system where the bulk properties are uniform in both the liquid (subsystem 1) and the gas phases (subsystem 2), the rate of change in the bulk fluid properties is slow relative to the timescale of inter- est, and all of the gradient in temperature and chemical com- position between the two fluids is isolated to an arbitrarily small interface region (Fig. 1).The difference in the properties across the interface is expressed using the A notation (A,, = a, -ul), and fluxes are defined as positive when heat and mass are transferred from the liquid to the gas (see Glossary for notation and units). This simplified model of the gas/liquid interface, although clearly unrealistic in detail, is useful for developing in a general sense the irreversible thermodynamic equations for exchange between the two phases.The system shown in Fig. 1 is, according to the nomencla- ture of Hasse," a discontinuous system, and as shown in the Appendix, the general form of the dissipation function Ta per unit area A for a discontinuous system can be written as T is the mean temperature across the interface, hi,2 is the partial molar enthalpy of species i in the gas phase, and pi is the chemical potential of species i. Ji is the molar flux of i per unit area from the liquid to the gas, and is the conduc- tive, or measurable, heat flux absorbed by the gas phase. The total heat transferred from the liquid to the gas, Jq,2+xihi, Ji, includes terms for both the conductive heat flux and the enthalpy flux associated with the mass exchange.gas (subsystem 2) ;" liquid (subsystem 1). Fig. 1 Schematic for heat and mass exchange in a generalized gas/ liquid system. The bulk properties of the liquid (subsystem 1) and gas (subsystem 2) phases are TI,Pi,l and T, ,Pi,z, respectively. The two subsystems exchange mass fluxes .Ii and a total heat flux J,, which includes components from both conduction and enthalpy transport, and the conductive heat fluxes into the liquid and gas are given by Jq,l and .Iq,2, respectively. Heat and mass fluxes are defined as posi-tive when the transport results in a decrease in the quantity in the liquid and/or an increase in the gas phase.J. CHEM. SOC. FARADAY TRANS.,1994, VOL. 90 Under most circumstances, the conductive heat loss from the liquid Jq,' does not balance the heat absorbed by the gas Jq,2, and the difference between the two is equal to Jq, 2 -Jq.1 = -1Avhi Ji (7)i A"hi, the heat of vaporization or latent heat of i (for solutes the heat of solution), is generally positive. When water, for example, evaporates from a surface (Ji> 0) the cooling of the bulk liquid Jq,l is balanced, at first approximation, by the latent heat flux carried by the water vapour flux away from the interface into the gas phase, and the conductive heat flux in the gas .Iq, is near zero.The general form of the dissipation function can be trans- formed to one more applicable to gas-liquid exchange by specifying the relationship for the chemical potential for each species between the two phases. For a dilute solute the chemical potential in the liquid and gas phases are given by15 pi,' = ,u;(sol, T,)+ RT, 1n(mi,,/mol kg-') (8) Pi,2 = PP(g, T2)+ RT2 WPi,2/atm) (9) where p'(s01) and po(g) are the standard states for an ideally dilute solute and ideal gas, respectively, mi.' is the molality (mole solute per kg solvent) of species i in solution, and Pi, is the partial pressure of species i in the air. To simplify the discussion any non-ideal behaviour in either the gas or solu- tion phases is neglected; a more complete treatment could be derived using activity coefficients and gas fugacities but that is beyond the scope of this paper. The partial pressure and molality of gas i can be related at equilibrium via Henry's law Pi = Kimi (10) where Ki is the Henry's law coefficient on a molality scale.Combining eqn. (8)-(lo), it can be shown that for constant T &(sol) -pp(g) = RT In K, (11) Similar relationships to eqn. (8) and (9) can be developed for the solvent using the chemical potential for the solvent po in an ideally dilute solution po(sol, T') = &(sol, Tl)+ RT, In xo (12) and Raoult's Law in place of Henry's law Po = P;xo (13) The chemical potential for the solvent is defined using the solvent mole fraction, xo,and the equilibrium solvent vapour pressure at Tl over pure solvent, P;.Note that the standard states for solute (Henry's law) and solvent (Raoult's law) are defined in a different manner and are applicable over different ranges. For an ideally dilute solute, the standard state chemi- cal potential is defined for mi = 1 where each solute molecule interacts as if in contact only with solvent molecules ;I5,l6 Henry's Law is applicable for small mi (mi-,0). In contrast, the chemical potential for a solvent is defined as the pure solvent at the specified P and T,and Raoult's law is valid for xo close to l.16,17 The standard state of an ideal gas is defined at a partial pressure of 1 atm and T = 273.15 K. The dissipation function for liquid-gas exchange can be found by substituting the expressions for the chemical poten- tials for trace gases and solvents into eqn.(6) and then sim- plifying using eqn. (1l), the relationship between &'(sol) and p;(g), and eqn. (10) (Henry's law) and eqn. (13)(Raoult's law) Using the van't Hoff relationship' h(s)p= relating the temperature derivative of chemical potential at constant pressure to the partial molar enthalpy, a relation- ship for A(&'/T)can be derived ClP(g, T2) PP(g, Tl) 'hdT (16)T2 Tl ---=-lr T2 The partial molar enthalpy in the gas phase, hi,B, is a slight function of temperature (%) = cp,i (17)P where cp, is the molar heat capacity of i at constant pressure, which for a small temperature range can be integrated about hi,2 Eqn.(16)can then be integrated to give: where eqn. (6) and the fact that: ln(T + AT)/T = AT/T; AT << T (20) have been used to cancel the two terms involving cP.l7Sub-stituting back into eqn. (14) leads to the final form for the dissipation function for exchange across the interface The contribution to the total heat transfer d@ from the enth- alpy of the transported material cancels out of the final equa- tion. Using the liquid, rather than the gas phase, as the reference system leads to the same result. Note that from the definition of Jq, and Ji as positive when flow is into the gas phase, both terms in eqn. (21) are always positive, as they must be since the internal entropy can only increase for an irreversible process.' For most systems, it can be assumed that the partial pres- sure difference directly across the gas/liquid interface is small relative to the mean partial pressure and eqn.(21) can be further simplified to To AT APi -= --J,,2 -RT C -Ji (22)A T i Pi 2.2 A1terna tive Reversible- path Approach The entropy change dS,,, for a system can be separated into two components dS,,, = dSi + dS, (23) where dS, is the entropy change due to the exchange of matter and heat between the system and the surroundings and dSi is the entropy change arising from internal pro- cesses.' ' For a reversible process, ds, is exactly zero, eqn. (l), while dS,, and thus dSsys, can be positive, negative or zero depending on the process. The entropy change of a closed system undergoing a reversible process can be computed from an integral over the process:16 J dS,,, = J %; reversible where dq,,, is the heat transferred reversibly from the sur- roundings to the system; ds,,, associated with a reversible process, therefore, is zero only for an adiabatic system.dS,,, should not be confused with the entropy change of the uni- verse dSuniv,which includes the entropy change for both the system and its surroundings and is zero for a reversible process. Owing to the fact that entropy is a state function and thus path independent, the dS, for an irreversible process required for the dissipation function, eqn. (2) can be determined from the initial and final states of the system alone; the computed value of dS, should be the same irrespective of the path used in the derivation. In the Appendix, the overall system is chosen to be closed and adiabatic, dS, = 0, and dSi is found as the sum of the subsystem entropy changes dSl and dS, from the irreversible heat and mass transfer.For a reversible path such as given in eqn. (25), dSi for each step is zero and the entropy change for the whole path dSsys, which is equal the dSi for the corresponding irreversible pr&&sses, is just the sum of the individual values of dS, for each step. Consider the simple system consisting of a pure liquid and J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 state variable and can therefore be expressed as the sum of the enthalpy changes for the individual steps of eqn.(25). Using eqn. (17) along with the fact that (8hi/aP),= 0 for an ideal gas AVhi = cp AT + 0 + Avhi,(TI,P~) (29) Inserting eqn. (7) and eqn. (29) into the entropy production eqn. (28) cancels the terms starting with c,, A,hi and leaving behind the same entropy production equation as pre- sented earlier from the more traditional approach. Again, the terms arising from the enthalpy transport cancel according to the final dissipation function. 2.3 Comparison with Previous Results Phillips' uses an alternative approach from that in the Appendix for calculating the entropy production for the irre- versible process of transferring a solute, in this case CO, gas, across the gas/liquid interface. The method involves construc- ting a reversible path by which a mole of solute is transferred from phase 1 to phase 2 via a gas phase.The temperature and partial pressure of the solute are adjusted from T,, P, (phase 1) to T, ,P, (phase 2) reversibly in the gas phase CO,(phase 1, P,, Tl)+CO,(gas, p,, TI) its vapour at different pressures and temperatures. A reversible path can be written for the condensation of an +'1,CO,(gas, 7i) infinitesimal amount of vapour dn dn(vapour, P, , T,) +dn(vapour, P, , Tl) -+ dn(vapour, PI, TI) --+ dn(liquid, P,, Tl) (25) The entropy change for the path is given as the sum of the dS, associated with each reversible step16 where the vapour is assumed to be an ideal gas and the tem- perature variation of c, is taken to be small.During reversible, isothermal condensation (gas transfer) a quantity of heat equal to A, hi times the mass flux is released, and this heat must be transferred to the surroundings to maintain a constant temperature. The third term in eqn. (26) reflects the heat of vaporization of the liquid to the vapour evaluated for constant TI and P,. The entropy production due to the heat flux between the liquid and vapour also needs to be included in eqn. (26). For isothermal, reversible heat transfer, the entropy change is equal to dS, = dq/T. If an amount of heat dq, is removed from the vapour and dq, added to the liquid, the entropy production is, therefore Combining eqn. (26) and (27) and rewriting in terms of fluxes J,, 2, J,,l and Ji and for small AT and AP [eqn. (20)] Recall from eqn.(7) that the heat flux from the vapour is related to the heat flux into the liquid via the heat of vapor- ization, J,,l = J,, ,+ A, hiJi. Like entropy, enthalpy is also a + CO,(gas, P, 9 T2) -+ CO,(phase 2, P,, T,) (30) The entropy changes computed for each individual step along the path are summed to arrive at the net entropy change for the irreversible process leading to the same end state. If done correctly the reversible path approach (Section 2.2) leads to the same dissipation function [eqn. (22)] as the traditional method. The dissipation function derived above for heat and mass fluxes across the gaspiquid interface [eqn. (22)] differs from that of Phillipst7-' Ta AT ATA T -c,, , -)Ji (31)T in two important respects: the inclusion of the extra term c,, AT/T and the definition of the heat flux J, .Both of these differences arise in part due to the treatment79' of the con- ductive heat fluxes Jq,l and Jq, and their relation to the heat of vaporization A, hi. In the derivation by Phillips,8 the entropy change attrib- uted to the final step of the path in eqn. (30), where vapour (or solute) is condensed onto the liquid with the same T and Pi,is set to zero; it is stated that 'the entropy lost by the [vapour] in the last step is similarly balanced by the entropy gained by transfer of the heat of solution to [the liquid], so these processes do not contribute to the rate of entropy production'. As discussed above, however, the entropy change associated with the condensation step is not zero, having a value of Avhi,(T1,P!)(T[eqn. (26)].The earlier derivation' also does not explicitly address the contribution to the total heat flux J, from the enthalpy of the transported mass, hiJi, and thus does not distinguish between the gas (J4,,) and liquid (J,,l) conductive heat fluxes. As a result, can- cellation of the heat capacity term in eqn. (28) does not occur leaving an extraneous term c, TJ, AT/T in the dissipation t Note that the dissipation equation from ref. 8 has been written in a slightly different form where the limit for small values of AT and APi has been taken. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 function [eqn. (9) in ref.81. The extra term propagates throughout the calculation, affecting the magnitude and even the sign of the thermodynamic forcing in the coupled case. The problems associated with the earlier derivation' are compounded further because the heat flux in the dissipation function'* is assigned to be Jq,l rather than J,, 2. The differ- ence between Jq,' and J,, eqn. (7)is quite large under most circumstances, and, as discussed in more detail below, this error leads to a considerable overestimation of the magnitude of the irreversible thermodynamic coupling between heat and mass fluxes for the gas/liquid interface. 3. Coupling of the Heat and Mass Fluxes Following eqn. (3), the dissipation function for exchange across the liquid/gas interface eqn.(22) can be expressed using the force-flux notation, where the driving forces are simply the fractional temperature and partial pressure differ- ences across the interface and where the fluxes are the conductive heat Jq, and mass Ji fluxes into the gas phase." For slowly evolving systems close to equilibrium the fluxes Jq,2 and Ji can be expressed as a linear sum of the force^^.'^ Xi (33) I Ji = Li,X, + 1LijXj (34) i The phenomenological coefficients L,, and Lijare the transfer coefficients for heat and mass across the interface in the uncoupled case. The cross-terms L,, and L,, are the transfer coefficients resulting from the coupling between the heat flux and the chemical potential difference and the mass flux and temperature difference across the interface, respectively.Addi- tional mass-mass coupling terms LiAi #j)can also arise due to coupling between the fluxes and forcing terms for different species. For a single-component system, eqn. (34) states that the evaporation and/or condensation rate Ji is controlled by two factors: (35) The corresponding conductive heat flux equation is The cross-effect or coupling coefficients Li, = Lqi can be related to the primary mass transfer coefficient Lii in the fol- lowing manner. Fix the temperature (and partial pressures of all other species in a multi-component system) as constant across the interface, i.e. X, = 0 and Xj+i= 0. Setting X,= 0 in eqn. (35) and eqn. (36) leads to L.=qfLii (37)qi where the proportionality term q:, the heat of transfer, is defined as the ratio of the conductive heat and mass fluxes in the gas phase for the case where AT = 0 qT = (38)(9)AT=O, APj#i=O The flux equation for Ji [eqn. (35)] can then be rewritten solely in terms of the uncoupled coefficient Liiand the heat of transfer qf (39)Pi Comparing eqn.(39) with the traditional, empirical gas exchange parametrization eqn. (5), one can see a strong simi- larity between the two formulations in that each can be decomposed into a thermodynamic and a kinetic component. Neglecting the cross-effect term in eqn. (39) for a moment, it can be shown that the two formulations are identical if one sets the phenomenological coefficient Lii equal to Pi Cip/RTK,. The effect of the irreversible thermodynamic coupling between the mass transfer and AT enters eqn.(39) as an additional term in the thermodynamic driving force for mass transfer. The magnitude of this perturbation depends on the size of AT/T and AP/P as well as the value of qf relative to RT. The heat of transfer qf (J mol-') reflects the conductive heat flux J,, set up in the gas phase by a mass transfer of a mole of species i under isothermal conditions. To a first approximation, q: is zero for gas-liquid exchange because the bulk of the heat released or absorbed during the phase transfer comes from the liquid rather than the gas phase; that is Jq,l x A, hi J, and J,, x 0 [eqn. (7)]. As shown in the next section, because of second-order effects a small fraction of the heat required for the phase transfer does come from the gas phase, and the qf for a generalized liquid/gas interface is about the same magnitude or slightly smaller than RT.Since for many systems of interest the fractional temperature differ- ence is smaller than that for partial pressure, we expect the irreversible cross-effects to appear as a minor correction to the traditional gas-exchange formulation. 4. Kinetic Theory of Evaporation and Condensation Irreversible thermodynamics by itself does not constrain the magnitude of the coupling effect across the gas/liquid inter- face; for that one must turn either to experimental and/or other theoretical results. Results from work on the kinetic theory of evaporation and condensation are used in this section to estimate the size of qf for a simple gas/liquid system.The kinetic theory calculations should be viewed as correct in general terms but not in detail since they are devel- oped from an idealized molecular model of the interface and gas phase, one much simpler than that found in real applica- tions. They do, however, indicate the direction and relative magnitude of the heat-mass coupling phenomena for the gas/ liquid interface. A liquid/vapour interface undergoing non-equilibrium phase change is characterized by a transition region, some- times called a kinetic or Knudsen layer, in the gas phase where the temperature and vapour pressure deviate, due to the presence of the liquid surface, from the values extrapo- lated to the interface from the gas phase" (Fig.2). The thick- ness of the kinetic layer is of the order of a few mean path lengths, and the details of the molecular behaviour in the layer can often be reduced to two simple parameters, the macroscopic jumps in temperature and partial pressure. These jumps are equal to the difference between the values extrapolated to the interface from the gas continuum, T(0) P,(O),and the liquid surface values, T,Pi,s, where Pi, is the saturation pressure at T,.'9-21 The macroscopic jumps are generally reported in fractional form molecular I layer I Ti I I I layer Iiquid/gas i nterfaceT(O)yPi01 Ts,Pi,s Fig. 2 Schematic of the kinetic or Knudsen layer near the gaspiquid interface.The partial pressure Piand temperature T in the kinetic layer deviate from their values, PJO)and T(O),extrapolated to surface from the gas continuum. The magnitude of this deviation is chara-terized by the macroscopicjumps E, and E~,i, the fractional property difference E, = [do) -aJa, . The macroscopic jumps for the kinetic layer can be com-puted, using kinetic gas theory, as a function of the heat and mass fluxes across the interfaces and are zero only when the vapour and liquid are in thermal and chemical equilibrium.20 The kinetic theory results for condensation and evapo-ration, derived from a microscopic, statistical mechanical view, can be shown to be equivalent to the macroscopic, irre-versible thermodynamic formulation for the gas/liquid inter-face.20The dissipation function Ta for a phase interface is, in full detail, quite complex,22 but it can be simplified greatly using several reasonable assumptions : a planar surface (no curvature terms), no viscous pressure effects, surface tension independent of temperature and composition and no sur-factants.The final two assumptions lead to no excess storage of energy or mass along the interface, and thus require that the fluxes of mass Ji and total heat J, be continuous across the interface. The dissipation function then reduces to a similar form as presented earlier :I8 Ta----E, Jq,2 -C RT E~,i Ji A I 4.1 Liquid-Vapour System: One Component The constitutive relations for the macroscopic jumps in the simplest system, the half-space problem involving fluxes between a pure liquid and its vapour, are given by Cipolla” as and 1 RT Ep,i = -Pi, piui Jq.2 -Bii -Pivi Ji (43) where vi is the most probable speed where Mi is the molecular weight.Bqq,Bqi,pi,, and Bii are the dimensionlessjump coefficients that relate the heat and mass fluxes to the jump The macroscopic jump equa-tions can be derived from the interface dissipation function [eqn. (41)] and are the reciprocal of the traditional flux equa-tions presented in Section 3 [eqn. (35) and (36)]. The Onsager relations hold for the jump coefficients (i.e. Bqi = &), and the matrix of jump coeffcents /3 can be related to a matrix of phenomenological coefficients 1 by 1 = 8-’, where both matrices are in non-dimensional form.” J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Evaporation or condensation from the liquid surface per-turbs the distribution of gas molecules in the kinetic layer near the interface, and Cipolla2* calculate the following numerical values for the jump coefficients /3 using a varia-tional principle on the linearized Boltzmann equation /344 = 1.030; /3ii = 2.125; /3,i = pi, = 0.447 (45) These jump coefficients are applicable for planar surfaces, slow evaporation/condensation rates, ’ and Maxwellian mol-ecules; correction factors may be required for non-monoatomic gases or different forms of molecular interactions leading to a possible reduction in the magnitude of E, and cp, .23 These /3 values do not necessarily hold for transport to and from small aerosol particles since the pertur-bation to the molecular distribution around the particle is much smaller in this case.24The /3 values are very similar to those reported by Labuntsov and Kryukov” and are within about 10% of the results from the more simple approach given by Barrett and Clement.24 From eqn.(42) and (43), it is apparent that for constant ,heat or mass fluxes the macroscopic jumps E, and E~ are ~ inversely proportional to the vapour pressure and the most probable speed, vi. The preferred systems for experimental study, therefore, are not surprisingly liquid metals such as mercury that have both a high molecular weight and low vapour pre~sure.~~,~’For comparison, the predicted macro-scopic temperature jumps for a conductive heat flux Jq, = lo00 W m-2 are 6 x and 4 x respectively, for water vapour at 289 K (Pi = 0.032 atm) and mercury at 400 K (Pi= 0.00132 atm).The jump equations can be inverted to give the relation-ship between the heat and mass fluxes and the macroscopic jumps (46) RT-J.= -1 iq E t -l..~.PiVi ti p. t (47) where for the liquid-vapour system of Cipolla et ~1.~’ I,, = 1.069; Iii = 0.518; l,i = li, = -0.225 (48) Note that the coefficients for the cross-effects, Eiq and lqi, are negative resulting in heat or mass fluxes opposite in direction, for the same sign jump, from that expected for the primary effects. For example, when the vapour pressure in the gas is higher than that of the liquid, .sp> 0, the mass flux is nega-tive, condensation onto the liquid surface, while the conduc-tive heat flux resulting from the cross-effect is positive; a pressure jump with no accompanying temperature jump leads to a net heating of the gas phase.The ratio of the heat and mass fluxes in this case is deter-mined by the heat of transport, which for the liquid vapour interface can be found from 4: = RT Iqi/lii 4: = -0.434RT (49) The heat flux due to coupling q:Ji can be compared with the latent heat flux A, hiJi associated with Ji. For most liquids, the heat of vaporization can be approximated to A, hi z 10RT (Trouton’s or cu. 20 times larger than @Ji. Note also that Ahi is generally positive while 4: is negative.Combining eqn. (49) and Trouton’s rule with the constraint that the latent heat flux is balanced by the vapour and liquid conductive heat fluxes, eqn. (7)leads to Jq,l = Ahi Ji -0.434RTJi x lORT Ji -0.43RTJi (50) J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Based on the kinetic theory results, about 4% of the heat required for the phase change comes from the vapour phase with the remaining 96% from the liquid. Note that this result does not hold for aerosols where, because of the small heat capacity and large thermal conductivity of the liquid droplet, the latent heat is supplied almost entirely from the gas phase.4 Another useful quantity that can be defined from the kinetic theory results is the ratio liq/lqqRT, which is equal to the ratio of the mass to heat flux driven by a temperature difference across the interface From eqn.(48), the mass-heat coupling parameter c? is -0.210/RT. When the vapour is warmer than the liquid but has the same partial pressure, the vapour is both cooled due to conduction and made more supersaturated by an evapo- rative flux generated from the non-equilibrium cross-effects. Interestingly, taking A" hi = lORT the liquid is also cooled by conduction, but since the heat capacity of liquids is generally much larger than that of gases the vapour will cool more rapidly. If the temperature contrast between the vapour and liquid phases is maintained by some outside heat source, the system will evolve to a steady-state where the cross-effect evaporative flux is balanced by condensation from the super- saturation and where the magnitude of the supersaturation E~,can be computed from the jump equations by setting Ji to zero.4.2 Liquid-Gas Mixture System: Multiple Components The analysis of Cipolla et a!.'' can be extended to include gas mixtures where some of the gases can be non-condensable or and where the pressure and molecular speed terms are now the total pressure P = XiPi and the mean speed u = Xixi, ui , respectively. The macroscopic jump coefficients B for a gas mixture are, with the exception of the mass coefficients Bii, altered only slightly if at all from their values in the pure liquid-vapour system, eqn. (45). The mass jump coefficient for species i, pii, varies inversely with both xi and xi, and the predicted jump E~,for a fixed mass flux is constant with Pi.That is, larger pressure jumps are required for the same mass flux when the partial pressure of i is smaller or when the ability of i to enter the liquid is reduced. The irreversible thermodynamic coupling effects for the multi-component system can be defined in the same fashion as for a single-component system with the parameters 4: and c? [eqn. (38) and (51)]. Additionally, coupling between the mass flux of one species and the pressure difference of another may also arise in the multi-component case. The mass-mass coupling is characterized by the parameter rn; (55) It is helpful to visualize the parameter dependences for the coupling parameters.Fig. 3 and 4 present the results, for a two-component mixture, of varying: (1) the mole fraction of vapour x1 in a vapour-gas mixture; and (2) the condensation coefficient x1 in an equimolar mixture, x1 = x2 = 0.5, x2 = 1. In both cases the total pressure P is kept constant and the molecular weights of the two gases are set equal, MI = M,. The modified The curves shown in Fig. 3 are valid for either a condensable only slightly soluble in the liquid pha~e.'~~~~~~~ or non-condensable second gas species. jump coefficients are: 16 Xi xi, and xk, are the molar fractions of species i and k in the gas phase and 6, is the Kronecker delta, 13~~(~+~)= 0. Ai/2 is the ratio of the thermal conductivity for species i to the total thermal conductivity, which Bedeaux et d.'*set equal to the mole fraction. xi is the condensation coefficient for i, the boundary condition for gas molecules at the interface reflec- ting the fraction of impinging molecules that condense onto the liquid surface.A value of x equal to 1.0means that all of the gas molecules condense on impact while x = 0 signifies a non-condensable gas that cannot cross into the liquid. The constitutive equations including multiple gas species become The heat of transfer q: does not vary with either the mole fraction or condensation coefficient of species 1(Fig. 3 and 4). The presence of additional gas species, therefore, does not invalidate the conclusions given above regarding the magni- tude or direction of the mass flux driven by a temperature difference across the interface.The mass-heat coupling parameter cT decreases with both x1 and x1 (Fig. 3 and 4) for similar reasons to the variation seen in pii. The interfacial 1o-~ I 1111111~ I I11111 I n- I 10-4 1o-2 lo-' 1oo X1 Fig. 4 Plot of dimensionless, irreversible coupling parameters, (a) -q:/RT, (b)m;l, (c) -c:/RT and (6)rnX us. condensation coefi- cient x1 for a two component mixture. The total pressure P is keptconstant and the molecular weights of the two gases are set equal, M, = M,. mass-mass coupling parameter mT2 has (Fig. 3) a maximum value of 0.043 at x1 = 1 and decreases linearly with xl. The mass-mass cross-effect flux of species 1 is at most 4.3%of J, and only when the mole fraction of species 1 is large.my, also decreases with x1 as would be expected (Fig. 4). The second mass-mass coupling parameter in Fig. 4, mf,, reflects the flux of species 2 forced by a cross-effect from species 1 and is inde- pendent of x1. 5. Discussion and Summary Recent theoretical ~ork~?~ has suggested that the transfer of mass between gas and liquid phases can be strongly modified by irreversible thermodynamic coupling of the mass flux with the temperature field. The irreversible coupling phenomena can be deduced from the form of the dissipation function for heat and mass transfer across the liquid/gas interface. The dissipation function has been rederived in Section 2 following two different methods, the traditional one presented, for example, in de Groot and Mazurg and Haase" and the reversible path approach outlined by Phillip~.~.~ The two techniques, when correctly implemented, lead to the result given by eqn.(22). An earlier derivation by Phillip~~.~ is shown to lead to an incorrect dissipation function in part because it does not properly treat the contribution to the total heat flux J, from the enthalpy of the transported mass, hiJi, and thus does not explicitly separate the gas (Jq, and liquid (Jq,1) conductive heat fluxes. The governing equation for gas-liquid mass flux can be divided into a kinetic and a thermodynamic component, and it is shown (Section 3) that the effect of irreversible thermody- namic processes is to modify the thermodynamic term to include an additional term q:AT/T in the mass transfer equation, eqn.(39). The heat of transfer q: is the ratio of the conductive heat to mass flux arising from the cross-terms in the irreversible thermodynamic formulation and is the char- acteristic parameter for understanding the coupling pheno- mena. The strength and direction of the temperature-driven mass flux is estimated for an idealized interface in Section 4 based on kinetic theory arguments for the transition region near a liquid surface. In this idealized system, q: is shown to have a value of -0.43R T for both pure vapours and gas mix- tures. The sign of 4: is such that the coupled or cross-effect mass flux forced by the temperature field occurs from the cold fluid to the warm, counter to the conductive heat flux.The relative weights of the pressure (RT) and cross-effect temperature (-0.43RT) terms in the mass-transfer equation, J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 eqn. (39), are comparable, but since for many systems, the fractional temperature difference AT/T will be smaller than that for pressure, the irreversible thermodynamic coupling effect should generally appear as a correction rather than as the dominant term in eqn. (39). The results presented here differ significantly from an earlier analysis of coupling effect^,^,' where the coupling effect is predicted to be of the same magnitude or larger than the uncoupled, pressure-driven mass flux.As discussed above, however, the incorrect replacement of Jq,,by Jq,lin the phenomenological flux equations, eqn. (35) and (36), leads to the erroneous conclusion2 that the heat of transfer 4: is equal to the enthalpy of vaporization AVhi.This error is very sig- nificant and invalidates many of the conclusions of the earlier ~ork.~+','~For example, the enthalpy of solution of many common gases in water is quite largei5 (10-20 kJ mol-'), a factor of 4 to 8 times RT at room temperature (RT29, = 2.5 kJ mol-'). Using AVhi for q:, therefore, leads to an over-estimation, by ca. 20 for evaporation/condensation and ca. 8-16 for gas exchange, of the importance of irreversible ther- modynamic phenomena for liquid-gas mass exchange. In addition, because AVhiis generally positive while 4: is nega-tive, the fluxes driven by cross-effects are in the opposite direction from those presented here. Real applications of mass-transfer between gas and liquid phases commonly involve mixtures in the gas phase, and the .~~kinetic theory results of Cipolla et ~1 offer a method for computing, at least for a very idealized molecular system, the size of irreversible effects for mixtures.An examination of eqn. (52) and Fig. 4, however, shows that the condensation coefficient xi plays an important role for slightly condensable gases and there has been little mention, to this point, of what is a reasonable range for this parameter. The flux of gas impinging on a liquid surface can be computed from kinetic theory, and xi can be written as the fraction of this kinetic flux that actually enters the liquid' Barrett and Clement24 argue based on experimental work on liquid metals25 that xi for vapour above a planar, liquid surface should be near 1.0.The discussion for trace gases, however, becomes more complicated. Noyes et aL6 estimate xi from laboratory measurements of the maximal transfer velocity eqn. (56). They find that for N, dissolving into various solvents, xi values range from 4 x lo-'' to 1 x and are approximately proportional to the equilibrium con- centration of N, in the solvent. The flii and c: values deter- mined for the gas mixture case are quite sensitive to the choice of xi, and this factor needs to be explored further..~~The kinetic theory results of Cipolla et ~1 are symmetric with respect to the direction of the mass flux Ji; that is the same values of fl are predicted for evaporation as for conden- sation. As discussed by Barrett and Clement,24 however, the mass flux during condensation to a planar surface can be greatly reduced due to the presence of non-condensable gas. The net mass flow towards the liquid surface leads to a build- up of non-condensable gas near the interface, and the resulting pressure gradient causes an additional impedence to the mass transfer, one that can overwhelm the impedance caused by the phase change alone.27 This phenomenon, which arises not in the kinetic layer but rather due to macroscopic effects in the gas-continuum region, can be appreciable in the two-surface problem where a non-condensable gas is trapped between an evaporating and a condensing ~urface.~~.~~ To examine this effect further requires a method for coupling the kinetic layer with the gas-continuum.13~'9 J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 For many practical applications, the major resistance or impedance for mass transfer between gas and liquid phases is not interfacial but rather is due to the formation of boundary layers in one or both of the fluids where the concentrations and temperatures deviate from their bulk values.' The inter- face-controlled system addressed in this paper is a special case of the more general mass-transfer problem in which the two bulk phases are assumed to be infinitely well mixed.8 The focus of this paper has been on the role of irreversible process at the interface itself, and the more general discussion of irre- versible thermodynamics in the boundary layers is presented e1~ewhere.l~Briefly, however, when the mass flux is domi- nated by the resistance of one or both of the boundary layers, the flux through the kinetic region is set external to the inter- face, and the jump equations, eqn.(53), then provide a steady- state estimate of the macroscopic interface jumps. Values of zp and E, are computed for typical air-sea ~0nditions.l~ Experimentally, irreversible coupling phenomena often appear in this fashion where the magnitude of the effect is calculated from steady-state property distributions forced by fixed mass and heat fluxes.25 Appendix Irreversible Thermodynamics and the Dissipation Function The derivation for the dissipation function To is included both for completeness and because it differs from that pre- sented elsewhere.8 A number of excellent references are avail- able for irreversible thermodynamics, and the reader is referred to these texts for a more detailed treat-ment.9,10,11,14,29The entropy production rate u can be for- mulated for either a single, continuous phase or for multiple, discontinuous (heterogeneous) phases.Consider an adia-batic, closed system made up of two subsystems, 1 and 2, separated by a membrane but which can exchange heat d@ and matter dn, (Fig.1). The subsystems are each assumed to be uniform or isotropic in internal properties (e.g. tem-perature, concentration). Let do be the total heat transported from subsystem 1 to subsystem 2, consisting of both the con- ductive or measurable heat dq and the enthalpy of the trans- ported matter hidni. Assuming only PV work (i.e. no electrical or magnetic work), the changes of the internal ener- gies dU of the two subsystems can be described by dU1 = TI dS1 -PI dV1 + 1pi.1 dni,, (All 1 dU2 = T2 dS, -P2 dV2 + 1pi, 2 dni, 2 (A2) 1 where T is absolute temperature, P is pressure, I/ is volume and pi is the chemical potential of species i. Eqn. (Al) and (A2) can be rearranged to solve for the entropy changes The internal energy changes dUl and dU2 can also be expressed using external variables dU, = d@l -Pl dV1; dU2 = d@2 -P2 dV2 (A5) For a closed, adiabatic system, the only exchanges of heat and matter are between the two subsystems d@= -d@1 = dcD2 (A6) and and the entropy change for each subsystem can then be written as The total entropy change dS = dSl + dS2 is or AT dS = --T2 d@ -i A(;) dni using T being a mean temperature and AT = T2 -T, The total heat term do can be expanded into the measur- able heat and enthalpy released from subsystem 1 and absorbed by subsystem 2 dcD = dql + 1hi,l dni = dq, + 1hi, dni (All) 1 i and the difference in the conductive heat transferred between the two subsystems is equal to dq2 -dq1 = 1(hi,l -hi, 2) dni (A 12) 1 The conductive heat terms dql and dq, are equal only when either the transfers of material dni are zero or when the molar enthalpies in the two subsystems are equal (for example, if both subsystems are gases, h, x h,).Note that enthalpies, hi, and thus the total heat @ do not have absolute values and must be referenced to a standard state. The general form for the dissipation function T dS/dt can be found by combining eqn. (A9) and (All) and differentiat- ing with respect to time (A13) where A is the area between the two subsystems, Ji is the molar flux of i per unit area from subsystem 1 to 2, and J,, is the measurable heat flux absorbed by subsystem 2. Eqn. (A13) is identical, barring different sign conventions, with the formulation of Haasel' for the time rate of entropy pro- duction for a discontinuous system.For subsystems where the system would be homogeneous if the membrane were to be removed (i.e. gas-gas or miscible liquid-liquid subsystems but not for the gas-liquid system), the term A(pJT) can be eliminated using where (pi),is the chemical potential of species i at constant temperature. Setting subsystem 2 as the reference state, eqn. (A 13) reduces to : and the dissipation function can be expressed simply as a function of the measurable heat and matter fluxes in one of the phases. In developing eqn. (A13)it is assumed that both phases are isotropic, with concentration or temperature gradients between the phases being isolated to the membrane separat- ing the two subsystems.In a real gas-liquid system, however, both phases often contain finite thickness boundary layers near the interface where properties diverge from their bulk values. Following the discussion above, a volumetric dissi- pation function Ta, (J s-' m-3) can be derived for a contin- uous region (single phase) of variable composition.' ',14 In one-dimension Ta = ---J,+ChiJi)-xTJi-(dL;) (A16) TId'(dz dz T where J, and Ji are defined as positive in the z direction. Following the discussion above, To, can be rewritten as: J dT dTa = -3 --1Ji -(p.)T dz dz IT This work was motivated by discussions of the NCAR/ NOAA CMDL carbon cycle group and by the encour-agement of D. Schimel and W. Large. I would also thank D.Glover, S. Green, R. Wanninkhof, and R. Keeling for their comments on earlier drafts of the manuscript. This work was funded by an NCAR Advanced Study Program Postdoctoral Fellowship. Glossary value of a in the liquid phase value of a in the gas phase value of a for solvent standard state of a a2 -a1 flux of a, positive from liquid to gas extrapolated interface value of a from gas contin- uum value of a at liquid surface macroscopic jump of a across gas/liquid interface area, m2 jump coefficient, various, dimensionless empirical transfer velocity, m s-' molar flux of i, mol m-2 s-l conductive heat flux, J m-s -' total heat flux, J m-2 s-' Henry's law coefficient, atm kg mol- ' phenomenological coefficient, various, dimension- less molecular weight, kg mol- pressure, atm partial pressure of i, atm gas constant, J mol-' K-' entropy, J K-internal entropy, J K-entropy flow from surroundings, J K-' absolute temperature, K internal energy, J volume, m3 thermodynamic driving force, various mass-heat coupling parameter, mol J-concentration of i, mol mP3 molar heat capacity, J mol-K-' specific heat capacity, J kg-'K-' molar enthalpy of i, J mol-' molality of i, mol kg- ' mass-mass coupling parameter mol of i, mol J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 conductive heat, J heat of transfer, J mol-most probable molecular speed, m s-' mole fraction of i heat of vaporization (solution), J mol-' total heat, J entropy production rate, J K-' s-volumetric entropy production rate, J K-' s-' m-3 m-' K-' thermal conductivity of i, J s-mass density, kg mP3 chemical potential of i, J mol-' chemical potential, constant T,J mol-' condensation coefficient of i References 1 P.V. Dankwerts, Gas-Liquid Reactions, McGraw-Hill, New York, 1970. 2 E. J. Davis, Aerosol Sci. Technol., 1983, 2, 121. 3 P. S. Liss and P. G. Slater, Nature (London), 1974, 247, 181. 4 P. S. Liss and L. Merlivat, in The Role of Air-Sea Exchange in Geochemical Cycling, ed. P. Buat-Menard. D. Reidel, Hingham, MA, USA, 1986, 113. 5 T. G. Theofanous, in Gas Transfer at Water Surfaces, ed. W. Brutsaert and G. H. Jirka, Reidel, Boston, USA, 1984. 6 R. M. Noyes, M. B. Rubin and P. G.Bowers, J. Phys. Chem., 1992,%, 10oO. 7 L. F. Phillips, Geophys. Res. 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Harris, Diffusion in Liquids, Butter-worths, London, 1984. Paper 3/05897B ;Received 30th September, 1993