Linear-Nonequilibrium Thermodynamics Theory for Coupled Heat and Mass Transport

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Linear-nonequilibrium thermodynamics theory for coupled heat and mass transport

Yasar Demirel University of Nebraska - Lincoln, [email protected]

Stanley I. Sandler University of Delaware, [email protected]

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Demirel, Yasar and Sandler, Stanley I., "Linear-nonequilibrium thermodynamics theory for coupled heat and mass transport" (2001). Yasar Demirel Publications. 8. https://digitalcommons.unl.edu/cbmedemirel/8

This Article is brought to you for free and open access by the Chemical and Biomolecular Research Papers -- Faculty Authors Series at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Yasar Demirel Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Published in International Journal of Heat and Mass Transfer 44 (2001), pp. 2439–2451 Copyright © 2001 Elsevier Science Ltd. Used by permission. www.elsevier.comllocate/ijhmt Submitted November 5, 1999; revised September 6, 2000.

Linear-nonequilibrium thermodynamics theory for coupled heat and mass transport

Yasar Demirel and S. I. Sandler

Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, DE 19716, USA Corresponding author — Y. Demirel

Abstract Linear-nonequilibrium thermodynamics (LNET) has been used to express the entropy generation and dissipation functions representing the true forces and flows for heat and mass transport in a multicomponent fluid. These forces and flows are introduced into the phenomenological equations to formulate the coupling phenomenon between heat and mass flows. The degree of the coupling is also discussed. In the literature such coupling has been formulated incompletely and sometimes in a confusing manner. The rea- son for this is the lack of a proper combination of LNET theory with the phenomenological theory. The LNET theory involves identifying the conjugated flows and forces that are related to each other with the phenomenological coefficients that obey the Onsager relations. In doing so, the theory utilizes the dissipation function or the entropy generation equation derived from the Gibbs relation. This derivation assumes that local thermodynamic equilibrium holds for processes not far away from the equilibrium. With this assumption we have used the phenomenological equations relating the conjugated flows and forces defined by the dissipation function of the irreversible transport and rate process. We have expressed the phenomenological equations with the resistance coefficients that are capable of reflecting the extent of the interactions between heat and mass flows. We call this the dissipation-phenomenological equation (DPE) approach, which leads to correct expression for coupled processes, and for the second law analysis.

1. Introduction ume fraction. Menshutina and Dorokhov [13] simulated drying in a heterogeneous multiphase system based on non- Simultaneous heat and mass transport plays a vital role equilibrium thermodynamics theory, and obtained the new in various physical, chemical and biological processes, general structures of thermodynamic forces. Mikhailov and hence a vast amount of work on the subject is available in Ozisik [14] employed the integral transform technique in the literature. Coupled heat and mass transfer occurs in sep- the development of the general solutions for various cou- aration by absorption, distillation, and extraction [1-4], in pled heat and mass diffusion problems. In a recent work, evaporation--condensation [5-7], in drying [8,9], in crys- Whitaker [15] investigated the coupledmultiphase trans- tallization and melting [10,11], and in natural convection port by the method of closure in which various integrals ap- [12]. These processes occur in important industrial appli- peared in the volume-averaged equations are solved. Braun cations. Various formulations and methodologies have been and Renz [6] adapted a numerical approach based on the suggested for describing combined heat and mass transport two-dimensional boundary layer equation for the multicom- problems. In drying, Guerrier et al. [8] investigated the cou- ponent diffusion interactions during condensation. Kaiping pling through the saturation pressure of the mixture, which and Renz [5] studied the thermal diffusion effects in partial is the function of both the temperature and the solvent vol- filmwise condensation experimentally and numerically.

2439 2440 Demirel & Sandler in International Journal of Heat and Mass Transfer 44 (2001)

Nomenclature z ratio of phenomenological coefficients, Equation a defined in Equation (59) (80) D diffusion coefficient wi mass fraction of component i DT thermal diffusion coefficient X thermodynamic force et total specific energy ep specific energy potential Greek symbols F mass force δ unit tensor h specific enthalpy ϕ ratio of irreversibility distribution j diffusion flow ΦS entropy generation rate J mass flow η ratio of flows J flow λ ratio of forces k thermal conductivity μ chemical potential l,L,K phenomenological coefficients ρ density m mass σ stress tensor M molar mass τ viscosity part of stress tensor p pressure Ψ dissipation function Q* heat of transport r degree of coupling Subscripts s specific entropy i, j, k substances ST Soret coefficient S entropy gen generation t time q heat related T absolute temperature p potential u specific internal energy T at constant temperature v specific volume, center-of-mass or barycentric s entropy related velocity u internal energy related

Investigation of the coupled processes in biological sys- dynamic equilibrium assumption, which is the extension tems has been expanding rapidly with the new concepts of thermodynamic relations in equilibrium systems to lo- such as biothermal engineering and biotransport. For ex- cal nonequilibrium subsystems. LNET evaluates the rate ample, laser applications in surgery, therapy and ophthal- of entropy generation resulting from the irreversibilities mology [16], thermal diffusivity of tissues [17], drug deliv- taking place in the processes. The Gouy-Stodola theorem ery [18], and the cryopreservation [19,20] are some of the states that the entropy generation is directly proportional to coupled heat and mass transfer processes in living systems. the loss of available free energy. The dissipated energy per Cryopreservation is the suspension of thermally driven re- unit time and per unit volume is called the dissipation func- action and transport processes that cause local mass trans- tion, and is the product of the rate of entropy generation fer due to the nonuniform pattern of ice formation and the and the absolute temperature. The dissipation function de- resulting concentration of solute within the living tissue. fines the conjugate forces and flows of the processes un- De Freitas et al. [19] utilized the linear-nonequilibrium der consideration. These flows are related to the forces in a thermodynamics (LNET) theory for the transport of water linear form with the phenomenological coefficients (PCs) and cryoprotectant in a multicellular tissue. The way mol- through phenomenological equations (PEs). The PCs obey ecules are organized, e.g. the protein folding, and biolog- the Onsager reciprocal relations, which states that the co- ical order [21-24], biological oscillation [25], membrane efficient matrix is symmetrical. As a direct consequence transport in biological systems [26] can be studied by the of the Onsager relations, the number of unknown coeffi- nonequilibrium thermodynamics analysis. Structured sta- cients in the PEs are reduced. The PCs are constants and tionary states of living systems need a constant supply of closely related to the transport coefficients of the rate pro- energy to maintain their states. The energy flow is cou- cesses and the coupling between them. The theory of the pled with other rate processes and the diffusion of matter. LNET based on the Onsager relations is highly developed Therefore, it is important to express the interaction in the by the work of Prigogine [33], Fitts [34], De Groot [35], transport and rate processes properly [19,27-32]. Katchalsky and Curran [27], Wisniewski et al. [36], Ca- LNET deals with the transport and rate processes, which plan and Essig [29], Eu [37], Kuiken [38], and by many are irreversible in nature. It is based on the local thermo- other researchers. Kedem [39] used LNET to represent the Linear-nonequilibrium thermodynamics theory for coupled heat and mass transport 2441 coupling between reactions and flows in anisotropic - bio fusion flow in the control volume method. In the LNET logical systems. Recently, Curtis and Bird [40] have gen- analysis, the transport and rate processes are expressed in eralized Fick’s law and the Maxwell-Stefan expression to the form of the PEs. These equations are capable of dis- include thermal, pressure, and forced diffusion in a mul- playing the interactions between the various transport and ticomponent fluid with the help of LNET. Miyazaki et al. rate processes through the cross-phenomenological coeffi- [41] developed a nonequilibrium thermodynamic model cients [29]. For the PEs to be useful to this end they must and discussed the proper definition for the internal energy have the conjugate flows and forces defined by the dissipa- of multicomponent systems. Ramshaw [42] developed a tion function resulting from the canonical expressions for phenomenological theory for multicomponent diffusion, open thermodynamic systems. Therefore, in this contribu- including thermal diffusion, in gas mixtures in which the tion we use the theory of LNET to express the PEs based species may have different temperatures. on the dissipation function, then define the coupling be- The LNET description of heat and mass transfer is com- tween the heat and mass transfer. We refer to this as the patible with the classical engineering approach based on dissipation-phenomenological equation (DPE) approach. the hydrodynamic theory of boundary layer, with transport This approach is especially useful if there is no large tem- coefficients and driving forces [43]. The theory of LNET perature and concentration differences in a system. Other- is utilized in the second law analysis, and the dissipation is wise, it may be advantageous to refer forces and flows to calculated from the entropy generation, or from the exergy the entropy generation. DPE approach is thermodynami- balance on the system [44]. The entropy generation ap- cally sound, and can be extended to analyze the coupling proach is especially important in terms of the process op- for the various irreversible flows and chemical reactions. timality as the individual processes contributing to the entropy generation can be identified and evaluated separately. 2. Local thermodynamic equilibrium Through the minimization of the entropy generation a thermodynamic optimum can be achieved for an existing de- Systems that are not in thermodynamic equilibrium are sign with a specified task [45]. To this end equipartition of nonhomogeneous systems in which at least some of the in- the entropy generation or of the thermodynamic forces can tensive parameters are functions of time and/or position. A also be used as the criterion for process optimum [46,47]. local state of a substance is considered as small elementary The second law analysis has especially found applications volumes of the system that is not in thermodynamic equi- in thermal engineering [45,48-50]. It is also used in sep- librium. These small volumes should contain a sufficient aration processes, and in design [2,51-55]. Many of these number of molecules for a macroscopic theory to be appli- studies also searched for a link between thermodynam- cable. When the local thermodynamic equilibrium holds, ics and economics. Sieniutycz and Shiner [56] reviewed the specific entropy and specific internal energy in each the various aspects of nonequilibrium thermodynamics to- volume element are evaluated as in a system at equilib- gether with the second law analysis and chemical engineer- rium, and the Gibbs relation and the Gibbs-Duhem relation ing comprehensively. San et al. [57], and Paulikakos and are applicable. For a nonequilibrium system, the pressure Johnson [58] used the control volume approach in the sec- has the same dependence on the specific internal energy u ond law analysis of combined heat and mass transfer. San and specific volume v, as in an equilibrium system, i.e., p et al. [57] used the analogy between heat and mass transfer, = (∂u/∂v) . The assumption of local thermodynamic equi- and evaluated the degree of coupling by an order of mag- s librium holds for great variety of rate processes in physics, nitude analysis for a binary mixture of ideal gases. Car- chemistry and biological system [29,33-36,56]. rington and Sun [59,60] utilized nonequilibrium thermodynamics in the second law analysis. However, they did not use the PEs to express the flows or the forces, and ne- 3. Phenomenological equations (PEs) glected the coupling between heat and mass transfer. The expressions developed by San et al. [57] and Bejan [61] are Potential gradients can produce irreversible changes, based on the equations for the entropy generation per unit and are called ‘forces’, and denoted by Xi (i = 1,2, ... ,n) for volume in a Newtonian Fluid as given by Hirschfelder et n gradients. The forces produce certain irreversible phe- al. [62]. nomena such as mass diffusion called the “flows” (fluxes), Consequently, the methodology of expressing and eval- and symbolized as Ji (i = 1,2, ... , n). Generally any force uating the coupling in heat and mass flows differ greatly can cause any flow Ji(Xi) (i = 1, 2, ... ,n). We know, for ex- in the literature. Sun and Carrington [63], and Carrington ample that a concentration gradient and a temperature gra- and Sun [60] noted shortcomings in the second law analy- dient could both produce mass flows as well as heat flows. sis and in expressing the coupling between heat and mass If the forces vanish, the system goes to an equilibrium transfer due to the use of the absolute mass instead of dif- state in which the flows are zero Ji(Xi = 0) = 0. The PEs 2442 Demirel & Sandler in International Journal of Heat and Mass Transfer 44 (2001) are based on the linear relationships between the flow and we can express the linear PEs in the form force, and any flow is the result of all the forces (5) (1) where the new PCs are defined as

The coefficientsL ik are called the PCs, and defined by (6a) (2) where L –1 is the inverse of the matrix L . The coefficients ik ik K have the units of force per unit of flow and represent the where Lik represents flow per unit force and has the char- generalized resistance. The following relation links the L acteristics of conductance. The coefficients with the same and K indices Lii, for example the diffusion coefficient, relate the conjugate force and generalized flow, namely the diffusion (6b) and the concentration gradient. The coefficients Lik with i ≠ k are known as the cross-coefficients, and related to the where |L| is the determinant of the L, and |L| is the princi- coupled transport coefficients, such as the thermal diffu- ik pal minor (see Appendix A). The matrix K also obeys the sion coefficient. Onsager’s fundamental theorem states Onsager relations, and satisfies the conditions [35]. that, provided a proper choice is made for the flows and forces the matrix of Lik is symmetric (6c) (3a)

The PCs are not functions of the thermodynamic forces These identities are known as the Onsager reciprocal rela- and the generalized flows. However, they can be functions tions, and can be derived from the principle of microscopic of the parameters of the local state T, p, and the concentra- reversibility, using the statistical thermodynamics [64]. tions w as well as the nature of the system [36,37,66]. Gambar and Markus [65] showed that the Onsager recipro- i When we consider a diffusion j which has a tensorial cal relations are the results of the global gauge symmetries i rank of 1, which is a vector, and a concentration gradient of the Lagrangian, which is related to the entropy produc- X also of tensorial rank 1, then the coefficient Lik is a sca- tion of the system considered. This means that the results k lar quantity that is consisted with the isotropic character of in general are valid for an arbitrary process. The explicit the system. So that the coefficients L do not vanish and equations for the Onsager relations depend on the nature ik the flow j and the force X can interact. We observe, there- of the system, and various kinds of coupling between flows i k fore the coupling or the cross effects between the various and forces are possible. Classical statistical mechanical diffusion flows, and between diffusion and heat flows. Ac- theory shows that the Onsager reciprocal relations are valid cording to the Curie-Prigogine principal [29], there will for diffusion and heat flow for a particular set of flows and be no coupling between scalar and vectorial quantities in forces [34]. All the PCs with the similar and dissimilar in- an isotropic system. However the scalar quantities, such dices must satisfy the conditions as chemical reactions, can cause the flow of a substance across an anisotropic membrane producing the active trans- (3b) port in biological systems [27,33]. (3c) 4. Dissipation function If there is no metastable equilibrium and all the forces and flows are independent, the inequality sign holds in The time derivative of the entropy is the rate of entropy Equations (3b) and (3c). generation. The LNET evaluates the rate of entropy gen- It is known that the forces are also functions of the flows eration for a specified task of a process. This evaluation

Xi(Jk) (k = 1,2, ... , n). Using the Maclaurin series expan- (as explained in Section 5) is based on the hypotheses of sion and considering the first-order approximation of lin- positive and definite entropy production and of the Gibbs ear relationships relation

(4) (7)

Linear-nonequilibrium thermodynamics theory for coupled heat and mass transport 2443 where u is the specific internal energy, v the specific vol- where XT and JT are the transpose of the respective vec- ume, p the pressure, μi the chemical potential and wi is the tors. Equation (15) shows that the dissipation function is mass fraction of substance i. This fundamental relation, quadratic in form for all forces and all flows. valid even away from thermostatic equilibrium, shows that The dissipation function or the entropy generation de- the entropy depends explicitly only on energy, volume and scribes the flows and the forces that are linearly depen- concentrations. If the entropy generation inside the system dent on each other through the pes. The form of the ex-

Si [33] is taken per unit time t and per unit volume v, it is pressions for the dissipation function does not uniquely called the volumetric rate of entropy generation ΦS, which determine the thermodynamic forces or generalized flows. is a positive and definite quantity For an open system, for example, we may define the energy flux in various ways. We may also define the diffusion (8) flows in several alternative ways depending on the choice of reference velocity. Thus we may transform the flows and The dissipation function, Ψ is given by the forces in various ways. This transformation will not af- fect the dissipation function if the forces and flows are re- (9) lated by PCs that obey the Onsager relations [36]. and shows that irreversibility causes the decrease in the 5. Balance equations of multicomponent fluids free energy, in agreement with the Gouy-Stodola theorem [45,67]. We consider a single phase of fluid mixture with n sub- The quantities Φ and Ψ may be expressed as the sum S stances subject to heat and mass transfer without chemi- of the products of the conjugate forces and flows stemming cal reactions. Using the assumption of the local thermody- from the second law of thermodynamics [33] namic equilibrium for such a system, balance equations for the mass, momentum, energy and entropy are necessary in (10) deriving the expressions for the entropy generation or the dissipation function. The notation used by Wisniewski et al. [36] has been adopted in the balance equations. (11 ) 5.1. The mass balance Here Ji and Ji′ are the generalized flows, and Xi and Xi′ are the thermodynamic forces. The quantities Φ and Ψ are S The mass flow of component i, J = ρ v , is a vector show- scalars, and the dot product of two vectors for the case of i i i ing the flow of a substance relative to a motionless coordi- heat and mass flows. The choice of thermodynamic forces nate system. The diffusion flow j relative to the mass aver- must be made so that in the equilibrium state when the i age velocity v is thermodynamic forces vanish (Xa = 0), the entropy generation must also vanish. The quantities Φ and Ψ’ depend on S (16) the mode of change between given states. If Equation (1) is introduced into the dissipation func- where the mass average velocity is defined by tion of Equation (11), we have

(12)

Similarly the entropy generation is expressed by Here vi and ρi are the velocity and density of substance i, while w is the mass fraction. The sum of diffusion flows of (13) i all the substances is zero and the PCs l and L are associated by (17)

lik = TLik (14) therefore of n diffusion flows, only n – 1 are independent. Equation (12) may be written in matrix form showing the The mass and diffusion flows, J and j of substance i are symmetry between the flows and the forces expressed in i i related by Equations (1) and (5) J = j + ρ v (18) (15) i i i 2444 Demirel & Sandler in International Journal of Heat and Mass Transfer 44 (2001)

The conservation equation for the substance i is The total specific energy e of a substance consists of the specific internal energy u, the specific kinetic energy ½ v2 ,

(19) and the specific potential energye p Using the substantial derivative of the density (26) and taking into account that the conservation equation for where v2 is the result of v · v. the total mass obtained from Equation (19) The conduction flux Jet of the total energy consists of the conduction flux J of the internal energy, the potential n u energy flux i∑ =1 epiji due to diffusion of substances, and the the mass conservation equation is obtained by replacing surface work — v· σ the density with the specific volumev = 1/ρ (27) (20) The rate of the change of total kinetic and potential energy The balance equation for the amount of substance i can per unit volume is [36] also be written in terms of the mass fraction, wi

(21)

5.2. The momentum balance (28) The Newton’s second law of motion states that the change in the momentum of a body is equal to all the where Fi is body force, and is associated with the specific forces acting on that body. The forces acting on the volume potential energy e of substance i by of fluid can be divided into a mass force F, and a surface pi force due to the stress tensor σ. If Fi is the force exerted per Fi + ∇epi = 0 unit mass of substance i, then We subtract Equation (28) from Equation (25) to obtain the (22) rate of change of the internal energy per unit volume is the force exerted per unit mass of the fluid. Using the (29) pressure force ∇p and the viscous force ∇· τ for the surface force, and the body force per unit volume ρF, the momentum balance equation is given in the substantial form If diffusion is slow, and the condition of local thermody- by namic equilibrium is satisfied, then the rate of change of (23) total energy of all components per unit volume is given by

The viscosity stress is negligible (τ = 0) in a fluid which is in mechanical equilibrium where the acceleration vanishes (dv/dt = 0). Consequently the pressure gradient is equal to the sum of the mass forces (30) ∇p – ρF = 0 (24) where u‾i is the partial molar internal energy of substance i. If Jq′ is the energy flow of pure heat conduction without 5.3. The energy balance a flow of internal energy due to diffusion of the substance, the total energy balance, Equation (25), becomes The local energy balance equation is that the rate of change of the total energy per unit volume is related to a convection fluxpev and a conduction flux Jet (31) (25) Linear-nonequilibrium thermodynamics theory for coupled heat and mass transport 2445

Comparison of this equation with Equation (27) yields and

(32)

The second term on the right-hand side of this equa- Equation (38) becomes tion is the net internal energy flow due to the diffusion of substances.

5.4. The entropy balance (39) The entropy balance equation in terms of the convection entropy flow psv, the conduction entropy flow JS, and the Using the relation between the chemical potential and en- entropy generation ΦS is given by thalpy given by

(33) (40) we can find the following relationship between the heat The entropy flux due to conduction is composed of two flux J ″ appearing in the entropy balance equation and the contributions: one is due to the heat flux in the entropy bal- q heat fluxJ q′ occurring in the energy balance, Equation (31) ance equation Jq″, and the other arising from the diffusion flow ji (41)

(34) ‾ where hi, v‾i are the partial molar enthalpy and velocity, respectively. where s‾ is the partial entropy of substance i. Inserting The definition of heat flow can be based on- amacro Equation (34) into Equation (33), and using the substantial scopic energy balance or entropy balance. The internal derivative gives energy of a substance results from the molecular kinetic energy and the potential energy of the intermolecular interactions. The change of kinetic energy is due to the diffu- (35) sion for the collision free movement of substances. When the kinetic energy changes as the result of molecular colli- Assuming that local thermodynamic equilibrium is valid, sions, it is defined as heat conduction. However definition we can use the Gibbs relation in terms of specific quantities of the potential energy are not uniquely associated with diffusion and conduction due to the complex interactions for each pair of substances. Fitts [34] called the J ″ as the sec- (36) q ond law heat flow which is related to the Jq′ by Equation (41). From Equations (35) and (39), the local entropy gen- where μi is the chemical potential of substance i. Differen- eration per unit volume is defined by ignoring the viscosity tiation of Equation (36) with respect to time yields stress, and given by

(37) (42)

After substituting Equations (29), (20), and (21) into Equa- With the following forces defined as: tion (37), we have (for heat transfer), (43)

(for mass transfer). (44) (38) Equation (42) expresses the entropy generation as the sum Using the following transformations of two distinctive contributions as a result of the product of the flows and the forces: These are the entropy production associated with heat transfer Φ = J X , and the entropy q u q n production due to mass transfer Φd = ∑i=1 ji Xi. Entropy 2446 Demirel & Sandler in International Journal of Heat and Mass Transfer 44 (2001) generation relation defines the independent forces and the coupling, namely the thermal diffusion (Soret effect) and flows to be used in the PEs. the heat flow due to the diffusion of substancei (Dufour effect), respectively. These effects are generally referred to the forces, namely to the gradients of temperature and com- 6. Heat and mass transfer coupling position. To this end we should take into account the fact that the roles of forces and flows are symmetric. For multi- In the DPE approach, we start with the dissipation func- component diffusion, the matrix L includes L and L . The tion Ψ = TΦ, which can be expressed using Equation (42), ii ij coefficients L are associated to the diffusion flow j aris- as ii i ing from its own chemical potential gradient of substance i, while the coefficients L define the part of j arising from (45) ij i the chemical potential gradients of substance j. The coefficients L result from the intermolecular interactions of the A transformation is useful for Equation (45). For that, ij dissimilar molecules. If the coefficients L are negative, firstly we use the total potential, which is the summation ij diffusion of j causes a diffusion flow ofi in the opposite di- of the chemical potential and the potential energy μ′ = μ + rection. In principle, each PC is uniquely determined, and e , and the thermodynamic force for the diffusion becomes p may be measured by a suitable experiment [34]. The thermal diffusivity may be related to a new quantity (46) called the heat of transport. To understand the concept consider expressing the thermodynamic force, using the resis-

tance type coefficientsK ij, from Equation (50) where ∇T μ′i is the gradient of the total potential at constant temperature, and given by

(52) Secondly, we express the internal energy fluxJ u in terms of J ″ using Equations (32) and (40) –1 q where Kji = Lij . This thermodynamic force can be inserted into Equation (47) (49), and we have

So that, with the help of Equations (46) and (47) and not- (53) ing that only n – 1 diffusion flowsj i are independent, Equa- tion (17), we have the form for the dissipation function in which all the forces are independent The heat of transport, Qi* of component i is expressed by

(48) (54)

We may now establish the PEs for the vector flows of Jq″ and can be used in the PEs to eliminate the coefficients Lqj ∇ ∇ and ji and the forces – lnT and – T (μi′ – μn′), and write or Ljq. For example after introducing the heat of transport, the linear relations between the flows and the forces Equation (53) becomes

(49) (55)

In an isothermal system where ∇ lnT = 0, Equation (55) yields (50) (56) Equations (49) and (50) make it possible to define uniquely the PCs that obey the Onsager reciprocal relations Equation (56) indicates that the heat of transport Qi* is the heat flow to keep temperature constant when diffusion flow (51) of substance i takes place. The heat of transport in Equa- tion (56) is based on J ″, and on the diffusion flow -re The coefficient L is associated to the thermal conduc- q qq ferred to the mass average velocity. So that the definition tivity, while the cross-coefficients Liq and Lqi define the Linear-nonequilibrium thermodynamics theory for coupled heat and mass transport 2447 of it varies with the choice of the heat flow and the diffusion flow. We may use the Onsager reciprocal relations to (64) relate the heat of transport to the thermal diffusion coefficients, and it is then possible to obtain experimental values of heats of transport. (65) For a multicomponent fluid at mechanical equilibrium (see Equation (24)), the Gibbs-Duhem relation at constant In Equations (64) and (65), we express the thermodynamic temperature yields forces as a function of conjugate flows which are easy to control experimentally. Kedem [39] also used a similar (57) form of the PEs to express active transport, that is the coupled processes of flows and chemical reactions in biologi- where the gradient of total potential in the absence of mass cal systems with anisotropic character. When the temperature gradient vanishes, (∇T = 0), the forces (Fi = 0) is given by diffusion flow is given by

(58) (66)

Equation (57) enables us to eliminate ∇ μ′ from Equa- T n Comparing Equations (61) and (66) yields the diffusion co- tion (48), and we have efficient, defined by

(59) (67) where aik = δik + wk /wn, and δik is the unit tensor. We can When the concentration gradient vanishes, ∇wi = 0, we now establish the following new set of PEs based on the have the thermal diffusion, in which mass flows are due to dissipation function of Equation (59) the gradient of temperature, and Equation (61) gives

(68) (60) where the thermal diffusion coefficient is defined by

(69)

(61) The thermal diffusion ratio kTiq relates these transport coef- ficientsD ij and DTiq Based on the rate of entropy generation form of Equations (48) and (14), the PEs of Equations (60) and (61) can also (70) be expressed as Due to the nonideality of the system, the gradient of chemi-

cal potential in Equation (67), and hence kTiq , may become (62) negative. This causes the flow of a substance to a hotter part in the system. Thermal diffusion is usually character-

ized by the Soret coefficientS Tiq, which is defined as the ratio of the thermal-diffusion coefficient to the ordinary dif- (63) fusion coefficient

Furthermore, Equations (60) and (61) can be transformed (71) to a new set of PEs by replacing the coefficients L with the coefficients K, given by Equation (6a). Since the K is the The Soret coefficient is a measure of the concentration gra- inverse matrix of L, the coefficients of K reflect the direct dient of the solute that can be maintained as a result of a interference when several flows interact in the system. We temperature gradient at a stationary state, where ji = 0 and now therefore consider the symmetry between the flows ∇T = constant. The Soret coefficient is also associated and forces, and Equations (60) and (61) become with the heat of transport Qi* as follows: 2448 Demirel & Sandler in International Journal of Heat and Mass Transfer 44 (2001)

The most widely adopted applications of Equation (74) (72) involve the entropy generation minimization with respect to certain parameters in searching a thermodynamically op-

The thermal diffusion factor α is given by α = Ts Tiq . timum operating conditions and/or a design for a process From Equation (60) we can express the Dufour effect, [56,57,61,67,69]. Before we do that we must evaluate the ˙ which is the flow of heat caused by only a concentration total entropy generation Sgen inside the volume of a given gradient (∇ln T = 0) system by

(73) (76)

When we compare Equation (73) with the combined Equa- tions (66) and (67), we can easily see that the Dufour coef- Prigogine [33] showed that for a certain value of the force for the heat flow, the rate of entropy generation becomes ficient is defined as DCi = Lqi /ρ, and due to the Onsager re- minimum at the stationary state where ji = 0. lations, is equal to thermal diffusion coefficientD Tiq . In a single component system, the thermomolecular We can also use Equations (64) and (65) in Equation pressure difference is the result of the coupling between (15) to express the dissipation function for coupled heat heat and mass flows. It is well known as the Knudsen ef- and mass flows fect in a gas, and fountain effect in a liquid system. When we have a membrane between gases or liquids, the cou- (77) pling occurs as the thermo-osmosis effect [27,33,64]. It is now straightforward to express the entropy genera- Here the first and the second terms on the right-hand side tion or the dissipation function. By substituting Equations of Equation (77) are the dissipations due to the heat and (62) and (63) into Equation (13), we can calculate the rate mass flows, respectively, while the third term denotes the of entropy generation dissipation due to the coupling. The degree of the coupling results from the relationships between the forces and the flows. From Equations (64) and

(65), we can define the ratio of forces (λ = ∇lnT/∇wi and the flows η( = Jq″/ji) for a binary mixture. Dividing Equa- tions (64) with (65), and further dividing both the numera- ½ tor and denominator by (KqqKjj) ji, we obtain

(74) (78) where l = l , and the coefficients of l are related to L by q j iq Equation (78) shows that the ratio of forces λ varies with Equation (14). The individual contributions related to heat the ratio of flows, η. As the quantity K /(K K )½ ap- and mass flows as well as the coupling between them are iq qq ij proaches zero, each flow becomes independent, and we represented by the first, second and third terms, respec- have the ratio of forces approaching λ → ± (K /K )η. If tively on the right-hand side of Equation (74). This equa- i qq ij K /(K K )½ approaches ±1, then the two forces are not as- tion is the basis of the second law analysis, and useful in iq qq ij sociated with the flows, and the ratio of forces approaches many ways. For example we can evaluate the individual a fixed value λ → ± (K /K )½. This case is the situation contributions in the entropy generation, hence determine qq ij where the matrix of the PCs becomes singular, and corre- the trade off between them. To this end the ratio of the irre- sponds to the equality sign in Equations (3c) and (6c). The versibility distribution ϕ in the volume of a system can be ratio obtained by (79) (75) indicates the degree of coupling, and it can be used as a When ϕ approaches to unity, the irreversibility due to heat base for comparison of systems with several coupled transfer becomes the dominating part in the entropy gener- forces. As the heat and mass flows are both the vectors, the ation [45,68]. The nature of the distribution of the entropy sign of r indicates the direction of forces of the substances. generation is also helpful in evaluating the optimality of a If K < 0 hence r > 0, and the flow of a substance may system with a required task [46,47]. iq drag another substance in the same direction; however it Linear-nonequilibrium thermodynamics theory for coupled heat and mass transport 2449 will flow in the opposite direction if Kiq > 0 and r < 0. For 7. Conclusions heat and mass flows the dissipation function defines the two limiting values of r between + 1 and –1 [29]. An in- The coupling between the flows of heat and mass is very complete coupling takes a value between these two limits. important in many industrial processes as well as in the The coupling may be further clarified by the ratio living systems, and it should be accounted for in a proper way. We have demonstrated that the theory of LNET can play crucial role in the proper definition of the coupled heat (80) and mass flows. To this end we have suggested the use of the resistance type of PCs in the PEs, in which the con- and Equation (78) becomes jugate forces and flows are identified by the dissipation function. We called this procedure as the DPE approach. (81) The dissipation function is the direct result of rigorous and sound theory of LNET based on the assumption of the local Equation (81) indicates that for the known values of z and thermodynamic equilibrium. We also have demonstrated η, the ratio of forces is determined by the degree of cou- the vital role played by the PEs in the second law analy- pling r. With complete coupling, λ is equal to z and r be- sis. Since the entropy production depends on the flows and comes +1 or –1. The minus sign of η stems from the situ- forces, the analysis of LNET defines the correct forces and ations where the differentiation of chemical potential with flows involved in an irreversible process. The role of the respect to concentration may have negative values due to PEs is the determination of the true transport coefficients, nonideality of the mixture. This complex behavior results and the interactions between the flows and forces. This role from solution nonideality of the mixture. has been formulated in this study. With the help of the z, we can define the reduced force ratio of zλ, and the reduced flow ratio η/z, and can relate them by Acknowledgments — Y.D. thanks the Center for Mo- lecular and Engineering Thermodynamics of the Uni- (82) versity of Delaware for the hospitality shown during his visit. This study has been funded by King Fahd Uni- Change of the reduced flow ratio with the reduced force ra- versity of Petroleum & Minerals under project #CHE/ tio in the range –1 < r < +1 is given in Figure 1. After Ca- SECOND-LAW/209. plan and Essig [29], two reference stationary states in the coupled processes can be defined as the level flow where

Xi = 0, and static head where Ji = 0. Example of systems at static head are an open circuit fuel cell and active transport Appendix A in a cell membrane, whereas the examples of systems at level flow are a short circuited fuel cell and salt and water We can express the phenomenological equations for two transport in kidneys. In an incompletely coupled systems, flows and two forces by a constant supply of energy is necessary to maintain these reference states without output or work. (A.1) (A.2)

Solving these equations for the thermodynamic forces yields (A.3)

(A.4) where

and

Figure 1. Change of reduced flowszλ with the reduced forces η/z. 2450 Demirel & Sandler in International Journal of Heat and Mass Transfer 44 (2001)

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