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Linear-Nonequilibrium Thermodynamics Theory for Coupled Heat and Mass Transport

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Linear-Nonequilibrium Thermodynamics Theory for Coupled Heat and Mass Transport University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Chemical and Biomolecular Research Papers -- Yasar Demirel Publications Faculty Authors Series 2001 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] Follow this and additional works at: https://digitalcommons.unl.edu/cbmedemirel Part of the Chemical Engineering Commons 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 dis- sipation 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 cou- pling phenomenon between heat and mass flows. The degree of the coupling is also discussed. In the lit- erature 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 dissipa- tion 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 as- sumption 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 phenom- enological equations with the resistance coefficients that are capable of reflecting the extent of the inter- actions between heat and mass flows. We call this the dissipation-phenomenological equation (DPE) ap- proach, 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
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