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On the basis of the nonlocal phenomenological relation between thermodynamic fluxes and forces in continuous systems, it is shown that the vectorial flux couples with the scalar force even in an isotropic system. This result has application to active transport in living organisms and to thermonuclear fusion research.

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... Thermodynamically coupled chemical reaction-transport systems control the behavior of many transport and rate processes in physical, chemical and biological systems, and require a through analysis accounting the induced flows by cross effects [1][2][3][4][5][6][7][8][9]. Many published work, including some recent ones [10][11][12], on reactiondiffusion systems mainly consider mathematically coupled nonlinear differential relationships. ...

... Another important thermodynamic coupling takes place between the hydrolysis of adenosine triphosphate and the molecular transport of substrates in active transport. The coupling between a scalar process of the hydrolysis and a vectorial process of the mass flow creates the molecular pumps responsible for uphill transport [1,14,15]. Therefore, incorporation of thermodynamic coupling into the modeling of reaction-diffusion systems, such as active transport, may be a vital step in describing these complex biochemical cycles. ...

Considerable work has been published on mathematically coupled nonlinear differential equations by neglecting thermodynamic coupling between heat and mass flows in reaction-transport systems. The thermodynamic coupling refers that a flow occurs without or against its primary thermodynamic driving force, which may be a gradient of temperature, or chemical potential, or reaction affinity. This study presents the modeling of thermodynamically coupled heat and mass flows of two components in a reaction-transport system with external heat and mass transfer resistances. The modeling equations are based on the linear nonequilibrium thermodynamics approach by assuming that the system is in the vicinity of global equilibrium. The modeling equations lead to unique definitions of thermodynamic coupling (cross) coefficients between heat and mass flows in terms of kinetic parameters and transport coefficients. These newly defined parameters need to be determined to describe coupled reaction-transport systems. Some representative numerical solutions obtained by MATLAB illustrate the effect of thermodynamic coupling coefficients on the change of temperature and mass concentrations in time and space.

... Linear flow-force relations are valid when the Gibbs free energy ranges less than 1.5 kJ mol −1 for chemical reactions [15,18]. However, some selected biological pathways occur at near GE conditions [2,3], and for some chemical reactions, the formalism of LNET can be used in wider ranges than usually expected [14,[25][26][27][28]. By conservation of mass, some flow-force relations of enzyme catalyzed and other chemical reactions can be described by a simple hyperbolic-tangent function. ...

... During a diffusion-controlled reaction, matter may be transported through an interface, which separates the reactants and the product. The progress of the reaction may be affected by the morphology of the interface with complicated structure, which controls the boundary conditions for the transport problem [27][28][29]. Morphological stability of interfaces in nonequilibrium systems may lead to self-organization and/or pattern-formation in biological, physical, chemical and geological systems [26,29]. Turing [1] demonstrated that even some simple reaction-diffusion systems could lead to spatial organizations due to instability of stationary structure depending on the activator-inhibitor interactions, control parameters and boundary conditions (see Appendix A). ...

Nonisothermal reaction-diffusion systems control the behavior of many transport and rate processes in physical, chemical and biological systems, such as pattern formation and chemical pumps. Considerable work has been published on mathematically coupled nonlinear differential equations by neglecting thermodynamic coupling between a chemical reaction and transport processes of mass and heat. This study presents the modeling of thermodynamically coupled system of a simple elementary chemical reaction with molecular heat and mass transport. The thermodynamic coupling refers that a flow occurs without or against its primary thermodynamic driving force, which may be a gradient of temperature or chemical potential or reaction affinity. The modeling is based on the linear nonequilibrium thermodynamics approach by assuming that the system is in the vicinity of global equilibrium. The modeling equations lead to unique definitions of cross-coefficients between a chemical reaction and heat and mass flows in terms of kinetic parameters, transport coefficients and degrees of coupling. These newly defined parameters need to be determined to describe some coupled reaction-transport systems. Some methodologies are suggested for the determination of the parameters and some representative numerical solutions for coupled reaction-transport systems are presented.

... Consideration of heat effects due to a chemical reaction, which is either a heat source or a sink, may be necessary for a through analysis of reaction diffusion systems (Aris, 1975; Froment and Bischoff, 1990; Dekker et al., 1995; Demirel and Sandler, 2002; Burghardt and Berezowski, 2003; Demirel, 2006; Xu et al., 2007; Sengers and de Zarate, 2007; de Zarate et al, 2007; Demirel, 2007; Demirel, 2008). Beside that, many catalytic reactions may take place with thermodynamically coupled heat and mass flows, and control the behavior of various physical, chemical, and biological systems (Aono, 1975; Prigogine, 1967; Kondepudi and Prigogine, 1999; Demirel and Sandler, 2001; Gas et al., 2003; Rose and Rose, 2005; Demirel, 2006; Demirel, 2007). Here, the thermodynamic coupling refers that a flow (i.e. ...

Considerable work published on chemical reaction-diffusion systems investigates mainly mathematically coupled nonlinear differential equations. This study presents the modeling of a simple elementary chemical reaction with thermodynamically and mathematically coupled heat and mass transport with external mass and heat transfer resistances. The thermodynamic coupling refers that a flow occurs without or against its primary thermodynamic driving force, which may be a gradient of temperature or chemical potential. The modeling is based on the linear nonequilibrium thermodynamics approach and phenomenological equations by assuming that the system is in the vicinity of global equilibrium. This approach does not need detailed coupling mechanisms. The modeling equations contain the cross coefficients controlling the coupling between heat and mass flows in terms of transport coefficients and surface conditions. These coefficients need to be determined for rigorous analysis of chemical reaction systems with thermodynamically coupled transport phenomena. Some representative numerical solutions of the modeling equations are presented to display the effect of coupling on concentration and temperatures in time and space for simple exothermic catalytic reactions.

... Biochemical nonequilibrium reaction systems operate with fluxes (material and energy), thermodynamic forces, multiple steady states, nonzero steady-state flux, and may be part of coupled transport and rate processes [21,31,36,40,41,79,82,97]. There is also a positive non-zero entropy production rate, which characterizes the nonequilibrium steady state (NESS). ...

Living cells represent open, nonequilibrium, self-organizing, and dissipative systems maintained with the continuous supply of outside and inside material, energy, and information flows. The energy in the form of adenosine triphosphate is utilized in biochemical cycles, transport processes, protein synthesis, reproduction, and performing other biological work. The processes in molecular and cellular biological systems are stochastic in nature with varying spatial and time scales, and bounded with conservation laws, kinetic laws, and thermodynamic constraints, which should be taken into account by any approach for modeling biological systems. In component biology, this review focuses on the modeling of enzyme kinetics and fluctuation of single biomolecules acting as molecular motors, while in systems biology it focuses on modeling biochemical cycles and networks in which all the components of a biological system interact functionally over time and space. Biochemical cycles emerge from collective and functional efforts to devise a cyclic flow of optimal energy degradation rate, which can only be described by nonequilibrium thermodynamics. Therefore, this review emphasizes the role of nonequilibrium thermodynamics through the formulations of thermodynamically coupled biochemical cycles, entropy production, fluctuation theorems, bioenergetics, and reaction-diffusion systems. Fluctuation theorems relate the forward and backward dynamical randomness of the trajectories or paths, bridge the microscopic and macroscopic domains, and link the time-reversible and irreversible descriptions of biological systems. However, many of these approaches are in their early stages of their development and no single computational or experimental technique is able to span all the relevant and necessary spatial and temporal scales. Wide range of experimental and novel computational techniques with high accuracy, precision, coverage, and efficiency are necessary for understanding biochemical cycles.

In many actual technological problems the system response is dictated by multi-physics events, which occur at different space–time-scales. Energy storage systems or active polymers are two illustrative real-life applications: they involve diffusion of chemically active species concurrent to mechanical deformations that take place in the same spatial domain. An abundant literature exists nowadays for this class of problems. There are, though, several major applications in which the motion of species is confined on advecting surfaces. The transport along membranes which advect and chemically respond only after getting into contact with an obstacle is a classical problem in cell mechanobiology or coatings, for instance. Modeling and simulations for such a class of problems is thoroughly investigated in the present note.

We show that when the thermodynamic fluxes are included as independent thermodynamic state variables of a generalized entropy, the original ONSAGER formulation may be directly used in the space of the fluxes. Therefore, the ONSAGER relations may be derived either in the space of the classical (slow) variables, by using a spatial FOURIER transformation, or in the space of the non-classical (fast) variables such as the physical fluxes, without need of any Fourier transform. Furthermore, we analyse the question of non-linear ONSAGER relations by studying one particular set of evolution equations of the fluxes, and considering the fluctuations of the fluxes around a non-equilibrium steady state. Comparison with kinetic theory is not completely conclusive, because of several open questions which we comment in the concluding remarks.

It is shown that vectorial phenomena couple thermodynamically with the scalar phenomena. This result is contrary to the well spread belief in the field of irreversible thermodynamics. Transport coefficients concerning the diffusion and the thermal conduction across a strong magnetic field are calculated in the presence of the deuteron-triton fusion reaction on the basis of the gas kinetic theory. When the reaction takes place, the diffusion coefficient increases and the thermal conductivity decreases. Effects of the reaction exceed those of the Coulomb collision as the temperature is high enough.

A nonlocal phenomenological equation is introduced for a multicomponent fluid where chemical or nuclear reactions are taking place. The reciprocity between the nonlocal linearcoefficients is examined closely. An approximation reduces the nonlocal equation to the ordinary phenomenological relation with correction terms which show clearly a coupling of the reaction with the diffusion and the thermal conduction in an isotropic system.

Nonisothermal reaction-diffusion systems control the behavior of many transport and rate processes in physical, chemical, and biological systems, such as pattern formation and molecular pumps. Considerable work has been published on mathematically coupled nonlinear differential equations by neglecting thermodynamic coupling between a chemical reaction and transport processes of mass and heat. The thermodynamic coupling refers that a flow occurs without or against its primary thermodynamic driving force, which may be a gradient of temperature, or chemical potential, or reaction affinity. Energy coupling in the membranes of living cells plays major role in the respiratory electron transport chain leading to synthesizing adenosine triphosphate (ATP). The ATP synthesis in turn, is matched and synchronized to cellular ATP utilization. Consequently, the hydrolysis of ATP is thermodynamically coupled to the transport of substrates. This study presents the modeling of thermodynamically coupled system of a simple elementary chemical reaction with molecular heat and mass transport. The modeling is based on the linear nonequilibrium thermodynamics (LNET) approach by assuming that the system is in the vicinity of global equilibrium. Experimental investigations revealed that LNET is capable of describing thermodynamically coupled processes of oxidative phosphorylation, mitochondrial H + pumps, and (Na + and K +)-ATPase. Moreover, the LNET formulation does not require the detailed mechanism of the coupling. The modeling equations lead to unique definitions of cross coefficients between a chemical reaction and heat and mass flows in terms of kinetic parameters, transport coefficients, and degrees of coupling, which are measurable. These newly derived cross coefficients need to be determined to describe some coupled reaction-transport systems. Some methodologies are suggested for the determination of the cross coefficients and some representative numerical solutions for coupled reaction-transport systems are presented. Such modeling may improve our understanding of active transport by molecular pumps.

DOI:https://doi.org/10.1103/PhysRev.81.1070

Calculations of the power balance in thermonuclear reactors operating under various idealized conditions are given. Two classes of reactor are considered: first, self-sustaining systems in which the charged reaction products are trapped and, secondly, pulsed systems in which all the reaction products escape so that energy must be supplied continuously during the pulse. It is found that not only must the temperature be sufficiently high, but also the reaction must be sustained long enough for a definite fraction of the fuel to be burnt.

Ahydrodynamic equation of motion for each component of a multicomponent fluid is derived on the basis of nonequilibrium thermodynamics. Special care has been directed to the choice of state variables. In some limiting cases, this equation leads to customary phenomenological equations, such as the equation for diffusion and the Navier-Stokes equation. The viscosity is a consequence of nonlocal coupling of forces and fluxes. The reciprocity between the linear coefficients is examined closely.

On nonlocal thermodynamics, inModem Developments in Thermodynamics

- A C Bringen
- A. C. Bringen