Balance equations and entropy production

Balance equations of extensive quantities describe a change in a system (except in rare gases and shock waves). These balance equations also contain intensive parameters specifying the local state of a continuous medium. Intensive parameters described by the macroscopic properties of the medium are based on the behavior of a large number of particles. 

It is necessary to consider the mechanics of a continuous medium to determine the thermodynamic state of a fluid. The properties of a fluid can be determined that are at rest relative to a reference frame or moving along with the fluid. Every nonequilibrium intensive parameter in a fluid changes in time and in space. 

Consider the temperature as a function of time and space T=T(t, x, y, z) the total differential of T is expressed as 

Dividing the total differential by the time differential, we obtain the total time derivative of T 

The partial time derivative of T, (dT/dt), shows the time rate of change of temperature of a fluid at a fixed position at constant x, y, and z 

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Entropy Balance Equation and Rate of Local Entropy Production... 

In order to obtain an additional balance equation for the microstructural parameter K, the principle of dissipation is utilized. The starting point is the entropy balance [Eq. (9)] with entropy density p t], entropy flux entropy supply a, and entropy production f >0. 

On combining the balance equation for energy assuming the absence of velocity gradients, the first law of thermodynamics and assuming Gibbs equation for entropy production for the case of local equilibrium, a the entropy production per unit volume per unit time due to the occurrence of irreversible processes in the system is given by... 

The balance equation for entropy can be derived using the conservation of energy and the balance equation for the concentrations. This gives us an explicit expression for entropy production or— which can be related to irreversible processes such as heat conduction, diffusion and chemical reactions—and the entropy current is- The formal entropy balance equation is... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

We use a short version of the seven-step method. The problem asks for the entropy and enthalpy changes accompanying a chemical reaction, so we focus on the balanced chemical equation and the thermodynamic properties of the reactants and products. 

Thermodynamic Governing Equations. Derivation of the expression for entropy production arising from mess transfer requires application of the fundamental balance equations. Potential and kinetic energy effects as well as momentum effects are neglected. With these assumptions the governing equations are given as follows ... 

The objective function (13) representing the total dissipation of kinetic energy of the flows at isothermal motion of fluid is proportional to the entropy production in the circuit and its transfer to the environment, i.e., proportional to the entropy accumulated by the isolated system (interconnection of the circuit and environment). The matrix equation (14) describes the first Kirchhoff law, which, as applied to hydraulic circuits, expresses the requirement for mass conservation. Equality (15) represents a balance between the energy generated and consumed in the circuit. 

Engineering systems mainly involve a single-phase fluid mixture with n components, subject to fluid friction, heat transfer, mass transfer, and a number of / chemical reactions. A local thermodynamic state of the fluid is specified by two intensive parameters, for example, velocity of the fluid and the chemical composition in terms of component mass fractions wr For a unique description of the system, balance equations must be derived for the mass, momentum, energy, and entropy. The balance equations, considered on a per unit volume basis, can be written in terms of the partial time derivative with an observer at rest, and in terms of the substantial derivative with an observer moving along with the fluid. Later, the balance equations are used in the Gibbs relation to determine the rate of entropy production. The balance equations allow us to clearly identify the importance of the local thermodynamic equilibrium postulate in deriving the relationships for entropy production. 

From the general entropy balance equation dS= dJS+ dxS, we conclude that for an incompressible and isothermal process, we have deS d,S. This relation shows the equality between the dissipated heat flow and internal entropy production and hence the loss of power is q = Eloss. Therefore, Eq. (b) becomes... 

The equation of change of entropy can be obtained by a balance [1, p. 372]. In a volume element, the accumulation of entropy with time arises from convective transport of entropy attached to matter, from inflow and outflow by molecular transport, and from local entropy production in the volume element itself. Thus, the equation of... 

At the end, we summarize the results of the model of a reacting mixture of fluids with linear transport properties from Sects. 4.5 and 4.6 (properties such as kinematics, stoichiometry and balances of mass, momentum and their moment, energy and entropy inequality are as in Sects. 4.2, 4.3 and 4.4). Constitutive equations, their properties and final form of entropy production are given in the end of Sect. 4.5 (from Eq. (4.156)), further thermodynamic quantities and properties are given at the... 

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