Energy Balance and Entropy Inequality

In Chap. 1 we postulate the First Law as (1.3) which gives the existence of internal energy fulfilling (1.5). Similarly as in Sect. 2.1 we can write (1.5) as a balance the time derivative of internal energy is equal to the sum of heating and power (cf. (2.1)) [11, 18, 22, 35]. This is applicable to the material volume of a (nonuniform) body or its arbitrary part consisting of a single substance. We postulate the existence of a 

We postulate also that u, q and Q are objective scalars (but see Rem. 21) then (3.97) is valid in all frames by Rem. 12 the first three integrals in (3.97) are objective as well as the remaining scalar 

Densities in (3.97) are field quantities but we assume that the heating surface density q depends, in excess, on the external normal n 

Then using the tetrahedron arguments (similarly as in deduction of (3.72)) we prove from (3.99), (3.97), that dependence on n is linear 

Exchange of radiation between distant parts of the same body is neglected q on the real surface of body is given as a boundary condition. Assuming the validity of such a balance for each part of the body, we use again the principle of solidification and again volume and surface densities pu, Q, q etc.) could be deduced from more plausible primitives. Cf. Rems. 7, 13 and 14. 

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A uniform thermodynamic system—(uniform) body—may be visualized as a block of single (i.e., pure or one-constituent) material, the mass of which is fixed (closed system) with properties depending only on time (and not on space). Therefore, the state and state functions change only in time. Results (1.5) and (1.42) (where it is possible to use time integration) may be expressed in the rate form as it was explained at the end of Sect. 1.4. Consequently, such forms of energy balance and entropy inequality in uniform systems are [1, 3, 4]... 

At the end of these Sects. 3.3 and 3.4 we note that energy balance and entropy inequality motivated by procedures like those in Chap. 1 together with generalization of frame indifference (plausible objectivity is postulated not only for motion (Sect. 3.2) but also, e.g., for power of surface and body forces or heating) permit to deduce balances in Sect. 3.3 (i.e., for mass, linear and angular momentum), internal energy, entropy and their objectivity, etc. For details see, e.g., [1, 22, 42, 43] and other works on modern thermomechanics [7, 8, 18, 20, 41]. 

Balance of Energy and Entropy Inequality in Reacting Mixture ... 

Conversely the results that energy and entropy are the state functions permit us to formulate their constitutive equations in rational thermodynamics. These, together with balances (say of energy (1.5), (2.1), etc.) and entropy inequality (like (1.42),... 

Balance of energy (2.1) and entropy inequality (2.2) may be applied also for more complicated models [59, 67], namely mixtures and multiphase systems. Similarly classical thermodynamics balances (2.1), (2.2) have given the equilibrium result (namely model A of Sect. 2.2) and we can then obtain classical results for chemical and phase equilibria we show them on simple examples in Sects. 2.4,2.5. 

The whole mass m of the system is constant and this closed system, exchanging only heat and volume work with its surroundings, is supposed to be described again by fields (time function) (2.3) and (2.4) added with fields of (positive) masses /ni(t), m2(t) of both constituents 1, 2 respectively. For closed systems (cf. Sects. 1.2, 2.1) the balance of energy (2.1) and entropy inequality (2.2) are valid but now together with the balance of mass... 

Summary. Energy balance containing heat transfer, and entropy inequality are typical thermodynamic conceptions. In fact, they constitute the (general forms of) First and Second Law of thermodynamics, respectively. Perhaps the most important for further development are the local energy balance in the form (3.107) and the Clausius-Duhem formulation of entropy inequality—(3.109). Introducing the (specific) free energy, (3.111), the latter is transformed to the reduced form (3.113). 

To discuss this property, we first rewrite local balances and rearrange balances of energy and entropy (inequality) into more appropriate forms ... 

The principle of admissibility demands also that reduced inequality (4.89) must be fulfilled in any admissible thermodynamic process (because (4.89) was constructed from all these balances, mainly those of energy and entropy inequality). Then, and this is the main idea of Coleman and Noll [124], inserting constitutive equations of the studied model into (4.89), the identical fulfilling of inequality obtained in this way at any admissible thermodynamic process permits to obtain further properties of the constitutive model (for this, it suffices to choose the thermodynamic processes appropriately). 

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... 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

In addition to the momentum balance equation (6), one generally needs an equation that expresses conservation of mass, but no other balance laws are required for so-called purely mechanical theories, in which temperature plays no role (as mentioned, balance of angular momentum has already been included in the definition of stress). If thermal effects are included, one also needs an equation for the balance of energy (that expresses the first law of thermodynamics energy is conserved) and an entropy inequality (that follows from the second law of thermodynamics the entropy of a closed system cannot decrease). The entropy inequality is, strictly speaking, not a balance law but rather imposes restrictions on the material models. 

When the temperature is not constant, the bulk heat transfer equation complements the system and involves Equations 5.240, 5.241, and 5.276. The heat transfer equation is a special case of the energy balance equation. It should be noted that more than 20 various forms of the overall differential energy balance for multicomponent systems are available in the literature." " The corresponding boundary condition can be obtained as an interfacial energy balance." - Based on the derivation of the buUc and interfaciaT entropy inequalities (using the Onsager theory), various constitutive equations for the thermodynamic mass, heat, and stress fluxes have been obtained. 

The construction of entropy and absolute temperature (even in nonequiUbrium) fulfilling entropy inequality is done by Silhavy s method in terms of the primitives work, heat and empirical temperature (for the latter, see AppendixA.1 cf. Zemansky cited in [17, p. 53]). Moreover, the existence of energy satisfying the energy balance will be also proved. 

Such a form of entropy inequality (1.42) and likewise the energy balance (1.5) will be used (in fact by further simplifications) in Chap. 2 where uniform systems without space gradients are treated The process is a time sequence of the states and we may expect the validity of (1.5), (1.42) for arbitrarily close time instants. Therefore we formulate these basic laws for the rate (time derivative) of the state functions (entropy, energy) with heatings (rate of heat exchange) and power, cf. (2.1),... 

Energy Balance, Entropy Inequality and Constitutive Principles, and Equations in Uniform Systems... 

Theorem A.5.5 (which is algebraic only) may be applied to the thermodynamics of our book, namely in the admissibility principle used on the models of differential type as we show in the examples below. The X are here the time or space derivatives of deformation and temperature fields other than those contained in the independent variables of the constitutive equations and therefore al a, /3, Aj, Aj, Bj are functions of these independent variables. Constraint conditions (A.99) usually come from balances (of mass, momentum, energy) and (A. 100) from the entropy inequality. 

The strength of I-Shih Liu method, therefore, manifests itself at the more complicated constraints [24, 26]. The most complicated case in our book—the reacting mixture with linear transport properties—with the use of entropy inequality and all balances (of mass, momentum, energy) as (A.IOO), (A.99), would be laborious. Therefore, to demonstrate the application of the I-Shih Liu s Theorem A.5.5, we choose relatively simple examples of the uniform fluid model B from Sect. 2.2 and the simple thermoelastic fluid from the end of Sect. 3.6. ... 

The example of the uniform fluid model B discussed in the Sects. 2.1,2.2 uses entropy inequality (Second Law) (2.2) and balance of energy (First Law) (2.1) as a constraint (balances of mass and momentum may be ignored because they are fulfilled trivially the mass of bodies is constant and velocity (and therefore also kinetic energy) is zero). 

In view of the success of the theory of micromorphic materials introduced by ERINGEN and his co-workers [8,9,10], it is reasonable to seek a thermomechanical theory of the mixture. Initially, we attempted to fomulate such a theory by use of the equations of balance proposed by ERINGEN supplemented by what appeared to be the relation between micromorphic fluids and liquid-gas mixtures. With these equations we proposed a set of constitutive equations and, by a method similar to that developed by COLEMAN and NOLL [11,12], considered restrictions on the constitutive equations starting from the entropy inequality [13,14]. This procedure leads to the result that the free energy density is influenced by the gradient of the quality of the gas phase. Standard results are obtained for the linear theory and, in the special case known as an ideal mixture, the constitutive equations are given explicitly. 

The equality of this equation represents a system at equilibrium where JT = A = 0. The work done by the controlling system dissipates as heat. This is in line with the first law of thermodynamics. The inequality in Eq. (11.4) represents the second law of thermodynamics. The cyclic chemical reaction in nonequilibrium steady-state conditions balances the work and heat in compliance with the first law and at the same time transforms useful energy into entropy in the surroundings in compliance with the second law. The dissipated heat related to affinity A under these conditions is different from the enthalpy difference AH° = (d(Aii°/T)/d(l/Tj). The enthalpy difference can be positive if the reaction is exothermic or negative if the reaction is endothermic. On the other hand, the A contains the additional energy dissipation associated with removing a P molecule from a solution with concentration cP and adding an S molecule into a solution with concentration cs. 

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