Abstract
In this chapter (see Sects. 7.1–7.6) we derive the macroscopic conservation laws of matter, momentum, energy and angular momentum for a multicomponent system subjected to conservative external forces and in which chemical reactions may occur. Then, in Sect. 7.7, the time growth of the macroscopic entropy of the system is derived, showing that it can be expressed as the product between thermodynamic fluxes and their conjugated thermodynamic forces. It should be noted that, as shown in Appendix E, these equations are consequences of the fundamental mechanical laws governing the motions of the constituent particles of the system.
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Notes
- 1.
Actually, v is the mean velocity of the fluid particles that are contained within the physical point-like volume defined in (1.6) (that is a volume large enough so as to neglect thermal fluctuations and yet small enough to neglect the effects of macroscopic gradients). Clearly, when we deal with a single component fluid, the average can be intended as mass or as molar average, as the two quantities are the same. As we will see in the next section, though, in multicomponent flows the two averages are different from each other and we will choose to define v as the mass-averaged velocity.
- 2.
Regular mixtures are composed of species that are rather similar to each other, so that they behave in some extent like mixtures of ideal gases, i.e. volume and enthalpy are additive quantities.
- 3.
See discussion in [2].
- 4.
Using Ricci’s tensor, the cross product between two vector a and b can be written as: (a×b) i =∑ j,k ϵ ijk a j b k . Note that ∑ m ϵ ijm ϵ mkl =δ ik δ jl −δ il δ jk and ∑ jk ϵ ijk ϵ jkl =2δ il .
References
de Groot, S.R., Mazur, P.: Non-Equilibrium Thermodynamics. Dover, New York (1984), Chap. III.3
Leal, L.G.: Laminar Flow and Convective Transport Processes, pp. 178–179. Butterworth, Stoneham (1992)
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Mauri, R. (2013). Balance Equations. In: Non-Equilibrium Thermodynamics in Multiphase Flows. Soft and Biological Matter. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5461-4_7
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