The Temperature Equation
In this chapter we carry out the classical procedure of thermodynamics of irreversible process [1],[2], to obtain the equation of the temperature in electrohydrodynamics. First, we review the thermodynamics of dielectric fluids at thermodynamic equilibriu
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Chapter 4
THE TEMPERATURE EQUATION
In this chapter we carry out the classical procedure of thermodynamics of irreversible process [1 ],[2], to obtain the equation of the temperature in electrohydrodynamics. First, we review the thermodynamics of dielectric fluids at thermodynamic equilibrium [3],[4]. The internal electric energy of polar fluids depends on temperature through the permittivity. Given that temperature and internal energy are intimately connected, we split the internal electrical energy for polar fluids into two terms. The term that depends on temperature is added to the term for the internal energy of the fluid in the absence of the electric field to form the total internal energy. The remainder term, which is of the form (1/2)EnpE 2 , where Enp is the permittivity at frequencies above the microwave region but below the short infra red region, is ascribed to the macroscopic electrical energy. The permittivity Enp only depends on the ionic and electronic molecular polarizabilities, and therefore it is independent of temperature. The assumption of local equilibrium allow us to extend the Gibbs relation to non-equilibrium processes, and the rate of variation of entropy can be calculated once the internal energy has been identified, since the production of the latter is equal to minus the production of kinetic, gravitational potential, and electrical energies (first principle of thermodynamics). The linearity between fluxes and forces, together with the Onsager reciprocity relations are used to determine the constitutive equations for the heat flux and electrical current density. The equation for the temperature is then derived from the entropy equation. The physics behind the terms of electrical origin is discussed, and their orders of magnitude estimated. Finally, an important simplification of the EHD equations, referred to as the Boussinesq approximation in fluid dynamics, is extended to the case of EHD.
4.1 Internalenergy For a fluid consisting of structureless point particles of mass mk, with interaction energy negligible compared to their kinetic energy, the total energy, U per unit volume averaging
A. Castellanos (ed.), Electrohydrodynamics © Springer-Verlag Wien 1998
63
Basic ConceptS- and Equations in Electrohydrodynamics over a small volume 8V is given by
(4.1) with Urn the kinetic energy per unit volume and Uo the internal energy per unit mass. For this simple fluid (ideal monatomic gas) the concept of temperature is introduced through the relation pU0 = (3/2)Nk 8 T with N the number of particles per unit volume, ka = 1.38 x w- 23 JK- 1 Boltzmann's constant, and T the absolute temperature. For ideal gases we have for the energy per unit mass
Uo
= cvT+ U~,
(4.2)
with U~ constant for any given gas, m the molecular mass, p the pressure, and cv the specific heat at constant volume. This specific heat is the energy added in the form of heat (keeping the volume constant) to increase the absolute temperature of the unit of mass by one degree. For liquids, at equilibrium, the interacti
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