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By Jean-Laurent Puebe

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Extra info for Fundamentals of Fluid Mechanics and Transport Phenomena

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N  I ; ) P p 1 pI. 38] p 1 We laid down that certain extensive variables were not exchanged between the sub-systems. 2, example 2), or by choosing different conditions according to certain ensembles of sub-systems. 2. 7a) whose P sub-systems exchange heat via a constant-volume process, such that the temperature Tp is the only variable intensive quantity of any of the sub-systems during the process. The energy Ep of each sub-system can be expressed as a function of its temperature and Thermodynamics of Discrete Systems 39 its specific heat capacity *p, which is assumed to be constant for the sake of simplicity: Ep * pT p The energy E of the system and its average temperature Tm can be obtained: E P ¦ * pT p p 1 · § P ¨ ¦ * ¸T p ¨p 1 ¸ m ¹ © *T m ; with : * P ¦ *p p 1 The average temperature of the out-of-equilibrium system can thus be written: Tm 1 * ¦ * pT p p In this particular case, the average temperature is the average of the temperatures weighted by the specific heat capacities.

Apart from some exceptions (shocks), we will only consider processes comprising a continuous series of states, described by variables which must be continuous functions of time. We will however allow situations with discontinuities (shocks, shockwaves, deflagration) which momentarily violate this continuity condition. 2. The notion of a system The notion of a system is a relatively vague one; it is in fact included in that of a process: a system is an entity which we consider during a process. As our considerations often take a differential form, the system is the principal part (zero order) on which we perform differential balances.

Definition An out-of-equilibrium system is characterized by a collection of intensive quantities whose values differ according to the sub-systems considered. It may be useful to characterize the system by a global intensive variable, which is an “average value” of the intensive variables of the sub-systems. In order to define this average value, we will refer to an “equivalent” equilibrium state of the system. Consider an out-of-equilibrium system S made up of P sub-systems Sp each of which is in instantaneous equilibrium (quasi-static transformations).

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