Entropy conservation equations

Replacing ds/dt by the entropy balance equation, du/dt by the energy conservation equation, and dpkidt by the mass balance equation, we have... 

In a shock wave the compression is tied to a change in entropy, the only source of which are the dissipative forces—viscosity and heat conductivity. In the calculation we obtain a negligible front depth and compression time in the shock wave. We emphasize that this is a result of the calculation, not an assumption necessary to write the conservation equations. 

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 conservation 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 conservation equations, respectively. These are often called macroscopic and microscopic balance equations. 

Hassanizadeh, S.M. and Gray, W.G. (1979) General conservation equations for multiphase systems 2. Mass, momenta, energy and entropy equations. Adv. Water Resources 2, 191-203... 

The condition v l(p/p) 1 is valid for low-speed flow specifically, the square of the Mach number [v /(dp/dp, s = entropy]—a measure of the ratio of the ordered kinetic energy to the random, thermal kinetic energy of the molecules—must be small. When equation (25) is an acceptable approximation, equations (22) (with dYi/dt = 0) through (24) constitute an appropriate set of conservation equations. In these cases it is not necessary to use equation (20) (except to compute the actual small pressure change, after the rest of the problem has been solved with p = constant). 

Equations (46), (48), (49), and (54) may be taken to be the governing differential conservation equations. Equations (48) and (49) are already in characteristic form they show that composition and entropy changes propagate only along streamlines. Equations (46) and (54) involve derivatives only of p and v the characteristics of these equations require further study. 

Equation (2.23) is completely analogous to Eq. (2.21), but there is some important difference. It is possible to conduct a thermodynamic process in which dU = 0 and Ti T2. Setting d5i + A.d52 = 0, we get from the first equality in Eq. (2.23) 71d5i + T2d52 = 0, X = T2/T1. Inserting this into the second equality ofEq. (2.23), we get the identity T = Ti. Therefore, it is necessary to obtain the factor A. from an independent relation, if we want to calculate some specific values of 5i and S2 where the energy would be stationary. However, we can state that for stationary energy df7 = 0, Ti -> T) implies d5i -d52, which means the conservation of entropy. ... 

The conservation equations for matter (2.4.3) and energy (2.4.10) provide constraints on quantities, and therefore they allow us to test for consistency in the specifications of a proposed process. For example, with these conservation laws we may be able to test whether the proposed outputs of matter or energy are consistent with the proposed inputs. However, the entropy balance (2.4.21) is not a conservation law, and therefore it does not provide a check on quantities or consistency. Instead, it provides a constraint on the direction of a proposed process. Some proposed processes can be performed in both forward and reverse directions, but many others can be performed in only one way. In the latter cases, the entropy balance can be used to identify the ranges of operating variables (temperatures, pressure, flow rates) that must be used to make a proposed process proceed in the desired direction. 

Neglecting relativistic effects, the total quantities of mass and energy are constant, or conserved, although it is possible to convert energy from one form to another. Therefore, it is possible to write mass and energy conservation relationships. Entropy, on the other hand, is generated in real processes. Thus, an entropy balance equation must include an entropy... 

In Chap. 6, we will examine various approaches for the determination of P and q for speciflc types of fluids. Finally, we complete our derivations of the transport equations by developing the equation for the conservation of entropy. 

Now, to obtain an entropy conservation equation, we can work with Eq. (5.9) modified to include the time dependence in a itself, or it is somewhat easier to work directly with the reduced Liouville equation, Eq. (3.20) or (3.24), for pairwise additive systems we choose the latter representation. 

For the energy and entropy conservation equations, we can write similar and equally important expressions, viz.. 

In the case of a reversible reaction, fj has no fixed sign, need not be monotonic, and sustained oscillations cannot be ruled out except by introducing entropy considerations. Suppose that there exists a positive definite form (t(u), vanishing only on the equilibrium manifold and satisfying a conservation equation of the type... 

Anderson and McFadden [8] listed a representative (but relatively simple) model that have been considered by a number of authors. The hydrodynamic equations governing inviscid, compressible flow of a single component fluid near its critical point are described by conservation equations for mass, momentum, and entropy (energy). 

This is an accounting or balance equation, but not a conservation equation. Mass and energy are conserved, entropy is not. Clausius made that distinction clear in his famous formulation, The energy of the universe is constant the entropy of the universe increases toward a maximum, which irreverent students have rephrased as You can t win you can t even break even ... 

If we restrict our attention to reversible processes, then entropy is conserved and this becomes a conservation equation. The increase of entropy in our system due to flow of matter in or out and heat flow in or out is exactly balanced by the decrease in entropy of the surroundings from which matter and heat flow into or out of our system. 

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