Entropy balances

The second method can be applied to mixtures as well as pure components. In this method the procedure is to find the final temperature by trial, assuming a final temperature and checking by entropy balance (correct when ASp t, = 0). As reduced conditions are required for reading the tables or charts of generalized thermodynamic properties, the pseudo critical temperature and pressure are used for the mixture. Entropy is computed by the relation. See reference 61 for details. ... 

For the generalized control volume shown in Figure 2-33, and entropy balance can be stated as follows ... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

If we consider a system of infinitesimal dimensions, then the entropy balance can be described directly by Eq. (2.2.13) ... 

Perform an isentropic expansion. That is, as the pressure decreases from P0 to the backpressure Pb (usually ambient pressure Pa), select intermediate values of pressure Pi. At each P1 find the temperature that keeps S constant 7 i. Solve for the vapor fraction x, using the entropy balance between planes 0 and 1 ... 

Some interesting aspects of the interface kinetics appear only when temperature and latent heat are included into the model, if the process of heat conductivity is governed by a classical Fourier law, the entropy balance equation takes the form Ts,= + x w where s = - df dr. Suppose for simplicity that equilibrium stress is cubic in strain and linear in temperature and assume that specific heat at fixed strain is constant. Then in nondimensional variables the system of equations takes the form (see Ngan and Truskinovsky, 1996a)... 

Unlike mass, which is conserved, leading to dp/dt= —V ], entropy is not conserved. That is, in addition to entropy flux that would lead to variation in entropy in a given volume, entropy is also produced from nowhere. Hence, entropy balance is written as... 

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

Entropy Balance Near the Reversible-Thermodynamic Equilibrium... 

Heat, momentum, mass, entropy balances at finite domain structure levels of solids and liquids, during deformation, melting, and solidification ... 

The concept of affinity introduced in the foregoing chapter (section 3.5) can apply to all the physicochemical changes that occur irreversibly. Let us now discuss the physical meaning of the affinity of chemical reactions. As mentioned in the foregoing, we have in Eq. 3.27 the fundamental inequality in entropy balance of irreversible processes as shown in Eq. 4.1 ... 

Gyftopoulos, E.P. and Widmer, T.F., "Availability Analysis The Combined Energy and Entropy Balance", Thermodynamics Second Law Analysis, A.C.S. Symposium Series, 122, 61-76, 1980. 

The calculation procedure consists of a determination of the air/fuel molar ratio that will result in a combustor outlet temperature of 2000F, followed by the calculation of enthalpy and entropy of flow streams at the various state points in the cycle. The net work output from the turbine is calculated by subtracting the compressor work requirements from the total turbine expansion work determined from the enthalpy balance. Irreversible entropy productions are calculated by a routine entropy balance around each process step. 

Substitution of Equation 6 into the entropy balance yields... 

Keller, A., "The Evaluation of Steam Power Plant Losses by Means of the Entropy Balance Diagram," Trans. ASME, 72, 949, (1959). 

For instance, a particle i shown in the figure as an open circle is transported from left to right by a driving force. Thus, when the particle i leaves the left side it releases an amount of heat and when it arrives to the right side it absorbs the same amount of heat if the phase and conditions are the same. An entropy balance at the right-hand side comprises both transport entropy and molar entropy. 

Assuming negligible kinetic and potential energy changes, mass, energy, and entropy balances yield... 

The entropy balance yields the rate of entropy production, Sprod... 

The entropy balance contains the nozzle and its surroundings, and we have... 

Entropy balance - Sout + Spioi = 0 - mxSx + m2S2 - m3S3 + Spmd = 0... 

Entropy balance -Sout +5 prod = +m2S2- f- + Sptod = 0... 

The formulation of linear nonequilibrium thermodynamics is based on the combination of the first and second laws of thermodynamics with the balance equations including the entropy balance. These equations allow additional effects and processes to be taken into account. The linear nonequilibrium thermodynamics approach is widely recognized as a useful phenomenological theory that describes the coupled transport without the need for the examination of the detailed coupling mechanisms of complex processes. 

In every nonequilibrium system, an entropy effect exists either within the system or through the boundary of the system. Entropy is an extensive property, and if a system consists of several parts, the total entropy is equal to the sum of the entropies of each part. Entropy balance is... 

Change in total entropy = Total entropy in Total entropy out + Total entropy produced Entropy balance in the rate form is given by... 

The general entropy balance relations for a control volume are given in terms of the rate of entropy change due to the heat transfer, mass flow, and entropy production... 

Exergy is a unifying concept of many forms of energy, such as heat, mechanical work, and chemical energy. We can derive the exergy Ex relation from the energy and entropy balances for the composite system shown in Figure 4.15... 

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