Entropy-balance equation

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)... 

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

Some of the molecular theories of multicomponent diffusion in mixtures led to expressions for mass flow of the Maxwell-Stefan form, and predicted mass flow dependent on the velocity gradients in the system. Such dependencies are not allowed in linear nonequilibrium thermodynamics. Mass flow contains concentration rather than activity as driving forces. In order to overcome this inconsistency, we must start with Jaumann s entropy balance equation... 

The mass energy and entropy balance equations are needed. The rate of change of probability density with time is... 

The entropy balance equation is derived assuming that the gas is at local equilibrium. We also assume that the suspension of Brownian particles in the heat bath may be a multicomponent ideal solution, and the thermodynamic potential is expressed by... 

From the general entropy balance equation dS= dJS+ dxS, we conclude that for an incompressible and isothermal process, we have deS d,S. This relation shows the equality between the dissipated heat flow and internal entropy production and hence the loss of power is q = Eloss. Therefore, Eq. (b) becomes... 

The remaining terms on the right of (6.2.13) must represent source terms if Eq. (6.2.13) is to be interpreted as an entropy balance equation d(p"5)/dt - - V JS + 9. Having thus identified - V Jg we can express 9 as the rate of entropy density generation locally as follows ... 

Entropy Balance Equation and Rate of Local Entropy Production... 

Table 4.1-1 gives several special cases of the entropy balance equation, on both a mass and a molar basis, for situations similar to those considered for the mass and energy balance equations in Tables 2.2-1,3.1-1, and 3.1-2. 

Be able to use the mass, energy, and entropy balance equations for mixtures (Secs. 

To derive the macroscopic entropy balance equation for mixtures we now %t. d — S, in Eq. 2.1-4 and make the same identification for the entropy flow terms as was made in Sec. 4.1, to obtain... 

The entropy balances are calculated using the steady-state entropy balance equation ... 

The conduction entropy flow consists of the heat flow J" and the diffusion flow j,. The J" is reduced heat flow that is the difference between the change in energy and the change in enthalpy due to matter flow. With the substantial derivative and using Eqn (3.46), we obtain the entropy balance equation based on a local thermodynamic equilibrium ... 

A fundamental aspect of our MD treatment is its acceptance or assumption of the existence, in each surface parallel to the electrode, of local thermodynamic functions and variables and of thermodynamic properties for all species whose individual molecules or ions are centered in that surface. This assumption is equivalent to the statement that we consider only systems for which the Gibbs entropy-balance equation holds locally. In some regions of the interphase between the bulk of the metal and the bulk of... 

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... 

Combining with the entropy balance. Equation 23.50, and defining destroyed exergy as irreversibility gives. 

The starting point of irreversible thermodynamics is the local form of the entropy balance equation. In a wide range of non-equilibrium processes, for instance heat conduction, thermo-diffusion non reactive mixture and thermoelectric effects are taking place in a volume V in static conditions, the entropy balance has the following form... 

The balance equation for entropy can be derived using the conservation of energy and the balance equation for the concentrations. This gives us an explicit expression for entropy production or— which can be related to irreversible processes such as heat conduction, diffusion and chemical reactions—and the entropy current is- The formal entropy balance equation is... 

For the sake of simplicity, we confine our attention to isothermal processes in which the local time derivative and spatial gradient of temperature are zero. In this case the entropy balance, equation 7, can be written in the following form ... 

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