The Entropy Balance Equation

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 

To obtain the explicit forms of J5 and a, we proceed as follows. For simplicity, we shall consider a system with no dissipation of kinetic energy due to convection or diffusion, and no external field. From the Gibbs relation Tds = du— Y2[tkdnk it follows that 

Now using the mole-number balance equation (15.3.10) and the balance equation (15.4.12) for the internal energy with 0(KE)/5r = 0, expression (15.5.2) can be written as 

This equation can be simplified and written in the form (15.5.1) by making the following observations. First, the affinity of reaction j is 

Through an elementary calculation using (15.5.4) and (15.5.5), we can rewrite (15.5.3) to arrive at the following equation for the entropy balance  

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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... 

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... 

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. 

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 ... 

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... 

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 ... 

We now deduce the expression of the entropy production rate Wg involved in the entropy balance equation ... 

The equations of thermodynamics at the interface can be used to find the entropy balance equation on the basis of the other balance equations. The process is identical to that used in the absence of fields. The equation is written thus ... 

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