Entropy balance equation for

Let us deduce the balance equation for entropy in a uniform solid body with a nonuniform temperature inside. In doing so, we shall ignore the possible volume variations due to the heat expansion (dv = 0). The flux of matter is also excluded in the solid (dci = 0). In this case, therefore. 

In irreversible thermodynamics entropy was introduced and discussed as a result of a need for quantification of the degree of irreversibility of a process. The theory thus explain the way in which the generation of disorder reflected by entropy change results in conversion of potentially useful work energy into practically useless thermal energy. In this discipline of thermodynamics the continuum balance equation for entropy plays a central role [32]. This equation expresses the fact that the entropy of a volume element changes with time for... 

Using this definition and Eq. 2.1-4, we have the following as the balance equation for entropy." ... 

A second restrictive equation results from the balance equation for entropy. For a system in stationary state the balance equation for entropy results in the following expression ... 

Thus, using the balance equation for entropy fluxes allows us to obtain a new scheme of determining the main parameters of discontinuous precipitation without Cahn s assumption about linear dependence of the driving force and the velocity of transformation boundary (the form of the dependence can be established in the process of calculation). The scheme may take into account some additional ways of energy dissipation in the transformation region, which result in the reduction of the boundary s mobility. Also, the system of problem solving may contain a more detailed description of the thermodynamic parameters and take into account additional kinds of driving forces. 

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... 

Enthalpy-pressure diagram for water-steam 813 Enthalpy term, energy balance equation 74 Entropy 3,28... 

The second law can be expressed in terms of the local balance equation for the entropy ... 

Thermal systems can be completely described using balance equations for mass, energy, and entropy in conjunction with thermophysical property relations and/or equations of state, equipment performance characteristics, thermokinetic or rate equations, and boundary/initial conditions. With the thermal system adequately described, it can be optimized by any current technique. Although the approach presented in this paper is not explicit in Second Law terms, it never-the-less will yield the optimal design and with the appropriate transformations, will yield any desired Second Law quantity. 

Use the values of standard molar entropies in Appendix K to calculate the entropy change at 25°C and one atmosphere pressure for the reaction of hydrazine with hydrogen peroxide. This explosive reaction has been used for rocket propulsion. Do you think the reaction is spontaneous The balanced equation for the reaction is... 

Another feature of the present theory is that it provides a formalism for deducing a complete mathematical representation of a phenomenon. Such a representation consists, typically, of (1) Balance equations for extensive properties (such as the "equations of change" for mass, energy and entropy) (2) Thermokinematic functions of state (such as pv = RT, for simple perfect gases) (3) Thermokinetic functions of state (such as the Fourier heat conduction equation = -k(T,p)VT) and (4) The auxiliary conditions (i.e., boundary and/or initial conditions). The balances are pertinent to all problems covered by the theory, although their formulation may differ from one problem to another. Any set of... 

We return to this problem in the next chapter after formulating the balance equation for an additional thermodynamic variable, the entropy. 

To complete our thermodynamic description of pure component systems, it is therefore necessary that we (1) develop an additional balance equation for a state variable and (2) incorporate into our description the unidirectional character of natural processes. In Sec. 4.1 we show that both these objectives can be accomplished by introducing a single new thermodynamic function, the entropy. The remaining sections of this chapter are concerned with illustrating the properties and utility of this new variable and its balance equation. 

In preparation for the discussion of how entropy changes accompanying a change of state can be computed, it is useful to consider the thermodynamic balance equations for a change of state of a closed system. The difference form of the energy balance is... 

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... 

In order to obtain an additional balance equation for the microstructural parameter K, the principle of dissipation is utilized. The starting point is the entropy balance [Eq. (9)] with entropy density p t], entropy flux entropy supply a, and entropy production f >0. 

Develop the balance equation for energy and entropy in open systems. 

On combining the balance equation for energy assuming the absence of velocity gradients, the first law of thermodynamics and assuming Gibbs equation for entropy production for the case of local equilibrium, a the entropy production per unit volume per unit time due to the occurrence of irreversible processes in the system is given by... 

Analyze and Plan In part (a) we must predict the value for AG° relative to that for AH° on fhe basis of fhe balanced equation for fhe reaction. In parf (b) we musf calculafe fhe value for AG° and compare wifh our qualifafive prediction. The free-energy change incorporafes bofh fhe change in enfhalpy and the change in entropy for fhe reaction (Equation 19.11), so under sfandard conditions ... 

To find the optimal profiles of the intensive variables in a chemical reactor, one solves the balance equations for mass, energy and momentum. Transport laws are also needed. The coupling of heat and mass is zero for fluxes that have different directions (that are perpendicular to and along the tube). Heterogeneous catalysis may require that coupling terms are taken along, as described in section 14.3.3. The local entropy production is found once the fluxes and forces are determined. An example of a set of such profiles is given in the next section. 

Generally, for an open thermodynamic system and its environment with which it has an exchange of matter, one can write the balance equation for the global variation of entropy [9-12]. 

A specific thermodynamic analysis is needed to derive expressions for the non-equilibrium properties of polymeric mixtures below the glass transition temperature. On the basis of the viscoleastic stress-strain relationship which has been considered in equation 3, specific relations may be obtained between equilibrium and nonequilibrium properties of the mixture. Indeed, when a mixture of one polymeric species and Np penetrant components is considered, the following balance equations for mass, energy and entropy may be written in local form, with respect to a polymer-fixed frame 11),... 

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