Entropy rate balance

The forms of the two balances depend on the particular circumstances of the process. As an illustration, we will consider a bulk flow process in steady state (Figure 2) in a chamber with a fixed volume. The energy and entropy rate balances are ... 

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. 

The rate of entropy generation and the lost work for each of the individual ] of the process are calculated by Eqs. (16.1) and (16.15). Since the flow rate of methane is not given, we take 1 kg of methane entering as a basis. The rates S, W m, and Q are therefore expressed not per unit of time but per kg of entering meth The heat transfer for the compression/cooling step is calculated by an enc balance ... 

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

Engineering systems mainly involve a single-phase fluid mixture with n components, subject to fluid friction, heat transfer, mass transfer, and a number of / chemical reactions. A local thermodynamic state of the fluid is specified by two intensive parameters, for example, velocity of the fluid and the chemical composition in terms of component mass fractions wr For a unique description of the system, balance equations must be derived for the mass, momentum, energy, and entropy. The balance equations, considered on a per unit volume basis, can be written in terms of the partial time derivative with an observer at rest, and in terms of the substantial derivative with an observer moving along with the fluid. Later, the balance equations are used in the Gibbs relation to determine the rate of entropy production. The balance equations allow us to clearly identify the importance of the local thermodynamic equilibrium postulate in deriving the relationships for entropy production. 

We all widely utilize aspects of the first law of thermodynamics. The first law mainly deals with energy balance regardless of the quality of that part of the energy available to perform work. We define first law efficiency or thermal efficiency as the ratio of the work output to total rate of heat input, and this efficiency may not describe the best performance of a process. On the other hand, the second law brings out the quality of energy, and second law efficiency relates the actual performance to the best possible performance under the same conditions. For a process, reversible work is the maximum useful work output. If the operating conditions cause excessive entropy production, the system will not be capable of delivering the maximum useful output. 

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... 

The entropy balance determines the rate of entropy production due to irreversibilities... 

Example 5.11 Hot fluid flow rate effect Consider two heat exchangers 1 and 2 operating at steady state and constant pressure with the same heat duty. The total entropy change of the cold fluid is the same for both heat exchangers and determined by the specified heat duty qs. There is no heat loss to the environment. The overall entropy balances for the heat exchangers are... 

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

At equilibrium, the affinities vanish (A] = 0,A2 = 0). Therefore, Jrl - Jt3 = 0 and. /r2. Jr3 0 and the thermodynamic equilibrium does not require that all the reaction velocities vanish they all become equal. Under equilibrium conditions, then, the reaction system may circulate indefinitely without producing entropy and without violating any of the thermodynamic laws. However, according to the principle of detailed balance, the individual reaction velocities for every reaction should also vanish, as well as the independent flows (velocities). This concept is closely related to the principle of microscopic reversibility, which states that under equilibrium, any molecular process and the reverse of that process take place, on average, at the same rate. 

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... 

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