Generalized Macroscopic Energy Balance

2 MACROSCOPIC ENERGY BALANCES 8.2.1 Generalized Macroscopic Energy Balance 

The energy balance is covered in detail in introductory courses on material and energy balances, and in later courses on thermodynamics. The following is a brief summary of the energy balance for chemically reacting systems. 

For a reactor in which one or more reactions are taking place. 

In formulating the energy balance in Eqn. (8-1), the kinetic and potential energy of the feed and product Streams, and of the reactor contents, have been neglected. In most cases, this is a valid assumption. ff 

The rate at which enthalpy enters the reactor is given by X Fg)Hgi. The summation 

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In this book, both material balances and energy balances are treated on a macroscopic basis, The general macroscopic energy balance for any system is as follows ... 

It is suggested that, in developing equations involving the macroscopic energy balance, the general equation (1.4.8) or (1.4.16) be simplified for the case under consideration. This approach will help to include all significant energy mechanisms in the final model. 

Differential momentum, mechanical-energy, or total-energy balances can be written for each phase in a two-phase flowing mixture for certain flow patterns, e.g., annular, in which each phase is continuous. For flow patterns where this is not the case, e.g., plug flow, the equivalent expressions can usually be written with sufficient accuracy as macroscopic balances. These equations can be formulated in a perfectly general way, or with various limitations imposed on them. Most investigations of two-phase flow are carried out with definite limits on the system, and therefore the balances will be given for the commonest conditions encountered experimentally. 

Integration over an arbitrary moving control volume, and use of the representation given by Eq. 2-22 leads to the general macroscopic total energy balance. 

While the Navier-Stokes equations and Bernoulli s equation, along with the macroscopic mass and linear momentum balances, were clearly available prior to 1888, it seems likely that the general macroscopic mechanical energy balance was not. Nevertheless, the fixed control volume... 

The temporal dependence of the fluctuations can be described by the overdamped equation of motion — the Landau-Khalatnikov equation [36, 37]. It can be understood as follows The equilibrium configuration of the order parameter is determined by the minimum of the free energy, 5T = f dVSf(T/,Vr/) = f dV 6f/6r])5r], which is satisfied for Sf/drj = 0. Here, S/Srj = d/dr] — V d/dVri). If the system is out of equilibrium, Sf/Sr] acts like a generalized elastic force which is balanced by viscous forces. When the macroscopic velocity can be neglected the viscosity is related solely to the rate of change of the order parameter Q,... 

Expressions of the conservation of mass, a particular chemical species, momentum, and energy are fundamental principles which are used in the analysis and design of any separation device. It is appropriate to formulate these laws first without specific rate expressions so that a clear distinction between conservation laws and rate expressions is made. Some of these laws contain a source or generation term, for example, for a particular chemical species, so that the particular quantity is not actually conserved. A conservation law for entropy can also be formulated which contributes to a useful framework for a generalized transport theory. Such a discussion is beyond the scope of this chapter. The conservation expressions are first presented in their macroscopic forms, which are applicable to overall balances on energy, mass, and so on, within a system. However, such macroscopic formulations do not provide the information required to size equiprrwnt. Such analyses usually depend on a differential formulation of the conservation laws which permits consideration of spatial variations of composition, temperature, and so on within a system. 

Annealing of semicrystalline polymers is a difficult process to understand. Both microscopic and macroscopic defects are reduced upon sample exposme to temperatures somewhat below the crystalline temperature. Polymer crystals are metastable, ie the lamellae thickness and lateral size are generally determined by the degree of supercooling balanced by the side and end free energies. Therefore, as a function of time, chain-folded polymer crystals thicken on annealing to minimize the number of folds. Although the exact mechanism has yet to be determined... 

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