Microscopic mechanical energy balance

The most fi equently used relationship in the design of flow systems is the macroscopic mechanical-cnergy balance. This equation is obtained by integrating the microscopic mechanical-energy balance over the volume of the system as shown by Bird et al. [9]. The balance is given by... 

A balance on mechanical energy can be written on a microscopic basis for an elemental volume by taking the scalar product of the local velocity and the equation of motion. After integration over the entire volume of the system the steady-state mechanical energy balance for a system with mass interchange with the surroundings becomes, on a per unit mass basis,... 

Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, totaf energy, and mechanical energy may be found in Whitaker (ibid.). Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. 

If one looks closer, using a microscope, at the point where the polymer film is detaching from the oxide surface, at the micrometer level (Fig. 3.9(b)), then it is evident that there is a crack traveling along the interface between the polymer and the glass. This crack is the mechanism by which the polymer material detaches from the surface. All brittle adhesive joints fail by cracking. This is a mechanism which involves elastic deformations and creation of new surfaces. It can be analyzed by the energy balance theory described in Cluster 7. 

Conservation laws at a microscopic level of molecular interactions play an important role. In particular, energy as a conserved variable plays a central role in statistical mechanics. Another important concept for equilibrium systems is the law of detailed balance. Molecular motion can be viewed as a sequence of collisions, each of which is akin to a reaction. Most often it is the momentum, energy and angrilar momentum of each of the constituents that is changed during a collision if the molecular structure is altered, one has a chemical reaction. The law of detailed balance implies that, in equilibrium, the number of each reaction in the forward direction is the same as that in the reverse direction i.e. each microscopic reaction is in equilibrium. This is a consequence of the time reversal syimnetry of mechanics. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

Many polymer blends or block polymer melts separate microscopically into complex meso-scale structures. It is a challenge to predict the multiscale structure of polymer systems including phase diagram, morphology evolution of micro-phase separation, density and composition profiles, and molecular conformations in the interfacial region between different phases. The formation mechanism of micro-phase structures for polymer blends or block copolymers essentially roots in a delicate balance between entropic and enthalpic contributions to the Helmholtz energy. Therefore, it is the key to establish a molecular thermodynamic model of the Helmholtz energy considered for those complex meso-scale structures. In this paper, we introduced a theoretical method based on a lattice model developed in this laboratory to study the multi-scale structure of polymer systems. First, a molecular thermodynamic model for uniform polymer system is presented. This model can... 

Point (microscopic) defects in contrast from the macroscopic are compatible with the atomic distances between the neighboring atoms. The initial cause of appearance of the point defects in the first place is the local energy fluctuations, owing to the temperature fluctuations. Point defects can be divided into Frenkel defects and Schottky defects, and these often occur in ionic crystals. The former are due to misplacement of ions and vacancies. Charges are balanced in the whole crystal despite the presence of interstitial or extra ions and vacancies. If an atom leaves its site in the lattice (thereby creating a vacancy) and then moves to the surface of the crystal, it becomes a Schottky defect. On the other hand, an atom that vacates its position in the lattice and transfers to an interstitial position in the crystal is known as a Frenkel defect. The formation of a Frenkel defect therefore produces two defects within the lattice—a vacancy and the interstitial defect—while the formation of a Schottky defect leaves only one defect within the lattice, that is, a vacancy. Aside from the formation of Schottky and Frenkel defects, there is a third mechanism by which an intrinsic point defect may be formed, that is, the movement of a surface atom into an interstitial site. Considering the electroneutrality condition for the stoichiometric solid solution, the ratio of mole parts of the anion and cation vacancies is simply defined by the valence of atoms (ions). Therefore, for solid solution M X, the ratio of the anion vacancies is equal to mJn. 

The application of microscopic reversibility to each molecular reactive collision in a chemical reaction system consisting of a statistically large assembly of molecules with a distribution of momenta and internal energy states is called the principle of detailed balance. Detailed balance requires one to write all elementary reactions as reversible, and it permits one to rule out some types of mechanisms, such as the cyclic sequence of the following equation ... 

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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