Mechanical Energy and the Conservation Principle

A young mother (of scientific inclination) notes that her son Junior s favorite playthings are a set of 37 building blocks. Part of each day s task is gathering up the blocks that Junior has strewn about the nursery (perhaps hidden under the bed or behind the closet door). Although the blocks are of many markings and colorations, and sometimes deviously hidden, she recognizes a useful block-conservation principle  

One day, she is surprised to find that only 36 blocks are present. However, she notices that Junior has been playing with a saw from his tool set, and she observes an uncommon quantity of sawdust strewn about She gathers and weighs the sawdust, and is delighted to find that it matches the weight of a test block. Thus, she modifies the block-conservation principle to read 

On still another day, she is again disheartened to discover that the left-hand side of (S2.13-3) gives only 36. However, she notices that Junior has been playing with matches, and she observes that the room temperature is slightly warmer than usual So she carefully measures the heat of combustion of a test block and calculates (from the known heat capacity of the nursery contents) the expected room temperature change, 

you must imagine that the first term is absent, and the block-conservation principle reads simply 

The mother wishes to demonstrate this wonderful principle to her husband. She weighs the sawdust, measures the sinkwater, and reads the thermometer to evaluate the successive terms in (S2.13-5), then announces proudly that the result is once again 37 blocks But, asks the husband, what exactly are the blocks you are talking about  

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To describe the theoretical dynamical and thermal behavior of the atmosphere, the fundamental equations of fluid mechanics must be employed. In this section these equations are presented in a relatively simple form. A more conceptual view will be presented in Section 3.6. The circulation of the Earth s atmosphere is governed by three basic principles Newton s laws of motion, the conservation of energy, and the conservation of mass. Newton s second law describes the response of a fluid to external forces. In a frame of reference which rotates with the Earth, the first fundamental equation, called the momentum equation, is given by ... 

In this section we will apply the laws of mechanical energy and the momentum conservation principle to processes of particle collisions. By collision, we mean any short interaction between particles. 

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. 

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. 

In Section 4.2.4, the governing equations of fluid mechanics for a turbulent flow are derived. Similarly, the governing equations for heat transfer and mass transfer can be derived from the principles of energy and mass conservation. In fact, the species conservation equation is an extension of the overall mass conservation (or the continuity) equation. For species i, it has the following form ... 

Experimentally, these principles emphasize dynamic measurements that make possible the separation of the dissipative and the conservative components of energy Incident upon the system. Dynamic mechanical analysis has been an Important area of research for over 40 years. Computer-controlled experimentation now makes It possible to apply analogous techniques to the measurement of many other thermodynamic stresses. One example currently under Investigation, dynamic photothermal spectroscopy. Is expected to provide a new approach to predicting the long-term effects of ultraviolet radiation on materials [39]. 

The first law of thermodynamics deals with energy conservation. We start with energy conservation in pure mechanical systems and extend this principle to thermodynamics. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

One of the pedagogically unfortunate aspects of quantum mechanics is the complexity that arises in the interaction of electron spin with the Pauli exclusion principle as soon as there are more than two electrons. In general, since the ESE does not even contain any spin operators, the total spin operator must commute with it, and, thus, the total spin of a system of any size is conserved at this level of approximation. The corresponding solution to the ESE must reflect this. In addition, the total electronic wave function must also be antisymmetric in the interchange of any pair of space-spin coordinates, and the interaction of these two requirements has a subtle influence on the energies that has no counterpart in classical systems. 

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