Conservation principles momentum

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 kinetics, Newton s second law, the principles of kinematics, conservation of momentum, and the laws of conservation of energy and mass are used to develop relationships between the forces acting on a body or system of bodies and the resulting motion. 

In collisions between two bodies the contact force and the duration of contact are usually unknown. However, the duration of contact is the same for both bodies, and the force on the first body is the negative of the force on the second body. Thus the net change in momentum is zero. This is called the principle of conservation of momentum. 

In fluid mechanics the principles of conservation of mass, conservation of momentum, the first and second laws of thermodynamics, and empirically developed correlations are used to predict the behavior of gases and liquids at rest or in motion. The field is generally divided into fluid statics and fluid dynamics and further subdivided on the basis of compressibility. Liquids can usually be considered as incompressible, while gases are usually assumed to be compressible. 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

Credit for the first recognizable statement of the principle of conservation of energy (heat plus work) apparently belongs to J. Robert Mayer (Sidebar 3.2), who published such a statement in 1842. Mayer also obtained a (slightly) improved estimate, approximately 3.56 J cal-1, for the mechanical equivalent of heat. Mayer had actually submitted his first paper on the energy-conservation principle two years earlier, but his treatment of the concepts of force, momentum, work, and energy was so confused that the paper was rejected. By 1842, Mayer had sufficiently straightened out his ideas to win publication,... 

The principle of conservation of momentum stales that for a dynamical... 

On one level it is a quantum effect, and can be described in terms of photon—phonon scattering. The incident NIR beam is a source of photons, and the energy from the piezotransducer provides a source of lattice phonons that propagate through the crystal. As in all collision processes, the twin principles of conservation of momentum and conservation of energy apply. The momentum of a quantum particle is linked to its wavevector by hk. The energy is linked to its frequency by hjj. 

The analysis of the conditions within a gas channel can also be assumed to be onedimensional given that the changes in properties in the direction transverse to the streamwise direction are relatively small in comparison to the changes in the stream-wise direction. In this section, we examine the transport in a fixed cross-sectional area gas channel. The principle conserved quantities needed in fuel cell performance modeling are energy and mass. A dynamic equation for the conservation of momentum is not often of interest given the relatively low pressure drops seen in fuel cell operation, and the relatively slow fluid dynamics employed. Hence, momentum, if of interest, is normally given by a quasi-steady model,... 

The engineering science of transport phenomena as formulated by Bird, Stewart, and Lightfoot (1) deals with the transfer of momentum, energy, and mass, and provides the tools for solving problems involving fluid flow, heat transfer, and diffusion. It is founded on the great principles of conservation of mass, momentum (Newton s second law), and energy (the first law of thermodynamics).1 These conservation principles can be expressed in mathematical equations in either macroscopic form or microscopic form. 

The conservation of momentum is one of the basic principles of mechanics and it is to be expected that momentum remains invariant under Lorentz transformation. The fact that the velocity of a moving body is observed to be different in relatively moving coordinate systems therefore implies that,... 

Applying the conservation of momentum principle for the x-direction to the control volume gives ... 

In order to determine the distributions of pressure, velocity, and temperature the principles of conservation of mass, conservation of momentum (Newton s Law) and conservation of energy (first law of Thermodynamics) are applied. These conservation principles represent empirical models of the behavior of the physical world. They do not, of course, always apply, e.g., there can be a conversion of mass into energy in some circumstances, but they are adequate for the analysis of the vast majority of engineering problems. These conservation principles lead to the so-called Continuity, Navier-Stokes and Energy equations respectively. These equations involve, beside the basic variables mentioned above, certain fluid properties, e.g., density, p viscosity, p conductivity, k and specific heat, cp. Therefore, to obtain the solution to the equations, the relations between these properties and the pressure and temperature have to be known. (Non-Newtonian fluids in which p depends on the velocity field are not considered here.) As discussed in the previous chapter, there are, however, many practical problems in which the variation of these properties across the flow field can be ignored, i.e., in which the fluid properties can be assumed to be constant in obtaining fire solution. Such solutions are termed constant... 

As was mentioned above, the Navier-Stokes equations are obtained by the appli-cation of the conservation of momentum principle to the fluid flow. The same control volume that was introduced above in the discussion of the continuity equation is considered and the conservation of momentum in each of the three coordinate directions is separately considered. The net force acting on the control volume in any of these directions is then set equal to the difference between the rate at which momentum leaves the control volume in this direction and the rate at which it enters in this direction. The net force arises from the pressure forces and the shearing forces acting on the faces of the control volume. The viscous shearing forces for two-dimensional flow (see later) are shown in Fig. 2.3. They are expressed in terms of the velocity field by assuming the fluid to be Newtonian and are then given by [4],[5] ... 

Thus, the presence of the fluctuating turbulent velocity components causes the momentum transfer rate to be different from p X (mean velocity)2 x dA. But in applying the momentum conservation principle to the control volume the presence of additional momentum transfer is the equivalent of an additional force on the face of the control volume in the opposite direction to the momentum transfer. Thus, the additional momentum transfer due to the fluctuating velocity leads to an equivalent stress, i.e., force per unit area, of value pu 2 and this is what is termed the turbulent stress on the face. 

The limiting forms of the equations that result from the application of these conservation principles to this control volume as dx -+ 0 give a set of equations governing the average conditions across the boundary layer. These resultant equations are termed the boundary layer momentum integral and energy integral equations. 

A flow is completely defined if the values of the velocity vector, the pressure, and the temperature are known at every point in the flow. The distributions of these variables can be described by applying the principles of conservation of mass, momentum, and energy, these conservation principles leading to the continuity, the Navier-Stokes, and the energy equations, respectively. If the fluid properties can be assumed constant, which is very frequently an adequate assumption, the first two of these equations can be simultaneously solved to give the velocity vector and pressure distributions. The energy equation can then be solved to give the temperature distribution. Fourier s law can then be applied at the surface to get the heat transfer rates. 

In order to find the relation between the wall shearing stress, rw, and the friction factor, /, for the fully developed flow, the conservation of momentum principle is applied to the control volume shown in Fig. 7.3. 

The analysis is, of course, based on the application of the conservation of mass, momentum, and energy principles. These conservation principles are applied to control volume spanning all or part of the liquid film such as that shown in Fig. 11.6 and, because two-dimensional flow is being assumed in the following analysis, a control volume with unit width will be considered. 

In any chemical or electrochemical process, the application of the conservation principles (specifically to the mass, energy or momentum) provides the outline for building phenomenological mathematical models. These procedures could be made over the entire system, or they could be applied to smaller portions of the system, and later integrated from these small portions to the whole system. In the former case, they give an overall description of the process (with few details but simpler from the mathematical viewpoint) while in the later case they result in a more detailed description (more equations, and consequently more features described). 

It is quite possible to take as an extreme case that a molecule may stick to the walls on a collision, later to be evaporated. This will not violate the momentum conservation principle if the evaporating molecule carries away, on the average, as much momentum as the original molecule brought. 

The mean pressure is simply P = F/dS. But for our conservation principle to hold, Aj (momentum) of the wall must be equal to twice the component of perpendicular momentum brought by all the molecules striking the surface in the time At. Let us set up coordinate axes at the surface element dS and investigate the collisions (Fig. VII.3). 

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