Forces momentum

A simplified schematic of a particle in a centrifuge is illustrated in Fig. 12-3. It is assumed that any particle that impacts on the wall of the centrifuge (at r2) before reaching the outlet will be trapped, and all others won t. (It might seem that any particle that impacts the outlet weir barrier would be trapped. However, the fluid circulates around this outlet corner, setting up eddies that could sweep these particles out of the centrifuge.) It is thus necessary to determine how far the particle will travel in the radial direction while in the centrifuge. To do this, we start with a radial force (momentum) balance on the particle ... 

In all cases the weight of all material within the control volume must be included in the force-momentum balance, although in many cases it will be a small force. Gravity is an external agency and it may be considered to act across the control surface. The momentum flows and all forces crossing the control surface must be included in the balance in the same way that material flows are included in a material balance. 

Rather than setting up a force-momentum balance for a particular flow problem as was done in Chapter 1, general equations, known as the Navier-Stokes equations, may be formulated. Before discussing the Navier-Stokes equations, it is necessary to consider some related matters. 

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,... 

Force, momentum, velocity and acceleration are examples of vector quantities (they have a direction and a magnitude) and are written in this book with an arrow over them. Other physical quantities (for example, mass and energy) which do not have a direction will be written without an arrow. The directional nature of vector quantities is often quite important. Two cars moving with the same velocity will never collide, but two cars with the same speed (going in different directions) certainly might ... 

As the rate of momentum transfer is equal to a force, momentum balances are equivalent to force balances. 

The setting up of the constitutive relation for a binary system is a relatively easy task because, as pointed out earlier, there is only one independent diffusion flux, only one independent composition gradient (driving force) and, therefore, only one independent constant of proportionality (diffusion coefficient). The situation gets quite a bit more complicated when we turn our attention to systems containing more than two components. The simplest multicomponent mixture is one containing three components, a ternary mixture. In a three component mixture the molecules of species 1 collide, not only with the molecules of species 2, but also with the molecules of species 3. The result is that species 1 transfers momentum to species 2 in 1-2 collisions and to species 3 in 1-3 collisions as well. We already know how much momentum is transferred in the 1-2 collisions and all we have to do to complete the force-momentum balance is to add on a term for the transfer of momentum in the 1-3 collisions. Thus,... 

Surface tension forces/momentum transport (dissipation)... 

The result is consecrated as the famous Einstein relation for the total relativistic energy of a system relating its d5mamical mass. Worth noting that to the same results one arrives when takes the force way first, namely employing the force-momentum relationship according with the 2" Newton law yet with the relativistic momentum... 

Vectors. Velocity, force, momentum, and acceleration are considered vectors since they have magnitude and direction. They are regarded as first-order tensors and are written in boldface letters in this text, such as v for velocity. The addition of the two vectors B -i- C by parallelogram construction and the subtraction of two vectors B — C is shown in Fig. 3.6-1. The vector B is represented by its three projectionsB, By, and Bj on the x, y, and z axes and... 

Applied physics is of necessity the oldest of all practical sciences, dating back to the first artificial use of an object by an early hominid. The basic practices have been in use by builders and designers for many thousands of years. With the development of mathematics and measurement, the practice of applied physics has grown apace, relying as it still does upon the application of basic concepts of vector properties (force, momentum, velocity, weight, moment of inertia) and the principles of simple machines (lever, ramp, pulley). 

J represent the flows, F the forces, and L the phenomenological coefficients, i and j denote the different flows and forces. Thus, concentration difference may be one force, temperature difference another force, momentum difference another force, etc., and the flows can be heat transfer, mass transfer, and momentum transfer. Onsager showed that from analysis of a positive definite matrix, the cross-coefficients in Equation (B.17) have to be equal. Thus ... 

Hence the FjA term is the flux of momentum (because force = differential form (converting FM to a shear stress r), then we obtain... 

Buoyancy forces Inertia forces Viscous forces Viscous forces Momentum diffusivity(kinematic viscosity) Thermal diffusivity Forced convection heat transfer Conduction heat transfer... 

In this section we first review general modeling principles, emphasizing the importance of the mass and energy conservation laws. Force-momentum balances are employed less often. For processes with momentum effects that cannot be neglected (e.g., some fluid and solid transport systems), such balances should be considered. The process model often also includes algebraic relations that arise from thermodynamics, transport phenomena, physical properties, and chemical kinetics. Vapor-liquid equilibria, heat transfer correlations, and reaction rate expressions are typical examples of such algebraic equations. 

Before ending this section on relativity theory, we reflect on a remark made by Lowdin [27] regarding the perihelion motion of Mercury. Describing a gravitational approach within the consmiction of special relativity, he demonstrated that the perihelion moved but that the effect was only half the correct value. The problem here is the fundamental inconsistency between the force-, momentum and the energy laws, while the discrepancy for so-called normal distances are almost impossible to observe directly since (1 - x(r)) 1. However using the present method to the classical constant of motion... 

Classical mechanics, introduced in the last chapter, is inadequate for describing systems composed of small particles such as electrons, atoms, and molecules. What is missing from classical mechanics is the description of wavelike properties of matter that predominates with small particles. Quantum mechanics takes into account the wavelike properties of matter when solving mechanical problems. The mathematics and laws of quantum mechanics that must be used to explain wavelike properties cause a dramatic change in the way mechanical problems must be solved. In classical mechanics, the mathematics can be directly correlated to physically measurable properties such as force, momentum, and position. In quantum mechanics, the mathematics that yields physically measurable properties is obtained from mathematical operations with an indirect physical correlation. 

In rotational motion other dynamic characteristics are also required, such as a force moment (torque) with regard to a motionless axis, a moment of inertia (MI) and an angular momentum, being in some respect analogous to the characteristics of linear motion (mass, force, momentum). 

Suppose that force F is arbitrarily applied to a body s point (in Figure 1.12 the body itself is again not shown). Divide the force vector into two components one parallel to the axis of rotation F11, and the other lying in the plane perpendicular to the rotation axis Fj. Only one of them (FJ influences the rotation, whereas F exerts pressure on the bearings in which an axis is fixed. The force momentum (torque) M in respect to an axis Oz is the value... 

This equation presents another form of Newton s second law for body rotation relative to the axis Oz the change of the angular momentum projection onto the axis Oz is equal to the sum of projections of all the force momentums applied to the body relative to the same axis. 

The rotation of the gyroscope s axis z relative to the vertical axis z is referred to as gyroscope precession. Under the action of the gravity force momentum the vector of the angular momentum L of the unbalanced gyroscope obtains an increment dL directed along... 

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