Mass force

It does not matter, from the fundamental point of view, how the pressure drop is generated in the filter. In the case of the centrifugal filters there is an additional phenomenon of the mass forces acting on the Hquid within the cake. The conventional filtration theory must be amended to include this effect (2). 

Removal of Cake by Mass Forces. This method of limiting cake growth employs mass or electrophoretic forces on particles, acting tangentially to or away from the filter medium. Only mass forces are considered here because the electrophoretic effects have been discussed previously. 

Mass, Force, and Weight. Weight is a force the weight of a body is the product of its mass and the acceleration due to gravity. 

Potential Length Volume Mass Force Pressure Energy... 

The added mass force accounts for the resistance of the fluid mass that is moving at the same acceleration as the particle. Neglecting the effect of the particle concentration on the virtual-mass coefficient, for a spherical particle, the volume of the added mass is equal to one-half of the particle volume, so that... 

Note that for liquid solid systems, Eq. (20) should also include the short-range lubrication forces and the effects of other forces such as the virtual mass force. But this is beyond the scope of this chapter. 

Let us make some connections to the results which came from the previous model development. First, if we compare (3.19)-(3.22) with (3.11)-(3.15), a natural identification of the solvent coordinate s in Sec. 3 is in fact just the fluctuating force SF on x at the transition state. (Note especially that this choice associates the solvent coordinate with a direct measure of the relevant solute-solvent interaction.) The solvent mass, force constant and frequencies in Sec. 3 would then be given molecular expressions via (3.19)-(3.21), while the solvent friction i (t) of Sec. 3 would be the friction per mass for Sf (3.22),... 

Almost every biological solution of low viscosity [but also viscous biopolymers like xanthane and dilute solutions of long-chain polymers, e.g., carbox-ymethyl-cellulose (CMC), polyacrylamide (PAA), polyacrylnitrile (PAN), etc.] displays not only viscous but also viscoelastic flow behavior. These liquids are capable of storing a part of the deformation energy elastically and reversibly. They evade mechanical stress by contracting like rubber bands. This behavior causes a secondary flow that often runs contrary to the flow produced by mass forces (e.g., the liquid climbs the shaft of a stirrer, the so-called Weissenberg effect ). 

When solving flow and heat transfer problems in polymer processing we must satisfy conservation of mass, forces or momentum and energy. Momentum and energy balances, in combination with material properties through constitutive relations, sometimes result in... 

Fa Carried mass force vector Pc Collisional stress tensor... 

Determine the initial acceleration of a free-falling glass bead with the density of 2,300 kg/m3 in the air and that of a rising spherical air bubble in a fluidized bed. The fluidized bed can be regarded as a pseudocontinuum medium with a density of 600 kg/m3. Discuss the significance of carried mass force for both cases. 

In the preceding equations, Fp can be expressed as a combination of local averaged drag force and virtual mass force [Anderson and Jackson, 1967]. 

A dispersion formula results, based on frequency dependency on masses, force constant and distance between the two masses, such as... 

In most of the more recent classical approaches [18], no allusion to Ehrenfest s (adiabatic) principle is employed, but rather the differential equations of motion from classical mechanics are solved, either exactly or approximately, subject to a set of initial conditions (masses, force constants, interaction potential, phase, and initial energies). The amount of energy, AE, transferred to the oscillator is obtained for these conditions. This quantity may then be averaged over all phases of the oscillating molecule. In approximate classical and semiclassical treatments, the interaction potential is expanded in a Taylor s series and only the first two terms are retained. 

Fluid motion may be described by applying Newton s second law to a particle. The momentum flow of a substance pvv is equal to the product of the mass flow pv and the barycentric velocity. Newton s second law of motion states that the change in the momentum of a body is equal to the resultant of all forces, mass force F and surface force a, acting on that body. If F, is the force exerted per unit mass of component i, we have... 

The mass forces may be the gravitational force, the force due to the rotational motion of a system, and the Lorentz force that is proportional to the vector product of the molecular velocity of component i and the magnetic field strength. The normal stress tensor a produces a surface force. No shear stresses occur (t = 0) in a fluid, which is in mechanical equilibrium. 

Hence, the pressure gradient is equal to the sum of the mass forces acting on the substance in a unit volume. 

For a system in mechanical equilibrium in which the pressure gradient is balanced by the mass forces, the Gibbs-Duhem relation becomes... 

From the classical mechanical point of view discussed in Sec. VI. 1, any system of N particles is uniquely defined by a knowledge of 6N independent pieces of information together with the description of the components of the system (masses, force fields, etc.). These QN quantities may be looked upon as the QN constants of integration implicit in Newton s differential equations of motion. 

The second important term is the virtual mass coefficient (Cv). When the dispersed phase accelerates (or decelerates) with respect to the continuous phase, the surrounding continuous phase has to be accelerated (or decelerated). For such a motion, additional force is needed, which is called added or virtual mass force. This force was given by the second term in Eq. (8). The constant Cy is called the virtual or added mass coefficient. It is difficult to estimate the value of Cv with the present status of knowledge. Therefore, many recommendations are available in the published literature. In an extreme case of potential flow, the value of Cy is 0.5. 

These difficulties can be understood by the following not rigorous computation. Let us consider in this domain, the flow of a Maxwell fluid of relaxation time X. The velocity field U and the stress tensor o are solutions of the following equations (as commonly accepted, inertia and mass forces are neglected) ... 

Before we get into these groups, let us first look at that term dimensions. To each property of matter that we can measure, we assign a dimension and a specific unit of measurement. Thus we have dimensions of length, mass, force, time, and temperature. Other entities or properties can be described by these above dimensions. For example, velocity is expressed as a length divided by time. Let us assign symbols for the basic dimensions. 

The unfortunate truth about people is how machinelike we can become. Gurdjieff observed how mass forces in cultures tend to make people more mechanical we will look at these forces in detail in other chapters. Right and wrong ways are set up and rules to implement them are established. Goodness then becomes a matter of following the rules. 

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