Force balances

For melt spinning, the surface tension of polymer melt typically is small compared to other forces, and it can be assumed that  

There are many different methods for calculating the drag coefficient. One of the most commonly used methods is to relate the drag coefficient to the Reynolds number (R) by  

Compared with F it is more convenient to measure the rheological force at the take-up device, i.e., z = L, where L is the distance between the take-up device and the spinneret. Therefore, the rheological force at a given point can be given by  

By using Equation 9.13, the rheological force at a given point in the spinning filament can be calculated from measurable quantities. The gradient of axial tension along the spiiming filament can be obtained from  

In melt spiiming, the tensile stress developed in the filament is a very important parameter and it largely determines the final stmcture and properties of the melt-spun filaments. The tensile stress (cr) at a given point is related to the rheological 

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The flow can be radial, that is, in or out through a hole in the center of one of the plates [75] the relationship between E and f (Eq. V-46) is independent of geometry. As an example, a streaming potential of 8 mV was measured for 2-cm-radius mica disks (one with a 3-mm exit hole) under an applied pressure of 20 cm H2 on QT M KCl at 21°C [75]. The i potentials of mica measured from the streaming potential correspond well to those obtained from force balance measurements (see Section V-6 and Chapter VI) for some univalent electrolytes however, important discrepancies arise for some monovalent and all multivalent ions. The streaming potential results generally support a single-site dissociation model for mica with Oo, Uff, and at defined by the surface site equilibrium [76]. 

The wettabiHty theory of adhesion is iaextricably related to the study of contact angles of Hquids on soHd surfaces. A force balance at the poiat of contact between the Hquid and the soHd can be written (3)... 

Equations 3 to 7 indicate the method by which terminal velocity may be calculated. Erom a hydrodynamic force balance that considers gravity, buoyancy, and drag, but neglects interparticle forces, the single particle terminal velocity is... 

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

In Figure 1, the force balance in Cartesian coordinates for a body not intersected by phase boundaries is... 

The most successful models are based on the finite element method. The flow is discretized into small subregions (elements) and mass and force balances are appHed in each. The result is a large system of equations, the solution of which usually gives the speed of the coating Hquid in each element, pressure, and the location of the unknown free surfaces. The smaller the elements, the more the equations which are often in the range of 10,000 to upward of 100,000. 

Electrokinetics. The first mathematical description of electrophoresis balanced the electrical body force on the charge in the diffuse layer with the viscous forces in the diffuse layer that work against motion (6). Using this force balance, an equation for the velocity, U, of a particle in an electric field... 

A particle falling under the action of gravity will accelerate until the drag force balances gravitational force, after which it falls at a constant terminal or free-settling velocity given by... 

Entrainment Flooding The early work of Souders and Brown [Ind. Eng. Chem., 26, 98 (1934)] based on a force balance on an average suspended droplet of hquid led to the definition of a capacity parameter C,i, ... 

Wilkinson et al. (op. cit.) make the following observation about the effect of gas density on bubble size The fad that the bubble size decreases shghtly for higher gas densities can be explained on the basis of a force balance. ... 

FIG. 19-65 Schematic representation of air hubhle-water-solid particle system (a) before, (h) after particle-bubble attachment, and (c) equilibrium force balance. 

Several additional studies [Winitzer, Sep. ScL, 8(1), 45 (1973) ibid., 8(6), 647 (1973) Maru, Wasan, and Kintner, Chem. Eng. Set., 26, 1615 (1971) and Rapacchietta and Neumann, J. Colloid Inteiface ScL, 59(3), 555 (1977)] which include body forces such as gravitational acceleration and buoyancy have been made. A typical example of a force balance describing suen a system (Fig. 22-39) is summarized in Eq. (22-41). 

For the gravity discharge case, the height of the fluid at maximum vacuum, which is the point at which air would begin to backflow into the tank, is determined by Eq. (26-54). Equation (26-55) calculates the corresponding vacuum in the tank s headspace at this hquid height. Since the drain nozzle is open to the atmosphere, this solution is a static force balance that is satisfied when the sum of the internal pressure and the remaining fluid head is equal to the atmospheric pressure. 

Complete wetting, i.e. spontaneous spreading should always be sought to maximize adhesion. This condition occurs when, with reference to Fig. 4, it is not possible to satisfy the horizontal force balance, i.e. ys > Vl + Ysl- The thermodynamic driving force for the spreading process is the spreading coefficient. 

Consider flow through a cake (Figure 3C) with the membrane located at a distance X from the filter plate. Neglecting all forces in the cake other than those created by drag and hydraulic pressure, a force balance from x to L gives ... 

Assuming laminar flow through the filter channels, the basic equation of filtration as obtained from a force balance is ... 

The result is a modified Euler number. You can prove to yourself that the pressure drop over the particle can be obtained by accounting for the projected area of the particle through particle size, S, in the denominator. Thus, by application of dimensional analysis to the force balance expression, a relationship between the dimensionless complexes of the Euler and Reynolds numbers, we obtain ... 

It is evident from Fig. 3.29 that there is a minimum fibre length which will permit the fibre to achieve its full load-carrying potential. The minimum fibre length in which the maximum fibre stress, ((rf)maxt be achieved is called the load transfer length, The value of may be determined from a simple force balance... 

Solution It is necessary to derive an expression for the inflation time. Referring to Fig. 5.22 where R is the radius of the inflated tube at any time, /, and hi is the wall thickness at this time, then a force balance on the shaded element gives (for unit length). 

The pressure difference (drop) between the ends of the element produces a force biptvP-, and considering the force balance,... 

The other forces affecting the particle are the gravitational force and friction hence the force balance is... 

Equation (14.83) gives us an estimate for the velocity c at various pipeline angles 5 when the corresponding values for vertical and horizontal lines are known. It is an estimate, of course, because its derivation was based on the force balance of a single particle and not on the balance of the mixture of particles as a whole. 

Modeling of the force was actually based on the force balance of a single solid particle extending the result to cover a larger set of solid particles, we obtained Eq. (14.105). This method of modeling is adequate for a homogeneous flow where the particles move separately, not as a new kind of solid-phase structure. Weber has written the friction force in the form... 

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