Particle force balance

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... 

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... 

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

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 ... 

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... 

An alternative approach to the representation of results for solid-liquid flow is to use the two-layer model which will be described in the following section. It will be seen that the coefficient of friction between the particles and the wall of the pipe is an important parameter in the model. It is suggested that its complete absence in equation 5.24 may be an important reason for the extent of the scatter. Unfortunately, it is a quantity which has been measured in only a very few investigations. It is interesting to note that the form of equation 5.19 was obtained by NEWITT et alP2) using a force balance similar to that... 

When a particle falls under the influence of gravity, it will accelerate until the combination of the frictional drag in the fluid and buoyancy force balances the opposing gravitational force. If the particle is assumed to be a rigid sphere, at this terminal velocity, a force balance gives3,4,7,8... 

Rosin et al. (1932) derived a simple expression for cyclone efficiency for a fixed cyclone diameter based on a force balance on the particle at the cyclone inlet, i.e.,... 

By definition, the terminal velocity of a particle (ut) is the superficial gas velocity which suspends an isolated particle without translational motion—i.e., the terminal free fall velocity for that particle. From force balance on the particle, the terminal velocity for an approximately spherical particle can be shown to be... 

The above six equations for continuity and force balance do not, however, afford a complete description of a heterogeneous particle-fluid system in which a dense phase and a dilute phase coexist. An additional constraint needs to be identified to account for the stability of the system. 

Using the nomenclature of Chapter 3 in Volume 2, a force balance on the particle in water gives ... 

A combination of adhesion and surface tension gives rise (pardon the pun) to capillary action. By its adhesion to the solid surface of the soil particles, the water wants to cover as much solid surface as possible. However, by the effect of surface tension, the water molecules adhering to the solid surface are connected with a surface him in which the stresses cannot exceed the surface tension. As water is attracted to the soil particles by adhesion, it will rise upward until attractive forces balance the pull of gravity (Figure 3.28). Smaller-diameter tubes force the air-water surface into a smaller radius, with a lower solid-surface-to-volume ratio, which results in a greater capillary force. Typical heights of capillary rise for several soil types are presented in Table 3.9. The practical relationship between normal subsurface water and capillary rise is presented in the following equation. 

Introducing the above drag expression Eq. (146) in the force balance particle equation of unified model (K14), we obtain... 

In a sedimenting suspension, however, the gravitational force on any individual particle is balanced by a combination of buoyancy and fluid friction forces, both of which are influenced by the flow field and pressure distribution in the vicinity of the particle. It is a somewhat academic exercise to apportion the total force between its two constituent parts, both of which are influenced by the presence of neighbouring particles, the concentration of which will determine the magnitude of their effect. 

For a single spherical particle settling at its terminal velocity hq, a force balance gives ... 

A force balance on an element of system containing unit volume of particles gives ... 

Measurements of the photophoretic force on crystalline ammonium sulfate particles were made by Lin and Campillo (1985) using an electrodynamic balance. The measurement procedure is identical to that for any such force, that is, the levitation voltage is measured in the absence of the photophoretic force and then when the force is exerted. A force balance yields an equation for the photophoretic force similar to Eq. (31) ... 

Both from deposition studies and force balances it can be derived that the optimum (aerodynamic) particle size lies between 0.5 and 7.5 pm. Within this approximate range many different subranges have been presented as most favourable, e.g. 0.1 to 5 pm [24], 0.5 to 8.0 pm [25], 2 to 7 pm [26] and 1-5 pm [27-29]. Particles of 7.5 pm and larger mainly deposit in the oropharynx [30] whereas most particles smaller than 0.5 pm are exhaled again [31]. All inhalation systems for drug delivery to the respiratory tract produce polydisperse aerosols which can be characterized by their mass median aerodynamic diameter (MMAD) and geometric standard deviation (oq). The MMAD is the particle diameter at 50% of the cumulative mass curve. 

Consider a bed of particulate solids or powder, say of a size similar to table salt. When a fluid, either a gas or a liquid, is passed upwards through the bed, the bed particles remain stationary or packed at low fluid velocities. This is a packed or fixed bed. If now the velocity of the fluid is increased, fhe particles will begin to separate and move away from one another the bed is said to expand. On increasing the velocity further, a point will be reached at which the drag force exerted by the fluid on a particle is balanced by the net weight of the particle. The particles are now suspended in the upward-moving stream of fluid. This is the point of minimum fluidization, or incipient fluidization, at and beyond which the bed is said to be fluidized. 

Substituting all the possible combinations of characteristics, i.e. values of p and q, info equation 1.10 gives rise to a number of differenf definitions of the mean size of a distribution. At minimum fluidization the drag force acting on a particle due to the flow of fluidizing gas over the particle is balanced by the net weight of fhe particle. The former is a function of surface area and the latter is proportional to particle volume. Consequently the surface-volume mean diameter, with p = 2 and = 3, is the most appropriate particle size to use in expressions for minimum fluidizing velocity. It is defined by equafion 1.11... 

Bubbles and drops tend to deform when subject to external fluid fields until normal and shear stresses balance at the fluid-fluid interface. When compared with the infinite number of shapes possible for solid particles, fluid particles at steady state are severely limited in the number of possibilities since such features as sharp corners or protuberances are precluded by the interfacial force balance. 

For a particle moving with steady terminal velocity Uj in a gravitational field, the drag force balances the difference between the weight and buoyancy ... 

From the analysis presented in the last two paragraphs, it is evident that the gravitational force acting upon the particle is used for the derivation of the equations for the terminal velocity and the pressure drop in the fluidized bed. Then, it is clear that the hydraulic density should be used in these equations as well as in any other equations that are derived from a similar force-balance analysis. For instance, this is the case of the Foscolo-Gibilaro criterion for determining the fluidization pattern (Section 3.8.2). 

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