Added mass effects

Drops accelerated by an air stream may split, as described in Chapter 12. For drops which do not split, measured drag coefficients are larger than for rigid spheres under steady-state conditions (R2). The difference is probably associated more with shape deformations than with the history and added mass effects discussed above. For micron-size drops where there is no significant deformation, trajectories may be calculated using steady-state drag coefficients (SI). 

Saltation of solids occurs in the turbulent boundary layer where the wall effects on the particle motion must be accounted for. Such effects include the lift due to the imposed mean shear (Saffman lift, see 3.2.3) and particle rotation (Magnus effect, see 3.2.4), as well as an increase in drag force (Faxen effect). In pneumatic conveying, the motion of a particle in the boundary layer is primarily affected by the shear-induced lift. In addition, the added mass effect and Basset force can be neglected for most cases where the particle... 

This added mass effect is modeled by introducing the virtual mass term, Fvm ... 

In bubbly flow simulations values ranging from 0.25 to 0.75 have been used for different bubble shapes and number densities of bubbles [65]. The added mass effect is obviously more important for light bubbles in liquids than for heavy solid particles or droplets in gas or liquid. 

Methods of construction for CWPs of 2.5-m (FRP) and 3-m (light-weight concrete) were demonstrated that are directly applicable to CWPs with diameters of 10 m or more. At-sea tests confirm that CWPs will be able to survive storms of maximum severity expected in a 100-year period in the tropical oceans. Lightweight concrete and FRP have been shown to be suitable for CWP constmction. Further information is desirable on added-mass effects of external water flow in lateral motions of laige-diameter pipes and on effects of multidimensional sea states on CWP dynamics. 

Depending on the dimensions and constmction of the composite cantilever, resonance frequencies were observed in the range of 0.7 to 1.2 MHz. The resonance spectra (Fig. IB) in vacuum ( 50 mTorr, 23.6 °C), in air (23.6 °C) and in phosphate buffered saline (lOmM, PBS 30 °C) are shown for the dominant high-order mode at 941.5 kHz in air (Q = 41). The parameter Q is a measure of peak sharpness and is equal to the ratio of resonance frequency to half-height peak width. When mounted in a specially constructed sample flow cell and PBS is flowed in at 0.4 to 1.0 mL/min, the resonant frequency decreased ( 50 kHz) due to added mass effect and the peak height decreased by 20 to 60%. At 1 mL/min, Q-value remained sufficiently high ( 23), that the resonant frequency can be measured with an accuracy of 20-40 Hz. Resonant frequency in... 

Gibilaro LG, Di Felice R, Foscolo PU. Added mass effects in fluidized beds application of the Geurst Wallis analysis of inertial coupling in two-phase flow. Chem Eng Sci 45 1561-1565, 1990. 

The BBOT equations have enabled the introduction of the added mass effect, which intervenes during the acceleration and deceleration phases of the particle in the fluid. In a transient regime, the added mass is essential for a bubble in a liquid pp Pf 1), because the bubble has to transmit momentum to the liquid in order to be able to accelerate itself. 

Table 9.2 Correction of predicted Md for two-phase model and added mass effects...

Introduction The term azeotropic distillation has been apphed to a broad class of fractional distillation-based separation techniques in that specific azeotropic behavior is exploited to effect a separation. The agent that causes the specific azeotropic behavior, often called the entrainer, may already be present in the feed mixture (a self-entraining mixture) or may be an added mass-separation agent. Azeotropic distillation techniques are used throughout the petro-... 

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... 

Two spherical particles, one of density 3000 kg/m3 and diameter 20. im, and the other of density 2000 kg/m3 and diameter 30 (im start settling from rest at the same horizontal level in a liquid of density 900 kg/m3 and of viscosity 3 mN s/m2. After what period of settling will the particles be again at the same horizontal level It may be assumed that Stokes Law is applicable, and the effect of added mass of the liquid moved with each sphere may be ignored. 

The effects of the mass of styrene (St) and MMA on polymer yields can be seen in Figs. 3 and 4. The concentration of PVPA or PSS-Na was kept constant and added mass of St or MMA was varied. 

The equations are now integrated, ignoring the effects of added mass which can be accounted for by replacing a by a and b by b. For the 7-direction, integrating equation 3.86 with respect to t ... 

For the Newton s law regime, R /pu2 is a constant and equal to 0.22 for a spherical particle. Therefore, substituting in equation 3.81 and putting i = 0 for vertical motion, and using the negative sign for downward motion (and neglecting the effect of added mass) ... 

In this solution, the effect of added mass is not taken into account. Allowance may be made by adjustment of the values of the constants in the equations as indicated in Section 3.7.3. 

The effects of added mass, which have not been taken into account in these equations, require the replacement of a by a and q by q, where ... 

Unless 7 1, all terms in Eq. (11-33) must be retained. Since Eq. (11-30) has no formal justification, the individual terms cannot definitely be ascribed to added mass or history effects. Even so, the relative magnitudes of the terms are of interest. Figure 11.7 shows the three terms for specific values of 7 and Rejs, expressed as fractions of the immersed particle weight. Added mass dominates initially history passes through a maximum and decays slowly steady drag increases monotonically to become the sole component at the terminal velocity. Both A and Ah depart from unity early in the motion. For smaller Rexs, history may be the dominant drag component for a brief period (02). 

Bl), based on a time-averaged form of Eq. (11-44) and applicable to any form of oscillation if the period is long compared with the response time of the fluid (see below). At the other extreme of simplification, Rschevkin (R5) showed that if all effects except added mass and pressure gradient are omitted from Eq. (11-44), a particle moves in phase with the fluid oscillations (i.e., P = 0) with amplitude ratio ... 

These piezoelectric crystal oscillators are very accurate mass sensors because their resonant frequencies can be measured precisely with relatively simple electronic circuitry. For certain quartz crystals, the resonant frequency is inversely related to the crystal thickness. A crystal resonating at 5 megahertz is typically 300 micrometers thick. If material is coated or adsorbed on the crystal surface, the resonant frequency will change (decrease) in proportion to the amount of material added. The effect of adsorbed mass on the oscillator frequency varies according to the operational mode of the device. In any case, interpretation of mass via changes in frequency or amplitude assumes that the coated films are rigidly elastic and infinitesimally thin (that is, an extension of the crystal). 

The effect of the microscopic kinematic viscosity at the crystal-liquid interface is particularly important. This can be seen in Fig. 4.9, where the discrepancy between predicted and measured frequency is apparent. The discrepancy is not related to the added mass of the glucose the model accounts for that difference. The deviation... 

The effect of the viscosity on the real part of the FPW phase velocity is obtained by substituting Af, for added mass per unit area m in Equation 3.74. The dependence of the attenuation coefficient of the FPW, a, is given by [72]... 

When a dispersed phase particle accelerates relative to the continuous phase, some part of the surrounding continuous phase also is accelerated. This extra acceleration of the continuous phase has the effect of added inertia or added mass (Fig. 4.4). 

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