Added mass

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-... 

Sugar increases density by adding mass without an appreciable change in volume. 

Before energy balance is calculated, we need to make mass balance. Figure 9.1 shows the material balance for ethanol and glycerol fermentation. Put simply, mass into the system is equal to mass out of die system. The mass of carbon dioxide is calculated by adding mass of dry cell, mass of glycerol, mass of edianol and mass of water at product stream and then subtracting die sum from die feed stream. As a result, die mass of carbon dioxide is defined. The heat of the reaction is calculated by the following equation ... 

Free and resonance vibrations do not permit the facile measurement of E or G over wide ranges in frequency at a given temperature, although with careful work, resonance responses can be examined at each of several harmonics (47,48). In general, to obtain three decades of frequency, the specimen dimensions and the magnitude of the added mass must be varied over a considerable range. 

Fig. 1. Single realization illustrating the time history for the events for the model shown in Fig. 2. The added mass always goes directly to the thin disk.

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... 

Tracing a kinetic redox path is a matter of redistributing mass among the basis entries, adding mass, for example, to oxidized basis entries at the expense of reduced entries. The stoichiometry of the mass transfer is given by the kinetic reaction 17.3, and the transfer rate is determined by the associated rate law (Eqn. 17.9, 17.12, or 17.21). 

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. 

To understand how an appropriate momentum equation can be derived, consider first a stationary tank into which solid masses are thrown, Figure 1.7a. Momentum is a vector and each component can be considered separately here only the x-component will be considered. Each mass has a velocity component vx and mass m so its x-component of momentum as it enters the tank is equal to mvx. As a result of colliding with various parts of the tank and its contents, the added mass is brought to rest and loses the x-component of momentum equal to mvx. As a result there is an impulse on the tank, acting in the x-direction. Consider now a stream of masses, each of mass m and with a velocity component vx. If a steady state is achieved, the rate of destruction of momentum of the added masses must be equal to the rate at which momentum is added to the tank by their entering it. If n masses are added in time t, the rate of addition of mass is nmJt and the rate of addition of x-component momentum is (nm/t)vx. It is convenient to denote the rate of addition of mass by Af, so the rate of addition of x-momentum is Mvx. 

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. 

Virtual mass = Mass of particle + Added mass. 

If allowance is made for the added mass and m is substituted for m in equation 3.78, then ... 

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 ... 

If allowance is made for added mass, a and b are substituted for a and b, respectively. Then ... 

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) ... 

If the added mass is taken into account, / remains unchanged, but c must be replaced by c in equations 3.97 and 3.101, where ... 

Equations 3.105 and 3.106 do not allow for added mass. If this is taken into account then ... 

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 ... 

Selected experiments from Fig. 3 were repeated, but in the presence of such Teflon shavings. Yield data were comparable. Importantly, the recovered catalyst residues became more compact and firmer, despite the added mass of the Teflon , suggestive of genuine physical adhesion or attraction. The catalyst residues were easier to manipulate, and less leaching occurred. 

The results of Perry et al. allow for the development of a diagnostic method that determines the limiting behavior from simple experimental data. In a similar analysis, Weber et al. added mass transport in the diffusion media explicitly and analyzed... 

Note 7 Notes 2 and 5 show that application of a sinusoidal uniaxial force to a Voigt-Kelvin solid of negligible mass, with or without added mass, results in an out-of-phase sinusoidal uniaxial extensional oscillation of the same frequency. 

Note 2 For a material specimen which behaves as Voigt-Kelvin solid under forced flexural oscillations with added mass at the point of application of the applied oscillatory force, Av is proportional to the loss modulus (E")... 

The first term of Eq. (11-11) is the Stokes drag for steady motion at the instantaneous velocity. The second term is the added mass or virtual mass contribution which arises because acceleration of the particle requires acceleration of the fluid. The volume of the added mass of fluid is 0.5 F, the same as obtained from potential flow theory. In general, the instantaneous drag depends not only on the instantaneous velocities and accelerations, but also on conditions which prevailed during development of the flow. The final term in Eq. (11-11) includes the Basset history integral, in which past acceleration is included, weighted as t — 5) , where (t — s) is the time elapsed since the past acceleration. The form of the history integral results from diffusion of vorticity from the particle. 

If added mass is also neglected, leaving only steady drag, ... 

The first term again represents drag in steady motion at the instantaneous velocity, with Cd an empirical function of Re as in Chapter 5. The other terms represent contributions from added mass and history, with empirical coefficients, Aa and Ah, to account for differences from creeping flow. From measurements of the drag on a sphere executing simple harmonic motion in a liquid, Aa and Ah appeared to depend only on the acceleration modulus according to ... 

There is no a prion justification for Eq. (11-30) since the form of Eq. (11-11) (and not simply the coefficients) depends on the assumption of creeping flow. Moreover, the form of the equation is open to criticism for example, momentum arguments suggest that the added mass term be written pVI2)d di U)/dt. However, Eq. (11-30) is the form for which Eqs. (11-31) and (11-32) were determined (05), and appears to give an accurate description of the motion of spheres from rest as demonstrated in Fig. 11.7. Curve 1 shows the predictions... 

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). 

Neglect of added mass and history simplifies calculation of unsteady motion considerably. However, for y characteristic of particles in liquids, this introduces substantial errors as illustrated by curve 4 in Fig. 11.7. The accuracy of the simplification improves as y and Re increase, but even for y as high as 10 trajectories calculated neglecting history and added mass substantially underpredict the duration of accelerated motion. Neglect of added mass causes the predicted trajectory to be in error from the start of the motion. Since it is the... 

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