Inviscid liquid

Biihhle-diameter correlation for air sparged into relatively inviscid liquids. Dt, = hiihhle diameter, D = orifice = gas velocity through sparging orifice, P = fluid density, and 1 = fluid viscosity. [From Can. J. Chem. Eng., 54,... 

When a gas is blown steadily through an orifice into an essentially inviscid liquid, a regular stream of bubbles is formed. A theoretical expression that relates the bubble volume V to volumetric gas flow rate G and gravitational acceleration g is the following ... 

Single gas bubbles in an inviscid liquid have hemispherical leading surfaces and somewhat flattened wakes. Their rise velocity is governed by Bernoulli s theory for potential flow of fluid around the nose of the bubble. This was first solved by G. I. Taylor to give a rise velocity Ug of ... 

The complex wave frequency Q (= ico — F) is related to k via a dispersion relation. For an inviscid liquid, Lamb s equation is well-known as a classical approximation for the dispersion relation [10]... 

Further, the constancy of bubble volume with flow rate at small flow rates which is observed by many investigators is true only for inviscid liquids having high surface-tension effects. If highly viscous liquids (n > 500 cp) are used, the bubble volume increases very fast with the flow rate. On increasing the... 

Bubble Formation in Inviscid Liquids Neglecting Surface Tension Effects... 

This equation closely resembles the empirical expression of Van Krevelen and Hoftijzer (VI) for the bubble formation in inviscid liquids, provided that the gas density is negligible compared to the liquid density. Their relationship... 

Fig. 11. Comparison of Kumar and Kuloor s model (K18) with the data and model of Davidson and Schuler (D9) for bubble formation in inviscid liquids.

This simplified equation is of the same form as the equations of Van Krevelen and Hoftijzer (VI) for air-water system and the theoretical equation of Davidson and Schuler (D9) for inviscid liquids. It is interesting to observe that the various equations differ only in the value of the constant although they are based on different mechanisms. 

Values of Bubble Volume Calculated by Rigorous [Eq. (28)] and Simplified [Eq. (29)] Equations for Inviscid Liquids without Surface-Tension Effect... 

When Q tends to zero, Eq. (31) simplifies to Eq. (30). If the surface-tension effect is negligible, then the right-hand side of Eq. (30) vanishes and Eq. (31) reduces to Eq. (22), which is applicable to inviscid liquids without surface tension. 

Thus, we conclude that the surface-tension effects can be neglected only at higher flow rates and not at lower ones. The error caused by neglect at low flow rates can be quite large, its magnitude depending on the orifice diameter. Equation (33) can be used for inviscid liquids both in the presence and absence of surface-tension effects. 

Fio. 12. Influence of surface tension on bubble volume for inviscid liquids. 

This model makes use of the same concepts as for inviscid liquids except that the viscous resistance is now taken into consideration. Further, the... 

The model for inviscid liquids is equally well applicable to viscous liquids also, provided that the resistance due to viscous drag is included in the analysis. As a first approximation, the viscous drag may be evaluated by a Stokes resistance term, since the bubble is not followed by a wake. Thus we proceed as before, first evaluating the force-balance bubble volume Kr and then the total bubble volume by reference to the detachment stage. 

Fig. 15. Comparison of the model (S3) with the data collected for bubble formation in inviscid liquids under constant pressure conditions.

The formation of bubbles at orifices in a fluidised bed, including measurement of their size, the conditions under which they will coalesce with one another, and their rate of rise in the bed has been investigated. Davidson el alP4) injected air from an orifice into a fluidised bed composed of particles of sand (0.3-0.5 mm) and glass ballotini (0.15 mm) fluidised by air at a velocity just above the minimum required for fluidisation. By varying the depth of the injection point from the free surface, it was shown that the injected bubble rises through the bed with a constant velocity, which is dependent only on the volume of the bubble. In addition, this velocity of rise corresponds with that of a spherical cap bubble in an inviscid liquid of zero surface tension, as determined from the equation of Davies and Taylor ... 

For conditions approaching constant pressure at the orifice entrance, which probably simulates most industrial applications, there is no independently verified predictive method. For air at near atmospheric pressure sparged into relatively inviscid liquids (11 100 cP), the correlation of Kumar et al. [Can. J. Chem. Eng., 54,503 (1976)] fits experimental data well. Their correlation is presented here as Fig. 14-92. 

As already noted, the foregoing calculations must be regarded as a guide only, since the films are intrinsically unstable, with waves being amplified as the liquid proceeds to the edge of the disc. It will be appreciated that this process proceeds more rapidly with relatively inviscid liquids. 

This requires numerical integration. As pointed out at the outset, these estimates of the mass transfer performance are likely to be conservative as the disturbance of the film by ripples has been neglected. This will reduce the exposure time significantly, particularly with inviscid liquids. 

Droplet formation occurs primarily through the surface tension and viscosity dominated breakup of these liquid threads due to symmetric (or dilational) waves as described by Rayleigh (6) for inviscid liquids and by Weber (J) for viscous fluids. Figure 3 shows the double pulsed image of the droplet formation process for No. 2 and SRC-II fuel sprays under identical atomizer conditions. These two photographs illustrate typical differences seen between these two fuels. 

Horizontal In-Shell Condensers The mean condensing coefficient for the outside of a bank of horizontal tubes is calculated from Eq. (5-93) for a single tube, corrected for the number of tubes in a vertical row. For undisturbed laminar flow over all the tubes, Eq. (5-97) is, for realistic condenser sizes, overly conservative because of rippling, splashing, and turbulent flow (Process Heat Transfer, McGraw-Hill, New York, 1950). Kern proposed an exponent of -Ve on the basis of experience, while Freon-11 data of Short and Brown General Discussion on Heat Transfer, Institute of Mechanical Engineers, London, 1951) indicate independence of the number of tube rows. It seems reasonable to use no correction for inviscid liquids and Kern s correction for viscous condensates. For a cylindrical tube bundle, where N varies, it is customary to take N equal to two-thirds of the maximum or centerline value. 

The frequency of gas bubbles which are formed steadily through an orifice in a fluidized bed has been studied by Harrison and Leung (H6). Their results show that the mechanism of chainlike bubble formation is the same as that in an inviscid liquid. If all of the excess gas above the minimum fluidization velocity passes through in the form of gas bubbles, the diameter of a sphere having the same volume as the originated bubble is represented by two equations (in units of cm/sec). Van Krevelen and Hoftijzer (V6) found that... 

Because the class of problems that we can currently study is restricted to flows in which the Reynolds number is very small, Re . 1, and because Pe = RePr, an obvious question is whether the combination Re PP 1 and Pe 1 is achievable in real systems. The key is to remember that the Prandtl number is an independent material parameter. For gases, for which Pr < 0(1), and for relatively inviscid liquids such as water that have Pr 0(1) or slightly larger, Pe will always be small when Re is small. On the other hand, viscous oils and greases can have Pr 103 — 106, and for these fluids Pe may be large, even though Re is small. This provides one clear motivation for studying heat transfer for the dual limit Re -PP 0(1) andPe > 0(1). 

Solid particle Translational flow of an ideal (inviscid) liquid Analytical, DBLA 0.7 [84,410]... 

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