Accelerated motion

While the particle is experiencing the accelerating motion as described above, heat is being transferred between it and the surrounding gas stream also in an unsteady state ... 

Torobin, L. B. and Gauvin, W. H. Can. J. Chem. Eng. 38 (1959) 129, 167, 224. Fundamental aspects of solids-gas flow. Part I Introductory concepts and idealized sphere-motion in viscous regime. Part II The sphere wake in steady laminar fluids. Part III Accelerated motion of a particle in a fluid. 

The range between these small and large particles is less well understood although some experimental studies have been reported (K9, Ul). Similar problems arise in interpretation as with accelerated motion (see Chapter 11). Measurements are commonly correlated by a turbulence-dependent drag coefficient, which contains a number of possible acceleration-dependent components. With fundamental understanding so poorly advanced, it is impossible to say to what extent results are specific to the experimental conditions employed. 

Equation (11-11) depends on neglect of inertial terms in the Navier-Stokes equation. Neglect of inertia terms is often less serious for unsteady motion than for steady flow since the convective acceleration term is small both for Re 0 (Chapters 3 and 4), and for small amplitude motion or initial motion from rest. The second case explains why the error in Eq. (11-11) can remain small up to high Re, and why an empirical extension to Eq. (11-11) (see below) describes some kinds of high Re motion. Note also that the limited diffusion of vorticity from the particle at high cd or small t implies that the effects of a containing wall are less critical for accelerated motion than for steady flow at low Re. 

The only rigid particle for which accelerated motion beyond the creeping flow range has been considered in detail is the sphere. Odar and Hamilton (06) suggested that Eq. (11-11) be extended to higher Re as ... 

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... 

As for steady motion, shape changes and oscillations may complicate the accelerated motion of bubbles and drops. Here we consider only acceleration of drops and bubbles which have already been formed formation processes are considered in Chapter 12. As for solid spheres, initial motion of fluid spheres is controlled by added mass, and the initial acceleration under gravity is g y - l)/ y + ) (El, H15, W2). Quantitative measurements beyond the initial stages are scant, and limited to falling drops with intermediate Re, and rising... 

Re- = dU i/v. These simplifications are only valid in Stokes flow, and can lead to substantial errors at higher Re [see, e.g., (R7)]. The effect of freestream turbulence can be included, via the correlations in Chapter 10, provided that the turbulence intensity can be estimated. Alternatively, one of the available correlations for drag in accelerated motion through a turbulent fluid can be used [see (C9)], although these are only applicable for limited ranges of experimental conditions. 

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