Developing flow hydrodynamically

Fully developed flow Hydrodynamically Developed Invariant Constant Developed Invariant Constant... 

Hydrodynamically Developing Flow. Hydrodynamically developing turbulent flow in concentric annular ducts has been investigated by Rothfus et al. [114], Olson and Sparrow [115], and Okiishi and Serouy [116]. The measured apparent friction factors at the inner wall of two concentric annuli (r = 0.3367 and r = 0.5618) with a square entrance are shown in Fig. 5.17 (r = 0.5618), where / is the fully developed friction factor at the inner wall. The values of/ equal 0.01,0.008, and 0.0066 for Re = 6000,1.5 x 104, and 3 x 104, respectively [114]. 

Hydrodynamically Developing Flow. Hydrodynamically developing flow in smooth parallel plate ducts with uniform velocity at the duct inlet has been analyzed by Deissler [92] by means of an integral method. The apparent friction factors/app in the hydrodynamic entrance are presented in Fig. 5.24. 

Thermally and Simultaneously Developing Flows. Hydrodynamically developing laminar flow in triangular ducts has been solved by different investigators as is reviewed by Shah and London [1]. Wibulswas [160] obtained a numerical solution for the problem of simultaneously... 

This expression has been used to correlate results obtained in a rectangular channel, Eq. (14) in Table VII, the hydrodynamic entrance length of the channel (Le = 0.0575 d Red) being too short to assure a fully developed flow. The results were still 24/ high, compared with Eq. (31) modified by a relaxation assumption ... 

It should be emphasized that these results are applicable only to fully developed flow. However, if the fluid enters a pipe with a uniform ( plug ) velocity distribution, a minimum hydrodynamic entry length (Lc) is required for the parabolic velocity flow profile to develop and the pressure gradient to become uniform. It can be shown that this (dimensionless) hydrodynamic entry length is approximately Le/D = 7VRe/20. 

In practice all real fluids have nonzero viscosity so that the concept of an inviscid fluid is an idealization. However, the development of hydrodynamics proceeded for centuries neglecting the effects of viscosity. Moreover, many features (but by no means all) of certain high Reynolds number flows can be treated in a satisfactory manner ignoring viscous effects. 

The thermal entrance region in a hydrodynamically fully developed flow in a rectangular duct may be studied by the use of the integral method. In this section, the uniform wall temperature and the uniform wall heat flux cases are discussed. The physical model is based on the following assumptions ... 

The energy equation for laminar and hydrodynamically developed flow is... 

The results presented so far in this section correspond to the regime of fully developed riser flow. Kuipers and van Swaaij (1996) applied the KTGF-based model developed by Nieuwland et al. (1996b,c) to study the effect of riser inlet configuration on the (developing) flow in CFB riser tubes and found that the differences in computed radial profiles of hydrodynamic key variables (i.e., gas and solids phase mass fluxes) rapidly disappear with increasing elevation in the riser tube. 

We start this chapter with a general physical description of internal flow, and the average velocity and average temperature. We continue with the discussion of the hydrodynamic, and thermal entry lengths, developing flow, and fully developed flow. We then obtain the velocity and temperature profiles for fully developed laminar flow, and develop relations for the friction factor and Nusselt nmnber. Hinally we present empirical relations for developing and full developed flows, and demonstrate their use. 

B Have a visual understanding of different flow regions in internal flow, such as Ihe entry and the fully developed flow regions, and calculate hydrodynamic and thermal entry lengths,... 

Fig. 3.34 Velocity and temperature profiles of a hydrodynamically and thermally developing flow...

Vanishing Prandtl numbers Pr = 0 can also mean that the thermal diffusivity is approaching infinity, whilst the viscosity remains finite. Then the flow is already thermally fully developed at the inlet but not yet hydrodynamically fully developed. As the Peclet number disappears, Pe = w m d/a = 0, X 1 = L/(dPe) = oo. The Nusselt number is equal to that for thermal fully developed flow Num = 3.6568. 

Nume is the Nusselt number for developing flow, Num is that for hydrodynamically fully developed laminar flow. 

In practice, it is difficult to introduce the radioactive nuclides into streaming carrier gas without disturbing the hydrodynamic patterns of the flow. Still, the analytical and engineering solutions to the problem of hydrodynamically developed flow proved useful in mathematical simulations of the gas-solid (thermo) chromatography (see Sect. 4.4). The two solutions are compared in the top of Fig. 3.6 their agreement is reasonable. 

Fig. 3.8 Top Distribution of a nonadsorbable gaseous tracer and that of the tracer carried by aerosol stream in a straight open cylindrical tube some time after injection of an infinitely thin plug. Hydrodynamically developed, diffusionally developing flow. Bottom Character of the resulting elution curves.

When looking for a reasonable approximation of p(t] ), one can consult the equations of Sect. 2.2. They describe diffusional deposition in channels under various flow regimes, when the tracer is evenly distributed over the inlet area. The solutions for both hydrodynamically and diffusionally developing flow directly apply to the problem of the first jump down the column. It is the common experimental situation... 

If the characteristic linear dimension of the flow field is small enough, then the measured hydrodynamic data differ from those predicted by the Navier-Stokes equations [79]. With respect to the value in macrocharmels, in microchannels (around 50 microns of section) (i) the friction factor is about 20-30% lower, (ii) the critical Reynolds number below which the flow remains laminar is lower (e.g., the change to turbulent flow occurs at lower linear velocities) and (iii) the Nusselt number, for example, heat transfer characteristics, is quite different [80]. The Nusselt number for the microchannel is lower than the conventional value when the flow rate is small. As the flow rate through the microchannel is increased, the Nusselt number significantly increases and exceeds the value for the fully developed flow in the conventional channel. These effects have been investigated extensively in relation to the development of more efficient cooling devices for electronic applications, but have clear implications also for chemical applications. 

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