Velocity slug-flow, profiles

Second, when a liquid is flowing in a long tube, a liquid velocity profile will develop, and bubbles near the center of the tube will rise more rapidly than those near the wall. In slug flow, all slugs rise with nearly the maximum velocity, but in dispersed bubble flow, the variations in speed tend to cancel and give a component to the bubble velocity due to the average liquid velocity. As before, from continuity, this mean liquid velocity at any cross section must be (Qo + Qt)A, so that the bubble rise velocity is... 

Consider a fully developed steady-state laminar flow of a constant-property fluid through a circular duct with a constant heat flux condition imposed at the duct wall. Neglect axial conduction and assume that the velocity profile may be approximated by a uniform velocity across the entire flow area (i.e., slug flow). Obtain an expression for the Nusselt number. 

Let us first consider the simple flat plate with a liquid metal flowing across it. The Prandtl number for liquid metals is very low, of the order of 0.01. so that the thermal-boundary-layer thickness should be substantially larger than the hydrodynamic-boundary-layer-thickness. The situation results from the high values of thermal conductivity for liquid metals and is depicted in Fig. 6-15. Since the ratio of 8/8, is small, the velocity profile has a very blunt shape over most of the thermal boundary layer. As a first approximation, then, we might assume a slug-flow model for calculation of the heat transfer i.e., we take... 

Suppose the fluid is highly conducting, such as a liquid metal. In this case, the thermal-boundary-layer thickness will be much greater than the hydrodynamic thickness. This is evidenced by the fact that the Prandtl numbers for liquid metals are very low, of the order of 0.01. For such a fluid, then, we might approximate the actual fluid behavior with a slug-flow model for energy transport in the thermal boundary layer, as outlined in Sec. 6-5. We assume a constant velocity profile... 

It is necessary to caution that the foregoing analysis is a highly idealized one, which has been used primarily to illustrate the effects of magnetic fields on heat transfer. A more realistic analysis would consider the variation of electrical conductivity of the fluid and take into account the exact velocity profile rather than the slug-flow model. A survey of more exact relations for heat transfer in MFD systems is given in Refs. 1 and 2. 

Nicklin et al. (N5) have shown that an equation similar to Eq. (5-3) is applicable to slug flow. A single slug rises at a velocity equal to 0.3SVgD. If Eq. (5-3) is modified to allow for a nonuniform velocity profile in the liquid regions between the slugs, an empirical equation results ... 

It can be noticed how the presence of Joule heating reduces the value of the Nusselt number dramatically, while a decrease in the value of the aspect ratio 5 dampens this effect. The value of the Nusselt number for a perfectly flat velocity profile (slug flow) for a rectangular duct, Nujf, is also plotted in Fig. 5 for the rectangular crosssection considered. It is evident that the values of the Nusselt number approach the corresponding Nusf when kDi increases, so much more so the lower the value of M. ... 

---