Hydrodynamic entrance region

When Amundson taught the graduate course in mathematics for chemical engineering, he always insisted that all boundary conditions arise from nature. He meant, I think, that a lot of simplification and imagination goes into the model itself, but the boundary conditions have to mirror the links between the system and its environment very faithfully. Thus if we have no doubt that the feed does get into the reactor, then we must have a condition that ensures this in the model. We probably do not wish to model the hydrodynamics of the entrance region, but the inlet must be an inlet. One merit of the wave model we have looked at briefly is that both boundary conditions apply to the inlet. 

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 flow velocities u and v are directed along and normal to the flow direction, x, y and z are the corresponding spatial coordinates, a is the angle of inclination, and h is the thickness of the liquid film. The hydrodynamic problem may be simplified by neglecting the entrance region at x=0 (Van den Bogaert Joos 1979),... 

Simultaneously developing flow is fluid flow in which both the velocity and the temperature profiles are developing. The hydrodynamic and thermal boundary layers are developing in the entrance region of the duct. Both the friction factor and Nusselt number vary in the flow direction. Detailed descriptions of fully developed, hydrodynamically developing, thermally developing, and simultaneously developing flows can be found in Shah and London [1] and Shah and Bhatti [2],... 

In this hydrodynamic entrance region, the apparent friction factor /app is employed to incorporate the combined effects of wall shear and the change in momentum flow rate due to the developing velocity profile. Based on the total axial pressure drop from the duct inlet (x = 0) to the point of interest, the apparent friction factor is defined as follows ... 

The incremental pressure drop number K(x) in the hydrodynamic entrance region is expressed as... 

TABLE 5.2 Axial Velocity and Pressure Drop in the Hydrodynamic Entrance Region of a Circular Duct [10]... 

Hydrodynamtcally Developing Flow. An analytical, close-form solution for hydrodynami-cally developing flow in rough circular ducts has been obtained by Zhiqing [87]. The velocity distribution in the hydrodynamic entrance region is given as... 

Hydrodynamically Developing Flow. Shah and London [1] summarize the solutions for the hydrodynamic development of laminar flow in concentric annuli. The apparent friction factor in the hydrodynamic entrance region, derived by Shah [103], is expressed as ... 

FIGURE 5.24 Turbulent flow apparent friction factors in the hydrodynamic entrance region of a parallel plate duct with uniform inlet velocity [45]. 

For equilateral triangular ducts having rounded corners with a ratio of the corner radius of curvature to the hydraulic diameter of 0.15, Campbell and Perkins [180] have measured the local friction factor and heat transfer coefficients with the boundary condition on all three walls over the range 6000 < Re < 4 x 104. The results are reported in terms of the hydrodynamically developed flow friction factor in the thermal entrance region with the local wall (Tw) to fluid bulk mean (Tm) temperature ratio in the range 1.1 < TJTm < 2.11, 6000 < Re < 4 x 10 and 7.45 [Pg.382]

Prakash and Liu [266] have numerically analyzed laminar flow and heat transfer in the entrance region of an internally finned circular duct. In this study, the fully developed / Re is compared with those reported by Hu and Chang [265] and Masliyah and Nandakumar [267]. The incremental pressure drop K(°°) and hydrodynamic entrance length L+hy together with /Re are given in Table 5.48, in which the term n refers to the number of fins, while / denotes the relative length of the fins. 

The behavior of a viscoelastic fluid in turbulent flow in the hydrodynamic entrance region of a rectangular channel can be estimated by assuming that the circular tube results are applicable provided that the hydraulic diameter replaces the tube diameter. 

Length Effect. The heat transfer coefficient can vary significantly in the entrance region of the laminar flow. For hydrodynamically developed and thermally developing flow, the local and mean heat transfer coefficients h, and h, for a circular tube or parallel plates are related as [19]... 

The ratio of the dimensionless axial distance (x/Dh) to the Reynolds number useful in the hydrodynamic entrance region... 

Mass Transfer Coefficient. - TTie length of the hydrodynamic entrance region and the mass transfer coefficient are calculated using the following equations. ... 

Basic Consideration - To predict accurate catalyst and gas temperature distributions, some basic calculations and considerations of effective area, flow condition, and the hydrodynamic entrance region are made. 

Hydrodynamic Entrance Region. - Flow conditions for almost all analyses described above are assumed to be fully developed laminar, although there would be a hydrodynamic entrance region for each channel. In this section, the effect of hydrodynamic entrance region is reviewed. The ratio of the local mass transfer coefficient for the entrance region to that for a fully... 

Cell Size Effect. - The mass transfer coefficient is inversely proportional to width of a cell for fully developed laminar region, and surface area is proportional to cell width. This means conversion through a cell is constant. However, as flow is proportional to the square of cell width as velocity is held constant, the conversion efficiency decreases as cell width increases. This tendency is shown in Figure 10. A large cell size may offer a moderate conversion rate and a longer hydrodynamic entrance region. These are favorable characteristics for a catalyst. On the other hand, for fully developed turbulent flow, the mass transfer coefficient increases with cell width to the 0.2 power, and the surface area is proportional to the width. This means conversion increases as the cell width increases to the 0.8 power. Therefore, the cell size effect for turbulent flow is rather small compared with laminar flow. However, the conversion efficiency decreases as cell width increases as is for laminar flow. 

The type of flow— laminar, turbulent, well-developed, or hydrodynamic entrance region—should be considered. 

The development of the thermal and dynamic boundary layers can be either coupled or uncoupled. When the thermal and dynamic boundary layers develop together, the flow is said to be a simultaneously developing flow. When only the dynamic boundary layer develops, with the fluid in thermal equilibrium with the walls, the flow is said to be a hydrodynamically developing flow. When the flow is hydrodynamically fully developed but the thermal boundary layer develops, the flow is said to be a hydrodynamically fully developed and thermally developing flow, this situation can be obtained in a microchannel with adiabatic walls in the region near the entrance (adiabatic preparation) and where heating at the walls starts beyond the hydrodynamic entrance region. 

For microchannels in which the flow is originated by means of electroosmosis, the hydrodynamic entrance region can start from an internal point of the channel where the zeta potential at the wall differs from zero. 

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