Hydrodynamic, fully developed, laminar flow

In the calculation of the velocity profile of a hydrodynamic, fully developed, laminar flow we will presume the flow to be incompressible and all properties to be constant. The velocity profile of a fully developed, tubular flow is only dependent on the radius r, wx = wx(r) and wy = 0. Therefore the acceleration term gdw /dt in the Navier-Stokes equation (3.59) disappears body forces are not present, so fc = 0. An equilibrium develops between the pressure and friction forces. Balancing the forces on an annular fluid element, Fig. 3.32, gives 

The axial pressure drop dp/dx is independent of the radial coordinate r it can also not be a function of the length x, as the left hand side of (3.220) only depends on the coordinate r. So that both sides agree, each must be constant. This means that the pressure can only change proportionally to the length x. Taking into account (dwx/dr)r=0 = 0 integration yields 

Hydrodynamic, fully developed, tubular flows were first studied by Hagen and Poiseuille. The flow is therefore also called Hagen-Poiseuille flow. We can see from (3.223), that the velocity at the tube axis is twice the mean velocity, wx(r = 0) = 2w m. From (3.222) we get the following for the pressure drop 

The mean velocity over a cross section is found from this to be 

The velocity profile of a fully developed flow is therefore 

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The Circular Tube Thermal-Entry-Length, with Hydrodynamically Fully Developed Laminar Flow... 

For a hydrodynamically fully developed laminar flow, the parabolic velocity profile is applicable. Hence,... 

Fig. 3.32 Force balance on a ring element in hydrodynamic, fully developed, laminar flow...

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

Ameel et al. [33] studied hydrodynamically fully developed laminar flow in circular chaimels when slip flow occurs. An Nu correlation in the slip flow regime was derived ... 

Equations (6) and (7) are valid only far from the inlet section of the microchannel (see developing flows and entrance region) where the velocity profile can be considered hydrodynamically fully developed. Laminar flow is designated as hydrodynamically fully developed when the fluid velocity distribution at a cross-section is of an invariant form independent of the axial distance z. 

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. 

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

Equation (17.27) describes a developing mass transfer boundary layer in a fully-developed laminar flow. However, if the mass transfer boundary spans the whole electrode gap, the Sherwood number attains a limiting value which is independent of flow hydrodynamics. The limiting Sherwood number is given, for a uniform mass flux at the electrode surface by [21]... 

The equations given in this chapter for Nu are only valid for a fully developed laminar flow, that is, without a hydrodynamic entrance region. Similar considerations can be made for the heat transfer in a tube with both a thermal and a hydrodynamic entrance region, whereby the differences in the values of Nu and Nu are mostly small (VDI, 2002). 

Figure 3.2.8 Local and average Nusselt number (Nu Nu) for laminar flow in a tube (fully developed laminar flow, that is, without hydrodynamic entrance region, see Section 3.2.1.2.1).

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

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. 

On the contrary, under the H boundary condition (see Convective Heat Transfer in Microchannels ) with a fully developed velocity profile hydrodynamically fully developed and thermally developing flow Fig. 3), the mean Nusselt number for circular microtubes as a functitMi of the dimensionless axial coordinate z for laminar flows can be calculated as follows ... 

Figure 3.2.23 Development of the velocity boundary layer of a fluid flowing in an empty tube. The velocity profile in the hydrodynamically fully developed region is parabolic in laminar flow (as shown) and somewhat blunt in turbulent flow (see Figure 3.2.22).

Just as for laminar flow, a minimum hydrodynamic entry length (Le) is required for the flow profile to become fully developed in turbulent flow. This length depends on the exact nature of the flow conditions at the tube entrance but has been shown to be on the order of Le/D = 0.623/VRe5. For example, if /VRe = 50,000 then Le/D = 10 (approximately). 

The hydrodynamic entry length is usually taken to be the distance from the lube entrance where the wall shear. stress (and thus the fficliou factor) reaches within about 2 percent of the fully developed value. In laminar flow, the hydrodynamic and thermal entry lengths are given approximately as (see Kays and Crawford (1993) and Shah and Bhatli (1987)]... 

At the entrance of a channel, the friction and heat flow rate are generally higher than downstream, where both the velocity and the temperature profiles are fully developed. Few experimental data exist for this region and most of smdies are analytical or numerical. For cylindrical mhes a laminar flow is hydrodynamically developed (within 5 %) if ... 

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. 

As a result of the development of the hydrodynamic and thermal boundary layers, four types of laminar flows occur in ducts, namely, fully developed, hydrodynamically developing, thermally developing (hydrodynamically developed and thermally developing), and simultaneously developing (hydrodynamically and thermally developing). In this chapter, the term fully developed flow refers to fluid flow in which both the velocity profile and temperature profile are fully developed (i.e., hydrodynamically and thermally developed flow). In such cases, the velocity profile and dimensionless temperature profile are constant along the flow direction. The friction factor and Nusselt number are also constant. 

For a circular duct with a diameter of 2a, the characteristics of laminar flow and heat transfer for four kinds of flows, namely, fully developed, hydrodynamically developing, thermally developing, and simultaneously developing, are outlined in the following sections. 

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. 

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