Fully Developed Laminar Flow

We consider steady-state laminar and fully developed thermal and hydrodynamic single-phase flow. 

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Friction Coefficient. In the design of a heat exchanger, the pumping requirement is an important consideration. For a fully developed laminar flow, the pressure drop inside a tube is inversely proportional to the fourth power of the inside tube diameter. For a turbulent flow, the pressure drop is inversely proportional to D where n Hes between 4.8 and 5. In general, the internal tube diameter, plays the most important role in the deterrnination of the pumping requirement. It can be calculated using the Darcy friction coefficient,, defined as... 

The laminar veloeity profile in Figure 8-2la is approximated by a series of annuli, within eaeh of whieh the veloeity is eonstant as illustrated in Figure 8-21b. Eaeh annulus is eonsidered to be a plug flow tubular reaetor having its own spaee veloeity. The veloeities of the fluid elements at different radii are given by the parabolie veloeity profile for fully developed laminar flow. The veloeity is expressed as... 

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

For single-phase gas flow in micro-channels of hydraulic diameter from 101 to 4,010 pm, in the range of Reynolds numbers Re < Recr, the Knudsen number 0.001 < Kn < 0.38, and the Mach number 0.07 < Ma < 0.84, the experimental friction factor agrees quite well with the theoretical one predicted for fully developed laminar flow. 

Hetsroni et al. (2005) evaluated the effect of inlet temperature, channel size and fluid properties on energy dissipation in the flow of a viscous fluid. For fully developed laminar flow in circular micro-channels, they obtained an equation for the adiabatic increase of the fluid temperature due to viscous dissipation ... 

The values of the Nusselt and the Poiseuille numbers for heat transfer and friction for fully developed laminar flows through specifled channels are presented in Table 7.1 (Shah and London 1978). 

Flow is typically laminar in microchannel devices, although not always rigorously so. Correlations for fully developed laminar flow in perfectly rectangular microchannels have been validated in the literature [33-35]. Transition and turbulent flows in a microchannel have no such consistent treatise, and are highly dependent upon channel shape, aspect ratio, and surface characteristics [36, 37]. 

Fully developed laminar flow in curved rectangular channels, J. Fluids Eng. 3 (1976) 42-48. 

The foregoing procedure can be used to solve a variety of steady, fully developed laminar flow problems, such as flow in a tube or in a slit between parallel walls, for Newtonian or non-Newtonian fluids. However, if the flow is turbulent, the turbulent eddies transport momentum in three dimensions within the flow field, which contributes additional momentum flux components to the shear stress terms in the momentum equation. The resulting equations cannot be solved exactly for such flows, and methods for treating turbulent flows will be considered in Chapter 6. 

Determine the shear stress distribution and velocity profile for steady, fully developed, laminar flow of an incompressible Newtonian fluid in a horizontal pipe. Use a cylindrical shell element and consider both sign conventions. How should the analysis be modified for flow in an annulus ... 

The velocity profile for steady, fully developed, laminar flow in a pipe can be determined easily by the same method as that used in Example 1.9 but using the equation of a power law fluid instead of Newton s law of viscosity. The shear stress distribution is given by... 

It is interesting to compare equations (6.32) and (6.33) with those for a fully developed laminar flow, equations (6.29) and (6.30). In Example 5.1, we showed that eddy diffusion coefficient in a turbulent boundary layer was linearly dependent on distance from the wall and on the wall shear velocity. If we replace the diffusion coefficient in equation (6.30) with an eddy diffusion coefficient that is proportional to hu, we get... 

For fully developed laminar flow, the shear stress at the wall of a circular duct is... 

Laminar Flow Development. In a typical application, it is not generally possible to sample isokinetically. Equations governing the collection efficiency (e.g., equation 1) apply only under laminar flow conditions. It is considered necessary therefore to leave a length of the tubing surface at the entrance deliberately uncoated to permit laminar flow to fully develop before actual collection occurs. For a simple tube, the minimum inlet length, Lh necessary to fully develop laminar flow (within about 98%) is given by (34) ... 

If fully-developed laminar flow is assumed, then (from Volume 1, Section 3.4.2) ... 

Steady-state, fully developed laminar flows of viscoelastic fluids in straight, constant-diameter pipes show no effects of viscoelasticity. The viscous component of the constitutive equation may be used to develop the flow rate-pressure drop relations, which apply downstream of the entrance region after viscoelastic effects have disappeared. A similar situation exists for time-dependent fluids. 

Example 7.8 Residence Time Distribution Functions in Fully Developed Laminar Flow of a Newtonian Fluid in a Pipe The velocity distribution... 

Solution This flow is z-axisymmetric. We, thus, select a cylindrical coordinate system, and make the following simplifying assumptions Newtonian and incompressible fluid with constant thermophysical properties no slip at the wall of the orifice die steady-state fully developed laminar flow adiabatic boundaries and negligible of heat conduction. 

Fig. 8.23. Experimental gas-liquid mass transfer coefficients for fully developed laminar flow (monolith section from 80—500 mm) = 25 cpsi square channels A = 50 cpsi square channels ...

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

Figure 8.2 Figure for fully developed laminar flow in conduits. 

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

Consider the fully developed laminar flow between two parallel plates at a distance 2a apart. Find the expression for the velocity profile and the friction factor. 

If the solution procedure is carried through as outlined above, the following is obtained for fully developed laminar flow through a pipe with constant wall temperature ... 

FULLY DEVELOPED LAMINAR FLOW IN A PLANE DUCT... 

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