Laminar flow in ducts

FULLY DEVELOPED LAMINAR FLOW IN DUCTS WITH OTHER CROSS-SECTIONAL SHAPES... 

This chapter has been concerned with the analysis of laminar flows in ducts with various cross-sectional shapes. If the flow is far from the inlet to the duct or from anything else causing a disturbance in the flow, a fully developed state is reached in many situations, the basic characteristics of the flow in this state not changing with distance along the duct. If the diffusion of heat down the duct can be neglected, which is true in most practical situations, it was shown that in such fully developed flows, the Nusselt number based on the difference between the local wall and bulk mean temperatures is constant. Values of the Nusselt number for fully developed flow in ducts of various cross-sectional shape were discussed. 

Shah and London [40] have compiled the heat-transfer and fluid-friction information for fully developed laminar flow in ducts with a variety of flow cross sections as shown in Table 6-1. In this table the following nomenclature applies ... 

Table 6-1 Heat Transfer and Fluid Friction for Fully Developed Laminar Flow In Ducts of Various Cross Sections.

For laminar flow in ducts running full and in open channels with various cross-sectional shapes other than circular, equations are given elsewhere (PI). 

Sherwood Number for Fully Developed Laminar Flow in Ducts of Different Cross Sections... 

Entrance flow is also accompanied by the growth of a boundary layer (Fig. 5b). As the boundary layer grows to fill the duct, the initially flat velocity profile is altered to yield the profile characteristic of steady-state flow in the downstream duct. For laminar flow in a tube, the distance required for the velocity at the center line to reach 99% of its asymptotic values is given by... 

The convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

TABLE 5-4 Values of Limiting Nusselt Number in Laminar Flow in Closed Ducts... 

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

The classical solution of the problem of steady laminar flow in straight ducts is based on a number of assumptions on flow conditions, Hetsroni et al. (2005)... 

In this case the problem of developed laminar flow in a straight duct reduces to integrating the equation... 

Nguyen TV (1992) Laminar heat transfer for thermal developing flow in ducts. Int J Heat Mass Transfer 35 1733-1741... 

The mass transfer coefficient is usually obtained from correlations for flow in non-porous ducts. One case is that of laminar flow in channels of circular cross-section where the parabolic velocity profile is assumed to be developed at the channel entrance. Here the solution of LfivfiQUE(7), discussed by Blatt et al.(H>, is most widely used. This takes the form ... 

This well-known relationship is valid for laminar flow in circular ducts, but it also sets the stage for more general scaling relationships in noncircular cross sections and turbulent flows. 

The mathematical analysis of flow in ducts of noncircular cross section is vastly more complex in laminar flow than for circular pipes and is impossible for turbulent flow. As a result, relatively little theoretical base has been developed for the flow of fluids in noncircular ducts. In order to deal with such flows practically, empirical methods have been developed. 

The analysis employed mass transfer coefficients for fully developed laminar flows in square ducts. 

Flow in Noncircular Ducts The length scale in the Nusselt and Reynolds numbers for noncircular ducts is the hydraulic diameter, D), = 4AJp, where A, is the cross-sectional area for flow and p is the wetted perimeter. Nusselt numbers for fully developed laminar flow in a variety of noncircular ducts are given by Mills (Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 307). For turbulent flows, correlations for round tubes can be used with D replaced by l. ... 

Each term has the dimensions of energy per unit of mass - in this case, ft-lbp/lbM. The factor, a, in the kinetic energy term, Av /2agc, corrects for the velocity profile across a duct. For laminar flow in a circular duct, the velocity profile is parabolic, and a = 1/2. If the velocity profile is flat, a = 1. For very rough pipes and turbulent flow, a may reach a value of 0.77 [10]. In many engineering applications, it suffices to let a = 1 for turbulent flow. 

This simple modification does not apply to laminar flow in noncircular ducts. 

Brown, G. M. Heat or mass transfer in a fluid in laminar flow in a circular duct or flat conduit. Amer. Inst. Chem. Eng. J. 6 (1960) 179—183... 

Gayev, Ye. (2001) Pulsating laminar flow in a duct with easily penetrable roughness near walls, Int. J. Fluid Mech. Research v. 28, No. 1-2, 164-172. 

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