Laminar duct flow development

TABLE 5.1 Asymptotic Nusselt Numbers for Laminar Duct Flow (With Fully Developed Velocity Profiles)... 

When the inlet length is expressed in terms of number of gap widths , the difference between the flow in a tube and the flow in an annulus of narrow gap differs only by 25% [(0.05 - 0.04)/0.05]. This situation is an indication that the growth of the laminar boundary layers from the wall to the center of the channel is similar in both cases. Because duct friction coefficients, a measure of momentum transfer, do not vary by more than a factor of 2 for ducts of regular cross sections when expressed in terms of hydraulic diameters, the use of the inlet length for tubes or parallel plates can be expected to be a reasonable approximation for the inlet lengths of other cross sections under laminar flow conditions. In the annular denuder, the dimensionless inlet length for laminar flow development, L, can be expressed as... 

This equation describes the temperature distribution in fully developed laminar plane duct flow when the wall heat flux is a constant. It can be written in terms of the specified wall heat flux. qw, by noting that when Eq. (4.77) is used to give the value of dT dy y = w in Eq. (4.71), the following is obtained ... 

Because the area of the duct per unit width is 2w and because, by definition, the mass flow rate is equal to the (density x area X mean velocity), this shows that the mean velocity in fully developed laminar plane duct flow is 6/4 (= 1.5) times the center line velocity. 

Laminar Flow. Based on the solutions for laminar boundary layer development over a flat plate and fully developed flow in circular and some noncircular ducts, /app Re can be correlated by the following equation ... 

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 ... 

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

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 Rectangular Duct Thermal-Entry Length, with Hydrodynamically Fully Developed Laminar Flow... 

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. 

FULLY DEVELOPED LAMINAR FLOW IN A PLANE DUCT... 

Next consider fully developed laminar flow through a plane duct whose wall temperature is kept constant As with pipe flow, for this boundary condition, Eq. (4.69) gives ... 

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. 

Water flows in a rectangular 5 mm x 10 mm duct with a mean bulk temperature of 20°C. If the duct wall is kept at a uniform temperature of 40°C and if fully developed laminar flow is assumed to exist, find the heat transfer rate per unit length of the duct. 

Consider a series of rectangular channels of width W, and height H, all with the same cross-sectional area whose walls are kept at a uniform temperature. The flow in these ducts can be assumed to be fully developed and laminar. 

Consider constant-property, fully developed laminar flow between two large parallel plates, i.e., in a wide plane duct. One plate is adiabatic and the other is isothermal and the velocity is high enough for viscous dissipation effects to be significant. Determine the temperature distribution in the flow. 

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.

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. ... 

For slip flow, the fully-developed velocity profile can be obtained from the momentum equations for laminar flow with constant thermophysical properties in a parallel-plate channel and a circular duct, respectively, as... 

Aparecido, J., and Cotta, R. (1990) Thermally Developing Laminar Flow inside Rectangular Ducts, Int. J. Heat andMass Transfer. Vol. 33, pp. 341-347. 

Santos, C.A.C., Medeiros, M.J., Cotta, RM., and Kaka9, S. (1998), Theoretical Analysis of Transient Laminar Forced Convection in Simultaneous Developing Flow in Parallel-Plate Channel, 7 AIAA/ASME Joint Themophysics and Heat Transfer Conference, AIAA Paper 97-2678, Albuquerque, New Mexico, June. Cheroto, S., Mikhailov, M.D., Kaka , S., and Cotta, R.M, (1999), Periodic Laminar Forced Convection -Solution via Symbolic Computation and Integral Trmsforms, Int. J. Thermal Sciences, V.38, no.7, pp.613-621. Kaka, S., Santos, C.A.C., Avelino, MR., and Cotta, R.M. (2001), Computational Solutions and Experimental Analysis of Transient Forced Convection in Ducts, Invited Paper, Int. J. of Transport Phenomena, V.3, pp. 1-17. 

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