Rectangular ducts, laminar flow

Theoretical flow equations were derived for the molecular flow region by Knudsen (Kl) as far back as 1909. These equations for molecular flow and Poiseuille s Law for laminar flow, were the basis for vacuum flow computation until the later years of World War II. Normand (Nl) was the first to translate these equations into practical forms for engineering applications. In this reference Normand gives useful empirical rules for applying Knudsen s equations to ducts of rectangular cross-section, non-uniform cross-section, baffles, elbows, etc. 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

A significant heat-transfer enhancement can be obtained when a nonckcular tube is used together with a non-Newtonian fluid. This heat-transfer enhancement is attributed to both the secondary flow at the corner of the nonckcular tube (23,24) and to the temperature-dependent non-Newtonian viscosity (25). Using an aqueous solution of polyacrjiamide the laminar heat transfer can be increased by about 300% in a rectangular duct over the value of water (23). 

The critical Reynolds number for transition from laminar to turbulent flow in noncirciilar channels varies with channel shape. In rectangular ducts, 1,900 < Re < 2,800 (Hanks and Ruo, Ind. Eng. Chem. Fundam., 5, 558-561 [1966]). In triangular ducts, 1,600 < Re < 1,800 (Cope and Hanks, Ind. Eng. Chem. Fundam., II, 106-117 [1972] Bandopadhayay and Hinwood, j. Fluid Mech., 59, 775-783 [1973]). 

The volumetric flow rate Q (leakage rate) of a fluid in laminar flow through a high aspect ratio (h/w 1) rectangular duct (i.e., the seal volume) of width, w, height, h, and path length, z, in the flow direction is given by ... 

Noncircular Channels Calculation of frictional pressure drop in noncircular channels depends on whether the flow is laminar or turbulent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter DH should be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraulic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraulic diameter for a circular pipe is DH = D, for an annulus of inner diameter d and outer diameter D,DH = D-d, for a rectangular duct of sides a, b, DH=ab/[2(a+b)]. The hydraulic radius Rh is defined as one-fourth of the hydraulic diameter. 

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

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. 

Figure 22 shows a schematic diagram of a channel electrode (ChE), which consists of an electrode embedded in the wall of a rectangular duct through which solution is made to flow under well-defined laminar steady-state conditions (Compton and Unwin, 1986 Cooper and Compton, 1998). 

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

For circular pipes, Rh = R- The reader is cautioned that some definitions of Rh omit the factor of 2 shown in Equation 3.22 so that the result must be multiplied by 2 for use in equations such as 3.18 and 3.19. The use of Rh is not recommended for laminar flow, but alternatives are available in the literature. Also, the method of false transients applied to PDEs in Chapter 16 can be used to calculate laminar velocity profiles in ducts with noncircular cross sections. For turbulent, low-pressure gas flows in rectangular ducts, the American Society of Heating, Refrigerating and Air Conditioning Engineers recommends use of an equivalent diameter defined as... 

Plot [nax/fl asafunctionofaspectratio, = ///W, for laminar flow in rectangular ducts. 

In this section, the friction factors and Nusselt numbers for fully developed, hydrodynami-cally developing, thermally developing, and simultaneously developing laminar flows in rectangular ducts are presented. 

TABLE 5.30 Nusselt Number for Fully Developed Laminar Flow in Rectangular Ducts With One Wall or More Walls Heating... 

TABLE 5.31 Fully Developed / Re, NuT, NuHj, and NuH2 for Laminar Flow in Rectangular Ducts With All Four Walls Transferring Heat [1]... 

Fully Developed Laminar Flow in Curved, Square, and Rectangular Ducts... 

FIGURE 5.44 Friction factor and Nusselt number for fully developed laminar flow in rectangular ducts with longitudinal thin fins from opposite walls [1]. 

Two types of corrugations (triangular and rectangular) in parallel plate ducts are displayed in the insets of Figs. 5.60 and 5.61, respectively. Sparrow and Charmchi [290] have obtained the solutions for fully developed laminar flow in these ducts. The flow in the duct is considered to be perpendicular to the plane of the paper. Both ducts are assumed to be infinite in the span-... 

Fully developed laminar flow and heat transfer in a parallel plate duct with spanwise-periodic rectangular corrugations at one wall have been investigated by Sparrow and Chukaev [291]. The end effect is also ignored in their analysis. The fully developed / Re is shown in Fig. 5.61, which is based on the results reported by Sparrow and Chukaev [291] and the extension by Shah and Bhatti [2]. The heat transfer characteristics for the three pairs of geometric parameters can be found in Sparrow and Chukaev [291]. 

N. M. Natarajan, and S. M. Lakshmanan, Laminar Flow in Rectangular Ducts Prediction of Velocity Profiles and Friction Factor, Indian J. Technol., (10) 435-438,1972. 

M. Tachibana, and Y. lemoto, Steady Laminar Flow in the Inlet Region of Rectangular Ducts, BulLJSME, (24/193) 1151-1158,1981. 

Taking note of Eqs. 10.56 and 10.58, the final equation describing the fully established laminar velocity profile of a power-law fluid flowing through a rectangular duct is given by... 

A number of analytical results are available for fully developed Nusselt values for the laminar flow of power law fluids in rectangular channels having aspect ratios ranging from 0 (i.e., plane parallel plates) to 1.0 (i.e., a square duct). Newtonian results (n = 1) are available for the T, HI, and H2 boundary conditions for the complete range of aspect ratios. Another limiting case for which many results are available is the slug or plug flow condition, which corresponds to n = 0. At other values of n, results are available for plane parallel plates and for the square duct. 

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