Laminar duct flow

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... 

Table 5. Correlations for Heat-Transfer and Darcy Friction Coefficients for Noncircular Laminar Duct Flow ...

Most plate heat exchanger designs fall into the viscous flow range. Considering only Newtonian fluids since most chemical duties fall into this category, in laminar ducted flow the flow can be said to be one of three types ... 

Kostic M (1994) On turbulent drag and heat transfer reduction phenomena and laminar heat transfer enhancement in non-circular duct flow of certain non-Newtonian fluid. Int J Heat Mass Transfer 37 133-147... 

Internal flows of the type here being considered occur in heat exchangers, for example, where the fluid may flow through pipes or between closely spaced plates that effectively form a duct Although laminar duct flows do not occur as extensively as turbulent duct flows, they do occur in a number of important situations in which the size of the duct involved is small or in which the fluid involved has a relatively high viscosity. For example, in an oil cooler the flow is usually laminar. Conventionally, it is usual to assume that a higher heat transfer rate is achieved with turbulent flow than with laminar flow. However, when the restraints on possible solutions to a particular problem are carefully considered, it often turns out that a design that involves laminar flow is the most efficient from a heat transfer viewpoint. 

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. 

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. 

Shah RK and London AL. Thermal boundary conditions and some solutions for laminar duct flow forced convection. J. Heat. Transf. 1974 96(C2) 159-165. 

Equipment designs based on indirect conduction usually transfer the heat from the primary heat transfer fluid to the intermediate wall within some kind of internal duct or channel. Transfer coefficients for these cases depend on the nature of the flow (laminar or turbulent) and the geometry of the duct or channel (short or long). Expressions for evaluating the transfer coefficients for these cases are available in standard texts. An expression for the convective thermal resistance can be generated similar to that derived for the conductive resistance ... 

Tomiyama A, Sou A, Zun I, Kanami N, Sakaguchi T (1995) Effects of Eotvos number and dimensionless liquid volumetric flux on lateral motion of a bubble in a laminar duct flow , In Serizawa A, Pukano T, Bataille J (eds) Advances in Multiphase Flow 1995, pp. 3-15, Elsevier... 

The pipe head loss may be calculated by the well known Darcy- Weisbach equation [94], valid for duct flows of any cross section and for laminar and turbulent flow ... 

Spurk (1997) derived closed-form solutions for the average flow velocity v between parallel plates and in full ducts. For laminar flow between parallel plates relationships between viscosity, geometry, pressure gradient and mean flow velocity are given as ... 

The heat transfer rate in the laminar duct flow is very sensitive to the thermal boundary condition. Hence, it is essential to carefully identify the thermal boundary condition in laminar flow. The heat transfer rate in turbulent duct flow is insensitive to the thermal boundary condition for most common fluids (Pr > 0.7) the exception is liquid metals (Pr < 0.03). Hence, there is generally no need to identify the thermal boundary condition in turbulent flow for all fluids except liquid metals. 

A careful observation of accurate experimental friction factors for all noncircular smooth ducts reveals that ducts with laminar/Re < 16 have turbulent/factors lower than those for the circular tube, whereas ducts with laminar/Re > 16 have turbulent/factors higher than those for the circular tube [48], Similar trends are observed for the Nusselt numbers. If one is satisfied within 15 percent accuracy, Eqs. 17.87 and 17.88 for/and Nu can be used for noncircular passages with the hydraulic diameter as the characteristic length in / Nu, and Re otherwise, refer to Table 17.16 for more accurate results for turbulent flow. 

Velocity Fluctuations and Reynolds Stresses Li and Olsen [9, 10] were the first researchers to measure profiles of velocity fluctuations in turbulent microchannel flow. They measured streamwise and transverse velocity fluctuations and Reynolds shear stresses for a range of Reynolds numbers spanning the laminar through fully turbulent regime. They found good agreement between their measured fluctuations and Reynolds shear stresses and values reported for macroscale turbulent duct flow. 

When considering thin layer flow in a rectangular duct (see Fig. la), a fully developed parabolic flow (laminar) can be assumed for conditions of the Reynolds number Re < 10 (with... 

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

He, C. and Ahmadi, G. (1998). Particle Deposition with Themophoresis in Laminar and Turbulent Duct Flows. Aerosol Sd. Technol., Vol. 29, pp. 525-546. 

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... 

Fig. 5. Entrance flows in a tube or duct (a) separation at sharp edge (b) growth of a boundary layer (illustrated for laminar flow).

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. 

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