Laminar boundary layer Nusselt number

The equation for the laminar Nusselt number Nut is obtained in a two-step procedure. In the first step, not only is the flow idealized as everywhere laminar, but the boundary layer is treated as thin. There results from this idealization the equation for the laminar thin-layer Nusselt number Nur. As already explained, natural convection boundary layers are generally not thin, so the second step is to correct Nur to account for thick boundary layers. This correction uses the method of Langmuir [175]. The corrected Nusselt number is the laminar Nusselt number Nuc. 

See also -> convection, -> Grashof number, - Hagen-Poiseuille, -> hydrodynamic electrodes, -> laminar flow, - turbulent flow, -> Navier-Stokes equation, -> Nusselt number, -> Peclet number, -> Prandtl boundary layer, - Reynolds number, -> Stokes-Einstein equation, -> wall jet electrode. 

The Nusselt number is the ratio of the tube diameter to the equivalent thickness of the laminar layer. Sometimes x is called the film thickness, and it is generally slightly greater than the thickness of the laminar boundary layer because there is some resistance to heat transfer in the buffer zone. 

In the laminar boundary layer regime, the Nusselt number relation is of the form of that for a vertical flat plate... 

Hence, Shaverage scales as the one-half power of Re for laminar boundary layer mass transfer across a solid-liquid interface, where Sc is appropriate. Obviously, the average Nusselt number scales as the one-half power of Re for... 

Inspection of the heat transfer (and of the local Nusselt number] along the circumference of a cylinder leads to the following conclusion. The overall heat transfer from cylinders (and other bodies] consists of three parts (i) the convective heat transfer in the laminar boundary layer of the front part, (ii) heat transfer from the rear part, where — depending on Re — separation and turbulences occur, and (Mi) a constant and small part representing the limit case of minimum heat loss to a stagnant surrounding [Re = 0, see text below Eq. (3.2.3)]. Hence, the mean Nu number can be expressed by a correlation equation of the form ... 

On the other hand, as shown on pages 143 to 151, in the flow of fluids across a cylindrical shape boundary-layer separation occurs, and a wake develops that causes form friction. No sharp distinction is found between laminar and turbulent flow, and a common correlation can be used for both low and high Reynolds numbers. Also, the local value of the heat-transfer coefficient varies from point to point around a circumference. In Fig. 12.5 the local value of the Nusselt number is plotted radially for all points around the circumference of the tube. At low Reynolds numbers, is a maximum at the front and back of the tube and a minimum at the sides. In practice, the variations in the local coefficient hg are often of no importance, and average values based on the entire circumference are used. 

Thin-Layer Approximation. Laminar analyses often make the further approximation that the boundary layer is so thin that when the simplified equations of motion are rewritten in terms of local surface coordinates, i.e., in terms of the x and y of Fig. 4.3a, several terms normally associated with curvature effects can be dropped. The Nusselt number equation, based on solutions to such laminar thin-layer equation sets, always takes the form... 

As a result of the development of the hydrodynamic and thermal boundary layers, four types of laminar flows occur in ducts, namely, fully developed, hydrodynamically developing, thermally developing (hydrodynamically developed and thermally developing), and simultaneously developing (hydrodynamically and thermally developing). In this chapter, the term fully developed flow refers to fluid flow in which both the velocity profile and temperature profile are fully developed (i.e., hydrodynamically and thermally developed flow). In such cases, the velocity profile and dimensionless temperature profile are constant along the flow direction. The friction factor and Nusselt number are also constant. 

The heat transfer phenomena and, in particular, the thermal boundary layer, are well understood for a cool, solid body immei l in a laminar hot gas stream in which no chemical reactions occur. The heat transfer can be predicted by the Nusselt number Nu = fi(Re, Pr). With some modifications, similar relationships hold for dissociation of gases, provided that Le, which describes the diffusion of the species, is close to unity... 

At 700°C and 1 atm this leads to a diffusion constant of 0.81 cm /s The flow field around a superheater tube is very complex involving both laminar and turbulent boundary layers and the estimation of the local boundary layer thicknesses (velocity, diffusion and thermal boundary layers) around the tube requires computer simulations with computational fluid dynamic (CFD) software packages. However, for this rough analysis an average value of the thermal boundary layer thickness around the tube is enough and can be estimated if the average Nusselt number around the tube is known... 

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