Rectangular ducts velocity profile

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... 

Channel techniques employ rectangular ducts through which the electrolyte flows. The electrode is embedded into the wall [33]. Under suitable geometrical conditions [2] a parabolic velocity profile develops. Potential-controlled steady state (diffusion limiting conditions) and transient experiments are possible [34]. Similar to the Levich equation at the RDE, the diffusion limiting current is... 

This section describes a spreadsheet to solve for the two-dimensional velocity profile in a rectangular duct. It also determines the factor /Re, given an aspect ratio. The spreadsheet is laid out to correspond to the mesh shown in Fig. D.5. The spreadsheet itself is shown in Fig. D.6. 

We developed a unified flow model that can accurately predict the volumetric flowrate, velocity profile, and pressure distribution in the entire Knudsen regime for rectangular ducts. The new model is based on the hypothesis that the velocity distribution remains parabolic in the transition flow regime, which is supported by the asymptotic analysis of the Burnett equations [1]. The general velocity slip boundary condition and the rarefaction correction factor are the two primary components of this unified model. 

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... 

The boundary conditions are zero velocity at the walls and zero slope at any planes of symmetry. Analytical solutions for the velocity profile in square and rectangular ducts are available but cumbersome, and a numerical solution is usually preferred. This is the reason for the transient term in Equation 16.7. A flat velocity profile is usually assumed as the initial condition. As in Chapter 8, is assumed to vary slowly, if at all, in the axial direction. For single-phase flows, u can vary in the axial direction due to changes in mass density and possibly to changes in cross-sectional area. The continuity equation is just AcUp = constant because the cross-channel velocity components are ignored. 

The method will be illustrated using Equation 16.7, with constant p and /r, to find the velocity profile in a rectangular duct. Equation 16.7 is converted to an ODE by using second-order approximations for the spatial derivatives. The result is... 

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. 

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... 

Equation 10.59 was solved by Schechter [49] using a variational principle and by Wheeler and Wissler [50] using a numerical method. Wheeler and Wissler also presented an approximate equation for the square duct geometry. Schechter reported approximate velocity profiles for a power law fluid flowing through rectangular ducts having aspect ratios 0.25, 0.50, 0.75, and 1.0. His results may be expressed as follows ... 

The Nu and / factors are also dependent upon the duct cross-sectional shape in laminar flow, and are practically independent of the duct shape in turbulent flow. The influence of variable fluid properties on Nu and / for fully developed laminar flow through rectangular ducts has been investigated by Nakamura et al. [57]. They concluded that the velocity profile is strongly affected by the p /p ratio, and the temperature profile is weakly affected by the p /pm ratio. They found that the influence of the aspect ratio on the correction factor (p ,/pm)m for the friction factor is negligible for p /pm < 10. For the heat transfer problem, the Sieder-Tate correlation (n = -0.14) is valid only in the narrow range of 0.4 < p /pm < 4. 

It can be noticed how the presence of Joule heating reduces the value of the Nusselt number dramatically, while a decrease in the value of the aspect ratio 5 dampens this effect. The value of the Nusselt number for a perfectly flat velocity profile (slug flow) for a rectangular duct, Nujf, is also plotted in Fig. 5 for the rectangular crosssection considered. It is evident that the values of the Nusselt number approach the corresponding Nusf when kDi increases, so much more so the lower the value of M. ... 

---