Channel flow boundary conditions

The boundary conditions required to solve the channel-flow boundary-layer problem are ... 

For flows in conventional channels, the flow dimensions are much larger than the molecular mean free path. Therefore, fluid properties are determined primarily by intermolecular colUsions. As the channel size is reduced, the molecular mean free path becomes comparable to channel size. Intermolecular colhsions lose their importance and the interactions between the fluid and the wall become significant. The derivations of the shp flow boundary conditions using the kinetic theory of gases will be shown based on the derivations of [2] and [3] and are explained in the following manuscript in this book [4]. Briefly, the first order velocity shp is given by ... 

The steady-state heat convection between two parallel plates and in circular, rectangular, and annular channels with viscous heat generation for both thermally developing and fully developed conditions is solved. Both constant wall temperature and constant heat flux boundary conditions are crmsidered. The velocity and the temperature distributions are derived from the momentum and energy equations, and the proper slip-flow boundary conditions are considered. 

It is important to note that stability limits in natural circulation systems arise before (and as a prelude to) CHF or DNB. Indeed, conventional forced flow CHF and DNB correlations cannot be applied to natural circulation and parallel channel systems if either the loss coefficients are unknown or not reported, or the appropriate constant pressure drop was not maintained or achieved in the tests. Throttling the inlet flow to set a flow boundary condition artificially stabilizes the channel. In actual plants, it is well known that the plant maintains a constant pressure drop, by having multiple parallel channels and/or a controlled downcomer hydrostatic head. 

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

It is noteworthy that several studies exhibit contradictory results for both the mechanical and thermal characteristics of the flow. This is generally due to differences in the many parameters that characterize these studies such as the geometry, shape and surface roughness of the channels, the fluid, the boundary conditions and the measuring methodology itself. These discrepancies indicate the need for extension of the experimental base to provide the necessary background to the theoretical model. 

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

The micro-channels utilized in engineering systems are frequently connected with inlet and outlet manifolds. In this case the thermal boundary condition at the inlet and outlet of the tube is not adiabatic. Heat transfer in a micro-tube under these conditions was studied by Hetsroni et al. (2004). They measured heat transfer to water flowing in a pipe of inner diameter 1.07 mm, outer diameter 1.5 mm, and 0.600 m in length, as shown in Fig. 4.2b. The pipe was divided into two sections. The development section of Lj = 0.245 m was used to obtain fully developed flow and thermal fields. The test section proper, of heating length Lh = 0.335 m, was used for collecting the experimental data. 

One particular characteristic of conduction heat transfer in micro-channel heat sinks is the strong three-dimensional character of the phenomenon. The smaller the hydraulic diameter, the more important the coupling between wall and bulk fluid temperatures, because the heat transfer coefficient becomes high. Even though the thermal wall boundary conditions at the inlet and outlet of the solid wall are adiabatic, for small Reynolds numbers the heat flux can become strongly non-uniform most of the flux is transferred to the fluid at the entrance of the micro-channel. Maranzana et al. (2004) analyzed this type of problem and proposed the model of channel flow heat transfer between parallel plates. The geometry shown in Fig. 4.15 corresponds to a flow between parallel plates, the uniform heat flux is imposed on the upper face of block 1 the lower face of block 0 and the side faces of both blocks... 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

The boundary conditions (10.12-10.14) correspond to the flow in a micro-channel with a cooled inlet and adiabatic receiver (an adiabatic pipe or tank, which is established at the exit of the micro-channel). Note, that the boundary conditions of the problem can be formulated by another way, if the cooling system has another construction, for example, as follows x = 0, Tl = IL.o, x = L, Tg = Tg.oo, when the inlet and outlet are cooled x = 0, dT /dx = 0, x = L, Tg = Tg.oo in case of the adiabatic inlet and the cooled outlet, etc. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

A number of authors have considered channel cross-sections other than rectangular [102-104]. Figure 2.17 shows some examples of cross-sections for which friction factors and Nusselt numbers were computed. In general, an analytical solution of the Navier-Stokes and the enthalpy equations in such channel geometries would be involved owing to the implementation of the wall boundary condition. For this reason, usually numerical methods are employed to study laminar flow and heat transfer in channels with arbitrary cross-sectional geometry. 

The effects of geometry and boundary conditions are usually interrelated, such as in flow redistribution among parallel channels. With common headers connected to the parallel channels, the flow distribution among channels is determined... 

The boundary conditions to be satisfied are that the lateral pressure difference between subchannels should be zero at the channel inlet and exit. Having passed once along the channel, this implies that iteration over the channel length may be necessary by using improved guesses of flow division between subchannels at the inlet. In practice, only one pass may be necessary, particularly for hydraulic model, in which lateral momentum transfer is neglected or only notionally included. Rowe (1969) has shown that for a single-pass solution to be stable and acceptable,... 

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