Laminar flows continued defined

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

In laminar flow, heat transfer occurs only by conduction, as there are no eddies to carry heat by convection across an isothermal surface. The problem is amenable to mathematical analysis based on the partial differential equations for continuity, momentum, and energy. Such treatments are beyond the scope of this book and are given in standard treatises on heat transfer, Mathematical solutions depend on the boundary conditions established to define the conditions of fluid flow and heat transfer. When the fluid approaches the heating surface, it may have an already completed hydrodynamic boundary layer or a partially developed one. Or the fluid may approach the heating surface at a uniform velocity, and both boimdary layers may be initiated at the same time. A simple flow situation where the velocity is assumed constant in all cross sections and tube lengths is called... 

Microfluidics handles and analyzes fluids in structures of micrometer scale. At the microscale, different forces become dominant over those experienced in everyday life [161], Inertia means nothing on these small sizes the viscosity rears its head and becomes a very important player. The random and chaotic behavior of flows is reduced to much more smooth (laminar) flow in the smaller device. Typically, a fluid can be defined as a material that deforms continuously under shear stress. In other words, a fluid flows without three-dimensional structure. Three important parameters characterizing a fluid are its density, p, the pressure, P, and its viscosity, r. Since the pressure in a fluid is dependent only on the depth, pressure difference of a few pm to a few hundred pm in a microsystem can be neglected. However, any pressure difference induced externally at the openings of a microsystem is transmitted to every point in the fluid. Generally, the effects that become dominant in microfluidics include laminar flow, diffusion, fluidic resistance, surface area to volume ratio, and surface tension [162]. 

Fig. 7.2. Schematic diagram of the apparatus. Each solution, one corresponding to one stable stationary state and the other to the other stationary state, is stored in one of two continuous-stirred tank reactors (CSTR) and pumped at a determined and variable rates through the laminar flow reactor (LFR), where they are brought in contact with each other in a sharp well-defined boundary. For the remainder of the definitions see the text. Prom [1]...

The continuous flow microreactor has been tested showing different phases of the reaction process between potassium permanganate and an alkaline solution of ethanol as a flmction of the flow rate At fast flow rates a laminar flow pattern of the separated liquids, which do not react with each other during the reaction time defined by the... 

The most important use of residence time theory is its application to equipment that is already bnilt and operating. It is usually possible to find a tracer together with injection and detection methods that will be acceptable to a plant manager. The RTD is measnred and then analyzed to understand system performance. In this section we focns on such uses. The washout function is assumed to have an experimental basis. Calculations using it will be numerical in nature or will be analytical procednres applied to a model that reproduces the data accurately. Data fitting is best done by nonlinear least squares using untransformed experimental measurements of W(t), F(t), or f(t) versus time, t. Eddy diffusion in a turbulent system justifies exponential extrapolation of the integrals that define the moments in Table 1-2. For laminar flow systems, washout experiments should be continued until at least five times the estimated valne for t. The dimensionless variance has limited usefnlness in laminar flow systems. 

The main difference of microstructured reactors from the classical continuous-flow reactors consists in a laminar flow regime of the fluids (liquid and gases). The laminar flow regime is defined by dimensionless number, that is, the Reynolds number Re (Equation 1.1), which depends on the velocity u, the density p, the traveled length I, and the viscosity r] of the fluid ... 

Consequently, numerical solution of the equations of change has been an important research topic for many decades, both in solid mechanics and in fluid mechanics. Solid mechanics is significantly simpler than fluid mechanics because of the absence of the nonlinear convection term, and the finite element method has become the standard method. In fluid mechanics, however, the finite element method is primarily used for laminar flows, and other methods, such as the finite difference and finite volume methods, are used for both laminar and turbulent flows. The recently developed lattice-Boltzmann method is also being used, primarily in academic circles. All of these methods involve the approximation of the field equations defined over a continuous domain by discrete eqnalions associated with a finite set of discrete points within the domain and specified by the user, directly or through an antomated algorithm. Regardless of the method, the numerical solution of the conservation equations for fluid flow is known as computational fluid dynamics (CFD). 

Fluid flow in small devices acts differently from those in macroscopic scale. The Reynolds number (Re) is the most often mentioned dimensionless number in fluid mechanics. The Re number, defined by pf/L/p, represents the ratio of inertial forces to viscous ones. In most circumstances involved in micro- and nanofluidics, the Re number is at least one order of magnitude smaller than unity, ruling out any turbulence flows in micro-/nanochannels. Inertial force plays an insignificant role in microfluidics, and as systems continue to scale down, it will become even less important. For such small Re number flows, the convective term (pu Vu) of Navier-Stokes equations can be dropped. Without this nonlinear convection, simple micro-/ nanofluidic systems have laminar, deterministic flow patterns. They have parabolic velocity... 

Turbulent flow is described by conservation equations of continuity and momentum, known as the Reynolds-averaged Navier-Stokes (RANS) equations. Laminar velocity terms in conservation equations are replaced by the steady-state mean components and time-dependent fluctuating components defined by Equation 6.100. 

The continuity equations for mass, x-direction momentum, chemical species and energy in the plane, stationary, laminar boundary layer flow have already been given as Eqs. (7.1) to (7.4). The stream function ij/, by means of which the mass continuity equation is automatically satisfied, is defined by Eqs. (7.5). Following the approaches of Lees (1956), Fay and Riddell (1958), and Chung (1965), self-similar solutions in the stagnation region are obtained via transformations from (x, y) co-ordinates to the two new variables... 

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