Fluid mechanics dimensionless flows

Heat Transfer In general, the fluid mechanics of the film on the mixer side of the heat transfer surface is a function of what happens at that surface rather than the fluid mechanics going on around the impeller zone. The impeller largely provides flow across and adjacent to the heat-transfer surface and that is the major consideration of the heat-transfer result obtained. Many of the correlations are in terms of traditional dimensionless groups in heat transfer, while the impeller performance is often expressed as the impeller Reynolds number. 

The behavior of the gas as it flows down the tube is controlled by fluid mechanics and a complete investigation wouldbe lengthy and outside the scope of this book. It is enough to say that the Reynolds number, which is a dimensionless parameter that characterizes the... 

In the previous section, the importance of the uniformity of the radial flow profile was established. In the present section, the fluid mechanical equations for all four flow configurations in Figure 1 are derived and solved for comparison. The development of equations closely follows the approach of Genkin et al. (1,2). Here we extend their work to include both radial and axial flow in the catalyst bed. Following our derivation in reference (16), the dimensionless equations for the axial velocity in the center-pipe for all four configurations are (the primes denote derivative with respect to the dimensionless axial coordinate). 

The energy consumption in agitation depends on the basic principles of fluid mechanics however, the flow patterns in a mixing vessel are much too complex for their rigorous application. Therefore, empirical relationships based on dimensionless groups are used. Here, because most fluid foods are non-Newtonian in nature, the... 

Incidentally, Bowes discussed in his book, Self-Heating Evaluating and Controlling the Hazards , some dimensionless numbers, such as the Grashof number, the Reynolds number and the Prandtl number, which are all used in order to discuss the convective flow in fluid mechanics. 

The Reynolds number is undoubtedly the most famous dimensionless parameter in fluid mechanics. It is named in honour of Osborne Reynolds, a British engineer who first demonstrated in 1883 that a dimensionless variable can be used as a criterion to distinguish the flow patterns of a fluid either being laminar or turbulent. Typically, a Reynolds number is given as follows ... 

In dealing with viscoelastic fluids, especially under turbulent flow conditions, it is necessary to introduce a dimensionless number to take account of the fluid elasticity [29-33], Either the Deborah or the Weissenberg number, both of which have been used in fluid mechanical studies, satisfies this requirement. These dimensionless groups are defined as follows ... 

Besides clarifying the strange shape of Fig. 6.2, Reynolds made the most celebrated application of dimensional analysis (Chap. 13) in the history of fluid mechanics. He showed that for smooth, circular pipes, for all newtonian fluids, and for all pipe diameters, the transition from laminar to turbulent flow occurs when the dimensionless group DVpIfjt, has a value of about 2000. Here D is the pipe diameter, V is the average fluid velocity in the pipe, p is the fluid density, and fi is the fluid viscosity. This dimensionless group is now called the Reynolds number For flows other than pipe flow, some other appropriate length is substituted for the pipe diameter in the Reynolds number, as discussed later. 

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... 

The problem addressed here is that of expressing mass transfer coefficients that apply to turbulent flow conditions in a tube in terms of appropriate dimensionless groups. To implement Step 1, both fluid mechanical and transport properties must be taken into account. The former determine the degree of turbulence or ability to form eddies, and hence the rate at which mass is transported to or from the tubular wall, whereas the transport parameter determines the rate at which mass is conveyed through the film adjacent to the interface. It is proposed to use velocity v, density p, and viscosity p as the fluid mechanical properties, as each of these parameters either promotes or resists the formation of eddies. Transport through the film is determined by only one parameter, the diffusivity of the conveyed species, hi addition to these factors, we expect pipe diameter to play a role because it determines the distance over which the mass is to be transported and plays a role as well in the degree of turbulence generated in the system. 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

In a hydrodynamically free system the flow of solution may be induced by the boundary conditions, as for example when a solution is fed forcibly into an electrodialysis (ED) cell. This type of flow is known as forced convection. The flow may also result from the action of the volume force entering the right-hand side of (1.6a). This is the so-called natural convection, either gravitational, if it results from the component defined by (1.6c), or electroconvection, if it results from the action of the electric force defined by (1.6d). In most practical situations the dimensionless Peclet number Pe, defined by (1.11b), is large. Accordingly, we distinguish between the bulk of the fluid where the solute transport is entirely dominated by convection, and the boundary diffusion layer, where the transport is electro-diffusion-dominated. Sometimes, as a crude qualitative model, the diffusion layer is replaced by a motionless unstirred layer (the Nemst film) with electrodiffusion assumed to be the only transport mechanism in it. The thickness of the unstirred layer is evaluated as the Peclet number-dependent thickness of the diffusion boundary layer. 

Obtain a dimensionless relation for the velocity profile in the neighbourhood of a surface for the turbulent flow of a liquid, using Prandtl s concept of a Mixing Length (Universal Velocity Profile). Neglect the existence of the buffer layer and assume that, outside the laminar sub-layer, eddy transport mechanisms dominate. Assume that in the turbulent fluid the mixing length Xe is equal to 0.4 times the distance y from the surface and that the dimensionless velocity u1 is equal to 5.5 when the dimensionless distance y+ is unity. 

The two examples, deliberately chosen for their simplicity, show that computational fluid dynamics facilitate a more in-depth examination of the local flow behavior of twin screw extruders. Local peaks in the mechanical and thermal stresses can be easily identified. By changing the geometry, stresses can be reduced and the quality of the polymer can thereby be optimized. Another application focus is the rapid determination of the dimensionless axis intercepts for the pressure build-up A, and A2 and for the power requirement B, and B2. The significance of these parameters has already been discussed in detail in the two previous chapters. 

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