Turbulent flow illustration

Pierce proposes and illustrates good agreement between the test data and the correlation for a smooth continuous curve for the Colburn factor over the entire range of Reynolds numbers for the laminar, transition, and turbulent flow regimes inside smooth tubes ... 

The dependence of the measured rise in fluid mixed-cup temperature on Reynolds number is illustrated in Fig. 3.12. The difference between outlet and inlet temperatures increases monotonically with increasing Re at laminar and turbulent flows. Under conditions of the given experiments, the temperature rise due to energy dissipation is very significant AT = 15—35 K at L/ i = 900—1,470 and Re = 2,500. The data on rising temperature in long micro-tubes can be presented in the form of the dependence of dimensionless viscous heating parameter Re/[Ec(L/(i)] on Reynolds number (Fig. 3.13). 

This classification of material behavior is summarized in Table 3-1 (in which the subscripts have been omitted for simplicity). Since we are concerned with fluids, we will concentrate primarily on the flow behavior of Newtonian and non-Newtonian fluids. However, we will also illustrate some of the unique characteristics of viscoelastic fluids, such as the ability of solutions of certain high polymers to flow through pipes in turbulent flow with much less energy expenditure than the solvent alone. 

The velocity field in turbulent flow can be described by a local mean (or time-average) velocity, upon which is superimposed a time-dependent fluctuating component or eddy. Even in one-dimensional flow, in which the overall average velocity has only one directional component (as illustrated in Fig. 6-3), the turbulent eddies have a three-dimensional structure. Thus, for the flow illustrated in Fig. 6-3, the local velocity components are... 

The Moody diagram illustrates the effect of roughness on the friction factor in turbulent flow but indicates no effect of roughness in laminar flow. Explain why this is so. Are there any restrictions or limitations that should be placed on this conclusion Explain. 

The simplest and most common device for measuring flow rate in a pipe is the orifice meter, illustrated in Fig. 10-7. This is an obstruction meter that consists of a plate with a hole in it that is inserted into the pipe, and the pressure drop across the plate is measured. The major difference between this device and the venturi and nozzle meters is the fact that the fluid stream leaving the orifice hole contracts to an area considerably smaller than that of the orifice hole itself. This is called the vena contracta, and it occurs because the fluid has considerable inward radial momentum as it converges into the orifice hole, which causes it to continue to flow inward for a distance downstream of the orifice before it starts to expand to fill the pipe. If the pipe diameter is D, the orifice diameter is d, and the diameter of the vena contracta is d2, the contraction ratio for the vena contracta is defined as Cc = A2/A0 = (d2/d)2. For highly turbulent flow, Cc 0.6. 

As discussed in Chapter 2, a fully developed turbulent flow field contains flow structures with length scales much smaller than the grid cells used in most CFD codes (Daly and Harlow 1970).29 Thus, CFD models based on moment methods do not contain the information needed to predict x, t). Indeed, only the direct numerical simulation (DNS) of (1.27)-(1.29) uses a fine enough grid to resolve completely all flow structures, and thereby avoids the need to predict x, t). In the CFD literature, the small-scale structures that control the chemical source term are called sub-grid-scale (SGS) fields, as illustrated in Fig. 1.7. 

At high Reynolds number, the velocity U(x, t) is a random field, i.e., for fixed time t = t the function U(x, D varies randomly with respect to x. This behavior is illustrated in Fig. 2.1 for a homogeneous turbulent flow. Likewise, for fixed x = x lJ(x. t) is a random process with respect to t. This behavior is illustrated in Fig. 2.2. The meaning of random in the context of turbulent flows is simply that a variable may have a different value each time an experiment is repeated under the same set of flow conditions (Pope 2000). It does not imply, for example, that the velocity field evolves erratically in time and space in an unpredictable fashion. Indeed, due to the fact that it must satisfy the Navier-Stokes equation, (1.27), U(x, t) is differentiable in both time and space and thus is relatively smooth. ... 

Thus, only the normal Reynolds stresses (i = j) are directly dissipated in a high-Reynolds-number turbulent flow. The shear stresses (i / j), on the other hand, are dissipated indirectly, i.e., the pressure-rate-of-strain tensor first transfers their energy to the normal stresses, where it can be dissipated directly. Without this redistribution of energy, the shear stresses would grow unbounded in a simple shear flow due to the unbalanced production term Vu given by (2.108). This fact is just one illustration of the key role played by 7 ., -in the Reynolds stress balance equation. 

Like the Kolmogorov scale in a turbulent flow, the Batchelor scale characterizes the smallest scalar eddies wherein molecular diffusion is balanced by turbulent mixing.3 In gas-phase flows, Sc 1, so that the smallest scales are of the same order of magnitude as the Kolmogorov scale, as illustrated in Fig. 3.1. In liquid-phase flows, Sc 1 so that the scalar field contains much more fine-scale structure than the velocity field, as... 

Equation (3.82) illustrates the importance of the scalar spectral energy transfer rate in determining the scalar dissipation rate in high-Reynolds-number turbulent flows. Indeed, near spectral equilibrium, 7 (/cd, 0 (like Tu(kDi, 0) will vary on time scales of the order of the eddy turnover time re, while the characteristic time scale of (3.82) is xn <[Pg.99]

In the paddle method, bulk Reynolds numbers range from Re = 2292 (25 rpm, 900 mL) up to Re = 31025 (200 rpm, 500 mL). In contrast, Reynolds numbers employing the basket apparatus range from Re = 231 to Re = 4541. These Reynolds numbers are derived from dissolution experiments in which oxygen was the solute [(10), Chapter 13.4.8] and illustrate that turbulent flow patterns may occur within the bulk medium, namely for flow close to the liquid surface of the dissolution medium. The numbers are valid provided that the whole liquid surface rotates. According to Levich (9), the onset of turbulent bulk flow under these conditions can then be assumed at Re 1500. 

What is turbulent flow We will use the simple illustration of a free-surface flow given in Figure 5.1 to describe the essential points of the turbulence phenomena. Turbulent open-channel flow can be described with a temporal mean velocity profile that reaches a steady value with turbulent eddies superimposed on it. These turbulent eddies are continually moving about in three dimensions, restricted only by the boundaries of the flow, such that they are eliminated from the temporal mean velocity profile, u in Figure 5.1. It is this temporal mean velocity profile that is normally sketched in turbulent flows. 

Figure 3.3. Various features of diffusion and convection associated with crystal growth in solution (a) in a beaker and (b) around a crystal. The crystal is denoted by the shaded area. Shown are the diffusion boundary layer (db) the bulk diffusion (D) the convection due to thermal or gravity difference (T) Marangoni convection (M) buoyancy-driven convection (B) laminar flow, turbulent flow (F) Berg effect (be) smooth interface (S) rough interface (R) growth unit (g). The attachment and detachment of the solute (solid line) and the solvent (open line) are illustrated in (b).

The complexities of turbulent flow are outside the province of this book. However, there are two further properties of laminar convective flow that are relevant to understanding the electrochemical situation. The first is easily understood by considering an excellent illustration of it—river flow. It is a matter of common observation that rivers (which flow convectively as a result of being pushed by gravity) move at maximum rale in the middle. At the river bank there is hardly any flow at all. This observation can be transferred to the flow of liquid through a pipe. The flow reaches a maximum velocity in the center. The liquid actually in contact with the walls of the pipe does not flow at all. The stationary layer is a few micrometers in thickness, about 1 % of the thickness of the diffusion layer set up by natural convection in an unstirred solution when an electrode reaction in steady state is occurring. 

In Chapter 21 on box models no distinction was made between a compound being present as a dissolved species or sorbed to solid surfaces (e.g., suspended particles, sediment-water interface). In Boxes 18.5 and 19.1, and also in Illustrative Example 19.6, we learned that several of the transport and transformation processes may selectively act on either the dissolved or the sorbed form of a constituent. For instance, a molecule sitting on the surface of a sedimentaiy particle at the lake bottom does not feel the effect of turbulent flow in the lake water, while the dissolved chemical species is passively moved around by the currents. In contrast, a molecule sorbed to a suspended particle (e.g., an algal cell) can sink through the water column because of gravity, unlike its dissolved counterpart. 

Inertial forces of the fluid increase with density and the square of velocity (pv2) while viscous forces decrease with increasing diameter of tube (nv/d) and increase with viscosity and velocity. High Reynolds numbers (Re>4000) result in turbulent flow with low Reynolds number (Re<2000) the flow is laminar. Laminar flow results from formation of layers of fluid with different velocities after a certain flow distance, as illustrated in Figure 2.10A. Flow at the walls is zero and increases approaching the center of the tubes. The laminar flow pattern results from layers of mobile phase with different velocities travelling parallel to each other. The maximum flow at the center is twice the average flow velocity of the fluid. Molecules in the fluid can exchange between fluid layers by molecular diffusion. Most open tubular columns operate under laminar flow conditions. 

FIGURE 5 Schematic illustration of Reynolds convention v = v + v for turbulent flow. 

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