Turbulent flow maximum velocity

In turbulent flow, properties such as the pressure and velocity fluctuate rapidly at each location, as do the temperature and solute concentration in flows with heat and mass transfer. By tracking patches of dye distributed across the diameter of the tube, it is possible to demonstrate that the liquid s velocity (the time-averaged value in the case of turbulent flow) varies across the diameter of the tube. In both laminar and turbulent flow the velocity is zero at the wall and has a maximum value at the centre-line. For laminar flow the velocity profile is a parabola but for turbulent flow the profile is much flatter over most of the diameter. 

The usual velocity distributions in a steady flow of liquid through a tube are shown in Figure 2.4. In either laminar or turbulent flow, the velocity at the tube wall is zero but is maximum at the tube axis. The ratio of the average velocity to the maximum velocity is 0.5 for laminar flow and approximately 0.8... 

Equation (5-62) predicts the point of maximiim velocity for laminar flow in annuli and is only an approximate equation for turbulent flow. Brighton and Jones [Am. Soc. Mech. Eng. Basic Eng., 86, 835 (1964)] and Macagno and McDoiigall [Am. Inst. Chem. Eng. J., 12, 437 (1966)] give more accurate equations for predicting the point of maximum velocity for turbulent flow. 

It is seen that it is important to be able to determine the velocity profile so that the flowrate can be calculated, and this is done in Chapter 3. For streamline flow in a pipe the mean velocity is 0.5 times the maximum stream velocity which occurs at the axis. For turbulent flow, the profile is flatter and the ratio of the mean velocity to the maximum... 

The flow of jets becomes turbulent at much lower Re numbers than channel flows. Calculating the stress from the mean velocity profiles does not reflect the true situation in turbulent flow. As in the case in most bioreactors, the maximum turbulent stress is determined by the turbulence, which can be calculated using Eqs. (2)-(4). It occurs in free jets after the nozzle, at the edge of the mixing zone. The following is generally valid ... 

For a single-phase turbulent flow the ratio of the maximum to the average flow velocity is approximately 1.2, and the value of Co may also be close to 1.2 for a bubbly flow. Zuber and Findlay (1965) pointed out that, as the mixture velocity increases, the value of the exponent increases and flatter profiles result. 

For the steady turbulent flow of a Newtonian fluid at high values of Re in a pipe of circular cross section, the mean velocity u is related to the maximum velocity vmix by the equation... 

Plot laminar and turbulent velocity profiles for steady state flow in a cylindrical pipe for a maximum velocity gm = 5 m/s using the radial positions 2r d - 0, 0.2, 0.4, 0.6 and 0.8. 

That is, the component of the slug velocity due to liquid upflow is very nearly the same as the maximum liquid velocity at the tube center for turbulent flow. 

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. 

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 2.10. Flow profiles in tubes and packed columns. (A) Laminar flow. r = tube radius, Vx =stream path velocity at radial position r. V = maximum flow velocity at tube center. (B) Turbulent flow.

The kinetic-energy terms of the various energy balances developed h include the velocity u, which is the bulk-mean velocity as defined by the equati u = m/pA Fluids flowing in pipes exhibit a velocity profile, as shown in Fi 7.1, which rises from zero at the wall (the no-slip condition) to a maximum the center of the pipe. The kinetic energy of a fluid in a pipe depends on actual velocity profile. For the case of laminar flow, the velocity profile parabolic, and integration across the pipe shows that the kinetic-ertergy should properly be u2. In fully developed turbulent flow, the more common in practice, the velocity across the major portion of the pipe is not far fro... 

Equation 9.11 is usually referred to as Poiseuille s law and sometimes as the Hagen-Poiseuille law. It assumes that the fluid in the cylinder moves in layers, or laminae, with each layer gliding over the adjacent one (Fig. 9-14). Such laminar movement occurs only if the flow is slow enough to meet a criterion deduced by Osborne Reynolds in 1883. Specifically, the Reynolds number Re, which equals vd/v (Eq. 7.19), must be less than 2000 (the mean velocity of fluid movement v equals JV, d is the cylinder diameter, and v is the kinematic viscosity). Otherwise, a transition to turbulent flow occurs, and Equation 9.11 is no longer valid. Due to frictional interactions, the fluid in Poiseuille (laminar) flow is stationary at the wall of the cylinder (Fig. 9-14). The speed of solution flow increases in a parabolic fashion to a maximum value in the center of the tube, where it is twice the average speed, Jv. Thus the flows in Equation 9.11 are actually the mean flows averaged over the entire cross section of cylinders of radius r (Fig. 9-14). 

The flow of gas in a vacuum system is of all three types successively when the pressure is being reduced from atmospheric to 1 tort or less. Turbulent flow occurs for a short time at the beginning of the evacuation when the gas velocity is high. Laminar flow takes place when the mean free path is small compared with the tube diameter under these conditions the gases near the tube wall are almost stationary, and those near the centre of the tube have the maximum velocity. As the pressure is reduced to about 1 toiT, molecular flow is first established (Fig. 10.1). 

An interesting point is the dependence of the turbulent Nu numbers on the boundary condition. For laminar flow it was shown that the dependence of Nu on the boundary condition rapidly fades away when the relative pitch is increased The difference in Nu between the two limiting boundary conditions is less than 30% for relative pitches larger than 1.1. Because of the flatter velocity profile in turbulent flow, the dependence on the boundary conditions is weak in turbulent flow, except for small relative pitches. An estimate of the maximum influence of the boundary condition on the turbulent Nu number can be obtained from the respective values for laminar flow. 

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