Gradient in turbulent flow

Using Bowen s method, develop a general scale-up procedure for predicting the pressure gradients in turbulent flow of this rock slurry. Estimate the pump power required for a flow rate of 0.45 m /s in a 400 mm diameter pipe, 500 m long. The pump has an efficiency of 60%. Take the density of slurry 1250 kg/m. ... 

In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between. 

In turbulent motion, the presence of circulating or eddy currents brings about a much-increased exchange of momentum in all three directions of the stream flow, and these eddies are responsible for the random fluctuations in velocity The high rate of transfer in turbulent flow is accompanied by a much higher shear stress for a given velocity gradient. 

Bubble and drop breakup is mainly due to shearing in turbulent eddies or in velocity gradients close to the walls. Figure 15.11 shows the breakup of a bubble, and Figure 15.12 shows the breakup of a drop in turbulent flow. The mechanism for breakup in these small surface-tension-dominated fluid particles is initially very similar. They are deformed until the aspect ratio is about 3. The turbulent fluctuations in the flow affect the particles, and at some point one end becomes... 

In turbulent flow, there is direct viscous dissipation due to the mean flow this is given by the equivalent of equation 1.98 in terms of the mean values of the shear stress and the velocity gradient. Similarly, the Reynolds stresses do work but this represents the extraction of kinetic energy from the mean flow and its conversion into turbulent kinetic energy. Consequently this is known as the rate of turbulent energy production ... 

Dodge and Metzner (1959) deduced the velocity profile from their measurements of flow rate and pressure gradient for turbulent flow of power law fluids in pipes. For the turbulent core, the appropriate equation is... 

Sleiched278 has indicated that this expression is not valid for pipe flows. In pipe flows, droplet breakup is governed by surface tension forces, velocity fluctuations, pressure fluctuations, and steep velocity gradients. Sevik and Park 279 modified the hypothesis of Kolmogorov, 280 and Hinze, 270 and suggested that resonance may cause droplet breakup in turbulent flows if the characteristic turbulence frequency equals to the lowest or natural frequency mode of an... 

For simple flows where the mean velocity and/or turbulent diffusivity depend only weakly on the spatial location, the Eulerian PDF algorithm described above will perform adequately. However, in many flows of practical interest, there will be strong spatial gradients in turbulence statistics. In order to resolve such gradients, it will be necessary to use local grid refinement. This will result in widely varying values for the cell time scales found from (7.13). The simulation time step found from (7.15) will then be much smaller than the characteristic cell time scales for many of the cells. When the simulation time step is applied in (7.16), one will find that Ni must be made unrealistically large in order to satisfy the constraint that Nf > 1 for all k. 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

What has been said above also holds for solid-fluid mass transfer. The concentration gradients for mass transfer from a solid phase to a fluid in turbulent flow should be analogous to the temperature gradients, such as shown in Figure 2.5. 

There is good experimental evidence (41, 61, 62, 69) that another type of quenching sometimes occurs, entirely in the gas phase and without the influence of walls. If the level of turbulence is too high, or if there arc very strong velocity gradients in the flow, it is possible that the flames may be overwhelmed. This may come about by dilution with cold gas more rapidly than it can be consumed, or the flame may be torn apart so that it cannot travel throughout the mixture. 

In turbulent flow, the velocity profile is nearly a straight line in the core region, and any significant velocity gradients occur in the viscous sublayer. 

In laminar flow, i is the molecular viscosity in that case Eq. (3) is a second-order linear two-dimensional PDE of the Poisson type. In turbulent flow, jl also depends on the velocity gradients (hence on the lateral position), and Eq. (3) then is quasilinear or nonlinear. 

The turbulent mechanism that carries motion, heat, or matter from one part of the fluid to another is absent in laminar flow. The agency of momentum transfer is the shear stress arising from the variations in velocity, that is, the viscosity. Similarly, heat and matter can only be transferred across streamlines on a molecular scale, heat by conduction and matter by diffusion. These mechanisms that are present but less important in turbulent flow are comparatively slow. Velocity, temperature, and concentration gradients are, therefore, much higher than in turbulent flow. 

The velocity at the wall increases more steeply in turbulent flow than in laminar. The shear stress and with that the resistances to flow are larger in turbulent flows than in laminar. Likewise the temperature and concentration gradients at the surface and therefore the heat and mass transfer rates are larger for turbulent flows than in laminar ones. Therefore turbulent flows are to be strived for in heat and mass transfer and for this reason they are present in most technical applications. However better heat and mass transfer has to be paid for by the increased power required for a pump or blower to overcome the resistances to flow. 

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