Velocity turbulent conditions

Pressure drop in catalyst beds is governed by the same principles as in any flow system. Consequently, at very low flow, pressure drop is directly proportional to velocity, and at very high flow, to the square of velocity. These conditions correspond to the laminar and turbulent regimes of the flow. 

If gas flows under turbulent conditions from a reservoir at a pressure P, through a horizontal nozzle, the velocity of flow M2. at the pressure Pi is given by ... 

In one-point models for turbulent mixing, extensive use of conditional statistics is made when developing simplified models. For example, in the PDF transport equation for /++ x, r), the expected value of the velocity fluctuations conditioned on the scalars appears and is defined by... 

Imagine two parallel plates of area A between which is sandwiched a liquid of viscosity r). If a force F parallel to the x direction is applied to one of these plates, it will move in the x direction as shown in Figure 4.1. Our concern is the description of the velocity of the fluid enclosed between the two plates. In order to do this, it is convenient to visualize the fluid as consisting of a set of layers stacked parallel to the boundary plates. At the boundaries, those layers in contact with the plates are assumed to possess the same velocities as the plates themselves that is, v = 0 at the lower plate and equals the velocity of the moving plate at that surface. This is the nonslip condition that we described in Chapter 2, Section 2.3. Intervening layers have intermediate velocities. This condition is known as laminar flow and is limited to low velocities. At higher velocities, turbulence sets in, but we do not worry about this complication. 

The third factors to control and keep constant are the gas pressure and superficial gas velocity. This probably will involve gas recirculation with either a small compressor, or through a hollow shaft or some other pumping device. As seen before, the bubble diameter, the mass transfer area, the gas hold-up, and the terminal bubble-rise velocity, all depend on the superficial velocity of the gas and the power input per unit volume. When these are kept constant, the various mass transfer resistances in the pilot plant and in the large unit will be the same, hence the global rate will be conserved. The last factor is the input power to the agitator. As required for mass transfer, the scale-up must be made on the basis of constant power input per unit volume. If turbulent conditions and geometrical similarity prevail, this rule imposes the following relationship ... 

Practical flotation processes, however, take place under conditions of turbulence. Turbulent flow, as opposed to laminar flow (see Section 6.1), is characterized by rapid, almost random, fluctuations in flow velocity. Turbulence helps keep the solid particles suspended, helps disperse the injected air phase into bubbles, and helps induce particle-bubble collisions and attachments. With regard to the role of turbulence in mineral flotation, a rule of thumb for suspension stability is the one-second criterion which states that the particles in a suspension are sufficiently well dispersed for flotation if individual particles do not remain settled at the bottom of the flotation vessel for longer than one second [53]. 

The flow cell translates time into distance and the combination of the three and varying the flow rates gave a range of observations from 0 to 30 s. SHG measurements of the static aqueous/dodecane interface were made at each port before and after the flow experiment to calibrate the observations from each port For a laminar (non-turbulent) flow, the two flow rates should be in the inverse ratio of the fluid viscosities this ratio for dodecane on water is 0.65 at 25°C, very close to the observed flow rate ratio of 0.67. The bulk flow rates for each liquid were measured by collecting the volume of liquid flowing in a known time. Since the cell operates under non-turbulent conditions, the velocity of each layer at the interface must be the same, but the average velocities of the two layers are different. Ideally a model of the flow conditions inside the cell would be used to accurately determine the velocity of the interface. Since this was not... 

The model based on terminal and slip velocity approach is rather tenuous. It breaks down as the density difference between liquid and solid approaches zero. Under highly turbulent conditions, an accurate estimation of slip velocity is rather difficult, and there is disagreement on whether or not the relative velocity between the solid and liquid alone is enough to obtain an accurate estimate of the mass-transfer coefficient. 

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