Channel flows

Here f denotes the fraction of molecules diffusely scattered at the surface and I is the mean free path. If distance is measured on a scale whose unit is comparable with the dimensions of the flow channel and is some suitable characteristic fluid velocity, such as the center-line velocity, then dv/dx v and f <<1. Provided a significant proportion of incident molecules are scattered diffusely at the wall, so that f is not too small, it then follows from (4.8) that G l, and hence from (4.7) that V v° at the wall. Consequently a good approximation to the correct boundary condition is obtained by setting v = 0 at the wall. ... 

Now suppose, as before, that distance is measured on a scale whose unit is comparable with the dimensions of che flow channel, and let v° be some characteristic mass mean velocity in the system, like introduced above... 

The idea of ultrafiltration has been extended ia recent years to the filtration of particles ia the micrometer and submicrometer range ia porous pipes, usiag the same cross-flow principle. In order to prevent blocking, thicker flow channels are necessary, almost exclusively ia the form of tubes. The process is often called cross-flow microfiltration but the term cross-flow filtration is used here. 

Oil reservoirs are layers of porous sandstone or carbonate rock, usually sedimentary. Impermeable rock layers, usually shales, and faults trap the oil in the reservoir. The oil exists in microscopic pores in rock. Various gases and water also occupy rock pores and are often in contact with the oil. These pores are intercoimected with a compHcated network of microscopic flow channels. The weight of ovedaying rock layers places these duids under pressure. When a well penetrates the rock formation, this pressure drives the duids into the wellbore. The dow channel size, wettabiUty of dow channel rock surfaces, oil viscosity, and other properties of the cmde oil determine the rate of this primary oil production. 

Wettabihty is defined as the tendency of one fluid to spread on or adhere to a soHd surface (rock) in the presence of other immiscible fluids (5). As many as 50% of all sandstone reservoirs and 80% of all carbonate reservoirs are oil-wet (10). Strongly water-wet reservoirs are quite rare (11). Rock wettabihty can affect fluid injection rates, flow patterns of fluids within the reservoir, and oil displacement efficiency (11). Rock wettabihty can strongly affect its relative permeabihty to water and oil (5,12). When rock is water-wet, water occupies most of the small flow channels and is in contact with most of the rock surfaces as a film. Cmde oil does the same in oil-wet rock. Alteration of rock wettabihty by adsorption of polar materials, such as surfactants and corrosion inhibitors, or by the deposition of polar cmde oil components (13), can strongly alter the behavior of the rock (12). 

When water is injected into a water-wet reservoir, oil is displaced ahead of the injected fluid. Injection water preferentially invades the small- and medium-sized flow channels or pores. As the water front passes, unrecovered oil is left in the form of spherical, uncoimected droplets in the center of pores or globules of oil extending through intercoimected rock pores. In both cases, the oil is completely surrounded by water and is immobile. There is htde oil production after injection water breakthrough at the production well (5). 

In an od-wet rock, water resides in the larger pores, oil exists in the smaller pores or as a film on flow channel surfaces. Injected water preferentially flows through the larger pores and only slowly invades the smaller flow channels resulting in a higher produced water oil ratio and a lower oil production rate than in the water-wet case. 

Cylinder Discharger Feeder Flow channel Hopper... 

A funnel flow bin typically exhibits a first-in/last-out type of flow sequence. If the material has sufficient cohesive strength, it may bridge over the outlet. Also, if the narrow flow channel empties out, a stable rathole may form. This stable rathole decreases the bin s five or usable capacity, causes materials to cake or spoil, and/or enhances segregation problems. Collapsing ratholes may impose loads on the stmcture that it was not designed to withstand. 

Many appHcations use screws with constant pitch to feed material from a slotted opening. The configuration shown in Figure 9a shows a constant pitch and constant diameter causing a preferential flow channel to form at the back (over the first flight) of the screw. This type of flow destroys the mass flow pattern and potentially allows some or all of the problems discussed about fiinnel flow. 

Fig. 9. Preferential flow channel caused by (a) a constant pitch screw feeder and (b) a belt feeder.

An improperly designed interface to the belt can cause soHds compaction, abrasive wear of the belt, and excessive power requited to move the belt. The preferential flow channel shown in Figure 9b withdraws material from one end of the outlet. Depending on the gate opening, this could be at the back... 

A potential problem for rotary valve usage is that they tend to pull material preferentially from the upside of the valve, which can affect the mass flow pattern. Another problem is that once soHd drops from the vane, the air or gas that replaces it is often pumped back up into the bin. In addition, air can leak around the valve rotor. Such air flows can decrease the soflds flow rates and/or cause flooding problems. A vertical section shown in Figure 13 can alleviate the preferential flow problem because the flow channel expands in this area, usually opening up to the full outlet. To rectify the countercurrent air flow problem, a vent line helps to take the air away to a dust collector or at least back into the top of the bin. 

The area of influence of a vibrating discharger is limited to a cylinder, the diameter of which is roughly equal to the top diameter of the discharger. Hence, if a vibrating discharger is mounted onto a conical hopper section, flow is confined predominantly to a central flow channel located directly above the discharger. This is tme unless the slope and smoothness of the static cone meet requirements for mass flow, or the diameter of the flow channel exceeds the critical rathole diameter for the material. 

Steam can also be injected into one or more weUs, with production coming from other weUs (steam drive). This technique is effective in heavy oil formations but has found Httle success during appHcation to tar sand deposits because of the difficulty in connecting injection and production weUs. However, once the flow path has been heated, the steam pressure is cycled, alternately moving steam up into the oil zone, then allowing oil to drain down into the heated flow channel to be swept to the production weUs. 

Plate and Frame. Plate-and-frame systems consist of plates (Fig. 15) each with a membrane on both sides. The plates have a frame around their perimeter which forms flow channels ca 1 mm wide between the plates when they are stacked. The stack is clamped between two end plates, sealing the frames together. 

Fig. 22. (a) Flow channel arrangement (b) flow channel triangular cross section where for angles of 90°, D q = ... 

Counter-current flow. Structured packings. Gauze-type with triangular flow channels, Bravo, Rocha, and Fair correlation... 

Perforated Plates and Screens A nonuniform velocity profile in turbulent flow through channels or process equipment can be smoothed out to any desired degree by adding sufficient uniform resistance, such as perforated plates or screens across the flow channel, as shown in Fig. 6-38. Stoker Ind. Eng. Chem., 38, 622-624 [1946]) provides the following equation for the effect of a uniform resistance on velocity profile ... 

In design of separating chambers, static vessels or continuous-flow tanks may be used. Care must be taken to protect the flow from turbulence, which coiild cause back mixing of partially separated fluids or which could cany unseparated hquids rapidly to the separated-hquid outlet. Vertical baffles to protect rising biibbles from flow currents are sometimes employed. Unseparated fluids should be distributed to the separating region as uniformly and with as little velocity as possible. When the bubble rise velocity is quite low, shallow tanks or flow channels should be used to minimize the residence time required. 

Often there are situations in which mass-flow bins cannot be installed for reasons such as space limitations and capacity requirements. Also, sometimes the product to be stored has an FF flow function that lies below the flow factorj, bridging takes place, and unassisted mass flow is not possible. To handle these situations, a number of flow assisters are available, the most desirable of which use a feeder and a short mass-flow hopper to enlarge the flow channel of a funnel-flow bin. The choice of feeder or flow assister should always be made as part of the storage-vessel analysis. The resulting systems are then usually as effective as the mass-flow types. 

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