Stream-tube model

The river can be divided into a series of longitudinal planes with no signihcant interaction, such that v = 0 and Sy = 0. (This is the assumption of the stream-tube computational models.)... 

Figure 27. Free surface computation for LDPE and LLDPE with stream-tube analysis and m Wagner model.

The ratio FWS/q can be reformulated to be better suited for use in other models such as e.g. the Advection- Dispersion- Matrix diffusion model. In that model, which can be formulated as a stream tube model in a porous medium the fractures are not modelled explicitly. Instead one may consider the rock as containing a number of fractures per m of rock. Each fracture has a FWS twice its size because both sides of the fracture are in contact with the water. An entity a can be defined that states the magnitude of FWS per m of rock. A given flowrate of water q mVs flowing through a cross section of rock Asr over a distance L will then be in contact with a FWS that is or A L That is the same as WL in equation (2). Thus a direct translation between the models is possible. 

Thiele, M. Blunt, M. Orr Jr., F. (1995A) Modeling flow in heterogeneous media using stream-tubes, I - Miscible and immiscible displacements. In Situ 19(3), 299-339. 

Wang, H.G. Wu, F.L. 2011. Method of calculation of a methane concentration field in gob areas with a knovra velodty field based on the model of stream tubes. Mining Science and Technology (China) 21(201 l)i277 280. 

A special kind of 2-D case is represented by the stream-tube models, used for... 

It has already been noted that transverse variation of velocity is a major factor in the mixing process. Therefore, models which neglect this variation will require an inflated coefficient such as Di,. This article will therefore focus on the so-called stream-tube models, which do include depth and velocity variations across the channel. However, solutions which do not consider such variations will also be presented, since sometimes data may not exist to enable use of the more complicated model. 

As discussed in Section II,G, stream-tube models generally allow one to incorporate lateral variation of depth and velocity into the model. Yotsukura and Cobb W6) showed that the solutions of Eq. (20) for = 1 are not too sensitive to variations in the specific distribution of h Uj Dy with respect to as long as its averaged value over the total river flow Q remains the same. This finding, coupled with their further experience, indicates that one can, for many natural channels, obtain a reasonable approximation by defining a constant diffusion factor as ... 

Any of the stream-tube solutions require a knowledge of the distribution of flow across the stream. This enables one to convert the results from the cumulative discharge system to a physical location in the stream cross section. The best way to establish this relationship is by velocity measurements in the section. However, this data is expensive to acquire and is usually not available. No really reliable analytical model exists today with which to predict the distribution in all cases. However, some empirical models have been presented. Slum (85) has used Manning s equation as a basis for defining the following relationship ... 

Several finite difference models have been presented. From the transport standpoint, the best two appear to be the stream-tube models by Holly 43) and... 

In rivers, it is preferable to use a 2-D model after vertical mixing has been achieved. The Holly (43) numerical model or the Yotsukura and Cobb (106) analytical model both represent stream-tube solutions. They incorporate channel geometry and transverse velocities. A 3-D model such as those by Prakash (73) and Benedict (J) are recommended for the 3-D phase prior to full vertical mixing. 

In principle, these models simulate the polymer-augmented water-flood performance in each streamtube at constant pressure drop. Fig. 5.70 illustrates the fractional-flow, saturation, and production profiles used in each streamtube. Performance of the pattern is determined by combining the displacement performance of each stream-tube at the same point in time. A model based on streamtube concepts, develops by the U.S. DOE, is available to the public and documented in a report. ... 

Sampling from pneumatic conveyors parallels gas sampling. The exception is that soflds loadings can be as high as 50 kg of soHds per kg of gas. Commercially available samplers extract particles directly from a transport line. Fixed position samplers are mounted directly on the pneumatic conveyor pipe. Devices are available which extract samples from the product stream by the projection of a sample tube iato the flow. Particles impact on the tube and fill the open cavity. The tube is then withdrawn, and an internal screw discharges the collected material (20). In another model, the RX Sampler (manufactured by Gustafson) (29), samples are withdrawn usiag compressed air. 

At the first level of detail, it is not necessary to know the internal parameters for all the units, since what is desired is just the overall performance. For example, in a heat exchanger design, it suffices to know the heat duty, the total area, and the temperatures of the output streams the details such as the percentage baffle cut, tube layout, or baffle spacing can be specified later when the details of the proposed plant are better defined. It is important to realize the level of detail modeled by a commercial computer program. For example, a chemical reactor could be modeled as an equilibrium reactor, in which the input stream is brought to a new temperature and pressure and the... 

Dispersion models, as just stated, are useful mainly to represent flow in empty tubes and packed beds, which is much closer to the ideal case of plug flow than to the opposite extreme of backmix flow. In empty tubes, the mixing is caused by molecular diffusion and turbulent diffusion, superposed on the velocity-profile effect. In packed beds, mixing is caused both by splitting of the fluid streams as they flow around the particles and by the variations in velocity across the bed. 

The continuous-stream flow-injection system (Figure 2) consisted of a gravity-feed electrolyte reservoir, a sample injection valve (Rheodyne, Model 50) fitted with a 30 /xL-sample loop, and a flow-through electrochemical detector cell. The channel diameter of the Teflon tubing for the stream was 0.8 mm. The tubing length from injector to detector was 10 cm. 

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