Flow function

In order to characterize this bonding tendency, the flow function of a material must be deterrnined. Data on flow function can be generated in a testing laboratory by measuring the cohesive strength of the bulk soHd as a function of consoHdation pressure appHed to it. Such strength is directly related to the abihty of the material to form arches and ratholes in bins and hoppers. 

A material s flow function is usually measured on the same tester as the wall friction angle, although the cell arrangement is somewhat different (Fig. 6). ConsoHdation values are easily controUed, and the cohesive strength of the bulk soHd is determined by measuring interparticle shear stresses while some predeterrnined normal pressure is being appHed. 

If most of the particles are less than ca 0.6 cm in size, flow obstmctions can occur by physical, chemical, or electrical bonds between particles. This cohesiveness is characterized by the bulk material s flow function. The forces acting to overcome a cohesive arch and cause flow are described by a hopper s flow factor, which can be obtained from the design charts (see Fig. 7). The minimum opening size required to prevent a cohesive arch from forming can be calculated from the comparison of the flow factor and flow function. 

Sizing the outlet of a fuimel flow bin involves consideration of both arching and ratholing. Minimum dimensions to overcome both can be calculated from the material s flow function. 

A powder s strength increases significantly with increasing previous compaction. The relationship between the unconfined yield stress/, or a powder s strength, and compaction pressure is described by the powder s flow function FE The flow function is the paramount characterization of powder strength and flow properties, and it is calculated from the yield loci determined from shear cell measurements. [Jenike, Storage and Flow of Solids, Univ. of Utah, Eng. Exp. Station Bulletin, no. 123, November (1964). See also Sec. 21 on storage bins, silos, and hoppers.]... 

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. 

Flow, control of, 265 Flow function on network, 258 Flow, optimal, method for, 261 Fock amplitude for one-particle system, 511 Fock space, 454 amplitudes, 570 description of photons, 569 representation of operators in, 455 Schrodinger equation in, 459 vectors in, 454 Focus, 326 weak, 328... 

If the amount of fluid within a fully saturated permeable medium is known as a function of position, the spatially resolved porosity distribution can be determined. If the medium is saturated with two fluids, and the signal from one can be distinguished, the fluid saturation can be determined. In this section, we will develop a method to determine the amount of a single observed fluid using MRI, and demonstrate the determination of porosity. In Section 4.1.4.3, we will demonstrate the determination of saturation distributions for use in estimating multiphase flow functions. 

The lack of a method to determine the spatial distributions of permeability has severely limited our ability to understand and mathematically describe complex processes within permeable media. Even the degree of variation of intrinsic permeability that might be encountered in naturally occurring permeable media is unknown. Samples with permeability variations will exhibit spatial variations in fluid velocity. Such variations may significantly affect associated physical phenomena, such as biological activity, dispersion and colloidal transport. Spatial variations in the porosity and permeability, if not taken into account, can adversely affect the determination of any associated properties, including multiphase flow functions [16]. 

Relative permeability and capillary pressure functions, collectively called multiphase flow functions, are required to describe the flow of two or more fluid phases through permeable media. These functions primarily depend on fluid saturation, although they also depend on the direction of saturation change, and in the case of relative permeabilities, the capillary number (or ratio of capillary forces to viscous forces). Dynamic experiments are used to determine these properties [32]. 

We present a general approach for estimating relative permeability and capillary pressure functions from displacement experiments. The accuracy with which these functions are estimated will depend on the information content of the measurements, and hence on the experimental design. We determine measures of the accuracy with which the functions are estimated, and use these measures to evaluate different experimental designs. In addition to data measured during conventional displacement experiments, we show that the use of multiple injection rates and saturation distributions measured with MRI can substantially increase the accuracy of estimates of multiphase flow functions. 

The fluid properties and porosity and permeability are determined independently. Boundary and initial conditions are specified for the particular experiment to be considered. With specified multiphase flow functions, the state equations, Eqs. (4.1.28, 4.1.5 and 4.1.6), can be solved for the transient pressure and saturation distributions, p (z,t) and s,(z,t), t= 1, 2. The values for F can then be calculated, which correspond to the measured data Y. 

The solution of these dynamic nonlinear differential equations is considerably more complex than the previous systems considered. In particular, stable solution methods are based on physically realistic multiphase flow functions that have the following properties relative permeability functions are non-negative, monotoni-cally increasing with their respective saturation, and are zero at vanishing saturations, and capillary pressure is monotonically increasing with respect to the saturation of the non-wetting phase. It is necessary that any iterative scheme for estimating the multiphase flow functions retain these characteristics at each step. 

In this case, the conventional experimental design is insufficient to ensure accurate estimates of all three multiphase flow functions [34]. We have considered two different modifications in the experimental design that can provide for improved estimates. These modifications can be incorporated separately, or together, thus representing a total of three different candidate experimental designs. 

We also use a linearized covariance analysis [34, 36] to evaluate the accuracy of estimates and take the measurement errors to be normally distributed with a zero mean and covariance matrix Assuming that the mathematical model is correct and that our selected partitions can represent the true multiphase flow functions, the mean of the error in the estimates is zero and the parameter covariance matrix of the errors in the parameter estimates is ... 

The estimation of flow functions from an actual experiment is reported next. A multi-rate primary drainage experiment was conducted on a Texas Cream limestone sample. Hexadecane was used as the oleic phase and deuterium oxide (D20) was used as the aqueous phase. Protons are imaged, so only the oil phase is observed. The pressure drop data, production data and saturation data are shown in Figures 4.1.11-... 

Our approach was demonstrated by determining multiphase flow functions from displacement experiments. Spatially resolved porosity and permeability distributions can be incorporated to mitigate errors encountered by assuming that the properties are uniform. We developed measures of the accuracy of the estimates and demonstrated improved experimental designs for obtaining more accurate estimates of the flow functions. One of the candidate experimental designs incorporated MRI measurements of saturation distributions conducted during the dynamic experiments. 

These methods provide unprecedented resolution of porosity and permeability for heterogeneous media, and substantially advance the reliability of estimates of multiphase flow functions. This work provides the foundation for further studies of heterogeneous media and important processes that occur within such media, including those with chemical and biological changes that are intimately affected by media structures. 

P. C. Richmond 1988, (Estimating Multiphase Flow Functions From Displacement Experiment), Ph.D. Thesis, Texas A M University. 

Di Carli MF, Prcevski P, Singh TP, Janisse J, Ager J, Muzik O et al. Myocardial blood flow, function, and metabolism in repetitive stunning. J Nucl Med 2000 41 1227-1234... 

Control volume method Finite element method Boundary element method and analytic element method Designed for conditions with fluxes across interfaces of small, well-mixed elements - primarily used in fluid transport Extrapolates parameters between nodes. Predominant in the analysis of solids, and sometimes used in groundwater flow. Functions with Laplace s equation, which describes highly viscous flow, such as in groundwater, and inviscid flow, which occurs far from boundaries. 

By measuring the force required to shear a bed of powder that is under various vertical loads, a relationship describing the cohesive strength of the powder as a function of the consolidating pressure can be developed (4). This relationship, known as a flow function, FF, can be analyzed to determine the minimum outlet diameters for bins to prevent arching and ratholing. 

If gravity discharge is used, the minimum outlet size required to prevent arching is dependent upon the flow pattern that occurs. Regardless of the flow pattern, though, the outlet size is determined with the powder s flow function, which is measured by way of the cohesive strength tests described earlier. 

Figure 8 Sample flow function (FF) and flow factor (ff), showing/crit at their intersection.

The flow function illustrated in Fig. 1 reveals a singular behaviour of the powders which may be associated with the process or with the high percentage of PVC. In fact, even if the curves were obtained by linear regression, a marked dispersion may be seen, particularly for the compaction of the mixture and, curiously, for the wet granulation powder. 

Regarding the flow function values, powders may be classed in two groups ... 

G(< )t) is also a derived function and is given by Figure 9. /c(ct1), the unconfined yield strength of the material, is determined by the flow function (FF) at the actual consolidating pressure al. The consolidation pressure al is a function of the head or height of powder above the outlet of the bin, as given by Janssen s equation ... 

In many respects, similar to the diffusion layer concept, there is that of the hydrodynamic boundary layer, <5H. The concept was due originally to Prandtl [16] and is defined as the region within which all velocity gradients occur. In practice, there has to be a compromise since all flow functions tend to asymptotic limits at infinite distance this is, to some extent, subjective. Thus for the rotating disc electrode, Levich [3] defines 5H as the distance where the radial and tangential velocity components are within 5% of their bulk values, whereas Riddiford [7] takes a figure of 10% (see below). It has been shown that... 

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