Pressure Drop-Flow Relationship

For greater concentrations of fine particles the suspension is more likely to be non-Newtonian, in which case the viscous properties can probably be adequately described by the power law or Bingham plastic models. The pressure drop-flow relationship for pipe flow under these conditions can be determined by the methods presented in Chapters 6 and 7. 

Figure 6.3 Analysis of the pressure drop - flow relationship for a compressible cake and Equation (6.28) becomes...

Siebes, M., D Argennio, D.Z. 1989. In-vitro study on the pressure drop -flow relationship of a compliant Coronary stenosis. Proc. lEEE-EMBS 11th Ann. Int. Conf, pp. 101-102. 

If the viscosity varies during flow for some reason (decreases with rising temperature or increases as a result of a chemical reaction such as polymerization), the linear Poiseuille P-vs-Q relation is violated and the pressure drop - flow rate curve may become nonmonotonic. This effect in polymerizing reactors can be explained by the fact that the most viscous products of a reaction are swept out of the reactor with increasing flow rate and are replaced. Instead, a reactor is refilled with a fresh reactive mixture of low viscosity. This leads to a decrease of the volume-averaged integral viscosity and therefore the pressure drop decreases. This can be illustrated by the following relationship ... 

Analogously to the well-behaved systems, the flow of the foaming mass should be described by a pressure drop-flow rate relationship (the foam flow characteristic). (In view of the fact that the flow is generated by volume expansion only, the foam flow characteristic, for a given formulation and a given set of geometrical and operational conditions, can be characterized by a (unique) pressure drop-nominal density function.)... 

The main value of data describing turbulent energy requirements is in the computation of pressure drop-flow rate characteristics for installed plant but there are also examples of performance evaluation using energy data -. As with laminar flow characteristics, although different, those for turbulent flow are relatively simple and easily described in terms of the friction factor-Reynolds number relationship used to describe empty tube. 

An expression for the pressure flow through an isosceles triangle was derived by Bird et al. [25] by following the variational principle due to von Helmholtz. Other expressions have been proposed by Kozicki et al. [26, 27], who used a simple geometric parameter method to predict the pressure drop-flow rate relationship in flow channels of arbitrary cross-section. Following Bird s approach, the output-pressure relationship for an isosceles triangle can be written as ... 

Another theoretical approach to cut size prediction that can be classified as another version of the residence-time theory is that of Trawinski. In direct analogy with gravity settling Trawinski used Stokes law, an effective clarification area and an average acceleration in a hydrocyclone to derive an expression for the cut size. The same author also proposed a rather simplistic correlation for the pressure drop-flow rate relationship. 

The correlations for the pressure drop-flow rate relationship change from paper to paper in the publications by Lynch and co-workers depending on the test system. Cyclones of 500 mm diameter were tested with silica and copper ore at concentrations of 15-65% by weight, whilst in another series of experiments , cyclones from 100 to 380 mm in diameter were tested with limestone at concentrations of 15-70% by weight. Finally, another series of tests at concentrations of limestone from 40 to 70% by weight are reported. 

Plitt took the data of Lynch and Rao and added his own, with smaller cyclones up to 150 mm diameter tested with silica flour, and used the data, 297 individual tests in total, to derive by regression analysis yet another correlation for the pressure drop-flow rate relationship, not dissimilar to the above equation. 

Note that for n = 1 and k = /i, Equations (4.7) and (4.8) reduce to the familiar Hagen-Poiseuille equation which describes the pressure drop-velocity relationship for the laminar flow of a Newtonian fluid. 

The gas flowing upward relative to the solids generates a frictional pressure drop. The relationship between the pressure drop per unit length (AP/Lg) and the relative velocity for a particular material is determined by the fluidization curve for that material. Normally, this fluidization curve is generated in a fluidization column where the solids are not flowing. However, the relationship also applies for solids flowing in a standpipe. 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

The relationship between the bore fluid pressure drop, AP and its flow rate is defined by Poiseuike s law ... 

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