Newtonian fluid phase

What is the slope of each of the following log-log graphs, where the first quantity appears on the vertical axis and the second quantity appears on the horizontal axis In all cases, the properties are evaluated in the incompressible Newtonian fluid phase. 

In this chapter, we focus on our efforts to model dispersed multiphase flows in which a discrete phase (consisting of solid particles, gas bubbles, or liquid droplets) is moving through, or is moved by, a continuous Newtonian fluid phase. Such flows appear frequendy in process equipment in the chemical, metallurgical, pharmaceutical, and food industries. Examples include fluidized bed reactors, spouted bed reactors, pneumatic conveyors, bubble column reactors, slurry reactors, and spray driers. Figure 1 shows a schematic overview of typical dispersed multiphase systems. 

In order to select the pipe size, the pressure loss is calculated and velocity limitations are estabHshed. The most important equations for calculation of pressure drop for single-phase (Hquid or vapor) Newtonian fluids (viscosity independent of the rate of shear) are those for the deterrnination of the Reynolds number, and the head loss, (16—18). 

Most non-Newtonian fluids are either two-phase systems, or single phase systems in which large molecules are in solution in a liquid which itself may be Newtonian or non-Newtonian, or are in the form of a melt. 

The Martinelli correlations for void fraction and pressure drop are used because of their simplicity and wide range of applicability. France and Stein (6 ) discuss the method by which the Martinelli gradient for two-phase flow can be incorporated into a choked flow model. Because the Martinelli equation balances frictional shear stresses cuid pressure drop, it is important to provide a good viscosity model, especially for high viscosity and non-Newtonian fluids. 

The principles of conservation of mass and momentum must be applied to each phase to determine the pressure drop and holdup in two phase systems. The differential equations used to model these principles have been solved only for laminar flows of incompressible, Newtonian fluids, with constant holdups. For this case, the momentum equations become... 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

Given the uncertainty in the form of the stress tensors, many authors have adopted a form analogous to single phase Newtonian fluid relating the stress terms to the pressure and viscosity of the fluid and particle phase, respectively. 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

Though most of the industrial fluids show non-Newtonian characteristics, the drop formation studies in them have not been reported. The results will very strongly depend on whether the non-Newtonian fluid forms the dispersed or continuous phase. 

Rheologically, LEH behaves as a non-Newtonian fluid and the viscosity of the product depends upon the particle density and presence of solutes and macromolecules in the dispersion phase of LEH (37,151). 

Perhaps the most important and striking features of high internal phase emulsions are their rheological properties. Their viscosities are high, relative to the bulk liquid phases, and they are characterised by a yield stress, which is the shear stress required to induce flow. At stress values below the yield stress, HIPEs behave as viscoelastic solids above the yield stress, they are shear-thinning liquids, i.e. the viscosity varies inversely with shear rate. In other words, HIPEs (and high gas-fraction foams) behave as non-Newtonian fluids. 

Ward and DallaValle (W2) studied the two-phase cocurrent flow of air and four non-Newtonian fluids in three horizontal pipes (I.D. 0.82 to 1.60 in.). Flow rates were such that both phases were in turbulent motion. The flow-behavior index of the liquids used was varied from 0.31 to 1.00,... 

Unsteady state phenomena have been stated to be of greater importance for non-Newtonian than for Newtonian materials and therefore warrant experimental investigation. The prediction of pressure drop for two-phase flow of a gas and a non-Newtonian fluid seems to be in a well-perfected state but requires extension to situations in which the liquid flow is laminar. Apparently no information is yet available on the problems of mixing, entrainment, and other similar relationships which are of importance if such contactors are to be designed for chemical rather than mechanical purposes. 

The characterisation of the viscosity is difficult for non-Newtonian fluids because the viscosity changes as a result of the flow process, which increases the shear rate. This is further complicated for two-phase fluids because the presence of bubbles will also affect the viscosity. The simpler methods to obtain G for high viscosity fluids make the simplifying assumptions that the fluid viscosity is equal to the liquid viscosity and that the fluid is Newtonian. 

It is not known whether high-pressure fluids are Newtonian fluids that behave according to the laws given by Eqns. (3.4-1), (3.4-2), and (3.4-3). With regards to diffusion problems, for example, the Fickian nature of diffusion may be rather the exception than the rule. The diffusivity often depends on solute concentration, not only in extraction with a supercritical gas [1] but also in ordinary low-pressure diffusion in the gas phase and in diffusion in a liquid in multicomponent systems and in porous media. 

FLOW. The rate at which zones migrate down the column is dependent upon equilibrium conditions and mobile phase velocity on the other hand, how the zone broadens depends upon flow conditions in the column, longitudinal diffusion, and the rate of mass transfer. Since there are various types of columns used in gas chromatography, namely, open tubular columns, support coated open tubular columns, packed capillary columns, and analytical packed columns, we should look at the conditions of flow in a gas chromatographic column. Our discussion of flow will be restricted to Newtonian fluids, that is, those in which the viscosity remains constant at a given temperature. 

For simple fluids, also known as Newtonian fluids, it is easy to predict the ease with which they will be poured, pumped, or mixed in either an industrial or end-use situation. This is because the shear viscosity or resistance to flow is a constant at any given temperature and pressure. The fluids that fall into this category are few and far between, because they are of necessity simple in structure. Examples are water, oils, and sugar solutions (e.g., honey unit hi.3), which have no dispersed phases and no molecular interactions. All other fluids are by definition non-Newtonian, so the viscosity is a variable, not a constant. Non-Newtonian fluids are of great interest as they encompass almost all fluids of industrial value. In the food industry, even natural products such as milk or polysaccharide solutions are non-Newtonian. 

Capillary forces in mixed fluid phase conditions are inversely proportional to the curvature of the interface. Therefore, menisci introduce elasticity to the mixed fluid, and mixtures of two Newtonian fluids exhibit global Maxwellian response. For more details see Alvarellos [1], his behavior is experimentally demonstrated with a capillary tube partially filled with a water droplet. The tube is tilted at an angle (3 smaller than the critical angle that causes unstable displacement. Then, a harmonic excitation is applied to the tube in the axial direction. For each frequency, the amplitude of the vibration is increased until the water droplet becomes unstable and flows in the capillary. Data in Figure 3 show a minimum required tube velocity between 40 and 50 Hz. This behavior indicates resonance of the visco-elastic system. The ratio of the relaxation time and characteristic time for pure viscous effect is larger than 11.64. 

For the Eulerian continuum modeling discussed in this chapter, it is assumed that the basic form of the Navier-Stokes equation can be applied to all phases. For some dense suspension cases, the particle phase may behave as a non-Newtonian fluid. In these cases, a simple extension of the Navier-Stokes equation may not be appropriate. 

Volume of the dispersed phase. Below a concentration of 50% aqueous phase and with 2-5% non-ionic emulsifier, emulsions behave as Newtonian fluids above 50% aqueous phase, emulsions become increasingly non-Newtonian (i.e., become shear rate-dependent and develop a yield value). 

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