Fluid-flow curves

Pseudoplastic fluids have no yield stress threshold and in these fluids the ratio of shear stress to the rate of shear generally falls continuously and rapidly with increase in the shear rate. Very low and very high shear regions are the exceptions, where the flow curve is almost horizontal (Figure 1.1). 

Thixotropy and Other Time Effects. In addition to the nonideal behavior described, many fluids exhibit time-dependent effects. Some fluids increase in viscosity (rheopexy) or decrease in viscosity (thixotropy) with time when sheared at a constant shear rate. These effects can occur in fluids with or without yield values. Rheopexy is a rare phenomenon, but thixotropic fluids are common. Examples of thixotropic materials are starch pastes, gelatin, mayoimaise, drilling muds, and latex paints. The thixotropic effect is shown in Figure 5, where the curves are for a specimen exposed first to increasing and then to decreasing shear rates. Because of the decrease in viscosity with time as weU as shear rate, the up-and-down flow curves do not superimpose. Instead, they form a hysteresis loop, often called a thixotropic loop. Because flow curves for thixotropic or rheopectic Hquids depend on the shear history of the sample, different curves for the same material can be obtained, depending on the experimental procedure. 

Figure 7-11. Power curve for the standard tank configuration. (Source Holland, F. A. and Bragg, R. Fluid Flow for Chemical Engineers, 2nd ed., Edward Arnold, 1995.)...

By integrating Eq. (13.35) step by step in time, the particle trajectory of the particle may be obtained. In the integration, the interaction between the particle and the wall may be approximated as being fully elastic however, when the particle hits the sidewall of the cyclone, the particle may be treated as being collected and the computation for the particle may terminated in order to save the computational time that may be required to track the particle to the bottom of the cyclone. If the particle trajectories for a range of particle diameters at different rates of fluid flow through the cyclone are determined, then the particle efficiency curve and the cut-off particle diameter of the cyclone may be obtained. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

The measurements are carried out at preselected shear rates. The resulting curves are plotted in form of flow-curves t (D) or viscosity-curves ti (D) and give information about the viscosity of a substance at certain shear rates and their rheological character dividing the substances in Newtonian and Non-Newtonian fluids. 

The static pressure difference will be independent of the fluid flow-rate. The dynamic loss will increase as the flow-rate is increased. It will be roughly proportional to the flow-rate squared, see equation 5.3. The system curve, or operating line, is a plot of the total pressure head versus the liquid flow-rate. The operating point of a centrifugal pump can be found by plotting the system curve on the pump s characteristic curve, see Example 5.3. 

An interwell chemical tracer study established fluid flow patterns within the pilot. Decline curve analysis showed that TFSA injection recovered more than 8150 +... 

The second category, time-dependent behaviour, is common but difficult to deal with. The best known type is the thixotropic fluid, the characteristic of which is that when sheared at a constant rate (or at a constant shear stress) the apparent viscosity decreases with the duration of shearing. Figure 1.21 shows the type of flow curve that is found. The apparent viscosity continues to fall during shearing so that if measurements are made for a series of increasing shear rates and then the series is reversed, a hysteresis loop is observed. On repeating the measurements, similar behaviour is seen but at lower values of shear stress because the apparent viscosity continues to fall. 

Relative permeability is the reduction of mobility between more than one fluid flowing through a porous media, and is the ratio of the effective permeability of a fluid at a fixed saturation to the intrinsic permeability. Relative permeability varies from zero to 1 and can be represented as a function of saturation (Figure 5.8). Neither water nor oil is effectively mobile until the ST is in the range of 20 to 30% or 5 to 10%, respectively, and, even then, the relative permeability of the lesser component is approximately 2%. Oil accumulation below this range is for all practical purposes immobile (and thus not recoverable). Where the curves cross (i.e., at an Sm of 56% and 1 - Sm of 44%), the relative permeability is the same for both fluids. With increasing saturation, water flows more easily relative to oil. As 1 - SI0 approaches 10%, the oil becomes immobile, allowing only water to flow. 

So far we have considered just the traditional steadystate design aspects of this fluid flow system. Now let us think about what would happen dynamically if we changed Fq. How wiU fcy) and Fyj vary with time Obviously F eventually has to end up at the new value of f,. We can easily determine from the steadystate design curve of Fig. 1.2 where h will go at the new steadystate. But what paths will h, ) and F(,f take to get to their new steadystates ... 

With the development of modern computation techniques, more and more numerical simulations occur in the literature to predict the velocity profiles, pressure distribution, and the temperature distribution inside the extruder. Rotem and Shinnar [31] obtained numerical solutions for one-dimensional isothermal power law fluid flows. Griffith [25], Zamodits and Pearson [32], and Fenner [26] derived numerical solutions for two-dimensional fully developed, nonisothermal, and non-Newtonian flow in an infinitely wide rectangular screw channel. Karwe and Jaluria [33] completed a numerical solution for non-Newtonian fluids in a curved channel. The characteristic curves of the screw and residence time distributions were obtained. 

Figure 11.6 The exit age distribution curve E for fluid flowing through a vessel also called the residence time distribution, or RTD.

An extensive class of non-Newtonian fluids is formed by pseudoplastic fluids whose flow curves obey the so-called power law ... 

These liquids are known as Ostwald-de Waele Fluids. Figure 8 depicts a typical course of such a flow curve. 

Figure 8 Typical course of the flow curve of an Ostwald-de Waele fluid obeying the so-called power law behavior.

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