Rheology Reynolds number

For a Newtonian fluid, the data for pressure drop may be represented on a pipe friction chart as a friction factor = (R/pu2) expressed as a function of Reynolds number Re = (udp/n). The friction factor is independent of the rheological properties of the fluid, but the Reynolds number involves the viscosity which, for a non-Newtonian fluid, is... 

A typical graph of drag ratio as a function of superficial air velocity is shown in Figure 5.5 in which each curve refers to a constant superficial liquid velocity. The liquids in question exhibited power law rheology and the corresponding values of the Metzner and Reed Reynolds numbers ReMR based on the superficial liquid velocity uL (see Chapter 3) are given. The following characteristics of the curves may be noted ... 

Because concentrated flocculated suspensions generally have high apparent viscosities at the shear rates existing in pipelines, they are frequently transported under laminar flow conditions. Pressure drops are then readily calculated from their rheology, as described in Chapter 3. When the flow is turbulent, the pressure drop is difficult to predict accurately and will generally be somewhat less than that calculated assuming Newtonian behaviour. As the Reynolds number becomes greater, the effects of non-Newtonian behaviour become... 

Flow of the liquid past the electrode is found in electrochemical cells where a liquid electrolyte is agitated with a stirrer or by pumping. The character of liquid flow near a solid wall depends on the flow velocity v, on the characteristic length L of the solid, and on the kinematic viscosity (which is the ratio of the usual rheological viscosity q and the liquid s density p). A convenient criterion is the dimensionless parameter Re = vLN, called the Reynolds number. The flow is laminar when this number is smaller than some critical value (which is about 10 for rough surfaces and about 10 for smooth surfaces) in this case the liquid moves in the form of layers parallel to the surface. At high Reynolds numbers (high flow velocities) the motion becomes turbulent and eddies develop at random in the flow. We shall only be concerned with laminar flow of the liquid. 

It should be noted that a dimensional analysis of this problem results in one more dimensionless group than for the Newtonian fluid, because there is one more fluid rheological property (e.g., m and n for the power law fluid, versus fi for the Newtonian fluid). However, the parameter n is itself dimensionless and thus constitutes the additional dimensionless group, even though it is integrated into the Reynolds number as it has been defined. Note also that because n is an empirical parameter and can take on any value, the units in expressions for power law fluids can be complex. Thus, the calculations are simplified if a scientific system of dimensional units is used (e.g., SI or cgs), which avoids the necessity of introducing the conversion factor gc. In fact, the evaluation of most dimensionless groups is usually simplified by the use of such units. 

It is convenient to distinguish between particle or fluid rotation about axes normal and parallel to the direction of relative motion. These two types of motion may be termed respectively top spin and screw motion (Til). Top spin is of more general importance since this corresponds to particle rotation caused by fluid shear or by collision with rigid surfaces. Workers concerned with suspension rheology and allied topics have concentrated on motion at low Re, while very high Reynolds numbers have concerned aerodynamicists. The gap between these two ranges is wide and uncharted, and we make no attempt to close it here. 

Toms BA (1948) Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds-numbers Proc 1st Intern Congr on Rheology North Holland Amsterdam II 135... 

Low Reynolds number flows with boundary integral representation have been used to describe rheological and transport properties of suspensions of solid spherical particles, as well as for numerical solution of different problems, including particle-particle interaction, the motion of a particle near a fluid interface or a rigid wall, the motion of particles in a container, and others. 

Equations for a number of non-Newtonian fluid types are available in the literature [213,359]. They tend to be somewhat unwieldy and require a knowledge of the fluid rheology. For power-law fluids in smooth pipes, the friction factor can be estimated by using a modified Reynolds number in Eq. (6.57). The Metzner-Reed modified Reynolds number, Re, is given by ... 

The most important case of this transition for chemical engineers is the transition from laminar to turbulent flow, which occurs in straight bounded ducts. In the case of Newtonian fluid rheology, this occurs in straight pipes when Re = 2100. A similar phenomenon occurs in pipes of other cross sections, as well and also for non-Newtonian fluids. However, just as the friction factor relations for these other cases are more complex than for simple Newtonian pipe flow, so the criteria for transition to turbulence cannot be expressed as a simple critical value of a Reynolds number. 

I would also like to list some of the challenges that will provide the foundation for where the profession has to go (Fig. 2). This is not meant to be comprehensive, but to suggest some of what we should be doing. This wish list derives from work Bob Brown and I have done on modeling flows of polymer fluids. The first item has to do with the need to understand the effects of polymer structure and rheology on flow transitions in polymeric liquids and on polymer processing operations. In the past, we ve studied extensively the behavior of Newtonian fluids and how Newtonian flows evolve as, say, the Reynolds number is varied. We have tools available to... 

The rheology of suspensions generally differs from fluids as a result of the hydrodynamic forces acting on the particles. The following figure illustrates this behavior in a print head. Since flows in print heads are in the range of low Reynolds numbers (Re 1-10 during drop formation in an ink channel with 350 m diameter), the velocity profile within (circular) capillaries is parabolic. This is indicated in Fig. 1. 

Sometimes, this expectation is not met. At high flow rates, there can be hydrodynamic instabilities that lead to secondary flows which ruin the rheological measurement. Such instabilities occur in Newtonian fluids, due, for example, to inertial effects, such as those in Poiseuille flow at Reynolds numbers exceeding 2000 (Drazin and Reid 1981). For some complex fluids, even at low Reynolds number there are instabilities that are driven by elastic effects (Larson 1992). 

The effect of fluid rheology on the power consumption of helical ribbon agitators has also been evaluated [54] and power consumption as a function of generalized Reynolds number for shear thinning but inelastic fluids defined. When shear thinning effects are small, and elasticity is negligible, deviations from the Newtonian... 

While many process fluids are Newtonian, some are non-Newtonian (as seen in Figure 9.7). For such cases, it is not sufQcient to use a single value for viscosity to determine the impeller Reynolds number. Concentrated slurries are typically non-Newtonian and particle-size dependent. They are frequently shear-thinning. Dilatant behavior seldom occurs. If the system is simply shear thinning, it is usually possible to describe its rheological behavior with a simple power-law relationship ... 

In the case of an inertia-free flow (at low Reynolds numbers) of a quasi-Newtonian power-law fluid with rheological index n close to 1 past a gas bubble, the drag coefficient can be calculated by the formula [190]... 

In conclusion, the development of comprehensive experimental and theoretical studies of DAL and rheology of bubble surfaces is needed. Having shown the importance to calculate the collision efficiency for very strong and very weak retardation of the bubble surface, the necessity of a detailed development of the theory of DAL of a bubble at moderate and high Reynolds numbers becomes quite obvious. 

The transition from laminar to turbulent flow in a non-Newtonian fluid depends on the rheological model used to describe it. The concepts of a critical friction factor or critical Reynolds number have been used to define the boundary. For a power law fluid the critical friction factor fCT is given by (96)... 

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